# The Projection Pressure for Asymptotically Weak Separation Condition and Bowen's Equation

Authors: Chenwei Wang; Ercai Chen
Journal: Discrete Dynamics in Nature and Society
 (2012)

---

## Abstract

Let{Si}i=1l be a weakly conformal iterated function system on ℝd with attractor K. Let π be the canonical projection. In this paper we define a new concept called “projection pressure” Pπ(φ) for φ∈C(Σ) and show the variational principle about the projection pressure under AWSC. Furthermore, we check that the zero of “projection pressure” still satisfies Bowen's equation. Using the root of Bowen's equation, we can get the Hausdorff dimension of the attractor K.


---

## Body

## 1.  Introduction

Let{Si:X→X}i=1l be a family of contractive maps on a nonempty closed set X⊂ℝd. Following Barnsley [1], we say that {Si}i=1l is an iterated function system (IFS) on X. Hutchinson [2] showed that there is a unique nonempty compact set K⊂X, called the attractor of {Si}i=1l, such that K=∪i=1lSi(K).There are many references to compute the Hausdorff dimension ofK or the Hausdorff dimension of multifractal spectrum, such as, [3–5]. Thermodynamic formalism played a significant role when we try to compute the Hausdorff dimension of K, especially the Bowen’s equation. Usually, we call PJ(tψ)=0 the Bowen’s equation, where PJ is the topological pressure of the map f:J→J, and ψ is the geometric potential ψ(z)=log|f′(z)|. The root of Bowen’s equation always approaches the Hausdorff dimension of some sets. In [6], Bowen first discovered this equation while studying the Hausdorff dimension of quasicircles. Later Ruelle [7], Gatzouras and Peres [8] showed that Bowen’s equation gives the Hausdorff dimension of J whenever f is a C1 conformal map on a Riemannian manifold and J is a repeller. According to the method for calculating Hausdorff dimension of cookie-cutters presented by