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\begin{document} \title [Power numerical radius inequalities] {{ Power numerical radius inequalities from an extension of Buzano's inequality }} \author[P. Bhunia]{Pintu Bhunia} \address{ {Department of Mathematics, Indian Institute of Science, Bengaluru 560012, Karnataka, India}} \email{pintubhunia5206@gmail.com; pintubhunia@iisc.ac.in} \thanks{Dr. Pintu Bhunia would like to thank SERB, Govt. of India for the financial support in the form of National Post Doctoral Fellowship (N-PDF, File No. PDF/2022/000325) under the mentorship of Professor Apoorva Khare} \thanks{} \subjclass[2020]{47A12, 47A30, 15A60} \keywords {Numerical radius, Operator norm, Bounded linear operator, Inequality} \date{\today} \maketitle \begin{abstract} Several numerical radius inequalities are studied by developing an extension of the Buzano’s inequality. It is shown that if $T$ is a bounded linear operator on a complex Hilbert space, then \begin{eqnarray*} w^n(T) &\leq& \frac{1}{2^{n-1}} w(T^n)+ \sum_{k=1}^{n-1} \frac{1}{2^{k}} \left\|T^k \right\| \left\|T \right\|^{n-k}, \end{eqnarray*} for every positive integer $n\geq 2.$ This is a non-trivial improvement of the classical inequality $w(T)\leq \|T\|.$ The above inequality gives an estimation for the numerical radius of the nilpotent operators, i.e., if $T^n=0$ for some least positive integer $n\geq 2$, then \begin{eqnarray*} w(T) &\leq& \left(\sum_{k=1}^{n-1} \frac{1}{2^{k}} \left\|T^k \right\| \left\|T \right\|^{n-k}\right)^{1/n} \leq \left( 1- \frac{1}{2^{n-1}}\right)^{1/n} \|T\|. \end{eqnarray*} Also, we deduce a reverse inequality for the numerical radius power inequality $w(T^n)\leq w^n(T)$. We show that if $\|T\|\leq 1$, then \begin{eqnarray*} w^n(T) &\leq& \frac{1}{2^{n-1}} w(T^n)+ 1- \frac{1}{2^{n-1}}, \end{eqnarray*} for every positive integer $n\geq 2.$ This inequality is sharp. \end{abstract} \section{\textbf{Introduction}} \noindent Let $\mathcal{B}(\mathcal{H})$ denote the $C^*$-algebra of all bounded linear operators on a complex Hilbert space $\mathcal{H}$ with usual inner product $\langle.,. \rangle$ and the corresponding norm $\|\cdot\|.$ Let $T\in \mathcal{B}(\mathcal{H})$ and let $|T|=(T^*T)^{1/2}$, where $T^*$ denotes the adjoint of $T.$ The numerical radius and the usual operator norm of $T$ are denoted by $w(T)$ and $\|T\|,$ respectively. The numerical radius of $T$ is defined as $$ w(T)=\sup \left\{|\langle Tx,x\rangle| : x\in \mathcal{H}, \|x\|=1 \right\}.$$ It is well known that the numerical radius defines a norm on $\mathcal{B}(\mathcal{H})$ and is equivalent to the usual operator norm. Moreover, \text{for every $T\in \mathcal{B}(\mathcal{H})$,} the following inequalities hold: \begin{eqnarray}\label{eqv} \frac12 \|T\| \leq w(T) \leq \|T\|. \end{eqnarray} The inequalities are sharp, $w(T)=\frac12 \|T\|$ if $T^2=0$ and $w(T)=\|T\|$ if $T$ is normal. Like as the usual operator norm, the numerical radius satisfies the power inequality, i.e., \begin{eqnarray}\label{power} w(T^n) \leq w^n(T) \end{eqnarray} \text{for every positive integer $n.$} One other basic property for the numerical radius is the weakly unitary invariant property, i.e., $w(T)=w(U^*TU)$ for every unitary operator $U\in \mathcal{B}(\mathcal{H}).$ To study more interesting properties of the numerical radius we refer to \cite{book,book2}. \noindent The numerical radius has various applications in many branches in sciences, more precisely, perturbation problem, convergence problem, approximation problem and iterative method as well as recently developed quantum information system. Due to importance of the numerical radius, many eminent mathematicians have been studied numerical radius inequalities to improve the inequalities \eqref{eqv}. Various inner product inequalities play important role to study numerical radius inequalities. The Cauchy-Schwarz inequality is one of the most useful inequality, which states that for every \text{$x,y \in \mathcal{H}$,} \begin{eqnarray}\label{cauchy} |\langle x,y\rangle| &\leq& \|x\| \|y\|. \end{eqnarray} A generalization of the Cauchy-Schwarz inequality is the Buzano's inequality \cite{Buzano}, which states that for $x,y,e \in \mathcal{H}$ with $\|e\|=1,$ \begin{eqnarray}\label{buzinq} |\langle x,e\rangle \langle e,y\rangle| &\leq& \frac{ |\langle x,y\rangle| +\|x\| \|y\|}{2}. \end{eqnarray} Another generalization of the Cauchy-Schwarz inequality is the mixed Schwarz inequality \cite[pp. 75--76]{Halmos}, which states that for every $x,y\in \mathcal{H}$ and $T\in \mathcal{B}(\mathcal{H}),$ \begin{eqnarray}\label{mixed} |\langle Tx,y\rangle|^2 &\leq& \langle |T|x,x\rangle \langle |T^*|y,y\rangle. \end{eqnarray} Using the above inner product inequalities various mathematicians have developed various numerical radius inequalities, which improve the inequalities \eqref{eqv}, see \cite{Bhunia_ASM_2022, Bhunia_LAMA_2022,Bhunia_RIM_2021,Bhunia_BSM_2021,Bhunia_ADM_2021,D08,Kittaneh_2003}. Also using other technique various nice numerical radius inequalities have been studied, see \cite{Abu_RMJM_2015,Bag_MIA_2020,Bhunia_LAA_2021,Bhunia_LAA_2019,Kittaneh_LAMA_2023, Kittaneh_STD_2005, Yamazaki}. Haagerup and Harpe \cite{Haagerup} developed a nice estimation for the numerical radius of the nilpotent operators, i.e., if $T^n=0$ for some positive integer $n\geq 2$, then \begin{eqnarray}\label{haag} w(T) &\leq& \cos \left(\frac{\pi}{n+1} \right) \|T\|. \end{eqnarray} \noindent In this paper, we obtain a generalization of the Buzano's inequality \eqref{buzinq} and using this generalization we develop new numerical radius inequalities, which improve the existing ones. From the numerical radius inequalities obtained here, we deduce several results. We deduce an estimation for the nilpotent operators like \eqref{haag}. Further, we deduce a reverse inequality for the numerical radius power inequality \eqref{power}. \section{\textbf{Numerical radius inequalities}} We begin our study with proving the following lemma, which is a generalization of the Buzano's inequality \eqref{buzinq}. \begin{lemma}\label{buz-extension} Let $x_1,x_2,\ldots,x_n,e \in \mathcal{H}$, where $\|e\|=1$. Then \begin{eqnarray*} \left| \mathop{\Pi}\limits_{k=1}^n \langle x_k,e\rangle \right| &\leq& \frac{ \left| \langle x_1,x_2\rangle \mathop{\Pi}\limits_{k=3}^n \langle x_k,e\rangle\right| + \mathop{\Pi}\limits_{k=1}^n \|x_k\|}{2}. \end{eqnarray*} \end{lemma} \begin{proof} Following the inequality \eqref{buzinq}, we have $$ |\langle x_1, e\rangle \langle x_2,e\rangle| \leq \frac{|\langle x_1, x_2\rangle |+ \|x_1\|\|x_2\|}{2}.$$ By replacing $x_2$ by $ \mathop{\Pi}\limits_{k=3}^n \langle x_k,e\rangle x_2$ and using $\left| \mathop{\Pi}\limits_{k=3}^n \langle x_2,e\rangle\right| \leq \mathop{\Pi}\limits_{k=3}^n \| x_k \|$, we obtain the desired inequality. \end{proof} Now using Lemma \ref{buz-extension} (for $n=3$), we prove the following numerical radius inequality. \begin{theorem}\label{th1} Let $T\in \mathcal{B}(\mathcal{H})$. Then \begin{eqnarray*} w(T) &\leq& \sqrt[3]{\frac14 w(T^3) + \frac14\left( \|T^2\|+ \|T^*T+TT^*\|\right) \|T\|}. \end{eqnarray*} \end{theorem} \begin{proof} Take $x\in \mathcal{H}$ and $\|x\|=1.$ From Lemma \ref{buz-extension} (for $n=3$), we have \begin{eqnarray*} |\langle Tx,x\rangle |^3 &=& |\langle Tx,x\rangle\langle T^*x,x\rangle\langle T^*x,x\rangle|\\ &\leq& \frac{|\langle Tx,T^*x\rangle \langle T^*x,x\rangle|+ \|Tx\| \|T^*x\|^2}{2}\\ &=& \frac{|\langle T^2x,x\rangle \langle T^*x,x\rangle|}{2}+ \frac{(\|Tx\|^2+ \|T^*x\|^2)\|T^*x\|}{4} \end{eqnarray*} Also, from the Buzano's inequality \eqref{buzinq}, we have \begin{eqnarray*} |\langle T^2x,x\rangle \langle T^*x,x\rangle| &\leq& \frac{|\langle T^2x,T^*x\rangle|+ \|T^2x\| \|T^*x\| }{2}\\ &=& \frac{|\langle T^3x,x\rangle|+ \|T^2x\| \|T^*x\| }{2}. \end{eqnarray*} Therefore, \begin{eqnarray*} |\langle Tx,x\rangle |^3 &\leq& \frac{|\langle T^3x,x\rangle|+ \|T^2x\| \|T^*x\|}{4} + \frac{(\|Tx\|^2+ \|T^*x\|^2)\|T^*x\|}{4}\\ &\leq& \frac14 w(T^3)+ \frac14 \|T^2\| \|T\|+\frac14 \|T^*T+TT^* \| \|T \|. \end{eqnarray*} Therefore, taking supremum over $\|x\|=1,$ we get the desired inequality. \end{proof} The inequality in Theorem \ref{th1} is an improvement of the second inequality in \eqref{eqv}, since $w(T^3)\leq \|T^3\|\leq \|T\|^3$, $\|T^2\|\leq \|T\|^2$ and $\|T^*T+TT^*\|\leq 2\|T\|^2.$ To show non-trivial improvement, we consider a matrix $T=\begin{bmatrix} 0&2&0\\ 0&0&1\\ 0&0&0 \end{bmatrix}.$ Then $w(T^3)=0$ and so $$ \sqrt[3]{ \frac14 w(T^3) + \frac14\left( \|T^2\|+ \|T^*T+TT^*\|\right) \|T\| } < \|T\|.$$ Next, using the Buzano's inequality \eqref{buzinq}, we obtain the following numerical radius inequality. \begin{theorem}\label{th2} Let $T\in \mathcal{B}(\mathcal{H}).$ Then \begin{eqnarray*} w(T) &\leq& \sqrt[3]{\frac12 w(TT^*T)+\frac14 \|T^*T+TT^*\| \|T\|}. \end{eqnarray*} \end{theorem} \begin{proof} Take $x\in \mathcal{H}$ and $\|x\|=1.$ By the Cauchy-Schwarz inequality \eqref{cauchy}, we have \begin{eqnarray*} |\langle Tx,x\rangle |^3 &\leq& \|Tx\|^2|\langle T^*x,x\rangle| = |\langle |T|^2x,x\rangle \langle T^*x,x\rangle|. \end{eqnarray*} From the Buzano's inequality \eqref{buzinq}, we have \begin{eqnarray*} |\langle Tx,x\rangle |^3 &\leq& \frac{|\langle |T|^2x,T^*x\rangle|+ \||T|^2x\| \|T^*x\|}{2}\\ &\leq& \frac{|\langle T|T|^2x,x\rangle |+ \||T|^2x\| \|T^*x\|}{2}\\ &\leq& \frac{|\langle T|T|^2x,x\rangle |+ \|T\| \|Tx\| \|T^*x\|}{2}\\ &\leq& \frac{|\langle T|T|^2x,x\rangle |}{2}+\frac{ (\|Tx\|^2+ \|T^*x\|^2)\|T\|}{4}\\ &\leq& \frac12 w(T|T|^2)+ \frac14 \|T^*T+TT^*\| \|T\|. \end{eqnarray*} Therefore, taking supremum over $\|x\|=1,$ we get the desired inequality. \end{proof} Clearly, for every $T\in \mathcal{B}(\mathcal{H})$, $$\sqrt[3]{\frac12 w(TT^*T)+\frac14 \|T^*T+TT^*\| \|T\|}\leq \sqrt[3]{\frac12 w(TT^*T)+\frac12 \|T\|^3} \leq \|T\|.$$ Also, using similar technique as Theorem \ref{th2}, we can prove the following numerical radius inequality. \begin{eqnarray}\label{eqn5} w(T)&\leq& \sqrt[3]{\frac12 w(T^*T^2)+ \frac12 \|T\|^3}. \end{eqnarray} And also replacing $T$ by $T^*$ in \eqref{eqn5}, we get \begin{eqnarray}\label{eqn6} w(T)&\leq& \sqrt[3]{\frac12 w(T^2T^*)+ \frac12 \|T\|^3}. \end{eqnarray} Consider a matrix $T=\begin{bmatrix} 0&2&0\\ 0&0&1\\ 0&0&0 \end{bmatrix}$. Then we see that $$ w(T^2 T^*)=1< w(T^*T^2)=2< w(TT^*T)=\sqrt{65}/2 .$$ Therefore, combining Theorem \ref{th2} and the inequalities \eqref{eqn5} and \eqref{eqn6}, we obtain the following corollary. \begin{cor}\label{cor2} Let $T\in \mathcal{B}(\mathcal{H})$. Then \begin{eqnarray*} w(T) &\leq& \sqrt[3]{\frac12 \min \Big(w(TT^*T), w(T^2T^*), w(T^*T^2) \Big) + \frac12 \|T\|^3}. \end{eqnarray*} \end{cor} Since $w(TT^*T)\leq \|T\|^3, \ w(T^2T^*)\leq \|T^2\| \|T\|\leq \|T\|^3,\ w(T^*T^2)\leq \|T^2\| \|T\|\leq \|T\|^3,$ the inequality in Corollary \ref{cor2} is an improvement of the second inequality in \eqref{eqv}. Now, Theorem \ref{th2} together with the inequalities \eqref{eqn5} and \eqref{eqn6} implies the following result. \begin{cor}\label{coor2} Let $T\in \mathcal{B}(\mathcal{H})$. If $w(T)=\|T\|,$ then \begin{eqnarray*} w(TT^*T)= w(T^2T^*)= w(T^*T^2)= \|T\|^3. \end{eqnarray*} \end{cor} Next, by using Lemma \ref{buz-extension} (for $n=3$), the Buzano's inequality \eqref{buzinq} and the mixed Schwarz inequality \eqref{mixed}, we obtain the following result. \begin{theorem}\label{th3} Let $T\in \mathcal{B}(\mathcal{H}).$ Then \begin{eqnarray*} w(T) &\leq& \sqrt[3]{\frac14 w(|T|T|T^*|)+ \frac14 \Big ( \|T^2\|+ \|T^*T+TT^*\| \Big)\|T\|}. \end{eqnarray*} \end{theorem} \begin{proof} Let $x\in \mathcal{H}$ with $\|x\|=1.$ From the mixed Schwarz inequality \eqref{mixed}, we have \begin{eqnarray*} |\langle Tx,x\rangle|^3 &\leq & \langle |T^*|x,x\rangle |\langle T^*x,x\rangle| \langle |T|x,x\rangle. \end{eqnarray*} Using Lemma \ref{buz-extension} (for $n=3$), we have \begin{eqnarray*} |\langle Tx,x\rangle|^3 &\leq & \frac{|\langle |T^*|x,T^*x\rangle \langle |T|x,x\rangle | + \||T^*|x\| \|T^*x\| \||T|x\| } {2}\\ &= & \frac{|\langle T |T^*|x,x\rangle | \langle |T|x,x\rangle + \||T^*|x\| \|T^*x\| \||T|x\| } {2}. \end{eqnarray*} By Buzano's inequality \eqref{buzinq}, we have \begin{eqnarray*} |\langle T |T^*|x,x\rangle | \langle |T|x,x\rangle &\leq& \frac{ | \langle T |T^*|x,|T|x\rangle |+ \|T |T^*|x\| \||T|x\| }{2}\\ &=& \frac{| \langle |T| T |T^*|x, x\rangle |+ \|T |T^*|x\| \||T|x\| }{2}.\\ \end{eqnarray*} Also by AM-GM inequality, we have \begin{eqnarray*} \||T^*|x\| \|T^*x\| \||T|x\| &\leq& \frac12(\||T^*|x\|^2 + \||T|x\|^2) \|T^*x\|\\ &=& \frac12 \langle (|T|^2+|T^*|^2)x,x\rangle \|T^*x\|. \end{eqnarray*} Therefore, \begin{eqnarray*} |\langle Tx,x\rangle|^3 &\leq & \frac{ |\langle |T| T |T^*|x, x\rangle |+ \|T |T^*|x\| \||T|x\| }{4} + \frac14 \langle (|T|^2+|T^*|^2)x,x\rangle \|T^*x\|\\ &\leq& \frac14 w(|T|T|T^*|)+ \frac14 \Big ( \|T |T^*|\|+ \|T^*T+TT^*\| \Big)\|T\|. \end{eqnarray*} From the polar decomposition $T^*=U|T^*|$, it is easy to verify that $\|T |T^*|\|=\|T^2\|.$ So, \begin{eqnarray*} |\langle Tx,x\rangle|^3 &\leq& \frac14 w(|T|T|T^*|)+ \frac14 \Big ( \|T^2 \|+ \|T^*T+TT^*\| \Big)\|T\|. \end{eqnarray*} Therefore, taking supremum over $\|x\|=1,$ we get the desired result. \end{proof} Now, combining Theorem \ref{th3} and Theorem \ref{th1}, we obtain the following corollary. \begin{cor}\label{cor3} Let $T\in \mathcal{B}(\mathcal{H}),$ then \begin{eqnarray*} w(T) \leq \sqrt[3]{\frac14 \min \Big( w(T^3), w(|T|T|T^*|) \Big)+ \frac14 \Big ( \|T^2\|+ \|T^*T+TT^*\| \Big)\|T\|}. \end{eqnarray*} \end{cor} Using similar technique as Theorem \ref{th3}, we can also prove the following inequality. \begin{eqnarray}\label{eqn7} w(T) &\leq& \sqrt[3]{\frac14 w(|T^*|T|T|)+ \frac38 \|T^*T+TT^*\| \|T\|}. \end{eqnarray} Clearly, the inequalities in Corollary \ref{cor3} and \eqref{eqn7} are stronger than the second inequality in \eqref{eqv}. And when $w(T)=\|T\|,$ then $$ w(|T|T|T^*|)= w(|T^*|T|T|) =w(T^3)=\|T\|^3.$$ Next theorem reads as follows: \begin{theorem}\label{th4} Let $T\in \mathcal{B}(\mathcal{H}).$ Then \begin{eqnarray*} w(T) &\leq& \sqrt[4]{ \Big( \frac12 w(T|T|) +\frac14 \|T^*T+TT^*\| \Big) \Big ( \frac12 w(T^*|T^*|)+ \frac14\|T^*T+TT^*\| \Big)}. \end{eqnarray*} \end{theorem} \begin{proof} Let $x\in \mathcal{H}$ with $\|x\|=1.$ From the mixed Schwarz inequality \eqref{mixed}, we have $$ |\langle Tx,x\rangle|^2 \leq \langle |T|x,x\rangle \langle |T^*|x,x\rangle.$$ Therefore, $$ |\langle Tx,x\rangle|^4 \leq \langle |T|x,x\rangle \langle |T^*|x,x\rangle |\langle Tx,x\rangle \langle T^*x,x\rangle|.$$ From the Buzano's inequality \eqref{buzinq}, we have \begin{eqnarray*} |\langle |T|x,x \rangle \langle T^*x,x\rangle | &\leq& \frac{|\langle |T|x, T^*x \rangle|+ \||T|x\| \|T^*x\|}{2}\\ &\leq& \frac12 |\langle T|T|x, x \rangle|+ \frac14 (\||T|x\|^2+ \|T^*x\|^2)\\ &\leq& \frac12 w(T|T|)+ \frac14 \|T^*T+TT^*\|. \end{eqnarray*} Similarly, \begin{eqnarray*} |\langle |T^*|x,x \rangle \langle Tx,x\rangle | &\leq& \frac{|\langle |T^*|x, Tx \rangle|+ \||T^*|x\| \|Tx\|}{2}\\ &\leq& \frac12 |\langle T^*|T^*|x, x \rangle|+ \frac14 (\||T^*|x\|^2+ \|Tx\|^2)\\ &\leq& \frac12 w(T^*|T^*|)+ \frac14 \|T^*T+TT^*\|. \end{eqnarray*} Therefore, $$|\langle Tx,x\rangle|^4 \leq \left(\frac12 w(T|T|)+ \frac14 \|T^*T+TT^*\|\right) \left( \frac12 w(T^*|T^*|)+ \frac14 \|T^*T+TT^*\|\right).$$ Taking supremum over $\|x\|=1,$ we get $$w^4(T) \leq \left(\frac12 w(T|T|)+ \frac14 \|T^*T+TT^*\|\right) \left( \frac12 w(T^*|T^*|)+ \frac14 \|T^*T+TT^*\|\right),$$ as desired. \end{proof} Again, using similar technique as Theorem \ref{th4}, we can prove the following result. \begin{theorem}\label{th5} Let $T\in \mathcal{B}(\mathcal{H}).$ Then \begin{eqnarray*} w(T) &\leq& \sqrt[4]{ \Big( \frac12 w(T|T^*|) +\frac12 \|T\|^2 \Big) \Big ( \frac12 w(T^*|T|)+ \frac12\|T\|^2 \Big)}. \end{eqnarray*} \end{theorem} Clearly, the inequalities in Theorem \ref{th4} and Theorem \ref{th5} are improvements of the second inequality in \eqref{eqv}. The inequalities imply that when $w(T)=\|T\|$, then $$w(T|T|)=w(T^*|T^*|) = w(T|T^*|)= w(T^*|T|)=\|T\|^2.$$ Now we consider the following example to compare the inequalities in Theorem \ref{th4} and Theorem \ref{th5}. \begin{example} Consider a matrix $T=\begin{bmatrix} 0&1&0\\ 0&0&1\\ 0&0&0 \end{bmatrix},$ then Theorem \ref{th4} gives $w(T)\leq \sqrt{ \frac{1}{2\sqrt{2}}+ \frac12 }$, whereas Theorem \ref{th5} gives $w(T)\leq \sqrt{ \frac{1}{4}+ \frac12}.$ Again, Consider $T=\begin{bmatrix} 0&1&0\\ 0&0&2\\ 0&0&0 \end{bmatrix},$ then Theorem \ref{th4} gives $w(T)\leq \sqrt{ \frac{\sqrt{17}+5}{4} }$, whereas Theorem \ref{th5} gives $w(T)\leq \sqrt{ \frac{5+5}{4} }.$ Therefore, we would like to note that the inequalities obtained in Theorem \ref{th4} and Theorem \ref{th5} are not comparable, in general. \end{example} \section{\textbf{Numerical radius inequalities involving general powers}} We develop a numerical radius inequality involving general powers $w^n(T)$ and $w(T^n)$ for every positive integer $n\geq 2$ and from which we derive nice results related to the nilpotent operators and reverse power inequality for the numerical radius. First we prove the following theorem. \begin{theorem}\label{th7} If $T\in \mathcal{B}(\mathcal{H}),$ then \begin{eqnarray*} |\langle Tx, x\rangle|^n &\leq& \frac{1}{2^{n-1}} \left|\langle T^nx, x\rangle \right|+ \sum_{k=1}^{n-1} \frac{1}{2^{k}} \left\|T^kx \right\| \left\|T^*x \right\|^{n-k}, \end{eqnarray*} for all $x\in \mathcal{H}$ with $\|x\|=1$ and for every positive integer $n\geq 2.$ \end{theorem} \begin{proof} We have \begin{eqnarray*} && |\langle Tx,x \rangle|^n \\ &=&|\langle Tx,x \rangle \langle T^*x,x \rangle \langle T^*x,x \rangle^{n-2} |\\ &\leq& \frac{\left|\langle Tx,T^*x \rangle \langle T^*x,x \rangle^{n-2} \right|+ \|Tx\| \|T^*x\|^{n-1} }{2} \,\, (\text{by Lemma \ref{buz-extension}})\\ &=& \frac{\left|\langle T^2x,x \rangle \langle T^*x,x \rangle \langle T^*x,x \rangle^{n-3} \right|+ \|Tx\| \|T^*x\|^{n-1} }{2}\\ &\leq & \frac{ \frac{\left|\langle T^2x,T^*x \rangle \langle T^*x,x \rangle^{n-3} \right|+ \|T^2x\| \|T^*x\|^{n-2} }{2}+ \|Tx\| \|T^*x\|^{n-1} }{2} \,\, (\text{by Lemma \ref{buz-extension}})\\ &= & \frac{\left|\langle T^3x,x \rangle \langle T^*x,x \rangle \langle T^*x,x \rangle^{n-4} \right|+ \|T^2x\| \|T^*x\|^{n-2} }{2^2}+\frac{ \|Tx\| \|T^*x\|^{n-1} }{2} \\ &\leq & \frac{\frac{\left|\langle T^3x,T^*x \rangle \langle T^*x,x \rangle^{n-4}\right|+ \|T^3x\| \|T^*x\|^{n-3} }{2} + \|T^2x\| \|T^*x\|^{n-2} }{2^2} +\frac{ \|Tx\| \|T^*x\|^{n-1} }{2} \\ && \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, (\text{by Lemma \ref{buz-extension}})\\ &= & \frac{\left|\langle T^4x,x \rangle \langle T^*x,x \rangle \langle T^*x,x \rangle^{n-5}\right|+ \|T^3x\| \|T^*x\|^{n-3} }{2^3} + \frac{ \|T^2x\| \|T^*x\|^{n-2} }{2^2} \\ && \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, +\frac{ \|Tx\| \|T^*x\|^{n-1} }{2}. \end{eqnarray*} Repeating this approach $(n-1)$ times (i.e., using Lemma \ref{buz-extension}), we obtain \begin{eqnarray*} |\langle Tx, x\rangle|^n &\leq& \frac{1}{2^{n-1}} \left|\langle T^nx, x\rangle \right|+ \sum_{k=1}^{n-1} \frac{1}{2^{k}} \left\|T^kx \right\| \left\|T^*x \right\|^{n-k}, \end{eqnarray*} as desired. \end{proof} The following generalized numerical radius inequality is a simple consequence of Theorem \ref{th7}. \begin{cor}\label{cor5} If $T\in \mathcal{B}(\mathcal{H}),$ then \begin{eqnarray}\label{pp} w^n(T) &\leq& \frac{1}{2^{n-1}} w(T^n)+ \sum_{k=1}^{n-1} \frac{1}{2^{k}} \left\|T^k \right\| \left\|T \right\|^{n-k}, \end{eqnarray} for every positive integer $n\geq 2.$ \end{cor} \begin{remark} (i) For every $T\in \mathcal{B}(\mathcal{H})$ and for every positive integer $n\geq 2$, \begin{eqnarray*} w^n(T) &\leq& \frac{1}{2^{n-1}} w(T^n)+ \sum_{k=1}^{n-1} \frac{1}{2^{k}} \left\|T^k \right\| \left\|T \right\|^{n-k} \\ &\leq& \frac{1}{2^{n-1}} w(T^n)+ \sum_{k=1}^{n-1} \frac{1}{2^{k}} \left\|T \right\|^{n} \\ &\leq& \frac{1}{2^{n-1}} \|T^n\|+ \sum_{k=1}^{n-1} \frac{1}{2^{k}} \left\|T \right\|^{n} \,\, \text{(by \eqref{eqv})}\\ &\leq& \frac{1}{2^{n-1}} \|T\|^n+ \sum_{k=1}^{n-1} \frac{1}{2^{k}} \left\|T \right\|^{n} = \|T\|^n. \end{eqnarray*} Therefore, the inequality \eqref{pp} is an improvement of the second inequality in \eqref{eqv}. \noindent (ii) From the above inequalities, it follows that if $w(T)=\|T\|$, then $$ w^n(T)=w(T^n)=\|T^n\|=\|T\|^n,$$ for every positive integer $n\geq 2.$ And so it is easy to see that when $w(T)=\|T\|$, then $$w(T)= \lim\limits_{n\to \infty} \|T^n\|^{1/n}= r(T),$$ where $r(T)$ denotes the spectral radius of $T.$ The second equality holds for every operator $T\in \mathcal{B}(\mathcal{H})$ and it is known as Gelfand formula for spectral radius. \noindent (iii) Taking $n=2$ in Corollary \ref{cor5}, we get $$ w^2(T) \leq \frac12 w(T^2)+ \frac12\|T\|^2,$$ which was proved by Dragomir \cite{D08}. \noindent (iv) Taking $n=2$ in Theorem \ref{th7}, we deduce that $$ w^2(T) \leq \frac12 w(T^2)+ \frac14 \|T^*T+TT^*\|,$$ which was proved by Abu-Omar and Kittaneh \cite{Abu_RMJM_2015}. \end{remark} Following Corollary \ref{cor5}, we obtain an estimation for the nilpotent operators. \begin{cor}\label{nilpotent} Let $T\in \mathcal{B}(\mathcal{H}).$ If $T^n=0$ for some least positive integer $n\geq 2$, then \begin{eqnarray*} w(T) &\leq& \left(\sum_{k=1}^{n-1} \frac{1}{2^{k}} \left\|T^k \right\| \left\|T \right\|^{n-k}\right)^{1/n} \leq \left( 1- \frac{1}{2^{n-1}}\right)^{1/n} \|T\|. \end{eqnarray*} \end{cor} Consider a matrix $T=\begin{bmatrix} 0&1&2\\ 0&0&3\\ 0&0&0 \end{bmatrix}.$ Then we see that Corollary \ref{nilpotent} gives $w(T)\leq \alpha \approx 3.0021$ and Theorem \ref{th1} gives $w(T) \leq \beta \approx 2.5546,$ whereas the inequality \eqref{haag} gives $w(T)\leq \gamma \approx 2.5811.$ Therefore, we conclude that for the nilpotent operators the inequality \eqref{haag} (given by Haagerup and Harpe) is not always better than the inequalities discussed here and vice versa. Finally, by using Corollary \ref{cor5} we obtain a reverse power inequality for the numerical radius. \begin{cor} Let $T\in \mathcal{B}(\mathcal{H})$. If $\|T\|\leq 1$, then \begin{eqnarray*} w^n(T) &\leq& \frac{1}{2^{n-1}} w(T^n)+ 1- \frac{1}{2^{n-1}}, \end{eqnarray*} for every positive integer $n\geq 2.$ This inequality is sharp. \end{cor} \begin{proof} From Corollary \ref{cor5}, we have \begin{eqnarray*} w^n(T) &\leq& \frac{1}{2^{n-1}} w(T^n)+ \sum_{k=1}^{n-1} \frac{1}{2^{k}} \left\|T^k \right\| \left\|T \right\|^{n-k}\\ &\leq& \frac{1}{2^{n-1}} w(T^n)+ \sum_{k=1}^{n-1} \frac{1}{2^{k}} \left\|T \right\|^{n}\\ &\leq& \frac{1}{2^{n-1}} w(T^n)+ \sum_{k=1}^{n-1} \frac{1}{2^{k}} \\ &=& \frac{1}{2^{n-1}} w(T^n)+ 1- \frac{1}{2^{n-1}}. \end{eqnarray*} If $T^*T=TT^*$ and $\|T\|=1,$ then \begin{eqnarray*} w^n(T) &=& \frac{1}{2^{n-1}} w(T^n)+ 1- \frac{1}{2^{n-1}}=1. \end{eqnarray*} \end{proof} \noindent {\bf{Data availability statements.}}\\ Data sharing not applicable to this article as no datasets were generated or analysed during the current study.\\ \noindent {\bf{Declarations.}}\\ \noindent {\bf{Conflict of Interest.}} The author declares that there is no conflict of interest. \end{document}

\begin{document} \title{Attractors of Sequences of Function Systems \\ and their relation to Non-Stationary Subdivision} \author[David Levin]{David Levin} \author[Nira Dyn]{Nira Dyn} \address{D. Levin, N. Dyn, School of Mathematical Sciences, Tel Aviv University, Israel} \author[P. V. Viswanathan]{Puthan Veedu Viswanathan} \address{P. V. Viswanathan, Department of Mathematics, Indian Institute of Technology, Delhi, India} \begin{abstract} Iterated Function Systems (IFSs) have been at the heart of fractal geometry almost from its origin, and several generalizations for the notion of IFS have been suggested. Subdivision schemes are widely used in computer graphics and attempts have been made to link fractals generated by IFSs to limits generated by subdivision schemes. With an eye towards establishing connection between non-stationary subdivision schemes and fractals, this paper introduces the notion of ``trajectories of maps defined by function systems" which may be considered as a new generalization of the traditional IFS. The significance and the convergence properties of `forward' and `backward' trajectories are studied. Unlike the ordinary fractals which are self-similar at different scales, the attractors of these trajectories may have different structures at different scales. \end{abstract} \maketitle \section{\bf Introduction}\label{sect1} The concept of Iterated Function system (IFS) was introduced by Hutchinson \cite{H} and popularized by Barnsley \cite{B1}. IFSs form a standard framework for describing self-referential sets such as fractals and provide a potential new method of researching the shape and texture of images. Due to its importance in understanding images, several extensions to the classical IFS such as Recurrent IFS, partitioned IFS and Super IFS are discussed in the literature \cite{B2,BEH,F}. Fractal functions whose graphs are attractors of suitably chosen IFS provide a new method of interpolation and approximation \cite{B1,PRM,MAN,PV1}. \par Subdivision schemes are efficient algorithmic methods for generating curves and surfaces from discrete sets of control points. A subdivision scheme generates values associated with the vertices of a sequence of nested meshes, by repeated application of a set of local refinement rules. These subdivision rules, usually linear, iteratively transform the vertices of a given mesh to vertices of a refined mesh. In recent years, the subject of subdivision has gained more popularity because of many new applications such as computer graphics. The reader may turn to \cite{CDM,DL,MP,PBP} for an introduction and survey of the mathematics of subdivision schemes and their applications. \par Being two different topics that had been developing independently and in parallel, the connections between subdivision and theory of IFS were sought after. Later it has been observed that there is a close connection between curves and surfaces generated by subdivision algorithms and self-similar fractals generated by IFSs \cite{SLG}. However, this relationship is established for stationary subdivision schemes. The relation between non-stationary subdivision and IFS remains obscure and unexplored. \par In this paper we target to establish the interconnection between the theory of IFS and non-stationary subdivision schemes. In this attempt, we introduce and study what we call "trajectories of a sequence of transformations". Trajectories generated by a sequence of function system maps may provide new attractor sets, generalizing fractal sets, and help us to link the theory of IFS with non-stationary subdivision schemes. \eject \section{\bf Preliminaries} For a nonspecialist, we mention here the concepts, notation and basic results concerning traditional IFS and provide a brief outline of subdivision. For a detailed exposition the reader may consult \cite{B1,H} and \cite{CDM,DL} respectively. \subsection{Basics of iterated function systems} Let $(X,d)$ be a complete metric space. For a function $f: X \to X$, we define the Lipschitz constant associated with $f$ by $$\text{Lip}(f) = \sup_{x,y \in X, x \neq y} \frac{d\big(f(x),f(y)\big)}{d(x,y)}.$$ A function $f$ is said to be Lipschitz function if $\text{Lip}(f) < + \infty$ and a contraction if $\text{Lip}(f) < 1$. Let $\mathbb{H}(X)$ be the collection of all nonvoid compact subsets of $X$. Then $\mathbb{H}$ is a metric space when endowed with the Hausdorff metric $$ h (B,C) = \max \big\{d(B,C), d(C,B)\big\},$$ where $d(B,C)= \sup_{b \in B} d(b,C)= \sup_{ b \in B} \inf_{c \in C} d(b,c)$. It is well-known that the metric space $\big(\mathbb{H}(X),h\big)$ is complete \cite{B2}. \begin{definition}\label{defIFS} An iterated function system, IFS for short, consists of a metric space $(X,d)$ and a finite family of continuous maps $f_i: X \to X$, $i \in \{1,2,\dots,n\}$. We denote such an IFS by $\mathcal{F}=\{X; f_i: i=1,2,\dots, n\}$. \end{definition} With the IFS $\mathcal{F}$ as above, one can associate a set-valued map referred to as Barnsley-Hutchinson operator. With a slight abuse of notation, we use the same symbol $\mathcal{F}$ for the IFS, the set of functions in the IFS, and for the Barnsley-Hutchinson operator defined below. Consider the function $\mathcal{F}: \mathbb{H}(X)\to \mathbb{H}(X)$ \begin{equation}\label{FX} \mathcal{F}(B) := \cup_{f \in \mathcal{F}} f(B),\ \ B\in \mathbb{H}(X), \end{equation} where $f(B):= \big\{f(b): b \in B \big\}$. The contraction constant of $\mathcal{F}$ is \cite{B2}: \begin{equation}\label{CLF} L_{\mathcal{F}}=\max_{i=1,2,\dots, n} \text{Lip}(f_i). \end{equation} If $f_i$ are contraction maps, the IFS is contractive. Therefore, by the Banach contraction principle we have \begin{theorem}\label{theoremIFS} Let $(X,d)$ be a complete metric space and $\mathcal{F}=\{X; f_i: i=1,2,\dots, n\}$ be an IFS with contraction constant $L_{\mathcal{F}}<1$. Then there exists a unique set $A_{\mathcal{F}}$, such that $\mathcal{F}(A_{\mathcal{F}}) = A_{\mathcal{F}}$. Furthermore, for every $B_0 \in \mathbb{H}(X)$ the sequence $B_{k+1} = \mathcal{F} (B_k)$ converges to $A_{\mathcal{F}}$ in $\mathbb{H}$. Also \cite{B2}, $$h(B_0,A_{\mathcal{F}}) = \frac{1}{1- L_{\mathcal{F}}}~ h(B_0, B_1).$$ \end{theorem} \begin{remark}\label{remarkIFS} \begin{enumerate} \item The set $A _{\mathcal{F}}$ appearing in the previous theorem is called the attractor of the IFS. The construction of $A _{\mathcal{F}}$ through iterations of the map $\mathcal{F}$ suggests the name iterated function system for $\mathcal{F}=\{X; f_i: i=1,2,\dots, n\}$. \item The result of Theorem \ref{theoremIFS} holds even if $\mathcal{F}$ is not a contraction map, but an $\ell$-term composition of $\mathcal{F}$, namely, $\mathcal{F}\circ \mathcal{F}\circ...\circ \mathcal{F}$ is a contraction map. The $\ell$-term composition is a contraction if all the compositions of the form \begin{equation}\label{lterm} f_{i_1}\circ f_{i_2}\circ \cdot\cdot\cdot f_{i_\ell},\ \ \ i_j\in\{1,2,...,n\}, \end{equation} are contractions. \end{enumerate} \end{remark} \subsection{Basics of subdivision schemes}\label{BOS} A subdivision scheme is defined by a collection of real maps called refinement rules relative to a set of meshes of isolated points $$N_0 \subseteq N_1 \subseteq \dots \subseteq \mathbb{R}^s.$$ Each refinement rule maps real vector values defined on $N_k$ to real vector values defined on a refined net $N_{k+1}$. Here we consider only scalar binary subdivision schemes, with $N_k=2^{-k}\mathbb{Z}^s$. Given a set of control points $p^{0}=\{p_j^0\in \mathbb{R}^m,\ \ j \in \mathbb{Z}^s\}$ at level $0$, a stationary binary subdivision scheme recursively defines new sets of points $p^k= \{p_j^k: j \in \mathbb{Z}^s\}$ at level $k \ge 1$, by the refinement rule \begin{equation}\label{sa} p_i^{k+1} = \sum_{j \in \mathbb{Z}^s} a_{i-2j} p_j^k,\ \ k\ge 0, \end{equation} or in short form, $$p^{k+1}=S_a p^k,\ \ k\ge 0.$$ The set of real coefficients $a= \{a_j: j \in \mathbb{Z}^s\}$ that determines the refinement rule is called the mask of the scheme. We assume that the support of the mask, $\sigma(a)= \{j \in \mathbb{Z}^s:a_j \neq 0\}$, is finite. $S_a$ is a bi-infinite two-slanted matrix with the entries $(S_a)_{i,j}=a_{i-2j}$.\\ A non-stationary binary subdivision scheme is defined formally as $$p^{k+1}=S_{a^{[k]}} p^k,\ \ k\ge 0,$$ where the refinement rule at refinement level $k$ is of the form \begin{equation}\label{sak} p_i^{k+1} = \sum_{j \in \mathbb{Z}^s} a_{i-2j}^{[k]} p_j^k,\ \ i\in \mathbb{Z}^s. \end{equation} In a non-stationary scheme, the mask $a^{[k]}:= \{a_j^{[k]}: j \in \mathbb{Z}^s\}$ depends on the refinement level. In univariate schemes $s=1$, there are two different rules in (\ref{sak}), depending on the parity of $i$. In this paper we refer to two definitions of convergent subdivision. The first is the classical one in subdivision theory \cite{DL}: \begin{definition}\label{C0conv}{\bf $C^0$-convergent subdivision}\\ A subdivision scheme is termed $C^0$-convergent if for any initial data $p^0$ there exists a continuous function $f:\mathbb{R}^s \to \mathbb{R}^m$, such that \begin{equation}\label{pktof} \lim_{k\to\infty}\sup_{i\in \mathbb{Z}^s}|p_i^{k}-f(2^{-k}i)|=0, \end{equation} and for some initial data $f\ne 0$. \end{definition} \begin{remark}\label{remarkC0} \begin{enumerate} \item The limit curve of a $C^0$-convergent subdivision is denoted by $p^\infty=S_a^\infty p^0$, and the function $f$ in Definition \ref{C0conv} specifies a parametrization of the limit curve. \end{enumerate} \end{remark} The analysis of subdivision schemes aims at studying the smoothness properties of the limit function $f$. For further reading see \cite{DL}. We introduce here a weaker type of convergence using a set distance approach, influenced by IFS convergence: \begin{definition}\label{hconv}{\bf $h$-convergent subdivision}\\ A subdivision scheme is termed $h$-convergent if for any initial data $p^0$ there exists a set $p^\infty\subset \mathbb{R}^m$, such that \begin{equation}\label{setlimit} \lim_{k\to\infty}h(p^k,p^\infty)=0, \end{equation} where $h$ is the Euclidian-Hausdorff metric on $\mathcal{R}^m$. The set $p^\infty$ is termed the $h$-limit of the subdivision scheme. \end{definition} It is clear that any $C^0$-convergent subdivision is also $h$-convergent. In both subjects, IFS and subdivision, one is interested in the limits of iterative processes. A connection between IFS and stationary subdivision is established in \cite{SLG}. In order to extend this connection to the case of non-stationary subdivision we investigate below the convergence properties of sequences of transformations in a metric space. \section{\bf Sequences of transformations and Trajectories} This section is intended to introduce trajectories induced by a sequence of transformations and establish some elementary properties. \par Let $(X,d)$ be a complete metric space. Consider a sequence of continuous transformations $\{T_i\}_{i \in N}$, $T_i: X \to X$. \begin{definition}{\bf Forward and backward procedures:} For the sequence of maps $\{T_i\}_{i \in N}$ we define forward and backward procedures \begin{enumerate} \item $ \Phi_k=T_k \circ T_{k-1} \circ \dots \circ T_1,$ \item $ \Psi_k=T_1 \circ T_2 \circ \dots \circ T_k.$ \end{enumerate} \end{definition} \begin{definition}{\bf Forward and backward trajectories:} Induced by the forward and the backward procedures, we define consequent forward and backward trajectories in $X$, starting from $x\in X$, $\{\Phi_k(x)\}$ and $\{\Psi_k(x)\}$, \begin{equation}\label{PhiPsi} \begin{aligned} \Phi_k(x)=T_k \circ T_{k-1} \circ \dots \circ T_1(x)=T_k\circ\Phi_{k-1}(x),\ \ k \in \mathbb{N},\\ \Psi_k(x)=T_1 \circ T_2 \circ \dots \circ T_k(x)=\Psi_{k-1}\circ T_k(x),\ \ k \in \mathbb{N}. \end{aligned} \end{equation} \end{definition} In the present section we study the convergence of both types of trajectories. Later on we demonstrate the application of both types to sequences of function systems and to subdivision. To state our next proposition, let us first introduce the following definition. \begin{definition} Two sequences $\{x_i\}_{i \in \mathbb{N}}$ and $\{y_i\}_{i \in \mathbb{N}}$ in a metric space $(X,d)$ are said to be asymptotically similar if $d(x_i,y_i) \to 0$ as $i \to \infty$. We denote this relation by \begin{equation} \{x_i\}\sim \{y_i\}. \end{equation} \end{definition} \begin{proposition}\label{Equivalence}{\bf Asymptotic similarity of trajectories}\\ Let $\{T_i\}_{i \in \mathbb{N}}$ be a sequence of transformations on $X$, where each $T_i$ is a Lipschitz map with Lipschitz constant $s_i$. If $\lim_{ k \to \infty} \prod_{i=1}^k s_i =0$, then for any $x,y\in X$, \begin{equation} \begin{aligned} \{\Phi_k(x)\}\sim \{\Phi_k(y)\},\\ \{\Psi_k(x)\}\sim \{\Psi_k(y)\}. \end{aligned} \end{equation} \end{proposition} Note that the condition $\lim_{k\to\infty} \prod_{i=1}^k s_i =0$ does not imply $\limsup_{k \to \infty} s_k <1$. \begin{proof} The proof is similar for the forward and the backward trajectories. Let $x,y\in X$ and consider the trajectories $\{\Psi_k(x)\}$ and $\{\Psi_k(y)\}$. Using the fact that $T_i$ is a Lipschitz map with Lipschitz constant $s_i$, we get \begin{equation}\label{prodsi} \begin{aligned} d\big(\Psi_k(x),\Psi_k(y)\big)\ \le\ & s_1 d(\big(T_2 \circ T_3 \circ \dots \circ T_k(x),T_2 \circ T_3 \circ \dots \circ T_k(y)\big)) \\ & \le s_1s_2d(\big(T_3 \circ T_4 \circ \dots \circ T_k(x),T_3 \circ T_4 \circ \dots \circ T_k(y)\big)) ... \\ & \le \big(\prod_{i=1}^k s_i\big) d(x,y), \end{aligned} \end{equation} from which the result follows. \end{proof} \begin{remark}\label{remarkT} The condition $\lim_{ k \to \infty} \prod_{i=1}^k s_i =0$ stated in Proposition \ref{Equivalence} does not guarantee convergence of the trajectories $\{\Phi_k(x)\}$. \end{remark} If $T_i=T$ $\forall i\in \mathbb{N}$, and $T$ is a Lipschitz map with Lipschitz constant $\mu<1$, then both types of trajectories are just the fixed-point iteration trajectories $\{T^k(x)\}$, where $T^k$ is the $k$-fold autocomposition of $T$ which converge to a unique limit for any starting point $x$. It is known from the Banach contraction principle that $\{T^k(x)\}$ converges to a unique limit irrespective of the starting point $x$. The question now arises regarding the convergence of general trajectories, i.e., which conditions guarantee the convergence of the forward and the backward trajectories. Having in mind the applications to fractal generation and to subdivision, we would like to know which trajectories yield new types of fractals or new types of limit functions. Let us start with the forward trajectories $\{\Phi_k(x)\}$. \begin{definition}\label{def3}{\bf Invariant set of $\{T_i\}$.}\\ We call $C\subseteq X$ an invariant set of a sequence of transformations $\{T_i\}_{i \in \mathbb{N}}$ if \begin{equation}\label{C} \forall~ x\in C,\ \ T_i(x)\in C,\ \ \forall~ i\in \mathbb{N}. \end{equation} \end{definition} \begin{lemma}\label{lemma3} Consider a sequence of transformations $\{T_i\}_{i \in \mathbb{N}}$. If there exists $q$ in $X$ such that for every $x\in X$ \begin{equation}\label{C2} d(T_i(x),q)\le \mu d(x,q)+M,\ \ 0\le\mu <1,\ \ M\in \mathbb{R}_+, \end{equation} then the ball of radius $\frac{M}{1-\mu}$ centered at $q$, $B\big(q,\frac{M}{1-\mu}\big)$, is an invariant set of $\{T_i\}_{i \in \mathbb{N}}$. \end{lemma} \begin{proof} For $x\in B\big(q,\frac{M}{1-\mu}\big)$ \begin{equation}\label{C3} d(T_i(x),q)\le \mu d(x,q)+M \le \mu \frac{M}{1-\mu}+M = \frac{M}{1-\mu}. \end{equation} \end{proof} \begin{remark}\label{remarkC} Under the conditions of Lemma \ref{lemma3}, any ball $B(q,R)$ with $R>\frac{M}{1-\mu}$ is also an invariant set of $\{T_i\}_{i \in \mathbb{N}}$. This follows since $M$ in (\ref{C2}) can be replaced by any $M^*>M$. \end{remark} \begin{example}\label{Ex1} Consider a sequence of affine transformations on ${\mathbb R}^m$ of the form \begin{equation}\label{Ex1eq} T_i(x)=A_ix+b_i,\ \ i\in \mathbb{N}, \end{equation} where $\{A_i\}$ are $m\times m$ matrices with $\|A_i\|_2\le \mu<1$, and $\|b_i\|_2\le M$. Then the conditions of Lemma \ref{lemma3} are satisfied with $q=0$, and thus $C=B\big(0, \frac{M}{1-\mu}\big)$ is an invariant set of $\{T_i\}_{i \in \mathbb{N}}$. \end{example} \begin{proposition}\label{forwardconvergence}{\bf Convergence of forward trajectories}\\ Let $\{T_i\}_{i \in \mathbb{N}}$ be a sequence of transformations on $X$, with a compact invariant set $C$, and assume $\{T_i\}_{i \in \mathbb{N}}$ converges uniformly on $C$ to a Lipschitz map $T$ with Lipschitz constant $\mu<1$. Then for any $x\in C$ the trajectory $\{\Phi_i(x)\}_{i \in \mathbb{N}}$ converges to the fixed-point $p$ of $T$, namely, \begin{equation} \lim_{k\to\infty}d(\Phi_k(x),p)=0. \end{equation} \end{proposition} \begin{proof} Denoting $\epsilon_i=\sup_{x\in C}d(T_i(x),T(x))$, $i\in \mathbb{N}$, it follows that \begin{equation}\label{epsto0} \lim_{i\to\infty}\epsilon_i=0. \end{equation} Since $T$ is a Lipschitz map with Lipschitz constant $\mu<1$, the fixed-point iterations $\{T^k(x)\}$ converge to a unique fixed-point $p\in X$ for any starting point $x$. It also follows that $C$ is an invariant set of $T$. Starting with $x\in C$, we have that $\{\Phi_k(x)\}\subseteq C$. Using the triangle inequality in $\{X,d\}$ and the Lipschitz property of $T$, we have \begin{equation} \begin{aligned} d(\Phi_{k+m}(x),T^m\Phi_k(x))= d(T_{k+m}\circ T_{k+m-1}\circ ...\circ T_{k+1}\circ \Phi_k(x),T^m\Phi_k(x))\le \\ d(T_{k+m}\circ T_{k+m-1}\circ ...\circ T_{k+1}\circ \Phi_k(x),T\circ T_{k+m-1}\circ ...\circ T_{k+1}\circ \Phi_k(x))+ \\ d(T\circ T_{k+m-1}\circ ...\circ T_{k+1}\circ \Phi_k(x),T^2\circ T_{k+m-2}\circ ...\circ T_{k+1}\circ \Phi_k(x))+ \\ ... \\ +d(T^{m-1}\circ T_{k+1}\circ \Phi_k(x),T^m\Phi_k(x))\le \\ \epsilon_{k+m} + \mu \epsilon_{k+m-1}+ \mu^2\epsilon_{k+m-2}+...+ \mu^{m-1}\epsilon_{k+1}\le \\ \max_{1\le i\le m}\{\epsilon_{k+i}\}\times {\frac {1} {1-\mu}}. \end{aligned} \end{equation} Now we use the relation \begin{equation} d(\Phi_{k+m}(x),p)\le d(\Phi_{k+m}(x),T^m\Phi_k(x))+d(T^m\Phi_k(x),p). \end{equation} The result follows by observing that for $k$ large enough $\max_{1\le i\le m}\{\epsilon_{k+i}\}$ can be made as small as needed (by (\ref{epsto0})), and for that $k$, for a large enough $m$, $d(T^m\Phi_k(x),p)$ is as small as needed. \end{proof} In Section \ref{IFS} we consider trajectories of transformations $\{T_i\}$ defined by function systems, and we look for the attractors of such trajectories. We refer to such systems as non-stationary function systems, and we apply them to generate new fractals. Proposition \ref{forwardconvergence} implies that in the case of forward trajectories, if $T_i\to T$ as $i\to \infty$, the limit of the forward trajectories is the attractor of the IFS corresponding to the limit function system, and hence not new. Let us now examine the backward trajectories $\{\Psi_k(x)\}$, and establish conditions for their convergence. \begin{proposition}\label{BTp2}{\bf Convergence of backward trajectories} Let $\{T_i\}_{i \in \mathbb{N}}$ be a sequence of transformations on $X$, with a compact invariant set $C$, and assume each $T_i$ is a Lipschitz map with Lipschitz constant $s_i$. If $\sum_{k=1}^\infty \prod_{i=1}^k s_i <\infty$, then the backward trajectories $\{\Psi_k(x)\}$, with $ \Psi_k=T_1 \circ T_2 \circ \dots \circ T_k,\ \ k \in \mathbb{N},$ converge for any starting point $x\in C$ to a unique limit in $C$. \end{proposition} \begin{proof} By (\ref{PhiPsi}) and the relation in (\ref{prodsi}) \begin{equation*} \begin{split} d\big(\Psi_{k+1}(x),\Psi_k(x) \big) = &~ d \big( \Psi_k (T_{k+1}(x)), \Psi_k(x) \big) \\ \le &~ \big(\prod_{i=1}^k s_i \big)d\big(T_{k+1}(x),x \big).\\ \end{split} \end{equation*} For $m,k \in \mathbb{N}$, $m>k$, we obtain \begin{equation}\label{e1} \begin{split} d\big(\Psi_m(x),\Psi_k(x)\big) \le &~ d\big(\Psi_m(x),\Psi_{m-1}(x)\big) + \dots + d\big(\Psi_{k+2}(x),\Psi_{k+1}(x) \big)+d\big(\Psi_{k+1}(x),\Psi_k(x) \big)\\ \le &~ \big(\prod_{i=1}^{m-1}s_i\big) d\big(T_m(x),x\big)+\dots + \big(\prod_{i=1}^{k+1}s_i\big) d\big(T_{k+2}(x),x \big)+ \big(\prod_{i=1}^{k}s_i\big) d\big(T_{k+1}(x), x \big). \end{split} \end{equation} For $i\in\mathbb{N}$, $T_i(x)\in C$ $\forall x\in C$, which implies that $d(T_i(x),x)\le M$ $\forall x\in C$, where $M$ is the diameter of $C$. Since $\sum_{k=1}^\infty \prod_{i=1}^k s_i < \infty$, Eq. (\ref{e1}) asserts that $d\big(\Psi_m(x),\Psi_k(x)\big) \to 0$ as $ k \to \infty$. That is, $\{\Psi_k(x) \}_{k \in \mathbb{N}}\subseteq C$ is a Cauchy sequence, and due to the completeness of $\{X,d\}$, it is convergent $\forall x\in C$. The uniqueness of the limit is derived by the equivalence of all trajectories as proved in Proposition \ref{Equivalence}. \end{proof} \begin{remark}\label{remark2} In view of (\ref{PhiPsi}, the result of Proposition \ref{BTp2} holds under the milder assumption that $C$ is an invariant set of $\{T_i\}_{i\ge I}$, for some $I\in\mathbb{N}$. \end{remark} \begin{remark}\label{remark3} {\bf Differences between forward and backward trajectories} \begin{enumerate} \item Note that if $T_i\to T$ and $T$ has Lipschitz constant $\mu<1$, then $$\sum_{k=1}^\infty \prod_{i=1}^k s_i < \infty ,$$ and both the forward and the backward trajectories converge. \item The condition $\lim_{k\to\infty}\prod_{i=1}^k s_i=0$ is sufficient for the asymptotic similarity result of both forward and backward trajectories. Under the stronger condition $\sum_{k=1}^\infty \prod_{i=1}^k s_i < \infty$ and the existence of a compact invariant set, we get convergence for the backward trajectories. \item In many cases, the backward trajectories converge, while the forward trajectories do not converge. To demonstrate this let the metric space be $\mathbb{R}$ with $d(x,y)=|x-y|$, and let us consider the simple sequence of contractive transformations $T_{2i-1}(x)=x/2$, $T_{2i}=x/2+c$, $i\ge 1$. The backward trajectories converge to the fixed point of $S_1=T_1\circ T_2$, which is $2c/3$. The forward trajectories have two accumulation points, which are the fixed point of $S_1$, i.e., $2c/3$, and the fixed point of $S_2=T_2\circ T_1$, which is $4c/3$. \end{enumerate} \end{remark} \section{\bf Trajectories of Sequences of Function Systems}\label{IFS} Generalizing the classical IFS we consider a sequence of function systems, SFS in short, and its trajectories. Let $(X,d)$ be a complete metric space. Consider an SFS $\{\mathcal{F}_i\}_{i\in \mathbb{N}}$ defined by $$ \mathcal{F}_i = \big\{X; f_{1,i}, f_{2,i}, \dots, f_{n_i,i} \big \},$$ where $f_{r,i}: X \to X$ are continuous maps. The associated set-valued maps are given by $$\mathcal{F}_i: \mathbb{H}(X) \to \mathbb{H}(X); \quad \mathcal{F}_i(A)= \cup_{r=1}^{n_i} f_{r,i}(A).$$ Denoting $s_{r,i}=\text{Lip}(f_{r,i})$, for $r=1,2,\dots,n_i$, we recall that as in (\ref{CLF}), the contraction factors of $\mathcal{F}_i$ in $(\mathbb{H}(X),h)$ is $L_{\mathcal{F}_i}=\max_{r=1,2,\dots, n_i} s_{r,i}\equiv s_i$. The traditional IFS theory deals with the attractor, namely, the set which is the `fixed-point' of a map $\mathcal{F}$. In this section we consider the trajectories of the SFS maps $\{\mathcal{F}_i\}_{i\in \mathbb{N}}$, which we refer to as forward and backward SFS trajectories \begin{equation} \Phi_k(A)=\mathcal{F}_k \circ \mathcal{F}_{k-1} \circ \dots \circ \mathcal{F}_1(A),\ \ \ \Psi_k(A)=\mathcal{F}_1 \circ \mathcal{F}_{2} \circ \dots \circ \mathcal{F}_k(A),\ \ k \in \mathbb{N}, \end{equation} respectively. As presented in Section \ref{sect1}, $\mathbb{H}(X)$, endowed with the Hausdorff metric $h$, is a complete metric space if $(X,d)$ is complete. The first observation is a corollary of Proposition \ref{Equivalence}: \begin{corollary}\label{IFSequiv}{\bf Asymptotic similarity of SFS trajectories}\\ Consider an SFS defined by $\mathcal{F}_i=\big\{X; f_{1,i}, f_{2,i}, \dots, f_{n_i,i} \big \}$, $i \in \mathbb{N}$, where $f_{r,i}: X \to X$ are Lipschitz maps. Further assume that the corresponding contraction factors $\{L_{\mathcal{F}_i}\}$ for the set-valued maps $\{\mathcal{F}_i\}$ on $(\mathbb{H}(X),h)$ satisfy $\lim_{ k \to \infty} \prod_{i=1}^k L_{\mathcal{F}_i} =0$. Then all the forward trajectories of $\{\mathcal{F}_i\}$ are asymptotically similar, and all the backward trajectories of $\{\mathcal{F}_i\}$ are asymptotically similar. \end{corollary} The next result is a corollary of Proposition \ref{forwardconvergence}: \begin{corollary}{\bf Convergence of forward SFS trajectories}\label{SFSforward}\\ Let $\{\mathcal{F}_i\}_{i\in \mathbb{N}}$ be as in Corollary \ref{IFSequiv}, with equal number of maps, $n_i=n$, and let $\mathcal{F}=\{X; f_r: r=1,2,\dots, n\}$. Assume that there exists $ C\subseteq X$, a compact invariant set of $\{f_{r,i}\}$ and that for each $r=1,2, \dots,n$, the sequence $\{f_{r,i}\}_{i\in \mathbb{N}}$ converges uniformly to $f_r$ on $C$ as $ i \to \infty$. Also assume that $\mathcal{F}$ has a contraction factor $L_{\mathcal{F}}<1$ . Then the forward trajectories $\{\Phi_k(A)\}$ converge for any initial set $A\subseteq C$ to the unique attractor of $\mathcal{F}$ . \end{corollary} \iffalse \begin{proof} Define $\tilde{\mathcal{F}}: \mathbb{H}(X) \to \mathbb{H}(X)$ by $$\tilde{\mathcal{F}}(A) = \cup_{r=1}^n \tilde{f_r} (A).$$ We have \begin{equation*} \begin{split} h\big(\mathcal{F}_i(A), \tilde{\mathcal{F}}(A)\big)=&~ h\Big(\cup_{r=1}^n f_{r,i} (A), \cup_{r=1}^n \tilde{f_r} (A)\Big)\\ \le &~ \max_{r=1,2,\dots,n} h \Big( f_{r,i} (A), \tilde{f_r} (A) \Big). \end{split} \end{equation*} Since $\{f_{r,i}\}_{i\in \mathbb{N}}$ converges uniformly to $\tilde{f_r}$ on $X$, Lemma \ref{nonstatIFSl1} in conjunction with the previous inequality implies that $\{\mathcal{F}_i\}_{i \in \mathbb{N}}$ converges $\tilde{\mathcal{F}}$ uniformly on $\mathbb{H}(X)$. Now the conclusion follows exactly as in Proposition \ref{nonstatIFSp2}. \end{proof} \fi \begin{remark}\label{remark4} The forward trajectories of the SFS in Corollary \ref{SFSforward} converge to the fractal set (attractor) associated with $\mathcal{F}$ (see \cite{B1}). This observation implies that forward trajectories of a converging SFS do not produce any new entities. \end{remark} Backward trajectories of SFS do not seem natural. However, as they converge under mild conditions, even if the SFS $\{\mathcal{F}_i\}_{i \in \mathbb{N}}$ does not converge to a contractive function system, their limits, or attractors, may constitute new entities, different from the known fractals which are self similar. \begin{corollary}\label{backSFS}{\bf Convergence of backward SFS trajectories} Let $\{\mathcal{F}_i\}_{i\in \mathbb{N}}$ and $\{L_{\mathcal{F}_i}\}$ be as in Corollary \ref{IFSequiv}. Assume there exists $ C\subseteq X$, a compact invariant set of $\{f_{r,i}\}$, $r=1,...,n_i$, $i\in \mathbb{N}$, and assume that $\sum_{k=1}^\infty \prod_{i=1}^k L_{\mathcal{F}_i}<\infty$. Then the backward trajectories $\{\Psi_k(A)\}$ converge, for any initial set $A\subseteq C$, to a unique set (attractor) $P\subseteq C$. \end{corollary} \section{\bf Hidden fractals} The fractal defined as the attractor of a single $\mathcal{F}=\{X; f_r: r=1,2,\dots, n\}$ has the property of self-similarity, i.e., its local shape is unchanged under certain contraction maps. The entities defined as the attractors of backward trajectories are more flexible. With a proper choice of $\{\mathcal{F}_i\}_{i\in \mathbb{N}}$ one can design different local behaviour under different contraction maps. Such a design relies on the observation that in a set defined by a sequence of contraction maps \begin{equation} \mathcal{G}_k(B)=\mathcal{F}_{1}\circ \mathcal{F}_{2}\circ \mathcal{F}_{3}\circ \cdot \cdot \cdot \circ \mathcal{F}_k(B), \end{equation} the first maps $\mathcal{F}_{1}$,$\mathcal{F}_{2}$,$\mathcal{F}_{3}$,... determine the global shape of the set, while the details of the local shape is determined by the last maps $\mathcal{F}_{k}$,$\mathcal{F}_{k-1}$,$\mathcal{F}_{k-2}$,.... To understand this note, e.g., that the set $\mathcal{F}_k(B)$ is undergoing a sequence of $k-1$ contraction maps. Therefore, its shape is not noticeable at larger scales. The arrangement of the set $\mathcal{G}_k(B)$ is finally fixed by the maps $\{f_{1,1},f_{1,2},...,f_{1,n}\}$ of $\mathcal{F}_{1}$. In general, if we scale by the contraction factor of $\Psi_k=\mathcal{F}_1\circ\mathcal{F}_2\circ ...\circ\mathcal{F}_k$, we shall see the behavior of the attractor of the backward trajectories of $\{\mathcal{F}_i\}_{i>k}$. \begin{example} As an example we consider an alternating sequence of maps $\{\mathcal{F}_i\}_{i\in \mathbb{N}}$, where for $10(j-1)<i\le 10j-5$, $\mathcal{F}_i$ is the function system generating cubic polynomial splines, and for $10j-5<i\le 10j$ it is the function system generating the Koch fractal. Both function systems are contractive of course. The forward trajectories do not converge (see Remark \ref{remark3}(3)), while any backward trajectory is rapidly converging. In Figure \ref{cubicKoch} we see on the left image of the global behavior of the limit which is a cubic spline behavior, and on the right image the local behavior near $x=0$, which is like the Koch fractal. In higher resolution we have smooth behavior again, and so on. Note that the scaling factor between the two images in Figure \ref{cubicKoch} is approximately $(1/2)^5$ which is the contraction factor of the first five mappings in $\{\mathcal{F}_i\}_{i\in \mathbb{N}}$. \begin{figure} \caption{The cubic-Koch attractor: "Smooth" in one scale and "Fractal" in another.} \label{cubicKoch} \end{figure} \end{example} \section{\bf IFS related to convergent stationary subdivision}\label{6} In this section we present IFS systems related to stationary subdivision schemes. The result in Subsections \ref{51}, \ref{52} are taken from \cite{SLG}. As in \cite{SLG} the discussion is restricted to the case $s=1$, i.e., curves in $\mathbb{R}^m$. \subsection{\bf $C^0$-convergent subdivision}\label{51} The connection between a $C^{0}$- convergent stationary subdivision for curves and IFS is presented in \cite{SLG}. In subdivision processes for curves ($s=1$) one starts with an initial control polygon $p^0$, and the limit curve depends upon $p^0\subset \mathbb{R}^m$. The attractor of the IFS does not depend upon the initial set. This dichotomy is resolved in \cite{SLG} by defining an IFS related to the subdivision operator $S$ which depends upon $p^0$. The resulting IFS then converges to the relevant subdivision limit from any initial starting set. To understand the extension to non-stationary subdivision, let us first elaborate the construction suggested in \cite{SLG} for the case of stationary subdivision for curves. As presented in Section \ref{BOS}, a stationary binary subdivision scheme for curves in the plane ($s=1,\ m=2$) is defined by two refinement rules that take a set of control points at level $k$, $p^k$, to a refined set at level $k+1$, $p^{k+1}$. For an infinite sequence $p^k$ this operation can be written in matrix form as \begin{equation}\label{S} p^{k+1}=Sp^k, \end{equation} where $S\equiv S_a$ is a two-slanted infinite martix with rows representing the two refinement rules, namely $S_{i,j}=a_{i-2j}$, and $p^k$ is a matrix with $m$ columns and an infinite number of rows. Given a finite set of control points, $\{p^0_j\in\mathbb{R}^m\}_{j=1}^n$ at level $0$, we are interested in computing the limit curve defined by these points. For a non-empty limit curve, $n$ should be larger than the support size $|\sigma(a)|$. We consider the sub-matrix of $S$ which operates on these points, and we cut from it two square $n\times n$ sub-matrices, $S_1$ and $S_2$, which define all the $n_1$ resulting control points at level $1$. Note that $S_1$ defines the transformation to the first $n$ points at level $1$, and $S_2$ defines the transformation to the last $n$ points at level $1$. Of course there can be an overlap between these two vectors of points, namely $n_1< 2n$. Some examples of these sub-matrices are given in \cite{SLG}. We provide below the explicit forms of $S_1$ and $S_2$: We distinguish two types of masks, an even mask, with $2\ell$ elements, $a_{-\ell+1},...,a_\ell$, and an odd mask with $2\ell+1$ elements, $a_{-\ell},...,a_\ell$. For both cases we assume $n>\ell+1$. For both the even and the odd masks \begin{equation}\label{S1} S_1=\{a_{i-2j}\}_{i=\ell+1,\ j=1}^{\ \ \ \ell+n,\ \ \ n}. \end{equation} $S_2$ is different for odd and even masks. For an even mask \begin{equation}\label{S2E} S_2=\{a_{i-2j}\}_{i=n-\ell+2,\ j=1}^{\ \ \ 2n-\ell+1,\ \ n}, \end{equation} and for an odd mask \begin{equation}\label{S2O} S_2=\{a_{i-2j}\}_{i=n-\ell+3,\ j=1}^{\ \ \ 2n-\ell+2,\ \ n}. \end{equation} Repeated applications of $S_1$ and $S_2$, define all the control points at all levels. Therefore, \begin{equation}\label{pinfty} \bigcup_{i_1,i_2,...,i_k\in \{1,2\}}S_{i_k},...,S_{i_2}S_{i_1}p^0 \to p^\infty,\ \ \ as\ \ k \to \infty, \end{equation} where $p^\infty$ is the set of points on the curve defined by the subdivision process starting with $p^0$. \begin{remark}\label{Union}{\bf Union of vectors of points}\\ $p^0$ is a vector of $n$ points in $\mathbb{R}^m$, and thus each $S_{i_k},...,S_{i_2}S_{i_1}p^0$ is a vector of $n$ points in $\mathbb{R}^m$, which we regard as a set of $n$ points in $\mathbb{R}^m$ . By $\bigcup S_{i_k},...,S_{i_2}S_{i_1}p^0$ we mean the set in $\mathbb{R}^m$ which is the union of all these sets. \end{remark} \begin{remark}\label{PTP}{\bf Parameterizing the points in $p^\infty$}\\ To order the points of the set $p^\infty$ we introduce the following parametrization. An infinite sequence $\eta=\{i_k\}_{k=1}^\infty$, $i_k\in\{1,2\}$ defines a vector of $n$ points in $\mathbb{R}^m$ \begin{equation} \lim_{k\to \infty}S_{i_k},...,S_{i_2}S_{i_1}p^0=(q_1,...,q_n)^t,\ \ q_i\in \mathbb{R}^m. \end{equation} In case of a $C^{0}$-convergent subdivision, the differences between adjacent points tend to zero \cite{DGL}. Therefore, all these $n$ points are the same point, \begin{equation}\label{samepoint} \lim_{k\to \infty}S_{i_k},...,S_{i_2}S_{i_1}p^0=(q_\eta,...,q_\eta)^t,\ \ q_\eta\in \mathbb{R}^m. \end{equation} We attach this point $q_\eta$ to the parameter value $x_\eta=\sum_{k=1}^\infty (i_k-1)2^{-k}\in [0,1]$. \end{remark} \subsection{\bf IFS related to stationary subdivision}\label{52} Here the metric space is $\{\mathbb{R}^n,d\}$ with $d(x,y)=\|x-y\|_2$, where $\|\cdot\|_2$ is the Euclidean norm. The observation (\ref{pinfty}) leads in \cite{SLG} to the definition of an IFS with two maps on $X=\mathbb{R}^n$ (row vectors) \begin{equation}\label{f1f2} f_r(A)=AP^{-1}S_r P,\ \ \ r=1,2, \end{equation} where $P$ is an $n\times n$ matrix defined as follows: \begin{enumerate} \item The first $m$ columns of $P$ are the $n$ given control points $p^0$, which are points in $\mathbb{R}^m$. \item The last column is a column of $1$'s. \item The rest of the columns are defined so that $P$ is non-singular. We assume here that the control points $p^0$ do not all lie on an $m-1$ hyper plane so that the first $m$ columns of $P$ are linearly independent, and that the column of $1$'s is independent of the first $m$ columns. \end{enumerate} This special choice of $P$, together with the special definition of $f_1,f_2$ in (\ref{f1f2}), yields the following essential observations: \begin{itemize} \item Since $S_1$ and $S_2$ have eigenvalue $1$, with right eigenvector $(1,1,...,1)^t$ which is also the last column of $P$, then \begin{equation}\label{PSP} P^{-1}S_rP =\left( \begin{array}{c|c} G_r & 0 \\ \hline v&1 \end{array} \right) , \ \ r=1,2, \end{equation} where $G_r$ are $(n-1)\times(n-1)$ matrices. Denoting by $Q^{n-1}$ the $n-1$ dimensional hyperplane (flat) of vectors of the form $(x_1,...,x_{n-1},1)$, it follows from (\ref{PSP}) that $f_r\ :\ Q^{n-1}\to Q^{n-1}$, $r=1,2$. \item By applying the IFS iterations to the set $A=P$, using equation (\ref{FX}), we identify the candidate attractor as \begin{equation}\label{Pinfty} P^\infty=\lim_{k\to\infty}\bigcup_{i_1,i_2,...,i_k\in \{1,2\}}S_{i_k},...,S_{i_2}S_{i_1}P. \end{equation} Similarly to Remark \ref{Union}, the rows of $P^\infty$ constitute a set of points in $\mathbb{R}^n$. By the structure of $P$, and in view of (\ref{pinfty}), we observe that $p^\infty$ is the set of points in $\mathbb{R}^m$ defined by the first $m$ components of the points (in $\mathbb{R}^n$) of $P^\infty$. \end{itemize} The above observations lead to the main result in \cite{SLG}, stated in the Theorem below. The original proof in \cite{SLG} of this theorem has a flaw. We provide here a proof which serves us later in the discussion on non-stationary subdivision. \begin{theorem}\label{ThSLG} Let $S_a$ be a $C^{0}$-convergent subdivision, and let $p^0$ be a sequence of initial control points. Define the IFS $\mathcal{F}=\{X; f_1,f_2\}$ on $Q^{n-1}$, with $f_1,f_2$ defined in (\ref{f1f2}) and $S_1,S_2$ defined in (\ref{S1})-(\ref{S2O}). Then the IFS converges to a unique attractor in $Q^{n-1}$, and the first $m$ components of the points of this attractor constitute the limit curve $p^\infty=S_a^\infty p^0$. \end{theorem} \begin{proof} Since all the eigenvalues of $S_1$ and $S_2$ which differ from $1$ are smaller than $1$, it follows that $\rho(G_r)<1$, $r=1,2$, where $\rho(G)$ is the spectral radius of $G$. This does not directly imply that the maps $f_1,f_2$ are contractive on $Q^{n-1}$. Following Remark \ref{remarkIFS}(2), to prove convergence of the IFS $\mathcal{F}$, we show that there exists an $\ell$-term composition of $\mathcal{F}$ is a contraction map. We notice that such an $\ell$-term composition of $\mathcal{F}$ is itself an IFS, with $2^\ell$ functions of the form \begin{equation}\label{ftau} f_\eta(A)=AP^{-1}S_{i_\ell}...S_{i_2}S_{i_1}P,\ \ \ \eta\in I_\ell, \end{equation} where $I_\ell=\{\eta=\{i_j\}_{j=1}^\ell$, $i_j\in\{1,2\}\}$. $S_a$ is $C^{0}$-convergent, thus by Definition \ref{C0conv} it is also uniformly convergent. It follows from (\ref{samepoint}) that for any $\epsilon>0$, there exists $ \ell=\ell(\epsilon)$ such that for any $\eta\in I_\ell$ \begin{equation}\label{stauP} S_{i_\ell}...S_{i_2}S_{i_1}P=Q_\eta+E_\eta, \end{equation} where $Q_\eta$ is an $n\times n$ matrix of constant columns, and $\|E_\eta\|_\infty<\epsilon$. The last column of $Q_\eta$ is $(1,1,...,1)^t$, and the last column of $E_\eta$ is the zero column. Recalling that the last column of $P$ is the constant vector of $1$'s, and since $P^{-1}P=I_{n\times n}$, it follows that \begin{equation}\label{Pm1stauP} P^{-1}S_{i_\ell}...S_{i_2}S_{i_1}P= \begin{pmatrix} 0&0&...&0&0\\ 0&0&...&0&0\\ . & . & . & . & .\\ . & . & . & . & .\\ 0&0&...&0&0\\ q_{\eta,1}&q_{\eta,2}&...&q_{\eta,n-1}&1\\ \end{pmatrix} +P^{-1}E_\eta= \left( \begin{array}{c|c} G_\eta & 0 \\ \hline q_{\eta}&1 \end{array} \right), \end{equation} Where $q_{\eta_j}(1,1,...,1)^t$ is the $j$-th column of $Q_\eta$. It follows that $\|G_\eta\|_2\le \epsilon\|P^{-1}\|_2$. Next we show that for $\epsilon$ small enough, $f_\eta$ is contractive with respect to the Euclidean norm in $Q^{n-1}$. Indeed, for $x,y\in \mathbb{R}^{n-1}$, $(x,1),(y,1)\in Q^{n-1}$, and \begin{equation}\label{fetaxy} d(f_\eta((x,1)),f_\eta((y,1)))=\|f_\eta((x,1)-(y,1))\|_2=\|f_\eta((x-y,0))\|_2=\|(x-y)^tG_\eta(x-y)\|_2. \end{equation} Choosing $\epsilon$ such that $\epsilon \|P^{-1}\|_2<1$, it follows that for all $ \eta\in I_{\ell(\epsilon)}$, the map $f_\eta$ is contractive on $Q^{n-1}$, and the IFS defined by $\mathcal{F}$ is convergent. \end{proof} \begin{remark} Theorem \ref{ThSLG} reveals the fractal nature of curves generated by subdivision. However, the self-similarity property of these curves is not achieved in $\mathbb{R}^m$. The self-similarity property is of $p^\infty$, as a set in $Q^{n-1}$. $p^\infty$ is the projection on $\mathbb{R}^m$ of this self similar entity in $Q^{n-1}$. \end{remark} \subsection{\bf A basis for convergent stationary subdivision} As presented above, and earlier in \cite{SLG}, the definition of an IFS for a $C^0$-convergent stationary subdivision involves the specific given control points $p^0$. We observe that it is enough to consider one basic IFS, and its attractor can serve as a basis for generating the limit of the subdivision process for any given $n$ control points $p^0$. Instead of the matrix $P$, we may define any other non-singular $n\times n$ matrix with a last column of $1$'s. We choose the matrix \begin{equation}\label{H} H= \begin{pmatrix} 1& 0& 0 & 0&...&1\\ 0 &1& 0& 0&... &1 \\ 0 &0& 1& 0&... &1 \\ \cdot&\cdot&\cdot&\cdot&\cdot&1\\ \cdot&\cdot&\cdot&\cdot&\cdot&1\\ 0 &0& 0&... &1&1 \\ 0 &0& 0&... &0&1 \\ \end{pmatrix}, \end{equation} and define the IFS with \begin{equation}\label{f1f2H} f_r(A)=AH^{-1}S_r H,\ \ \ r=1,2, \end{equation} As shown above, the attractor of this IFS is the union of $n\times n$ matrices \begin{equation}\label{Hinfty} \mathcal{H}^\infty=\lim_{k\to\infty}\bigcup_{i_1,i_2,...,i_k\in \{1,2\}}S_{i_k},...,S_{i_2}S_{i_1}H. \end{equation} In view of Remark \ref{Union}, $\mathcal{H}^\infty\subset Q^{n-1}$. For any given control points $p^0$ we can simply calculate $p^\infty$ as the set \begin{equation}\label{pinftyH} p^\infty=\mathcal{H}^\infty H^{-1}p^0. \end{equation} \iffalse \subsection{\bf A class of $h$-convergent subdvision} The IFS machinery enables us to identify a class of $h$-convergent subdivision schemes (Definition \ref{hconv}). Using the above idea on a basis for stationary subdivision we can present an almost inverse theorem to Theorem \ref{ThSLG}: \begin{theorem}\label{thH} Let $S_a$ be a subdivision scheme, and assume there exists a non-singular matrix $H$ such that the IFS defined by (\ref{f1f2H}) is contractive in some $Q\subseteq \mathbb{R}^n$. Then, $S_a$ is $h$-convergent. \end{theorem} \begin{proof} The proof follows directly from (\ref{Hinfty}),(\ref{pinftyH}). \end{proof} The next result presents a specific wide class of such $h$-convergent subdivision schemes. The proof involves an extension of the IFS construction in \cite{SLG}. \begin{theorem}\label{thsvv} Let $S_a$ be a subdivision scheme such that $S_1$ and $S_2$ have a common invariant subspace $V$ in $\mathbb{R}^n$, $dim(V)=\ell$, such that \begin{equation}\label{svv} S_iv=v,\ \ \ \forall v\in V,\ \ \ \ i=1,2, \end{equation} and each has $n-\ell$ eigenvalues satisfying $|\lambda|<1$. Then $S_a$ is $h$-convergent. \end{theorem} \begin{proof} The proof requires an appropriate definition of the matrix $H$ appearing in the construction of the IFS in (\ref{f1f2H}). W.l.o.g. we assume $n>m+\ell$, and let $v_1,...,v_\ell$ be a basis of $V$. Define $H$ to be an $n\times n$ non-singular matrix with $v_1,...,v_\ell$ as its last $\ell$ columns. Since $S_1$ and $S_2$ have eigenvalue $1$, with eigenvectors $v_1,...,v_\ell$ which are also the last columns of $H$, then \begin{equation}\label{HSH2} H^{-1}S_rH= \left({\bf B}_{n\times n-\ell} \, \middle| \, \frac{{\bf 0}_{(n-\ell)\times\ell}}{I_{\ell\times\ell}} \right) \end{equation} Denoting by $Q^{n-\ell}$ the $n-\ell$ dimensional affine subspace of vectors of the form $(x_1,...,x_{n-\ell},1,...,1)$, it follows from (\ref{HSH2}) that $f_r:Q^{n-\ell}\to Q^{n-\ell}$, $r=1,2$, where $f_1$, $f_2$ are defined by (\ref{f1f2H}). Furthermore, since all the other eigenvalues of $S_1$ and $S_2$ are smaller than $1$, the maps $f_1,f_2$ are contractive on $Q^{n-\ell}$. Using Theorem \ref{thH} we conclude that $S_a$ is $h$-convergent. \end{proof} \begin{remark} The above theorem covers the case where $S_1$ and $S_2$ have eigenvalue $1$, with eigenvector $(1,1,...,1)^t$, and all other eigenvalues of $S_1$ and $S_2$ are smaller than $1$, and yet $S_a$ is not $C^0$-convergent. This happens if $\rho(S_1,S_2)\ge 1$, where $\rho(S_1,S_2)$ denotes the joint spectral radius of $S_1$ and $S_2$ (see e.g. \cite{DL}). Returning to Remark \ref{PTP}, if $S_a$ is $h$-convergent and not $C^0$-convergent, one can not guarantee the assignment of a unique point to a given parameter $x_\eta\in[0,1]$. Hence, the set $p^\infty$ may not be parameterizable, or is not representing a $C^0$ curve. \end{remark} \fi \section{\bf SFS trajectories associated with non-stationary subdivision} This research was motivated by the idea to adapt the framework of the previous section to non-stationary subdivision processes. In binary non-stationary subdivision, as shown in (\ref{sak}), the refinement rules may depend upon the refinement level, and can be written in matrix form as \begin{equation}\label{Sk} p^{k+1}=S^{[k]}p^k, \end{equation} where each $S^{[k]}\equiv S_{a^{[k]}}$ is a ``two-slanted" matrix. As demonstrated in \cite{DL1}, non-stationary subdivision processes can generate interesting limits which cannot be generated by stationary schemes, e.g., exponential splines. Interpolatory non-stationary subdivision schemes can generate new types of orthogonal wavelets, as shown in \cite{DKLR}. In the following we discuss the possible relation between non-stationary subdivision processes and SFS processes. A necessary condition for the convergence (to a continuous limit) of a stationary subdivision scheme is the {\bf constants reproduction property}, namely, \begin{equation}\label{Seeqe} Se=e, \ \ \ \ e=(...,1,1,1,1,1,...)^t . \end{equation} As explained in Section \ref{6}, this condition is used in \cite{SLG} in order to show that the maps defined in (\ref{f1f2}) are contractive on $Q^{n-1}$. This condition is not necessarily satisfied by converging non-stationary subdivision schemes. It is also not a necessary condition for the construction of SFS related to non-stationary subdivision. \subsection{\bf Constructing SFS mappings for non-stationary subdivision} In the following we assume that the supports of the masks $a^{[k]}$, $|\sigma(a^{[k]})|$, are of the same size, which is at most the number of initial control points. As in the stationary case, for a given set of control points, $\{p^0_j\}_{j=1}^n$, we define for each $k$ the two square $n\times n$ sub-matrices of each $S^{[k]}$, $S^{[k]}_1$ and $S^{[k]}_2$, in the same way as for a stationary scheme, by equations (\ref{S1}), (\ref{S2E}), (\ref{S2O}). The points generated by the subdivision process are obtained by applying $S^{[1]}_1$ and $S^{[1]}_2$, to the initial control points vector $p^0$, and then applying $S^{[2]}_1$ and $S^{[2]}_2$ to the two resulting vectors, and so on. The set of points generated at level $k$ of the subdivision process is given by \begin{equation}\label{nspk} p^k=\bigcup_{i_1,i_2,...,i_k\in \{1,2\}}S^{[k]}_{i_k},...,S^{[2]}_{i_2}S^{[1]}_{i_1}p^0\ . \end{equation} If the subdivision is $C^0$-convergent or $h$-convergent, then \begin{equation}\label{pktopinfty} p^k\to p^\infty\ \ \ as\ \ k\to\infty, \end{equation} in the sense of Definitions \ref{C0conv}, \ref{hconv} respectively. Here $p^\infty$ is the set of points defined by the non-stationary subdivision process starting with $p^0$. Now we define the SFS $\{\mathcal{F}_k\}$, where $\mathcal{F}_k=\big\{X; f_{1,k}, f_{2,k} \big \}$, with the level dependent maps \begin{equation}\label{f1f2k} f_{r,k}(A)=AP^{-1}S^{[k]}_r P,\ \ \ r=1,2, \end{equation} where $P$ is the $n\times n$ matrix defined as in the stationary case. \begin{remark}\label{QorR} If the non-stationary scheme satisfies the constant reproduction property at every subdivision level, then all the mappings in the SFS map $Q^{n-1}$ into itself (by (\ref{PSP})). If not, then the mappings are considered as maps on $\mathbb{R}^n$. \end{remark} Let us now follow a forward trajectory and a backward trajectory of $\Sigma\equiv\{\mathcal{F}_k\}$, starting from $A\subset \mathbb{R}^n$: $$\mathcal{F}_k(A)=f_{1,k}(A)\cup f_{2,k}(A)=AP^{-1}S^{[k]}_1P\cup AP^{-1}S^{[k]}_2P,$$ and $$\mathcal{F}_{j}(\mathcal{F}_k(A))=f_{1,j}(AP^{-1}S^{[k]}_1P\cup AP^{-1}S^{[k]}_2P)\cup f_{2,j}(AP^{-1}S^{[k]}_1P\cup AP^{-1}S^{[k]}_2P).$$ We note that $$f_{r,j}(AP^{-1}S^{[k]}_iP)=AP^{-1}S^{[k]}_iPP^{-1}S^{[j]}_rP=AP^{-1}S^{[k]}_iS^{[j]}_rP.$$ Therefore, $$\mathcal{F}_{j}(\mathcal{F}_k(A))=\bigcup_{r,i\in\{1,2\}}AP^{-1}S^{[k]}_iS^{[j]}_rP.$$ In the same way it follows that at the $k$th step of a forward trajectory of $\Sigma$ we generate the set \begin{equation}\label{FASSP} \mathcal{F}_{k}\circ\mathcal{F}_{k-1}\circ ...\circ\mathcal{F}_{2}\circ\mathcal{F}_1(A)=\bigcup_{i_1,i_2,...,i_k\in \{1,2\}}AP^{-1}S^{[1]}_{i_1},...,S^{[k-1]}_{i_{k-1}}S^{[k]}_{i_k}P. \end{equation} Similarly, the set generated at the $k$th step of a backward trajectory is \begin{equation}\label{ASSP} \mathcal{F}_{1}\circ\mathcal{F}_2\circ ...\circ\mathcal{F}_{k-1}\circ\mathcal{F}_k(A)=\bigcup_{i_1,i_2,...,i_k\in \{1,2\}}AP^{-1}S^{[k]}_{i_k},...,S^{[2]}_{i_2}S^{[1]}_{i_1}P. \end{equation} For the special backward trajectory with $A=P$ we obtain \begin{equation}\label{PSSP} \mathcal{F}_{1}\circ\mathcal{F}_2\circ ...\circ\mathcal{F}_{k-1}\circ\mathcal{F}_k(P)=\bigcup_{i_1,i_2,...,i_k\in \{1,2\}}S^{[k]}_{i_k},...,S^{[2]}_{i_2}S^{[1]}_{i_1}P. \end{equation} If the non-stationary subdivision scheme is either $C^0$-convergent or $h$-convergent, then, in view of (\ref{nspk}), it follows that the first $m$ components in this special trajectory converge to the limit $p^\infty$ of $\{S_{a^{[k]}}\}$, starting with $p^0$. The challenging question is finding for which classes of non-stationary schemes {\bf all} the backward trajectories converge to the same limit. As we show later, and as explained in Remark \ref{remark4}, forward trajectories of $\Sigma$ are less interesting. \subsection{\bf Attractors of forward and backward SFS trajectories for non-stationary subdivision} We consider forward and backward SFS trajectories for several cases of non-stationary subdivision schemes: \begin{itemize} \item[Case (i)] A $C^{0}$-convergent non-stationary scheme $\{S_{a^{[k]}}\}$. \item[Case (ii)] A non-stationary scheme $\{S_{a^{[k]}}\}$ satisfying the constants reproduction property, with masks of the same support, converging to a mask $a$ of a $C^{0}$-convergent subdivision, i.e., $\sigma(a^{[k]})=\sigma(a)$, and \begin{equation}\label{aktoa} \lim_{k\to \infty}a^{[k]}_j=a_j,\ \ \ j\in\sigma(a). \end{equation} \item[Case (iii)] A non-stationary scheme $\{S_{a^{[k]}}\}$ with masks $\{a^{[k]}\}$ satisfying the constants reproduction property, and corresponding $\{\mathcal{F}_k\}$ satisfying $\sum_{\ell=1}^\infty \prod_{k=1}^\ell L_{\mathcal{F}_k}<\infty$. \end{itemize} In Case (i) we do not assume that the non-stationary subdivision scheme reproduces constants, nor do we assume that the masks $\{a^{[k]}\}$ converge to a limit mask. Therefore, the associated SFS maps do not necessarily map $Q^{n-1}$ to itself. We do assume that the non-stationary scheme is $C^0$-convergent. \begin{theorem}\label{nsprop2} Let $\{S_{a^{[k]}}\}$ be a non-stationary $C^0$-convergent subdivision scheme, and let $\Sigma=\{\mathcal{F}_k\}_{k=1}^\infty$ be the SFS defined in (\ref{f1f2k}). Then the backward trajectories of $\Sigma$ starting with $A\subset Q^{n-1}$ converge to a unique attractor. The first $m$ components of the points of this attractor constitute the limit curve (in $\mathbb{R}^m$) of the non-stationary scheme defined in (\ref{nspk})-(\ref{pktopinfty}). \end{theorem} \begin{proof} Here we consider the SFS as mappings from $\mathbb{R}^n$ to itself. Since $\{S_{a^{[k]}}\}$ converges, it immediately follows from (\ref{PSSP}) that the backward trajectory of $\Sigma$ initialized with $A=P$ converge. We would like to show that all the backward trajectories of $\Sigma$ initialized with an arbitrary set of points $A\subset Q^{n-1}$ converge to the same limit. We recall that the first $m$ columns of $P$ are the control points $p^0$. Starting the backward trajectory of $\Sigma$ with $A=P$, it follows, as discussed in Remark \ref{PTP}, that an infinite sequence $\eta=\{i_k\}_{k=1}^\infty$, $i_k\in\{1,2\}$, defines a vector of $n$ equal points in $\mathbb{R}^m$ \begin{equation} q=\lim_{k\to \infty}S^{[k]}_{i_k},...,S^{[2]}_{i_2}S^{[1]}_{i_1}p^0=(q_\eta,...,q_\eta)^t,\ \ q_\eta=(q_{\eta,1},...,q_{\eta,m}), \end{equation} attached to a parameter value $x_\eta=\sum_{k=1}^\infty (i_k-1)2^{-k}$. Starting the backward trajectory with a general set $A$ in $Q^{n-1}$, and following the same sequence $\sigma$, it follows from (\ref{ASSP}) that the limit is the $n\times m$ matrix $AP^{-1}q$. We recall that the last column of $P$ is a constant vector of $1$'s. Since each column of $q$ is a constant vector of length $n$, and since $P^{-1}P=I_{n\times n}$, it follows that \begin{equation}\label{pm1q} P^{-1}q= \begin{pmatrix} 0&0&...&0\\ 0&0&...&0\\ & . & & . \\ & . & & . \\ 0&0&...&0\\ q_{\eta,1}&q_{\eta,2}&...&q_{\eta,m}\\ \end{pmatrix}. \end{equation} For any row vector of the form $r=(r_1,r_2,...,r_{n-1},1)\in Q^{n-1}$, it follows from (\ref{pm1q}) that $rP^{-1}q=q_\eta$. If $A$ represents a set of $N$ points in $Q^{n-1}$, i.e., the $n$th element in each row of $A$ is $1$, it follows that $AP^{-1}q$ represent $N$ copies of the same point $q_\eta$. That is, for any sequence of indices $\eta$, the limit of the corresponding trajectory is the same for any initial $A\subset Q^{n-1}$, and it is the limit point of the non-stationary subdivision attached to the parameter value $x_\eta$. Comparing the trajectories displayed in (\ref{ASSP}) and (\ref{PSSP}), it follows that \begin{equation}\label{limeqlim} \lim_{k\to\infty}\mathcal{F}_{1}\circ\mathcal{F}_2\circ ...\circ\mathcal{F}_{k-1}\circ\mathcal{F}_k(A)=AP^{-1} \lim_{k\to\infty}\mathcal{F}_{1}\circ\mathcal{F}_2\circ ...\circ\mathcal{F}_{k-1}\circ\mathcal{F}_k(P). \end{equation} Interchanging the order of $\lim_{k\to\infty}$ and $\bigcup_{i_1,i_2,...,i_k\in \{1,2\}}$ we conclude that both trajectories converge to the same limit for any $A\subset Q^{n-1}$. \end{proof} In Case (ii) we consider a non-stationary scheme $\{S_{a^{[k]}}\}$ with masks converging to a mask $a$, \begin{equation}\label{aktoa2} \lim_{k\to \infty}a^{[k]}_j=a_j,\ \ \ j\in\sigma(a), \end{equation} with $S_a$ a convergent stationary scheme. Thus \begin{equation}\label{fikktofi} \lim_{k\to \infty} f_{r,k}=f_r,\ \ \ r=1,2. \end{equation} Following Corollaries \ref{SFSforward} and \ref{backSFS}, we are now ready to discuss the convergence of forward and backward trajectories of $\Sigma\equiv\{\mathcal{F}_k\}$. \begin{corollary}\label{nscor1}{\bf Forward trajectories of $\{\mathcal{F}_k\}$:} Let $\{S_{a^{[k]}}\}$ have the constant reproducing property, with masks $\{a^{[k]}\}$ of the same support size converging to the mask of a $C^{0}$-convergent subdivision scheme $S_a$. Then the forward trajectories of the SFS $\{\mathcal{F}_k\}$ defined above converge to the attractor $P^\infty$ of the IFS related to $S_a$. \end{corollary} \begin{proof} Let $\mathcal{F}$ be the IFS related to $S_a$, and let $\{\mathcal{F}_k\}$ be the SFS related to the non-stationary scheme $\{S_{a^{[k]}}\}$. Following the proof of Theorem \ref{ThSLG}, there exists an $\ell$ such that the $\ell$-term composition of $\mathcal{F}$, namely, $\mathcal{G}=\mathcal{F}\circ\mathcal{F}\circ ... \circ\mathcal{F}$, is a contraction map. Let \begin{equation} \mathcal{G}_k=\mathcal{F}_{k\ell}\circ\mathcal{F}_{k\ell-1}\circ ... \circ\mathcal{F}_{(k-1)\ell+1},\ \ k\ge 1. \end{equation} Thus, $\mathcal{G}_k\to \mathcal{G}$ as $k\to\infty$, and $\exists K$ such that the maps $\{\mathcal{G}_k\}_{k\ge K}$ are contractive. In order to apply Corollary \ref{SFSforward} we need to show the existence of an invariant set $C$ for the maps $\{\mathcal{G}_k\}$. Applying Example \ref{Ex1} we derive the existence of an invariant set $C_K$ for the maps $\{\mathcal{G}_k\}_{k\ge K}$. $C_K$ is a ball of radius $r$ in $Q^{n-1}$, centered at $q=(0,0,...,0,1)^t$. By Remark \ref{remarkC}, any ball of radius $R>r$, centered at $q$, is also an invariant set of $\{\mathcal{G}_k\}_{k\ge K}$. Using this observation in Corollary \ref{SFSforward}, implies that all forward trajectories of $\{\mathcal{G}_k\}_{k\ge K}$ converge from any set in $Q^{n-1}$ to the attractor of $\mathcal{G}$. In particular, for any set $A\in Q^{n-1}$, we can start the forward trajectory of $\{\mathcal{G}_k\}_{k\ge K}$ with the set \begin{equation} \mathcal{G}_{K-1}\circ\mathcal{G}_{K-2}\circ ...\circ\mathcal{G}_{2}\circ\mathcal{G}_1(A), \end{equation} and conclude that all forward trajectories of $\{\mathcal{G}_k\}_{k\ge 1}$ converge from any point in $Q^{n-1}$ to the attractor of the IFS related to $S_a$. \end{proof} \begin{remark} \begin{enumerate} \item It is important to note that in case the non-stationary scheme does not reproduce constants, the result in Corollary \ref{nscor1} does not necessarily hold. To see this it is enough to consider the simple case where $S_i^{[k]}=S_i$, $i=1,2$, for $k\ge 2$, and only $S_1^{[1]}$ and $S_2^{[1]}$ are different, and the corresponding $S_{a^{[1]}}$ does not reproduce constants. Then, in view of the expression (\ref{FASSP}), the forward trajectory with $A=P$ converges to $S_1^{[1]}P^\infty\cup S_2^{[1]}P^\infty\ne P^\infty$, where $P^\infty$ is the attractor corresponding to the stationary subdivision with $S_1$ and $S_2$. \item The important conclusion from the above corollary is that forward trajectories of an SFS related to a non-stationary subdivision with masks converging to the mask of a $C^0$-convergent subdivision do not produce any new attractors. On the other hand, the backward trajectories related to such non-stationary subdivision schemes do generate new interesting curves. See e.g. \cite{DL}. \item Under the conditions of Corollary \ref{nscor1}, it is proved in \cite{CDMM} that the non-stationary subdivision $\{S_{a^{[k]}}\}$ is $C^0$- convergent. Therefore, by Theorem \ref{nsprop2} the backward trajectories of $\Sigma$ starting with $A\subset Q^{n-1}$ converge to a unique attractor. This result follows from Corollary \ref{backSFS} as well. \end{enumerate} \end{remark} In case (iii), the mask of the subdivision schemes $\{S_{a^{[k]}}\}$ do not have to converge to a mask of a $C^0$-convergent subdivision scheme. We still assume here that the non-stationary scheme reproduces constants, i.e., $(1,1,...,1)^t$ is an eigenvector of $S_1^{[k]}$ and $S_2^{[k]}$ with eigenvalue $1$, for $k\ge 1$. Let us denote by $\mu(S_{a^{[k]}})$ the maximal absolute value of the eigenvalues of $S_1^{[k]}$ and $S_2^{[k]}$ which differ from $1$. \begin{corollary}\label{nscor5} Consider a constant reproducing non-stationary scheme $\{S_{a^{[k]}}\}$ and let $\{\mathcal{F}_k\}_{k=1}^\infty$ be the SFS defined by (\ref{f1f2k}). If $\ \sum_{\ell=1}^\infty \prod_{k=1}^\ell L_{\mathcal{F}_k}<\infty$ then: \begin{enumerate} \item All the backward trajectories of $\{\mathcal{F}_k\}$ converge to a unique attractor in $Q^{n-1}$. \item The first $m$ components of this attractor constitute the $h$-limit (in $\mathbb{R}^m$) of the scheme applied to the initial control polygon $p^0$. \end{enumerate} \end{corollary} The proof follows directly from Corollary \ref{backSFS}. \subsection{Numerical Examples} \begin{example}(Case (i) and case (ii)) For our first example we consider a non-stationary subdivision which produces exponential splines. It is convenient to view the mask coefficients $\{a_i\}$ of a subdivision scheme as the coefficients of a Laurent polynomial $$a(z)=\sum_ia_iz^i.$$ The subdivision mask for generating cubic polynomial splines is $$a(z)={(1+z)^4\over 8}={1\over 8}+{1\over 2}z+{3\over 4}z^2+{1\over 2}z^3+{1\over 8}z^4.$$ Following \cite{SLG}, the corresponding matrices $P$, $S_1$ and $S_2$, for $n=5$, are $$ P= \begin{pmatrix} x_1&y_1&1&0&1\\ x_2&y_2&0&1&1\\ x_3&y_3&0&0&1\\ x_4&y_4&0&0&1\\ x_5&y_5&0&0&1\\ \end{pmatrix}, \ \ \ S_1= \begin{pmatrix} {1\over 2}&{1\over 2}&0&0&0\\ {1\over 8}&{3\over 4}&{1\over 8}&0&0\\ 0&{1\over 2}&{1\over 2}&0&0\\ 0&{1\over 8}&{3\over 4}&{1\over 8}&0\\ 0&0&{1\over 2}&{1\over 2}&0\\ \end{pmatrix}, \ \ \ S_2= \begin{pmatrix} 0&{1\over 2}&{1\over 2}&0&0\\ 0&{1\over 8}&{3\over 4}&{1\over 8}&0\\ 0&0&{1\over 2}&{1\over 2}&0\\ 0&0&{1\over 8}&{3\over 4}&{1\over 8}\\ 0&0&0&{1\over 2}&{1\over 2}\\ \end{pmatrix}. $$ A related non-stationary subdivision is defined by the sequence of mask polynomials \begin{equation}\label{ak} a^{[k]}(z)={b_k(1+z)(1+c_kz)^3}, \ \ \ \text{with} \ \ c_k=\text{exp}({\lambda 2^{-k-1}}),\ b_k=1/(1+c_k)^3. \end{equation} The non-stationary subdivision $\{S_a^{[k]}\}$ generates exponential splines with integer knots, piecewise spanned by $\{1,e^{\lambda x},xe^{\lambda x},x^2e^{\lambda x}\}.$ The matrices $S^{[k]}_1,S^{[k]}_2$ are $$ S^{[k]}_1=b_k \begin{pmatrix} 3c_k^2+c_k^3&1+3c_k&0&0&0\\ c_k^3&3(c_k+c_k^2)&1&0&0\\ 0&3c_k^2+c_k^3&1+3c_k&0&0\\ 0&c_k^3&3(c_k+c_k^2)&1&0\\ 0&0&3c_k^2+c_k^3&1+3c_k&0\\ \end{pmatrix}, $$ $$ S^{[k]}_2=b_k \begin{pmatrix} 0&3c_k^2+c_k^3&1+3c_k&0&0\\ 0&c_k^3&3(c_k+c_k^2)&1&0\\ 0&0&3c_k^2+c_k^3&1+3c_k&0\\ 0&0&c_k^3&3(c_k+c_k^2)&1\\ 0&0&0&3c_k^2+c_k^3&1+3c_k\\ \end{pmatrix}. $$ We observe that $\lim_{k\to \infty} c_k=1$, and thus $\lim_{k\to\infty}a^{[k]}= a$. The conditions for both Corollary \ref{nscor1} and Theorem \ref{nsprop2} are satisfied, and both forward and backward trajectories of $\{\mathcal{F}_k\}$ converge. The attractors of both forward and backward trajectories, for $\lambda=3$, are presented in Figure \ref{nsfig}. The symmetric set is in Figure \ref{nsfig} is the attractor of the forward trajectory, which is a segment of the cubic polynomial B-spline, and the non-symmetric set is the attractor of the backward trajectory, and it is a part of the exponential B-spline. \begin{figure} \caption{Left: Forward trajectory limit - cubic spline\\ Right: Backward trajectory limit - exponential spline.} \label{nsfig} \end{figure} \end{example}. \begin{example}(Case (iii)). As we have learnt from Corollary \ref{backSFS}, backward SFS trajectories may converge under quite mild conditions. In particular, an SFS derived from a non-stationary subdivision process, may converge even if it is not asymptotically equivalent to a converging stationary process. Let us consider the random non-stationary 4-point interpolatory subdivision process defined by the Laurent polynomials \begin{equation}\label{randomns} a^{[k]}(z)=-w_k(z^{-3}+z^3)+(0.5+w_k)(z^{-1}+z)+1, \end{equation} where $\{w_k\}_{k=1}^\infty$ are randomly chosen in an interval $I$. For the constant sequence $w_k=w$, this is the Laurent polynomial representing the stationary $4$-point scheme presented in \cite{DGL}. This random 4-point subdivision has been considered in \cite{Levin}, and it is shown there that the scheme is $C^1$ convergent for $w_k\in [\epsilon, 1/8-\epsilon]$. Here we study the convergence for a larger interval $I$. We define the SFS $\mathcal{F}_k=\big\{\mathbb{R}^n; f_{1,k}, f_{2,k} \big \}$ where $f_{1,k}, f_{2,k}$ are define by (\ref{f1f2k}) with the corresponding matrices $S^{[k]}_1,S^{[k]}_2$ $$ S^{[k]}_1= \begin{pmatrix} 0&1&0&0&0&0\\ -w_k&0.5+w_k&0.5+w_k&-w_k&0&0\\ 0&0&1&0&0&0\\ 0&-w_k&0.5+w_k&0.5+w_k&-w_k&0\\ 0&0&0&1&0&0\\ 0&0&-w_k&0.5+w_k&0.5+w_k&-w_k\\ \end{pmatrix}, $$ $$ S^{[k]}_2= \begin{pmatrix} -w_k&0.5+w_k&0.5+w_k&-w_k&0&0\\ 0&0&1&0&0&0\\ 0&-w_k&0.5+w_k&0.5+w_k&-w_k&0\\ 0&0&0&1&0&0\\ 0&0&-w_k&0.5+w_k&0.5+w_k&-w_k\\ 0&0&0&0&1&0\\ \end{pmatrix}, $$ and $$ P= \begin{pmatrix} 0&2&1&0&0&1\\ 1&1&0&1&0&1\\ 2&1&0&0&1&1\\ 3&2&0&0&0&1\\ 2&4&0&0&0&1\\ 1&4&0&0&0&1\\ \end{pmatrix}. $$ \end{example} Considering Corollary \ref{backSFS} about the convergence of backward SFS trajectories, we need the existence of a compact invariant set of $\{f_{r,i}\}$, and that $\sum_{k=1}^\infty \prod_{i=1}^k L_{\mathcal{F}_i}<\infty$. By numerical simulations we observe that for this example $\sum_{k=1}^\infty \prod_{i=1}^k L_{\mathcal{F}_i}<\infty$ is satisfied if $\{w_k\}$ are chosen according to a uniform random distribution in $I=[-b,b]$, with $0<b<0.86$. We further conclude that for $\{w_k\}\in I$ there exists $m$ such that for any $i\in \mathbb{N}$, $\prod_{i=k}^{k+m-1} L_{\mathcal{F}_i}<\mu<1$. Using Example \ref{Ex1} we can verify that there exists a compact invariant set of the linear maps $\{\mathcal{A}_i\}$, where $$\mathcal{A}_i=\mathcal{F}_i\circ \mathcal{F}_{i+1}\circ ....\circ \mathcal{F}_{i+m-1}.$$ By Corollary \ref{backSFS}, this guarantees the convergence of the backward trajectories of $\{A_{km}\}$ to a unique attractor, and this implies the convergence of the backward trajectories of $\{\mathcal{F}_i\}$. Figures \ref{randw1}, \ref{randw2}, \ref{randw3} depict the convergence of the backward trajectories $\{\Psi_k(A)\}$ of $\{\mathcal{F}_i\}$ for $w_k\in [-0.2,0.2]$, $w_k\in [-0.4,0.4]$, $w_k\in [-0.8,0.8]$, respectively, and for $k=10,12,14$. \begin{figure} \caption{$w_k\in [-0.2,0.2]$; \ \ Backward trajectories:\ \ $\Psi_k(A)$, $k=10,12,14$.} \label{randw1} \end{figure} \begin{figure} \caption{$w_k\in [-0.4,0.4]$; \ \ Backward trajectories:\ \ $\Psi_k(A)$, $k=10,12,14$.} \label{randw2} \end{figure} \begin{figure} \caption{$w_k\in [-0.8,0.8]$; \ \ Backward trajectories:\ \ $\Psi_k(A)$, $k=10,12,14$.} \label{randw3} \end{figure} \end{document}

\begin{document} \title{Generic rigidity of reflection frameworks} \author{Justin Malestein\thanks{Temple University, \url{justmale@temple.edu}} \and Louis Theran\thanks{Institut für Mathematik, Diskrete Geometrie, Freie Universität Berlin, \url{theran@math.fu-berlin.de}}} \date{} \maketitle \begin{abstract} \begin{normalsize} We give a combinatorial characterization of generic minimally rigid reflection frameworks. The main new idea is to study a pair of direction networks on the same graph such that one admits faithful realizations and the other has only collapsed realizations. In terms of infinitesimal rigidity, realizations of the former produce a framework and the latter certifies that this framework is infinitesimally rigid. \end{normalsize} \end{abstract} \section{Introduction} \seclab{intro} A \emph{reflection framework} is a planar structure made of \emph{fixed-length bars} connected by \emph{universal joints} with full rotational freedom. Additionally, the bars and joints are symmetric with respect to a reflection through a fixed axis. The allowed motions preserve the \emph{length} and \emph{connectivity} of the bars and \emph{symmetry} with respect to some reflection. This model is very similar to that of \emph{cone frameworks} that we introduced in \cite{MT11}; the difference is that the symmetry group $\mathbb{Z}/2\mathbb{Z}$ acts on the plane by reflection instead of rotation through angle $\pi$. When all the allowed motions are Euclidean isometries, a reflection framework is \emph{rigid} and otherwise it is \emph{flexible}. In this paper, we give a \emph{combinatorial} characterization of minimally rigid, generic reflection frameworks. \subsection{The algebraic setup and combinatorial model} Formally a reflection framework is given by a triple $(\tilde{G},\varphi,\tilde{\bm{\ell}})$, where $\tilde{G}$ is a finite graph, $\varphi$ is a $\mathbb{Z}/2\mathbb{Z}$-action on $\tilde{G}$ that is free on the vertices and edges, and $\tilde{\bm{\ell}} = (\ell_{ij})_{ij\in E(\tilde{G})}$ is a vector of non-negative \emph{edge lengths} assigned to the edges of $\tilde{G}$. A \emph{realization} $\tilde{G}(\vec p,\Phi)$ is an assignment of points $\vec p = (\vec p_i)_{i\in V(\tilde{G})}$ and a representation of $\mathbb{Z}/2\mathbb{Z}$ by a reflection $\Phi\in \operatorname{Euc}(2)$ such that: \begin{eqnarray}\eqlab{lengths-1} ||\vec p_j - \vec p_i||^2 = \ell_{ij}^2 & \qquad \text{for all edges $ij\in E(\tilde{G})$} \\ \eqlab{lengths-2} \vec p_{\varphi(\gamma)\cdot i} = \Phi(\gamma)\cdot\vec p_i & \qquad \text{for all $\gamma\in \mathbb{Z}/2\mathbb{Z}$ and $i\in V(\tilde{G})$} \end{eqnarray} The set of all realizations is defined to be the \emph{realization space} $\mathcal{R}(\tilde{G},\varphi,\bm{\ell})$ and its quotient by the Euclidean isometries $\mathcal{C}(\tilde{G},\varphi,\bm{\ell}) = \mathcal{R}(\tilde{G},\varphi,\bm{\ell})/\operatorname{Euc}(2)$ to be the configuration space. A realization is \emph{rigid} if it is isolated in the configuration space and otherwise \emph{flexible}. As the combinatorial model for reflection frameworks it will be more convenient to use colored graphs. A \emph{colored graph} $(G,\bm{\gamma})$ is a finite, directed \footnote{For the group $\mathbb{Z}/2\mathbb{Z}$, the orientation of the edges do not play a role, but we give the standard definition for consistency.} graph $G$, with an assignment $\bm{\gamma} = (\gamma_{ij})_{ij\in E(G)}$ of an element of a group $\Gamma$ to each edge. In this paper $\Gamma$ is always $\mathbb{Z}/2\mathbb{Z}$. There is a standard dictionary \cite[Section 9]{MT11} associating $(\tilde{G},\varphi)$ with a colored graph $(G,\bm{\gamma})$: $G$ is the quotient of $\tilde{G}$ by $\Gamma$, and the colors encode the covering map via a natural map $\rho : \pi_1(G,b) \to \Gamma$. In this setting, the choice of base vertex does not matter, and indeed, we may define $\rho : \HH_1(G, \mathbb{Z})\to \mathbb{Z}/2\mathbb{Z}$ and obtain the same theory. \subsection{Main Theorem} We can now state the main result of this paper. \begin{theorem}[\reflectionlaman]\theolab{reflection-laman} A generic reflection framework is minimally rigid if and only if its associated colored graph is reflection-Laman. \end{theorem} The \emph{reflection-Laman graphs} appearing in the statement are defined in \secref{matroid}. Genericity has its standard meaning from algebraic geometry: the set of non-generic reflection frameworks is a measure-zero algebraic set, and a small \emph{geometric} perturbation of a non-generic reflection framework yields a generic one. \subsection{Infinitesimal rigidity and direction networks} As in all known proofs of ``Maxwell-Laman-type'' theorems such as \theoref{reflection-laman}, we give a combinatorial characterization of a linearization of the problem known as \emph{infinitesimal rigidity}. To do this, we use a \emph{direction network} method (cf. \cite{W88,ST10,MT10,MT11}). A \emph{reflection direction network} $(\tilde{G},\varphi,\vec d)$ is a symmetric graph, along with an assignment of a \emph{direction} $\vec d_{ij}$ to each edge. The \emph{realization space} of a direction network is the set of solutions $\tilde{G}(\vec p)$ to the system of equations: \begin{eqnarray} \eqlab{dn-realization1} \iprod{\vec p_j - \vec p_i}{\vec d^{\perp}_{ij}} = 0 & \qquad \text{for all edges $ij\in E(\tilde{G})$} \\ \eqlab{dn-realization2} \vec p_{\varphi(\gamma)\cdot i} = \Phi(\gamma)\cdot\vec p_i & \qquad \text{for all $\gamma\in \mathbb{Z}/2\mathbb{Z}$ and $i\in V(\tilde{G})$} \end{eqnarray} where the $\mathbb{Z}/2\mathbb{Z}$-action $\Phi$ on the plane is by reflection through the $y$-axis. A reflection direction network is determined by assigning a direction to each edge of the colored quotient graph $(G,\bm{\gamma})$ of $(\tilde{G},\varphi)$ (cf. \cite[Lemma 17.2]{MT11}). Since all the direction networks in this paper are reflection direction networks, we will refer to them simply as ``direction networks'' to keep the terminology manageable. A realization of a direction network is \emph{faithful} if none of the edges of its graph have coincident endpoints and \emph{collapsed} if all the endpoints are coincident. A basic fact in the theory of finite planar frameworks \cite{W88,ST10,DMR07} is that, if a direction network has faithful realizations, the dimension of the realization space is equal to that of the space of infinitesimal motions of a generic framework with the same underlying graph. In \cite{MT10,MT11}, we adapted this idea to the symmetric case when all the symmetries act by rotations and translations. As discussed in \cite[Section 1.8]{MT11}, this so-called ``parallel redrawing trick'' \footnote{This terminology comes from the engineering community, in which the basic idea has been folklore for quite some time.} described above does \emph{not} apply verbatim to reflection frameworks. Thus, we rely on the somewhat technical (cf. \cite[Theorem B]{MT10}, \cite[Theorem 2]{MT11}) \theoref{direction-network}, which we state after giving an important definition. Let $(\tilde{G},\varphi,\vec d)$ be a direction network and define $(\tilde{G},\varphi,\vec d^{\perp})$ to be the direction network with $(\vec d^\perp)_{ij} = (\vec d_{ij})^\perp$. These two direction networks form a \emph{special pair} if: \begin{itemize} \item $(\tilde{G},\varphi,\vec d)$ has a faithful realization. \item $(\tilde{G},\varphi,\vec d^\perp)$ has only collapsed realizations. \end{itemize} \begin{theorem}[\linkeddirectionnetworks]\theolab{direction-network} Let $(G,\bm{\gamma})$ be a colored graph with $n$ vertices, $2n-1$ edges, and lift $(\tilde{G},\varphi)$. Then there are directions $\vec d$ such that the direction networks $(\tilde{G},\varphi,\vec d)$ and $(\tilde{G},\varphi,\vec d^\perp)$ are a special pair if and only if $(G,\bm{\gamma})$ is reflection-Laman. \end{theorem} Briefly, we will use \theoref{direction-network} as follows: the faithful realization of $(\tilde{G},\varphi,\vec d)$ gives a symmetric immersion of the graph $\tilde{G}$ that can be interpreted as a framework, and the fact that $(\tilde{G},\varphi,\vec d^\perp)$ has only collapsed realizations will imply that the only symmetric infinitesimal motions of this framework correspond to translation parallel to the reflection axis. \subsection{Notations and terminology} In this paper, all graphs $G=(V,E)$ may be multi-graphs. Typically, the number of vertices, edges, and connected components are denoted by $n$, $m$, and $c$, respectively. The notation for a colored graph is $(G,\bm{\gamma})$, and a symmetric graph with a free $\mathbb{Z}/2\mathbb{Z}$-action is denoted by $(\tilde{G},\varphi)$. If $(\tilde{G},\varphi)$ is the lift of $(G,\bm{\gamma})$, we denote the fiber over a vertex $i\in V(G)$ by $\tilde{i}_\gamma$, with $\gamma\in \mathbb{Z}/2\mathbb{Z}$, and the fiber over a directed edge $ij$ with color $\gamma_{ij}$ by $\tilde{i}_\gamma \tilde{j}_{\gamma+\gamma_{ij}}$. We also use \emph{$(k,\ell)$-sparse graphs} \cite{LS08} and their generalizations. For a graph $G$, a \emph{$(k,\ell)$-basis} is a maximal $(k,\ell)$-sparse subgraph; a \emph{$(k,\ell)$-circuit} is an edge-wise minimal subgraph that is not $(k,\ell)$-sparse; and a \emph{$(k,\ell)$-component} is a maximal subgraph that has a spanning $(k,\ell)$-graph. Points in $\mathbb{R}^2$ are denoted by $\vec p_i = (x_i,y_i)$, indexed sets of points by $\vec p = (\vec p_i)$, and direction vectors by $\vec d$ and $\vec v$. Realizations of a reflection direction network $(\tilde{G},\varphi,\vec d)$ are written as $\tilde{G}(\vec p)$, as are realizations of abstract reflection frameworks. Context will always make clear the type of realization under consideration. \section{Reflection-Laman graphs} \seclab{matroid} In this short section we introduce the combinatorial families of sparse colored graphs we use. \subsection{The map $\rho$} Let $(G,\bm{\gamma})$ be a $\mathbb{Z}/2\mathbb{Z}$-colored graph. Since all the colored graphs in this paper have $\mathbb{Z}/2\mathbb{Z}$ colors, from now on we make this assumption and write simply ``colored graph''. We recall two key definitions from \cite{MT11}. The map $\rho : \HH_1(G, \mathbb{Z})\to \mathbb{Z}/2\mathbb{Z}$ is defined on cycles by adding up the colors on the edges. (The directions of the edges don't matter for $\mathbb{Z}/2\mathbb{Z}$ colors. Similarly, neither does the traversal order.) As the notation suggests, $\rho$ extends to a homomorphism from $\HH_1(G, \mathbb{Z})$ to $\mathbb{Z}/2\mathbb{Z}$, and it is well-defined even if $G$ is not connected. \subsection{Reflection-Laman graphs} Let $(G,\bm{\gamma})$ be a colored graph with $n$ vertices and $m$ edges. We define $(G,\bm{\gamma})$ to be a \emph{reflection-Laman graph} if: the number of edges $m=2n-1$, and for all subgraphs $G'$, spanning $n'$ vertices, $m'$ edges, $c'$ connected components with non-trivial $\rho$-image and $c'_0$ connected components with trivial $\rho$-image \begin{equation}\eqlab{cone-laman} m'\le 2n' - c' - 3c'_0 \end{equation} This definition is equivalent to that of \emph{cone-Laman graphs} in \cite[Section 15.4]{MT11}. The underlying graph $G$ of a reflection-Laman graph is a $(2,1)$-graph. \subsection{Ross graphs and circuits} Another family we need is that of \emph{Ross graphs} (see \cite{BHMT11} for an explanation of the terminology). These are colored graphs with $n$ vertices, $m = 2n - 2$ edges, satisfying the sparsity counts \begin{equation}\eqlab{ross} m'\le 2n' - 2c' - 3c'_0 \end{equation} using the same notations as in \eqref{cone-laman}. In particular, Ross graphs $(G,\bm{\gamma})$ have as their underlying graph, a $(2,2)$-graph $G$, and are thus connected \cite{LS08}. A \emph{Ross-circuit} \footnote{The matroid of Ross graphs has more circuits, but these are the ones we are interested in here. See \secref{reflection-22}.} is a colored graph that becomes a Ross graph after removing \emph{any} edge. The underlying graph $G$ of a Ross-circuit $(G,\bm{\gamma})$ is a $(2,2)$-circuit, and these are also known to be connected \cite{LS08}, so, in particular, a Ross-circuit has $c'_0=0$, and thus satisfies \eqref{cone-laman} on the whole graph. Since \eqref{cone-laman} is always at least \eqref{ross}, we see that every Ross-circuit is reflection-Laman. Because reflection-Laman graphs are $(2,1)$-graphs and subgraphs that are $(2,2)$-sparse are, in addition, Ross-sparse, we get the following structural result. \begin{prop}[\xyzzy][{\cite[Proposition 5.1]{MT12},\cite[Lemma 11]{BHMT11}}]\proplab{ross-circuit-decomp} Let $(G,\bm{\gamma})$ be a reflection-Laman graph. Then each $(2,2)$-component of $G$ contains at most one Ross-circuit, and in particular, the Ross-circuits in $(G,\bm{\gamma})$ are vertex disjoint. \end{prop} \subsection{Reflection-$(2,2)$ graphs}\seclab{reflection-22} The next family of graphs we work with is new. A colored graph $(G,\bm{\gamma})$ is defined to be a \emph{reflection-$(2,2)$} graph, if it has $n$ vertices, $m=2n-1$ edges, and satisfies the sparsity counts \begin{equation} \eqlab{ref22a} m' \le 2n' - c' - 2c'_0 \end{equation} using the same notations as in \eqref{cone-laman}. The relationship between Ross graphs and reflection-$(2,2)$ graphs we will need is: \begin{prop} \proplab{ross-adding} Let $(G,\bm{\gamma})$ be a Ross-graph. Then for either \begin{itemize} \item an edge $ij$ with any color where $i \neq j$ \item or a self-loop $\ell$ at any vertex $i$ colored by $1$ \end{itemize} the graph $(G+ij,\bm{\gamma})$ or $(G+\ell,\bm{\gamma})$ is reflection-$(2,2)$. \end{prop} \begin{proof} Adding $ij$ with any color to a Ross $(G,\bm{\gamma})$ creates either a Ross-circuit, for which $c'_0=0$ or a Laman-circuit with trivial $\rho$-image. Both of these types of graph meet this count, and so the whole of $(G+ij,\bm{\gamma})$ does as well. \end{proof} It is easy to see that every reflection-Laman graph is a reflection-$(2,2)$ graph. The converse is not true. \begin{prop}\proplab{reflection-laman-vs-reflection-22} A colored graph $(G,\bm{\gamma})$ is a reflection-Laman graph if and only if it is a reflection-$(2,2)$ graph and no subgraph with trivial $\rho$-image is a $(2,2)$-block. $\qed$ \end{prop} Let $(G,\bm{\gamma})$ be a reflection-Laman graph, and let $G_1,G_2,\ldots,G_t$ be the Ross-circuits in $(G,\bm{\gamma})$. Define the \emph{reduced graph} $(G^*,\bm{\gamma})$ of $(G,\bm{\gamma})$ to be the colored graph obtained by contracting each $G_i$, which is not already a single vertex with a self-loop (this is necessarily colored $1$), into a new vertex $v_i$, removing any self-loops created in the process, and then adding a new self-loop with color $1$ to each of the $v_i$. By \propref{ross-circuit-decomp} the reduced graph is well-defined. \begin{prop}\proplab{reduced-graph} Let $(G,\bm{\gamma})$ be a reflection-Laman graph. Then its reduced graph is a reflection-$(2,2)$ graph. \end{prop} \begin{proof} Let $(G,\bm{\gamma})$ be a reflection-Laman graph with $t$ Ross-circuits with vertex sets $V_1,\ldots,V_t$. By \propref{ross-circuit-decomp}, the $V_i$ are all disjoint. Now select a Ross-basis $(G',\bm{\gamma})$ of $(G,\bm{\gamma})$. The graph $G'$ is also a $(2,2)$-basis of $G$, with $2n-1 - t$ edges, and each of the $V_i$ spans a $(2,2)$-block in $G'$. The $(k,\ell)$-sparse graph Structure Theorem \cite[Theorem 5]{LS08} implies that contracting each of the $V_i$ into a new vertex $v_i$ and discarding any self-loops created, yields a $(2,2)$-sparse graph $G^+$ on $n^+$ vertices and $2n^+ - 1 - t$ edges. It is then easy to check that adding a self-loop colored $1$ at each of the $v_i$ produces a colored graph satisfying the reflection-$(2,2)$ counts \eqref{ref22a} with exactly $2n^+ -1$ edges. Since this is the reduced graph, we are done. \end{proof} \subsection{Decomposition characterizations} A \emph{map-graph} is a graph with exactly one cycle per connected component. A \emph{reflection-$(1,1)$} graph is defined to be a colored graph $(G,\bm{\gamma})$ where $G$, taken as an undirected graph, is a map-graph and the $\rho$-image of each connected component is non-trivial. \begin{lemma}\lemlab{reflection-22-decomp} Let $(G,\bm{\gamma})$ be a colored graph. Then $(G,\bm{\gamma})$ is a reflection-$(2,2)$ graph if and only if it is the union of a spanning tree and a reflection-$(1,1)$ graph. \end{lemma} \begin{proof} By \cite[Lemma 15.1]{MT11}, reflection-$(1,1)$ graphs are equivalent to graphs satisfying \begin{equation} \eqlab{ref11a} m' \le n' - c'_0 \end{equation} for every subgraph $G'$. Thus, \eqref{ref22a} is \begin{equation}\eqlab{ref22redux} m' \le (n' - c'_0) + (n' - c' - c'_0) \end{equation} The second term in \eqref{ref22redux} is well-known to be the rank function of the graphic matroid, and the Lemma follows from the Edmonds-Rota construction \cite{ER66} and the Matroid Union Theorem. \end{proof} In the next section, it will be convenient to use this slight refinement of \lemref{reflection-22-decomp}. \begin{prop}\proplab{reflection-22-nice-decomp} Let $(G,\bm{\gamma})$ be a reflection-$(2,2)$ graph. Then there is a coloring $\bm{\gamma}'$ of the edges of $G$ such that: \begin{itemize} \item The $\rho$-image of every subgraph in $(G,\bm{\gamma}')$ is the same as in $(G,\bm{\gamma})$. \item There is a decomposition of $(G,\bm{\gamma}')$ as in \lemref{reflection-22-decomp} in which the spanning tree has all edges colored by the identity. \end{itemize} \end{prop} \begin{proof} It is shown in \cite[Lemma 2.2]{MT10} that $\rho$ is determined by its image on a homology basis of $G$. Thus, we may start with an arbitrary decomposition of $(G,\bm{\gamma})$ into a spanning tree $T$ and a reflection-$(1,1)$ graph $X$, as provided by \lemref{reflection-22-decomp}, and define $\bm{\gamma}'$ by coloring the edges of $T$ with the identity and the edges of $X$ with the $\rho$-image of their fundamental cycle in $T$ in $(G,\bm{\gamma})$. \end{proof} \propref{reflection-22-nice-decomp} has the following re-interpretation in terms of the symmetric lift $(\tilde{G},\varphi)$: \begin{prop}\proplab{reflection-laman-decomp-lift} Let $(G,\bm{\gamma})$ be a reflection-$(2,2)$ graph. Then for a decomposition, as provided by \propref{reflection-22-nice-decomp}, into a spanning tree $T$ and a reflection-$(1,1)$ graph $X$: \begin{itemize} \item Every edge $ij\in T$ lifts to the two edges $i_0j_0$ and $i_1j_1$. (In other words, the vertex representatives in the lift all lie in a single connected component of the lift of $T$.) \item Each connected component of $X$ lifts to a connected graph. \end{itemize} \end{prop} \section{Special pairs of reflection direction networks} \seclab{direction-network} We recall, from the introduction, that for reflection direction networks, $\mathbb{Z}/2\mathbb{Z}$ acts on the plane by reflection through the $y$-axis, and in the rest of this section $\Phi(\gamma)$ refers to this action. \subsection{The colored realization system} The system of equations \eqref{dn-realization1}--\eqref{dn-realization2} defining the realization space of a reflection direction network $(\tilde{G},\varphi,\vec d)$ is linear, and as such has a well-defined dimension. Let $(G,\bm{\gamma})$ be the colored quotient graph of $(\tilde{G},\varphi)$. To be realizable at all, the directions on the edges in the fiber over $ij\in E(G)$ need to be reflections of each other. Thus, we see that the realization system is canonically identified with the solutions to the system: \begin{eqnarray}\eqlab{colored-system} \iprod{\Phi(\gamma_{ij})\cdot\vec p_j - \vec p_i}{\vec d_{ij}} = 0 & \qquad \text{for all edges $ij\in E(G)$} \end{eqnarray} From now on, we will implicitly switch between the two formalisms when it is convenient. \subsection{Genericity} Let $(G,\bm{\gamma})$ be a colored graph with $m$ edges. A statement about direction networks $(\tilde{G},\varphi,\vec d)$ is \emph{generic} if it holds on the complement of a proper algebraic subset of the possible direction assignments, which is canonically identified with $\mathbb{R}^{2m}$. Some facts about generic statements that we use frequently are: \begin{itemize} \item Almost all direction assignments are generic. \item If a set of directions is generic, then so are all sufficiently small perturbations of it. \item If two properties are generic, then their intersection is as well. \item The maximum rank of \eqref{colored-system} is a generic property. \end{itemize} \subsection{Direction networks on Ross graphs} We first characterize the colored graphs for which generic direction networks have strongly faithful realizations. A realization is \emph{strongly faithful} if no two vertices lie on top of each other. This is a stronger condition than simply being faithful which only requires that edges not be collapsed. \begin{prop}\proplab{ross-realizations} A generic direction network $(\tilde{G},\varphi,\vec d)$ has a unique, up to translation and scaling, strongly faithful realization if and only if its associated colored graph is a Ross graph. \end{prop} To prove \propref{ross-realizations} we expand upon the method from \cite[Section 20.2]{MT11}, and use the following proposition. \begin{prop}\proplab{reflection-22-collapse} Let $(G,\bm{\gamma})$ be a reflection-$(2,2)$ graph. Then a generic direction network on the symmetric lift $(\tilde{G},\varphi)$ of $(G,\bm{\gamma})$ has only collapsed realizations. \end{prop} Since the proof of \propref{reflection-22-collapse} requires a detailed construction, we first show how it implies \propref{ross-realizations}. \subsection{Proof that \propref{reflection-22-collapse} implies \propref{ross-realizations}} Let $(G,\bm{\gamma})$ be a Ross graph, and assign directions $\vec d$ to the edges of $G$ such that, for any extension $(G+ij,\bm{\gamma})$ of $(G,\bm{\gamma})$ to a reflection-$(2,2)$ graph as in \propref{ross-adding}, $\vec d$ can be extended to a set of directions that is generic in the sense of \propref{reflection-22-collapse}. This is possible because there are a finite number of such extensions. For this choice of $\vec d$, the realization space of the direction network $(\tilde{G},\varphi,\vec d)$ is $2$-dimensional. Since solutions to \eqref{colored-system} may be scaled or translated in the vertical direction, all solutions to $(\tilde{G},\varphi,\vec d)$ are related by scaling and translation. It then follows that a pair of vertices in the fibers over $i$ and $j$ are either distinct from each other in all non-zero solutions to \eqref{colored-system} or always coincide. In the latter case, adding the edge $ij$ with any direction does not change the dimension of the solution space, no matter what direction we assign to it. It then follows that the solution spaces of generic direction networks on $(\tilde{G},\varphi,\vec d)$ and $(\widetilde{G+ij},\varphi,\vec d)$ have the same dimension, which is a contradiction by \propref{reflection-22-collapse}. $\qed$ \subsection{Proof of \propref{reflection-22-collapse}} It is sufficient to construct a specific set of directions with this property. The rest of the proof gives such a construction and verifies that all the solutions are collapsed. Let $(G,\bm{\gamma})$ be a reflection-$(2,2)$ graph. \paragraph{Combinatorial decomposition} We apply \propref{reflection-22-nice-decomp} to decompose $(G,\bm{\gamma})$ into a spanning tree $T$ with all colors the identity and a reflection-$(1,1)$ graph $X$. For now, we further assume that $X$ is connected. \paragraph{Assigning directions} Let $\vec v$ be a direction vector that is not horizontal or vertical. For each edge $ij\in T$, set $\vec d_{ij} = \vec v$. Assign all the edges of $X$ the vertical direction. Denote by $\vec d$ this assignment of directions. \begin{figure} \caption{Schematic of the proof of \propref{reflection-22-collapse}: the $y$-axis is shown as a dashed line. The directions on the edges of the lift of the tree $T$ force all the vertices to be on one of the two lines meeting at the $y$-axis, and the directions on the reflection-$(1,1)$ graph $X$ force all the vertices to be on the $y$-axis.} \label{fig:ref-22-collapse} \end{figure} \paragraph{All realizations are collapsed} We now show that the only realizations of $(\tilde{G},\varphi,\vec d)$ have all vertices on top of each other. By \propref{reflection-laman-decomp-lift} $T$ lifts to two copies of itself, in $\tilde{G}$. It then follows from the connectivity of $T$ and the construction of $\vec d$ that, in any realization, there is a line $L$ with direction $\vec v$ such that every vertex of $\tilde{G}$ must lie on $L$ or its reflection. Since the vertical direction is preserved by reflection, the connectivity of the lift of $X$, again from \propref{reflection-laman-decomp-lift}, implies that every vertex of $\tilde{G}$ lies on a single vertical line, which must be the $y$-axis by reflective symmetry. Thus, in any realization of $(\tilde{G},\varphi,\vec d)$ all the vertices lie at the intersection of $L$, the reflection of $L$ through the $y$-axis and the $y$-axis itself. This is a single point, as desired. \figref{ref-22-collapse} shows a schematic of this argument. \paragraph{$X$ does not need to be connected} Finally, we can remove the assumption that $X$ was connected by repeating the argument for each connected component of $X$ separately. $\qed$ \subsection{Special pairs for Ross-circuits} The full \theoref{direction-network} will reduce to the case of a Ross-circuit. \begin{prop}\proplab{ross-circuit-pairs} Let $(G,\bm{\gamma})$ be a Ross circuit with lift $(\tilde{G},\varphi)$. Then there is an edge $i'j'$ such that, for a generic direction network $(\tilde{G'},\varphi,\vec d')$ with colored graph $(G-i'j',\bm{\gamma})$: \begin{itemize} \item The solution space of $(\tilde{G'},\varphi,\vec d')$ induces a well-defined direction $\vec d_{ij}$ between $i$ and $j$, yielding an assignment of directions $\vec d$ to the edges of $G$. \item The direction networks $(\tilde{G},\varphi,\vec d)$ and $(\tilde{G},\varphi,(\vec d)^\perp)$ are a special pair. \end{itemize} \end{prop} Before giving the proof, we describe the idea. We are after sets of directions that lead to faithful realizations of Ross-circuits. By \propref{reflection-22-collapse}, these directions must be non-generic. A natural way to obtain such a set of directions is to discard an edge $ij$ from the colored quotient graph, apply \propref{ross-realizations} to obtain a generic set of directions $\vec d'$ with a strongly faithful realization $\tilde{G}'(\vec p)$, and then simply set the directions on the edges in the fiber over $ij$ to be the difference vectors between the points. \propref{ross-realizations} tells us that this procedure induces a well-defined direction for the edge $ij$, allowing us to extend $\vec d'$ to $\vec d$ in a controlled way. However, it does \emph{not} tell us that rank of $(\tilde{G},\varphi,\vec d)$ will rise when the directions are turned by angle $\pi/2$, and this seems hard to do directly. Instead, we construct a set of directions $\vec d$ so that $(\tilde{G},\varphi,\vec d)$ is rank deficient and has faithful realizations, and $(\tilde{G},\varphi,\vec d^\perp)$ is generic. Then we make a perturbation argument to show the existence of a special pair. The construction we use is, essentially, the one used in the proof of \propref{reflection-22-collapse} but turned through angle $\pi/2$. The key geometric insight is that horizontal edge directions are preserved by the reflection, so the ``gadget'' of a line and its reflection crossing on the $y$-axis, as in \figref{ref-22-collapse}, degenerates to just a single line. \subsection{Proof of \propref{ross-circuit-pairs}} Let $(G,\bm{\gamma})$ be a Ross-circuit; recall that this implies that $(G,\bm{\gamma})$ is a reflection-Laman graph. \paragraph{Combinatorial decomposition} We decompose $(G,\bm{\gamma})$ into a spanning tree $T$ and a reflection-$(1,1)$ graph $X$ as in \propref{reflection-laman-decomp-lift}. In particular, we again have all edges in $T$ colored by the identity. For now, we \emph{assume that $X$ is connected}, and we fix $i'j'$ to be an edge that is on the cycle in $X$ with $\gamma_{i'j'}\neq 0$; such an edge must exist by the hypothesis that $X$ is reflection-$(1,1)$. Let $G' = G\setminus i'j'$. Furthermore, let $T_0$ and $T_1$ be the two connected components of the lift of $T$. For a vertex $i \in G$, we denote the lift in $T_0$ by $i_0$ and the lift in $T_1$ by $i_1$. We similarly denote the lifts of $i'$ and $j'$ by $i_0', i_1'$ and $j_0', j_1'$. \paragraph{Assigning directions} The assignment of directions is as follows: to the edges of $T$, we assign a direction $\vec v$ that is neither vertical nor horizontal. To the edges of $X$ we assign the horizontal direction. Define the resulting direction network to be $(\tilde{G},\varphi,\vec d)$, and the direction network induced on the lift of $G'$ to be $(\tilde{G'},\varphi,\vec d)$. \paragraph{The realization space of $(\tilde{G},\varphi,\vec d)$} \figref{ross-circuit-special-pair} contains a schematic picture of the arguments that follow. \begin{lemma}\lemlab{RC-proof-1} The realization space of $(\tilde{G},\varphi,\vec d)$ is $2$-dimensional and parameterized by exactly one representative in the fiber over the vertex $i$ selected above. \end{lemma} \begin{proof} In a manner similar to the proof of \propref{reflection-22-collapse}, the directions on the edges of $T$ force every vertex to lie either on a line $L$ in the direction $\vec v$ or its reflection. Since the lift of $X$ is connected, we further conclude that all the vertices lie on a single horizontal line. Thus, all the points $\vec p_{j_0}$ are at the intersection of the same horizontal line and $L$ or its reflection. These determine the locations of the $\vec p_{j_1}$, so the realization space is parameterized by the location of $\vec p_{i'_0}$. \end{proof} Inspecting the argument more closely, we find that: \begin{lemma} In any realization $\tilde{G}(\vec p)$ of $(\tilde{G},\varphi,\vec d)$, all the $\vec p_{j_0}$ are equal and all the $\vec p_{j_1}$ are equal. \end{lemma} \begin{proof} Because the colors on the edges of $T$ are all zero, it lifts to two copies of itself, one of which spans the vertex set $\{\tilde{j_0} : j\in V(G)\}$ and one which spans $\{\tilde{j_1} : j\in V(G)\}$. It follows that in a realization, we have all the $\vec p_{j_0}$ on $L$ and the $\vec p_{j_1}$ on the reflection of $L$. \end{proof} In particular, because the color $\gamma_{i'j'}$ on the edge $i'j'$ is $1$, we obtain the following. \begin{lemma}\lemlab{RC-proof-5} The realization space of $(\tilde{G},\varphi,\vec d)$ contains points where the fiber over the edge $i'j'$ is not collapsed. \end{lemma} \begin{figure} \caption{Schematic of the proof of \propref{ross-circuit-pairs}: the $y$-axis is shown as a dashed line. The directions on the edges of the lift of the tree $T$ force all the vertices to be on one of the two lines meeting at the $y$-axis. The horizontal directions on the connected reflection-$(1,1)$ graph $X$ force the point $\vec p_{j_0}$ to be at the intersection marked by the black dot and $\vec p_{j_1}$ to be at the intersection marked by the gray one.} \label{fig:ross-circuit-special-pair} \end{figure} \paragraph{The realization space of $(\tilde{G}',\varphi,\vec d)$} The conclusion of \lemref{RC-proof-1} implies that the realization system for $(\tilde{G},\varphi,\vec d)$ is rank deficient by one. Next we show that removing the edge $i'j'$ results in a direction network that has full rank on the colored graph $(G',\bm{\gamma})$. \begin{lemma}\lemlab{RC-proof-2} The realization space of $(\tilde{G},\varphi,\vec d)$ is canonically identified with that of $(\tilde{G}',\varphi,\vec d)$. \end{lemma} \begin{proof} In the proof of \lemref{RC-proof-1}, that $X$ lifts to a connected subgraph of $\tilde{G}$ was not essential. Because a horizontal line is preserved by the reflection, realizations will take on the same structure provided that $X$ lifts to a subgraph with two connected components. Removing $i'j'$ from $X$ leaves a graph $X'$ with this property since $X'$ is a tree. It follows that the equation corresponding to the edge $i'j'$ in \eqref{colored-system} was dependent. \end{proof} \paragraph{The realization space of $(\tilde{G},\varphi,\vec d^\perp)$} Next, we consider what happens when we turn all the directions by $\pi/2$. \begin{lemma}\lemlab{RC-proof-3} The realization space of $(\tilde{G},\varphi,\vec d^\perp)$ has only collapsed solutions. \end{lemma} \begin{proof} This is exactly the construction used to prove \propref{reflection-22-collapse}. \end{proof} \paragraph{Perturbing $(\tilde{G},\varphi,\vec d)$} To summarize what we have shown so far: \begin{itemize} \item[(a)] $(\tilde{G},\varphi,\vec d)$ has a $2$-dimensional realization space parameterized by $\vec p_{i'_0}$ and identified with that of a full-rank direction network on the Ross graph $(G',\bm{\gamma})$. \item[(b)] There are points $\tilde{G}(\vec p)$ in this realization space where $\vec p_{i'_0}\neq \vec p_{j'_1}$. \item[(c)] $(\tilde{G},\varphi,\vec d)$ has a $1$-dimensional realization space containing only collapsed solutions. \end{itemize} What we have not shown is that the realization space of $(\tilde{G},\varphi,\vec d)$ has \emph{faithful} realizations, since the ones we constructed all have many coincident vertices. \propref{ross-realizations} will imply the rest of the theorem, provided that the above properties hold for any small perturbation of $\vec d$, since some small perturbation of \emph{any} assignment of directions to the edges of $(G',\bm{\gamma})$ has only faithful realizations. \begin{lemma}\lemlab{RC-proof-4} Let $\vec{ \hat d'}$ be a perturbation of the directions $\vec d'$ on the edges of $G'$. If $\vec{ \hat d'}$ is sufficiently close to $\vec d'$ , then there are realizations of the direction network $(\tilde{G}',\varphi,\vec{ \hat d'})$ such that $\vec p_{i'_0}\neq \vec p_{j'_1}$. \end{lemma} \begin{proof} The realization space is parameterized by $\vec p_{i'_0}$, and so $\vec p_{j'_1}$ varies continuously with the directions on the edges and $\vec p_{i'_0}$. Since there are realizations of $(\tilde{G}', \varphi, \vec d)$ with $\vec p_{i_0} \neq \vec p_{j_1}$, the Lemma follows. \end{proof} \lemref{RC-proof-4} implies that any sufficiently small perturbation of the directions assigned to the edges of $G'$ gives a direction network that induces a well-defined direction on the edge $i'j'$ which is itself a small perturbation of $\vec d_{i'j'}$. Since the ranks of $(\tilde{G'},\varphi,\vec d')$ and $(\tilde{G},\varphi,\vec d^\perp)$ are stable under small perturbations, this implies that we can perturb $\vec d'$ to a $\vec{\hat d'}$ that is generic in the sense of \propref{ross-realizations}, while preserving faithful realizability of $(\tilde{G},\varphi,\hat{\vec d})$ and full rank of the realization system for $(\tilde{G},\varphi,\hat{\vec d}^\perp)$. The Proposition is proved for when $X$ is connected. \paragraph{$X$ need not be connected} The proof is then complete once we remove the additional assumption that $X$ was connected. Let $X$ have connected components $X_1, X_2,\ldots,X_c$. For each of the $X_i$, we can identify an edge $(i'j')_k$ with the same properties as $i'j'$ above. Assign directions to the tree $T$ as above. For $X_1$, we assign directions exactly as above. For each of the $X_k$ with $k\ge 2$, we assign the edges of $X_k\setminus (i'j')_k$ the horizontal direction and $(i'j')_k$ a direction that is a small perturbation of horizontal. With this assignment $\vec d$ we see that for any realization of $(\tilde{G},\varphi,\vec d)$, each of the $X_k$, for $k\ge 2$ is realized as completely collapsed to a single point at the intersection of the line $L$ and the $y$-axis. Moreover, in the direction network on $\vec d^\perp$, the directions on these $X_i$ are a small perturbation of the ones used on $X$ in the proof of \propref{reflection-22-collapse}. From this is follows that, in any realization $(\tilde{G},\varphi,\vec d^\perp)$, is completely collapsed and hence full rank. We now see that this new set of directions has properties (a), (b), and (c) above required for the perturbation argument. Since that argument makes no reference to the decomposition, it applies verbatim to the case where $X$ is disconnected. $\qed$ \subsection{Proof of \theoref{direction-network}} The easier direction to check is necessity. \paragraph{The Maxwell-direction} If $(G,\bm{\gamma})$ is not reflection-Laman, then it contains either a Laman-circuit with trivial $\rho$-image, or a violation of $(2,1)$-sparsity. If there is a Laman-circuit with trivial $\rho$-image, the Parallel Redrawing Theorem \cite[Theorem 4.1.4]{W96} in the form \cite[Theorem 3]{ST10} implies that this subgraph has no faithful realizations for $(G,\varphi,\vec d)$ only if it does in $(G,\varphi,\vec d^\perp)$ if rank-deficient. A violation of $(2,1)$-sparsity implies that the realization system \eqref{colored-system} of $(\tilde{G},\varphi,\vec d^\perp)$ has a dependency, since the realization space is always at least $1$-dimensional. \paragraph{The Laman direction} Now let $(G,\bm{\gamma})$ be a reflection-Laman graph and let $(G',\bm{\gamma})$ be a Ross-basis of $(G,\bm{\gamma})$. For any edge $ij \notin G'$, adding it to $G'$ induces a Ross-circuit which contains some edge $i'j'$ having the property specified in \propref{ross-circuit-pairs}. Note that $G' - ij +i'j'$ is again a Ross-basis. We therefore can assume (after edge-swapping in this manner) for all $ij \notin G'$ that $ij$ has the property from \propref{ross-circuit-pairs} in the Ross-circuit it induces. We assign directions $\vec d'$ to the edges of $G'$ such that: \begin{itemize} \item The directions on each of the intersections of the Ross-circuits with $G'$ are generic in the sense of \propref{ross-circuit-pairs}. \item The directions on the edges of $G'$ that remain in the reduced graph $(G^*,\bm{\gamma})$ are perpendicular to an assignment of directions on $G^*$ that is generic in the sense of \propref{reflection-22-collapse}. \item The directions on the edges of $G'$ are generic in the sense of \propref{ross-realizations}. \end{itemize} This is possible because the set of disallowed directions is the union of a finite number of proper algebraic subsets in the space of direction assignments. Extend to directions $\vec d$ on $G$ by assigning directions to the remaining edges as specified by \propref{ross-circuit-pairs}. By construction, we know that: \begin{lemma}\lemlab{laman-1} The direction network $(\tilde{G},\varphi,\vec d)$ has faithful realizations. \end{lemma} \begin{proof} The realization space is identified with that of $(\tilde{G'},\varphi,\vec d')$, and $\vec d'$ is chosen so that \propref{ross-realizations} applies. \end{proof} \begin{lemma}\lemlab{laman-2} In any realization of $(\tilde{G},\varphi,\vec d^{\perp})$, the Ross-circuits are realized with all their vertices coincident and on the $y$-axis. \end{lemma} \begin{proof} This follows from how we chose $\vec d$ and \propref{ross-circuit-pairs}. \end{proof} As a consequence of \lemref{laman-2}, and the fact that we picked $\vec d$ so that $\vec d^\perp$ extends to a generic assignment of directions $(\vec d^*)^\perp$ on the reduced graph $(G^*,\bm{\gamma})$ we have: \begin{lemma} The realization space of $(\tilde{G},\varphi,\vec d^\perp)$ is identified with that of $(\tilde{G^*},\varphi,\vec (d^*)^\perp)$ which, furthermore, contains only collapsed solutions. \end{lemma} Observe that a direction network for a single self-loop (colored $1$) with a generic direction only has solutions where vertices are collapsed and on the $y$-axis. Consequently, replacing a Ross-circuit with a single vertex and a self-loop yields isomorphic realization spaces. Since the reduced graph is reflection-$(2,2)$ by \propref{reduced-graph} and the directions assigned to its edges were chosen generically for \propref{reflection-22-collapse}, that $(\tilde{G},\varphi,\vec d^\perp)$ has only collapsed solutions follows. Thus, we have exhibited a special pair, completing the proof. $\qed$ \paragraph{Remark} It can be seen that the realization space of a direction network as supplied by \theoref{direction-network} has at least one degree of freedom for each edge that is not in a Ross basis. Thus, the statement cannot be improved to, e.g., a unique realization up to translation and scale. \section{Infinitesimal rigidity of reflection frameworks} \seclab{reflection-laman-proof} Let $(\tilde{G},\varphi,\bm{\ell})$ be a reflection framework and let $(G,\bm{\gamma})$ be the quotient graph. The configuration space, which is the set of solutions to the quadratic system \eqref{lengths-1}--\eqref{lengths-2} is canonically identified with the solutions to: \begin{eqnarray}\eqlab{colored-lengths} ||\Phi(\gamma_{ij})\cdot \vec p_j - \vec p_i||^2 = \ell^2_{ij} & \qquad \text{for all edges $ij\in E(G)$} \end{eqnarray} where $\Phi$ acts on the plane by reflection through the $y$-axis. (That ``pinning down'' $\Phi$ does not affect the theory is straightforward from the definition of the configuration space: it simply removes rotation and translation in the $x$-direction from the set of trivial motions.) \subsection{Infinitesimal rigidity} Computing the formal differential of \eqref{colored-lengths}, we obtain the system \begin{eqnarray}\eqlab{colored-inf} \iprod{\Phi(\gamma_{ij})\cdot \vec p_j - \vec p_i}{\vec v_j - \vec v_i} = 0 & \qquad \text{for all edges $ij\in E(G)$} \end{eqnarray} where the unknowns are the \emph{velocity vectors} $\vec v_i$. A standard kind of result (cf. \cite{AR78}) is the following. \begin{prop}\proplab{ar-direction} Let $\tilde{G}(\vec p,\Phi)$ be a realization of an abstract framework $(\tilde{G},\varphi,\bm{\ell})$. If the corank of the system \eqref{colored-inf} is one, then $\tilde{G}(\vec p)$ is rigid. \end{prop} Thus, we define a realization to be \emph{infinitesimally rigid} if the system \eqref{colored-inf} has maximal rank, and \emph{minimally infinitesimally rigid} if it is infinitesimally rigid but ceases to be so after removing any edge from the colored quotient graph. By definition, infinitesimal rigidity is defined by a polynomial condition in the coordinates of the points $\vec p_i$, so it is a generic property associated with the colored graph $(G,\bm{\gamma})$. \subsection{Relation to direction networks} Here is the core of the direction network method for reflection frameworks: we can understand the rank of \eqref{colored-inf} in terms of a direction network. \begin{prop}\proplab{rigidity-vs-directions} Let $\tilde{G}(\vec p,\Phi)$ be a realization of a reflection framework with $\Phi$ acting by reflection through the $y$-axis. Define the direction $\vec d_{ij}$ to be $\vec \Phi(\gamma_{ij})\cdot \vec p_j - \vec p_i$. Then the rank of \eqref{colored-inf} is equal to that of \eqref{colored-system} for the direction network $(G,\bm{\gamma},\vec d^{\perp})$. \end{prop} \begin{proof} Exchange the roles of $\vec v_i$ and $\vec p_i$ in \eqref{colored-inf}. \end{proof} \subsection{Proof of \theoref{reflection-laman}} The, more difficult, ``Laman direction'' of the Main Theorem follows immediately from \theoref{direction-network} and \propref{rigidity-vs-directions}: given a reflection-Laman graph \theoref{direction-network} produces a realization with no coincident endpoints and a certificate that \eqref{colored-inf} has corank one. $\qed$ \subsection{Remarks} The statement of \propref{rigidity-vs-directions} is \emph{exactly the same} as the analogous statement for orientation-preserving cases of this theory. What is different is that, for reflection frameworks, the rank of $(G,\bm{\gamma},\vec d^{\perp})$ is \emph{not}, the same as that of $(G,\bm{\gamma},\vec d)$. By \propref{reflection-22-collapse}, the set of directions arising as the difference vectors from point sets are \emph{always non-generic} on reflection-Laman graphs, so we are forced to introduce the notion of a special pair as in \secref{direction-network}. \end{document}

\begin{document} \title{Controlled deflection of cold atomic clouds and of Bose-Einstein condensates} \author{N. Gaaloul} \affiliation{Laboratoire de Spectroscopie Atomique, Mol\'{e}culaire et Applications, D\'{e}partement de Physique, Facult\'{e} des Sciences de Tunis, Universit\'{e} Tunis El Manar, 2092 Tunis, Tunisia.} \affiliation{Laboratoire de Photophysique Mol\'{e}culaire du CNRS, Universit\'{e} Paris-Sud, B\^{a}timent 210, 91405 Orsay Cedex, France.} \author{A. Jaouadi} \affiliation{Laboratoire de Spectroscopie Atomique, Mol\'{e}culaire et Applications, D\'{e}partement de Physique, Facult\'{e} des Sciences de Tunis, Universit\'{e} Tunis El Manar, 2092 Tunis, Tunisia.} \affiliation{Laboratoire de Photophysique Mol\'{e}culaire du CNRS, Universit\'{e} Paris-Sud, B\^{a}timent 210, 91405 Orsay Cedex, France.} \author{L. Pruvost} \affiliation{Laboratoire Aim\'{e} Cotton du CNRS, Universit\'{e} Paris-Sud, B\^{a}timent 505, 91405 Orsay Cedex, France.} \author{M. Telmini} \affiliation{Laboratoire de Spectroscopie Atomique, Mol\'{e}culaire et Applications, D\'{e}partement de Physique, Facult\'{e} des Sciences de Tunis, Universit\'{e} Tunis El Manar, 2092 Tunis, Tunisia.} \author{E. Charron} \affiliation{Laboratoire de Photophysique Mol\'{e}culaire du CNRS, Universit\'{e} Paris-Sud, B\^{a}timent 210, 91405 Orsay Cedex, France.} \abstract{We present a detailed, realistic proposal and analysis of the implementation of a cold atom deflector using time-dependent far off-resonance optical guides. An analytical model and numerical simulations are used to illustrate its characteristics when applied to both non-degenerate atomic ensembles and to Bose-Einstein condensates. Using for all relevant parameters values that are achieved with present technology, we show that it is possible to deflect almost entirely an ensemble of $^{87}$Rb atoms falling in the gravity field. We discuss the limits of this proposal, and illustrate its robustness against non-adiabatic transitions.} \pacs{37.10.Gh} } \maketitle \section{Introduction} \label{sec:Intro} Optical and magnetic fields are extremely efficient tools used for the controlled manipulation of large ensembles of cold atoms\,\cite{Adams_1994,Balykin_1995}. In the past fifteen years, cold matter waves have shown great possibilities in the context of linear atom optics, when phase-space densities are sufficiently low that the effect of collisions can be neglected. Dipole and radiation-pressure forces have for instance allowed the achievement of various optical manipulations such as atomic focusing, diffraction or interference\,\cite{Berman_1997,Meystre_2001}. Many efforts have been recently devoted to the experimental implementation of atomic beam splitters with magnetic\,\cite{Muller_2000,Cassettari_2000,Muller_2001,Hommelhoff_2005} or optical\,\cite{Houde_2000,Hansel_2001,Dumke_2002} potentials. These different experimental investigations were accompanied by various theoretical studies\,\cite{Stickney_2003,Kreutzmann_2004,Bortolotti_2004,Gaaloul_2006,Zhang_2006}. These devices are obviously of clear interest for atom interferometry experiments. After the advent of Bose-Einstein condensation (BEC) in 1995\,\cite{Anderson_1995,Davis_1995}, different setups were designed in order to split and recombine a BEC\,\cite{Shin_2004,Wang_2005,Schumm_2005}. In this case, the experimental implementation is even more difficult since inter-atomic interactions due to high atomic densities in the wave-guides can sometimes not only induce the fragmentation of the BEC\,\cite{Stickney_2002,Gaaloul_2007}, but also affect the overall coherence of the system\,\cite{Chen_2003}. In a recent paper we have derived a semi-classical mo\-del for the description of the splitting dynamics of a cold atomic cloud in such a device\,\cite{Gaaloul_2006}. This setup involves two crossing far off-resonant dipole guides [see Fig.\,\ref{fig:f1}(a)], and we have shown that a simple variation of the laser beam intensities allows to control the splitting ratio in the two guides. In the present paper, we first show that if the vertical guide is switched off when the atomic cloud reaches the crossing point, this device becomes an efficient coherent atom deflector. We then extend this study to the quantum degenerate regime, in order to demonstrate the efficiency of this deflection setup with Bose-Einstein condensates. \begin{figure} \caption{(a) Schematic representation of the proposed optical deflector for cold atoms. The right inset is a magnification of the crossing region. The vertical position of the crossing point is $z=-h$, and the total transverse width of the oblique guide is equal to $2\ell_1$. (b) Timing of the magnetic and optical trap (MOT) and of the vertical (V) and oblique (O) guides used in this setup. $t_c=\sqrt{2h/g}$ corresponds to the time at which the Rb atoms reach the crossing height $z=-h$.} \label{fig:f1} \end{figure} As illustrated in Fig.\,\ref{fig:f1}(a), we use a setup involving two crossing far off-resonant dipole guides similar to the one of Ref.\,\cite{Houde_2000}. A large ensemble of $^{87}$Rb atoms is initially trapped and cooled around the position $z=0$ in Fig.\,\ref{fig:f1}(a). This trap is switched off at time $t=0$, while a vertical far off-resonant laser beam, crossing the cloud close to its center, is switched on. A significant portion of the atoms, falling due to gravity, is captured and guided in this vertical wave-guide\,\cite{Houde_2000}. When the center of the guided cloud reaches a given height $z=-h$, at time $t=t_c$, the vertical laser beam is switched off while a second oblique guide is switched on. This timing sequence is illustrated schematically in Fig.\,\ref{fig:f1}(b). The durations of the switching-on and -off procedures are supposed to be much shorter than the typical time scale of the fall dynamics. In spite of the high velocities achieved in this vertical fall, we will show that this setup allows for the implementation of an efficient deflector since the atoms can be deviated from their initial trajectory with no significant loss. This scheme is used both with a thermal cloud of atoms and with an atomic condensate after rescaling the whole problem due to the difference in size of condensates compared to cold atomic clouds. The outline of the paper is as follows: in Sec.\,\ref{sec:Pot} we discuss the properties of $^{87}$Rb atoms that are relevant for our analysis. We also give the values of typical laser parameters that realize this atom deflector. We describe briefly our semi-classical numerical model in Sec.\,\ref{sec:Model-cold}. In Sec.\,\ref{sec:Results-cold} we give the results of our numerical investigations on the performance of this setup with cold atomic clouds ($T\sim 10\,\mu$K). We show that a high efficiency ($\geqslant 90\%$) can be achieved with large deflection angles. We also discuss the adiabaticity of the deflection process. We then present in Sec.\,\ref{sec:Bec} a full quantum model designed to treat the dynamics of a BEC falling in the gravity field in the presence of these time-dependent guiding potentials. We then present the results of the numerical simulations with BECs, demonstrating the efficiency of the proposed setup in the quantum degenerate domain. Our conclusions are finally summarized in the last section. \section{Guiding potentials} \label{sec:Pot} During the guiding process and in the case of a large detuning, the atoms are subjected to a dipole force induced by the dipole potential \begin{equation} \label{eq:pot} {\cal U}({\textbf{\em r}}) = \frac{\hbar\Gamma}{2}\,\frac{I({\textbf{\em r}})/I_s}{4\delta/\Gamma}\,, \end{equation} where $\delta=\omega_L-\omega_0$ denotes the detuning between the laser frequency $\omega_L$ and the atomic transition frequency $\omega_0$. $I_s$ is the saturation intensity, and $\Gamma$ the natural linewidth of the atomic transition\,\cite{Phillips_1992,Grimm_2000}. The atomic dynamics is supposed to take place in the $(x,z)$ plane defined by the two guides (see Fig.\,\ref{fig:f1}(a)) thanks to a strong confinement applied in the $y$-direction. The transverse intensity distribution of the TEM$_{00}$ vertical laser beam of power $P_0$ is approximated by the Gaus\-sian-like form \begin{equation} \label{eq:int} \begin{array}{lccl} \textrm{if }|x| \leqslant \ell_0\textrm{ : } & I_0(x) & = & \displaystyle\frac{2P_0}{\pi w_0^2}\,\sin^2\left(\frac{\pi}{2}\,\frac{x-\ell_0}{\ell_0}\right)\,,\\ \textrm{if }|x| > \ell_0\textrm{ : } & I_0(x) & = & 0\,, \end{array} \end{equation} where the size $\ell_0$ of the vertical guide is simply related to the laser waist $w_0$ by the relation \begin{equation} \label{eq:size} \ell_0 = w_0 \sqrt{2\ln2} \sim 1.18\,w_0\,. \end{equation} This sinus-squared shape, which is often used in time-dependent calculations\,\cite{Giusti-Suzor_1995}, is very close to the ideal Gaussian intensity distribution, except for the absence of the extended wings of the true Gaussian shape which lengthen the calculations without noticeable contribution to the physical processes. With this sinus-squared convention, the guiding region ($|x| \leqslant \ell_0$) is also well defined. The trapping potentials associated with the vertical and oblique laser guides are thus expressed as \begin{subequations} \label{eq:U0U1} \begin{eqnarray} \label{eq:U0} {\cal U}_0(x) & = & -U_{0} \sin^{2}\left(\frac{\pi}{2}\,\frac{x -\ell_0}{\ell_0}\right)\textrm{~~for }|x| \leqslant \ell_0\\ \label{eq:U1} {\cal U}_1(x,z) & = & -U_{1} \sin^{2}\left(\frac{\pi}{2}\,\frac{x'-\ell_1}{\ell_1}\right)\textrm{ for }|x'| \leqslant \ell_1 \end{eqnarray} \end{subequations} where $x'$ denotes the rotated coordinate $x' = x\cos\gamma+(z+h)\sin\gamma$. Typical laser powers $P_0 \sim 5-30$\,W for a Nd:YAG laser operating at 1064\,nm with laser waists of about $100-300\,\mu$m yield potential depths of about $5-250$\,$\mu$K. With these laser parameters, the $^{87}$Rb transition to consider is the D$_1$\,: 5$^2$S$_{1/2}\rightarrow$\,5$^2$P$_{1/2}$, with a decay rate $\Gamma/2\pi \simeq 5.75$\,MHz, a saturation intensity $I_s \simeq 4.5$\,mW/cm$^2$ and a detuning $\delta/2\pi \simeq -95.4$\,THz. With these conditions, the Rayleigh range $z_R = \pi w_0^2 / \lambda$ is about 3\,cm, thus allowing us to neglect the divergence of the beam on a length up to about 1\,cm. \section{Semi-classical model for cold atoms} \label{sec:Model-cold} The guided atomic dynamics can be followed by solving numerically the time-dependent Schr\"{o}dinger equation for the atomic translational coordinates, taking into account the effect of the gravity field, and choosing realistic values for all laser parameters. We adopt a semi-classical approach where the $z$ coordinate is described classically, following \begin{subequations} \label{eq:z(t)} \begin{eqnarray} t \leqslant t_c :\;\; z_{cl}(t) & = & -gt^2/2\\ t > t_c :\;\; z_{cl}(t) & = & -g\big[t_c+(t-tc)\cos\gamma\big]^2/2\,, \end{eqnarray} \end{subequations} where $t_c=(2h/g)^{1/2}$ is the time at which the atoms reach the crossing point (position \mbox{$z=-h$}). These equations of motion are obtained under the assumption of energy conservation for a classical particle which is perfectly deflected, and which therefore follows the paths blazed initially by the vertical beam and later on by the oblique guide. The other dimension $x$ is treated at the quantum level. This semi-classical approach was compared to the experimental study\,\cite{Houde_2000} in Ref.\,\cite{Gaaloul_2006}. In this approach, the two-dimensional guiding potentials\,(\ref{eq:U0U1}) can be replaced by the one-dimensional time-dependent potential \begin{subequations} \label{eq:U(x,t)} \begin{eqnarray} t \leqslant t_c :\;\; {\cal U}(x,t) & = & {\cal U}_0(x)\\ t > t_c :\;\; {\cal U}(x,t) & = & {\cal U}_1(x,z_{cl}(t))\,, \end{eqnarray} \end{subequations} and the quantum dynamics is now summarized in the one-dimensional time-dependent Hamiltonian \begin{equation} \label{eq:H(x,t)} \hat{\mathcal{H}}(x,t) = -\frac{\hbar^{2}}{2m}\frac{\partial^{2}}{\partial x^{2}} + {\cal U}(x,t)\,, \end{equation} where $m$ denotes the $^{87}$Rb atomic mass. The time-de\-pen\-dent Schr\"{o}dinger equation \begin{equation} \label{eq:TDSE} i\hbar \frac{\partial }{\partial t}\varphi (x,t)=\hat{\mathcal{H}}(x,t)\;\varphi (x,t)\,, \end{equation} is then solved using the numerical split operator technique of the short-time propagator\,\cite{Feit_1983}, assuming that the atom is initially ($t=0$) in a well defined eigenstate $v$, of energy $E_v$, of the potential\,(\ref{eq:U0}) created by the vertical laser beam. In addition, it was shown in Ref.\,\cite{Gaaloul_2006} that the deflection probability obtained for the initial classical conditions $z(0)=0$ and $\dot{z}(0)=0$ is very close to the probability averaged over the entire atomic cloud. We therefore use these initial classical conditions in the present study. At the end of the propagation, the final wave function $\varphi (x,t_{f})$ is analyzed spatially, in order to extract the deflection efficiency $\eta_D$. An averaging procedure over the set of all possible initial states finally allows to calculate the total deflection probability $\langle\eta_D\rangle$ of the entire atomic cloud (see Sec.\,\ref{sec:multiv} hereafter for details). \section{Numerical Results for cold atoms} \label{sec:Results-cold} \subsection{Case of a single initial state} In this study, the value of the position $h$ of the crossing point between the two guides is the main parameter which controls the efficiency of the deflector. Indeed, for large values of $h$ the atoms reach the crossing point with a large kinetic energy $E_c=mgh$, and they will not be deflected if this energy exceeds by far the binding energy in the oblique guide. In order to predict precisely the largest value of the height $h$ allowing for atomic deflection, one should compare the kinetic energy gained by the atoms along the direction $x'$ transverse to the oblique guide at the position $z=-h-\ell_1/\sin\gamma$ [see Fig.\,\ref{fig:f1}(a)] with the binding energy $U_1-E_v$. The energy $E_v$ denotes here the energy of the initial vibrational state $v$. One can effectively expect that the deflection will fail if \begin{equation} \label{eq:critere} m g \left( h + \frac{\ell_1}{\sin\gamma} \right) \sin^2\gamma > U_1-E_v\,. \end{equation} In this expression, the $\sin^2\gamma$ factor originates from the fact that the transverse direction $x'$ of the deflecting beam makes an angle $\gamma$ with the fall direction $z$. The validity of this simple prediction is illustrated in Fig.\,\ref{fig:f2}, which represents the deflection probability $\eta_D$ as a function of $h$ [Fig.\,\ref{fig:f2}(a)] and of $w_1=\ell_1/\sqrt{2\ln2}$ [Fig.\,\ref{fig:f2}(b)], all other parameters being fixed. These probabilities are calculated numerically for the initial state $v=0$ and for $v=2094$, whose energy is about halfway in the optical potential $(E_v \simeq -U_0/2)$. In both graphs, the frontiers defined by the inequality\,(\ref{eq:critere}) are indicated by vertical dashed arrows. By comparison with the ``exact'' value obtained from the solution of the time-dependent Schr\"odinger equation\,(\ref{eq:TDSE}), one can notice that these frontiers correspond to a deflection probability of 50\%. This energy criterion, which simply compares the atomic kinetic energy with the binding energy in the oblique guide, can thus be used safely to predict the efficiency of this setup. \begin{figure} \caption{Deflection probability $\eta_D$ as a function (a) of the falling distance $h$ [see Fig.\,\ref{fig:f1}] and (b) of the waist $w_1$ of the oblique laser beam. The deflection angle is equal to $\gamma = 10\,$deg. These results are for a single initial state\,: $v=0$ (solid line with red circles) or $v=2094$ (solid line with green squares). In both graphs, the dashed blue arrows mark the positions at which a deflection efficiency of 50\% is expected according to inequality\,(\ref{eq:critere}). The laser parameters have been chosen such that $U_0=U_1=30\,\mu$K, and $w_0=100\,\mu$m. This corresponds to $P/\delta \simeq 3.1 \times 10^{-4}\,$W/GHz. In graph (a) the oblique laser waist is $w_1=100\,\mu$m and in graph (b) the height $h$ is equal to 9.03\,mm for $v=0$ and to 4.18\,mm for $v=2094$.} \label{fig:f2} \end{figure} One can also notice in Fig.\,\ref{fig:f2}(a) the different variations of $\eta_D$ with $h$ for $v=0$ and for $v=2094$. The different behavior of these two vibrational levels comes from the fact that $v=0$ is associated with a well localized atomic wavefunction, deeply bound in an almost harmonic potential, while $v=2094$ is entirely delocalized over a large spatial range $|x|\leqslant\ell_0/2$, since its energy is about halfway in the potential. As a consequence, $v=0$ satisfies fully the conditions imposed by the Ehrenfest theorem\,\cite{Ehrenfest_1927} and its evolution can be described classically, while $v=2094$ shows a quantum behavior. For $v=0$, as soon as the inequality\,(\ref{eq:critere}) is satisfied, the deflection probability falls to zero, in agreement with the usual dynamics of a classical particle. On the other hand, the stationnary state $v=2094$ can be seen as a coherent superposition of incoming and outgoing wave packets characterized by a rather broad kinetic energy distribution of width $\Delta E_c \sim U_0/2$. The packet moving in the $+x$ direction will be easily captured by the oblique guide, while the packet moving in the opposite direction easily avoids this wave guide. These two different dynamics are not much affected by the exact value of the falling height $h$, and this explains the very slow variation of $\eta_D$ with $h$ in Fig.\,\ref{fig:f2}(a) for $v=2094$. The variation of $\eta_D$ with $w_1$ [see Fig.\,\ref{fig:f2}(b)] is also opposite for $v=0$ and $v=2094$. The case $v=0$ can again be interpreted classically\,: when $w_1$ increases, the possibility is open for the atoms to fall from a higher distance $d=(h+\ell_1/\sin\gamma)$, thus gaining a larger kinetic energy. This explains the decrease of $\eta_D$ with $w_1$ for $v=0$. This variation is just reversed in the case of $v=2094$. Here, the initial wave function is characterized by a large typical size $\Delta x \sim \ell_0$. An efficient deflector can thus only be obtained if the size of the oblique wave guide remains of the order of, or is higher than, this typical size $\ell_0$. Consequently, for $v=2094$, when $w_1$ decreases below $w_0$, the deflection probability decreases, as seen in Fig.\,\ref{fig:f2}(b). In addition, it is worth noting that, due to the $\sin^2\gamma$ factor in the inequality\,(\ref{eq:critere}), it is possible to induce an efficient deflection of atoms with relatively large kinetic energies using modest laser powers, as long as the angle $\gamma$ remains small. For instance, in the case $v=0$ shown in Fig.\,\ref{fig:f2}(a) for $\gamma = 10\,$deg, an almost perfect deflection is obtained for $h=8.5$\,mm, even though the total kinetic energy of the atom reaches then about $E_c \sim 900\,\mu$K, {\it i.e.} 30 times the depth of the oblique wave guide. A larger deflection angle could be achieved easily and with a very high efficiency by simply adding a succession of several deflection setups, each one inducing a small deflection of about 10\,degrees. An important issue for the preservation of the coherence properties of an atomic cloud is the adiabaticity of the process. Previous theoretical studies have shown that similar beam splitter setups are able to conserve the coherence of the system even for a thermal distribution of atoms with an average energy far exceeding the level spacing of the transverse confinement\,\cite{Kreutzmann_2004}. This behavior results from the fact that non-adiabatic transitions are induced by the time derivative operator $d/dt$ which does not couple states of opposite parities, thus preventing nearest neighbor transitions\,\cite{Hansel_2001}. In comparison, transitions to other states presenting the same parity as the initial state are also not favored since they involve larger energy differences\,\cite{Zhang_2006}. As shown in Fig.\,\ref{fig:f3}, this robustness to non-adiabatic transitions is also present in our deflection scheme. This figure represents the probability distributions $\left|\varphi (x,t_{f})\right|^2$ calculated 7\,mm below the crossing point $z=-h$ for the initial state $v=0$, with $h=2$\,mm [Fig.\,\ref{fig:f3}(a)], $h=7$\,mm [Fig.\,\ref{fig:f3}(b)], and $h=9$\,mm [Fig.\,\ref{fig:f3}(c)]. The vibrational distributions obtained in the oblique guide after deflection are also shown in the small insets of Fig.\,\ref{fig:f3}(a) and\,\ref{fig:f3}(b). Even though the kinetic energy of the atoms exceeds the average vibrational spacing in the trap, the initial state $v=0$ is preserved at 99.1\% for $h=2$\,mm, and at 50.3\% for $h=7$\,mm. Indeed, in the first case, only $v=2$ is slightly populated, while the first five even vibrational levels are populated in the second case. It is only when the falling height $h$ approaches the limit given by the inequality\,(\ref{eq:critere}) that the population of the initial state $v=0$ is almost entirely redistributed to higher excited states, as seen in the wave function shown Fig.\,\ref{fig:f3}(c). \begin{figure} \caption{Atomic probability distributions $\left|\varphi (x,t_{f})\right|^2$ as a function of the transverse coordinate $x$ at the end of the propagation, for (a) $h=$2\,mm, (b) $h=$7\,mm and (c) $h=9$\,mm. The laser parameters are identical to the one of Fig.\,\ref{fig:f2}, with $v=0$ and $w_1=100\,\mu$m. Note that, for the sake of clarity, the horizontal axis has been broken in panel\,(c). The small insets in panels (a) and (b) represent the vibrational distributions in the oblique guide at the end of the propagation.} \label{fig:f3} \end{figure} \subsection{Case of an initial vibrational distribution} \label{sec:multiv} Realistically, an atomic cloud of typical size $\sigma_0$ and temperature $T_0$ can be described as a statistical mixture of trapped vibrational states, represented by the density matrix \begin{equation} \label{eq:thermal} \rho(\sigma_0,T_0) = \sum_{v} c_v(\sigma_0,T_0) \; | v \rangle \langle v |\,, \end{equation} where the coefficients $c_v(\sigma_0,T_0)$ are involved functions of the cloud parameters $\sigma_0$ and $T_0$ and of the wave guide parameters $U_0$ and $w_0$ (see equation (16) in reference\,\cite{Gaaloul_2006} for instance). The calculation of the total deflection probability of the entire cloud therefore requires to average incoherently the deflection probability of each possible initial vibrational level $v$, taking into account the weight functions $c_v(\sigma_0,T_0)$. It is also worth noting that typical initial vibrational distributions $P(v)=|c_v(\sigma_0,T_0)|^2$ are relatively flat when $k_B T \sim U_0$, except for the lowest energy levels which are usually more populated\,\cite{Gaaloul_2006}. In the calculation, we include all populated vibrational states. \begin{figure} \caption{Deflection probability $\eta_D$ as a function of the initial vibrational level $v$. (a) The deflection angle is equal to $\gamma=10\,$deg, and the solid line with red circles stands for $U_1=30\,\mu$K while the solid line with green squares is for $U_1=25\,\mu$K. (b) The depth of the oblique wave guide is equal to $U_1=30\,\mu$K, and the solid line with red circles stands for $\gamma=10\,$deg while the solid line with green squares is for $\gamma=13\,$deg. The falling height is $h=4$\,mm and the oblique laser waist is equal to $w_1=100\,\mu$m. All other parameters are identical to the one of Fig.\,\ref{fig:f2}.} \label{fig:f4} \end{figure} Fig.\,\ref{fig:f4} shows the variation of the deflection efficiency with the initial vibrational level $v$ for a series of different laser parameters. The transverse trapping potential associated with the vertical wave guide supports about 5000 vibrational states when $U_0=30\,\mu$K and $w_0=100\,\mu$m. One can notice the general tendency of measuring a lower deflection probability when $v$ increases, in perfect agreement with the variation expected from the energy criterion\,(\ref{eq:critere}). In addition, one can notice that increasing $U_1$ [Fig.\,\ref{fig:f4}(a)] or decreasing $\gamma$ [Fig.\,\ref{fig:f4}(b)] increases the deflection probability of any initial state. In Fig.\,\ref{fig:f4}, the vertical dashed arrows indicate the limits defined by the inequality\,(\ref{eq:critere}), which are again in good agreement with the numerical values. One can also notice that the highest levels $v \simeq 5000$ are not deflected. This is due to the fact that atoms trapped in these levels, whose energies are very close to the threshold, are easily lost during the deflection process. \begin{figure} \caption{Total deflection probability $\left<\eta_D\right>$ of an atomic cloud of $^{87}$Rb of size $\sigma_0=0.15$\,mm at temperature $T_0=10\,\mu$K. The laser parameters have been chosen such that $U_0=30\,\mu$K, $w_0=200\,\mu$m, $w_1=158\,\mu$m, and $h=4$\,mm (a) or $h=1$\,mm (b).} \label{fig:f5} \end{figure} Fig.\,\ref{fig:f5} represents the averaged deflection probability $\langle \eta_D \rangle$ as a function of the deflection angle $\gamma$ and of the potential depth $U_1$ of the oblique laser guide, for a thermal input state of size $\sigma_0=0.15\,$mm and temperature $T_0=10\mu$K, with $h=4\,$mm [Fig.\,\ref{fig:f5}(a)] and $h=1\,$mm [Fig.\,\ref{fig:f5}(b)]. Realistic values have been chosen for all laser parameters, close to the one used in the experimental study\,\cite{Houde_2000}, and the coefficients $c_v(\sigma_0,T_0)$ of Eq.\,(\ref{eq:thermal}) were calculated following Ref.\,\cite{Gaaloul_2006}. One can notice a rapid decrease of $\langle \eta_D \rangle$ when $U_1$ decreases and when $\gamma$ increases. However, an almost complete deflection (93.8\%) is still observed in the case $h=1\,$mm with $\gamma=25\,$deg and $U_1=120\,\mu$K, even though the total kinetic energy of the atoms reaches then about $E_c \sim 100\,\mu$K at the crossing point, all trapped states being significantly populated initially. For $\gamma=10\,$deg, the deflection efficiency reaches 99.8\%. We have also verified that decreasing the temperature of the initial cloud increases significantly the deflection efficiency since it suppresses the population of the highest trapped states, for which the deflection process is less efficient [see Fig.\,\ref{fig:f4}]. It is also worth noting that since the deflection process is less efficient for the highest trapped levels, it could also be used to selectively separate the lowest energy levels of the trap. Since it behaves very well for the lowest trapped states, we expect that this setup will prove useful with Bose-Einstein condensates. We therefore derive in the next section a quantum model aimed at the description of the dynamics of a Bose gas in such a deflection setup. \section{Deflection of Bose-Einstein condensates} \label{sec:Bec} \subsection{Theoretical model} \label{sec:Theo-BEC} From the theoretical point of view, in the case of a low density the dynamics of the macroscopic wave function $\Psi(\mathbi{r},t)$ of a Bose-Einstein condensate can be accurately described by the mean-field Gross-Pitaevskii equation\,\cite{Gross_1961,Pitaevskii_1961,Gross_1963}. In three dimensions and in the presence of both a time-dependent external potential $V(\mathbi{r},t)$ and the gravity field this equation reads \begin{equation} \label{eq:3DGPE} i\hbar\frac{\partial\Psi}{\partial t} = \Big[ -\frac{\hbar^2}{2m} \nabla^2_{\!r} + V(\mathbi{r},t) + mgz + NU_0\left|\Psi\right|^2 \Big]\Psi\,, \end{equation} where $U_0=4\pi\hbar^2a_0/m$ is the scattering amplitude and $a_0$ the $s$-wave scattering length. $N$ denotes the condensate number and $N U_0 \left|\Psi(\mathbi{r},t)\right|^2$ is the mean field interaction energy. The three-dimensional coordinate is denoted by \mbox{$\mathbi{r} \equiv (x,y,z)$}. In the absence of a trapping potential in the $z$-di\-rec\-tion, the condensate will not only expand but also fall around the average classical height $z_{cl}(t)=-gt^2/2$. Since the de Broglie wavelength of the BEC is no more negligible compared to the characteristic distances of the problem, a quantum treatment of this direction is necessary unlike the thermal atoms case discussed in the first part of the paper. The simulation of the fall dynamics is thus greatly facilitated when done in the moving frame \mbox{$\mathbi{R} \equiv (X,Y,Z)$}, where \begin{equation} \label{eq:frame} \mathbi{R}=\mathbi{r}+\frac{1}{2}\,gt^2\,\mathbi{u}_z\,, \end{equation} using the unitary transformation \begin{equation} \label{eq:tranf} \Xi(\mathbi{R},t) = \exp\left[\,i\;\frac{mgt}{\hbar}\left(z+\frac{gt^2}{6}\right)\right] \Psi(\mathbi{r},t)\,. \end{equation} Indeed, applying this transformation yields a simplified Gross-Pitaevskii equation \begin{equation} \label{eq:3DGPEs} i\hbar\frac{\partial\Xi}{\partial t} = \Big[ -\frac{\hbar^2}{2m} \nabla^2_{\!R} + V(\mathbi{R},t) + NU_0\left|\Xi\right|^2 \Big] \Xi\,, \end{equation} where the gravitational term $mgz$ has vanished. Following the variational approach of Ref.\,\cite{Salasnich_2002}, we now assume (as in Sec. \ref{sec:Model-cold} and \ref{sec:Results-cold}) a strong harmonic confinement in the perpendicular $Y$-direction, with \begin{equation} \label{eq:poty} V(\mathbi{R},t) = \frac{1}{2} m \omega_\perp^2 Y^2 + V_\parallel(X,Z,t)\,, \end{equation} where $V_\parallel(X,Z,t)$ denotes the optical guiding potential \begin{subequations} \label{eq:potpar} \begin{eqnarray} t \leqslant t_c :\;\; V_\parallel(X,Z,t) & = & {\cal U}_0(X)\\ t > t_c :\;\; V_\parallel(X,Z,t) & = & {\cal U}_1(X,Z-gt^2/2)\,. \end{eqnarray} \end{subequations} The confinement along the perpendicular direction $Y$ is supposed to be much stronger than along the parallel directions $X$ and $Z$, thus yielding the conditions \begin{equation} \label{eq:confine} \omega_\perp \gg \left[\frac{4U_{0}}{mw_{0}^2}\right]^{\frac{1}{2}} \quad\textrm{and}\quad \omega_\perp \gg \left[\frac{4U_{1}}{mw_{1}^2}\right]^{\frac{1}{2}}\,. \end{equation} The condensate dynamics is now followed using the appropriate ansatz\,\cite{Salasnich_2002,Jackson_1998} \begin{equation} \label{eq:trialwf} \Xi(\mathbi{R},t) = \Phi(X,Z,t)\;f\big(Y|\Omega\big)\,, \end{equation} where \begin{equation} \label{eq:trialwfG} f\big(Y|\Omega\big) = \frac{e^{-\frac{1}{2}\frac{Y^2}{\Omega^2}}}{\pi^{\frac{1}{4}}\Omega^{\frac{1}{2}}}\,. \end{equation} This choice amounts to assume a Gaussian shape of the wave function in the $Y$-direction, characterized by a time-dependent width $\Omega(X,Z,t)$. This width varies slowly along the parallel directions, thus implying \begin{equation} \nabla^2_{\!R} f \simeq \partial^2 f/\partial Y^2\,. \end{equation} It has been shown that this choice is well justified not only in the limit of weak interatomic couplings but also with large condensate numbers\,\cite{Perez-Garcia_1996,Perez-Garcia_1997,Parola_1998}. An effective two-dimensional non-linear wave equation is then derived using the quantum least action principle\,\cite{Schiff_1968,Salasnich_2002} for $\Phi(X,Z,t)$ \begin{eqnarray} \label{eq:2DGPE} i\hbar\frac{\partial\Phi}{\partial t} & = & \Big[ -\frac{\hbar^2}{2m} \nabla^2_{\parallel} + V_\parallel + \frac{N\,U_0}{\sqrt{2\pi}}\,\frac{\left|\Phi\right|^2}{\Omega}\nonumber\\ & & \qquad + \frac{1}{4}\left(\frac{\hbar^2}{m\Omega^2} + m\omega_\perp^2\Omega^2\right) \Big] \Phi\,. \end{eqnarray} This equation describes the condensate dynamics in the $(X,Z)$ plane, with an accuracy which goes beyond the usual two-dimensional Gross-Pitaevskii equation. It takes into account the influence of the dynamics along the perpendicular direction on the evolution of $\Phi(X,Z,t)$ with the introduction of the width parameter $\Omega(X,Z,t)$. Note the difference by a factor $1/2$ in the last two terms of Eq.(\ref{eq:2DGPE}) when compared to Eq.(25) of Ref.\,\cite{Salasnich_2002}, due to a misprint in Ref.\,\cite{Salasnich_2002}. The least action variational principle also yields the following quartic equation governing the evolution of this parameter \begin{equation} \label{eq:Omega} \left(\frac{1}{2}m\omega_\perp^2-\frac{2V_\parallel}{w^2(t)}\right)\Omega^4-\frac{N\,U_0}{2\sqrt{2\pi}}\left|\Phi\right|^2\,\Omega-\frac{\hbar^2}{2m}=0\,, \end{equation} where $w(t)=w_0$ for $t \leqslant t_c$ and $w(t)=w_1$ for $t > t_c$. This last equation was obtained assuming $\Omega(X,Z,t) \ll w(t)$ for all $X$, $Z$ and $t$, in agreement with the strong confinement in $Y$. Compared to Eq.(26) of Ref.\,\cite{Salasnich_2002}, the additional term $2V_\parallel/w^2(t)$ is a small correction due to the $Y$-dependence of the TEM$_{00}$ laser intensity profile. The time-dependent wave equation\,(\ref{eq:2DGPE}) is solved numerically using the splitting technique of the short-time propagator\,\cite{Feit_1983}, while the quartic equation\,(\ref{eq:Omega}) for $\Omega$ is solved at each time step and for each coordinate grid point $(X,Z)$ using an efficient numerical algorithm\,\cite{Hacke_1941}. The initial wave function is obtained using the imaginary time relaxation technique\,\cite{Kosloff_cpl}, and the three-dimensional condensate wave function $\Psi(\mathbi{r},t)$ is reconstructed at the end of the propagation by inverting the transformations\,(\ref{eq:frame}) and\,(\ref{eq:tranf}). \begin{figure} \caption{(a) and (c)\,: Atomic density $|\Phi(x,z,t)|^2$ in arbitrary units as a function of $x$ and $z$ at times $t=0$ (upper graph) and $t=5.2\,$ms (lower graph). (b) and (d)\,: Width parameter $\Omega(x,z,t)$ in $\mu$m as a function of $x$ and $z$ at times $t=0$ (upper graph) and $t=5.2\,$ms (lower graph). In each sub-plot, the color scale is defined on the right hand side of the graph. The guiding potentials are defined by $U_0=2.2\,\mu$K [$P/\delta \simeq 2.3 \times 10^{-5}\,$W/GHz], $U_1=8.8\,\mu$K [$P/\delta \simeq 9.2 \times 10^{-5}\,$W/GHz], and $w_0=w_1=300\,\mu$m. The deflection angle is $\gamma=50\,$deg and the condensate number is $N=5 \times 10^4$. In the $y-$direction, the trapping frequency $\omega_y$ is assumed to be 10 times larger than $\omega_x$. The crossing height is $z=-h=-10\,\mu$m and the time step for the split operator numerical propagation is $\delta t=1\,\mu$s.} \label{fig:f6} \end{figure} \subsection{Numerical Results} \label{sec:Res-BEC} A typical BEC dynamics is illustrated in Fig.\,\ref{fig:f6}, which shows a surface plot of the atomic density $|\Phi(x,z,t)|^2$ and of the width parameter $\Omega(x,z,t)$ at time $t=0$ [upper part, labels (a) and (b)] and during the deflection process, at time $t=5.2\,$ms [lower part, labels (c) and (d)]. In this numerical example, the condensate number is $N=5 \times 10^4$. The crossing height $z=-h$ is reached at time $t_c=1.43\,$ms, and the deflection angle is fixed at the value $\gamma=50\,$deg. In the lower graphs the transparent oblique line indicates the direction of deflection corresponding to this angle. \begin{figure} \caption{Condensate deflection efficiency $\eta_D$ as a function of (a) the crossing height $h$, (b) the deflection angle $\gamma$ and (c) the ratio of binding energies $U_1/U_0$, equivalent to the ratio of laser powers $P_1/P_0$. In these three graphs, the potential depth $U_0$ is fixed at 2.2\,$\mu$K, and $w_0=w_1=300\,\mu$m. The condensate number is $N=5 \times 10^4$. In (a) $U_1=8.8\,\mu$K and $\gamma=60\,$deg. In (b) $U_1=8.8\,\mu$K and $h=10\,\mu$m. In (c) $h=10\,\mu$m and $\gamma=50\,$deg (blue line with circles) and $\gamma=70\,$deg (red line with squares). The dashed blue arrow in the upper graph marks the position at which a deflection efficiency of 50\,\% is expected according to Eq.(\ref{eq:critere}).} \label{fig:f7} \end{figure} One can first notice in Fig.\,\ref{fig:f6}(b) and (d) that even for a relatively small condensate number, a Gaussian ansatz for $f(Y|\Omega)$ can only be used if the width $\Omega$ is a free parameter which can take different values at different positions $x$ and $z$. This result is in agreement with the numerical studies of Salasnich \emph{et al}\,\cite{Salasnich_2002}. The inset (a) of Fig.\,\ref{fig:f6} shows the symmetric ground state wave function of the initial condensate prepared in a parabolic trapping potential of equal frequencies $\omega_x=\omega_z$. At time $t=5.2\,$ms [insets (c) and (d) of Fig.\,\ref{fig:f6}], the condensate wave function has expanded during the fall dynamics, and has been efficiently deflected along the direction $\gamma=50\,$deg by the oblique laser guide. In our numerical simulations, we propagate the condensate wave function well after the crossing point between the vertical and oblique laser beams has been rea\-ched, and we obtain the deflection efficiency $\eta_D$ by calculating the condensate number in the oblique trapping potential at the end of the propagation. Fig.\,\ref{fig:f7} shows the variation of the condensate deflection efficiency with the crossing height $h$ [inset (a)], the deflection angle $\gamma$ [inset (b)] and the ratio of laser powers $P_1/P_0=U_1/U_0$ [inset (c)]. The other parameters are given in the figure caption. The simple energy criterium\,(\ref{eq:critere}) is still shown to be relatively accurate with Bose-Einstein condensates since in Fig.\,\ref{fig:f7}(a), a deflection efficiency of 50\,\% is expected for a falling height $h = 114\,\mu$m according to Eq.(\ref{eq:critere}), and the numerical simulation yields a deflection probability of 55\,\%. The variation of the deflection efficiency $\eta_D$ with $h$, $\gamma$ and $U_1/U_0$ is also found to be very similar to the one obtained with cold atomic clouds. One can finally notice that large deflection angles can be reached when $U_1 > U_0$, with almost no atom loss for instance when $\gamma = 50\,$deg and $U_1 = 4 U_0$. This high deflection efficiency can be explained by the small size of the BEC and the weak spread in velocities compared to cold atoms. \section{Conclusion} \label{sec:Conclusion} In summary, we have presented a detailed analysis of the implementation of an optical deflector for cold atomic clouds and for Bose-Einstein condensates. Our analysis is quite close to the experimental conditions, and is clearly within the reach of current technology. We have shown how to create a high performance deflector using two crossing laser beams which are switched on and off in a synchronized way. We have found that a 10\,$\mu$K cloud of Rubidium atoms can be deflected by 25\,degrees with an efficiency of about 94\%, and by 10\,degrees with an efficiency exceeding 99\%. A succession of such deflecting setups at this small angle could also be implemented in order to achieve larger deflection angles with high fidelities. We have shown that this device is robust against non-adiabatic transitions, an undesirable effect which could have led to heating processes. A high degree of control can therefore be achieved with such quantum systems, opening some possibilities for a range of applications. We have also derived an original approach treating the dynamics of a Bose-Eintein condensate in the gravity field using the quantum least action principle in a moving frame. This model was used to demonstrate the high efficiency of this deflection setup with quantum degenerate gases since deflection angles of up to about 50\,degrees can be implemented with no significant atom loss. \end{document}

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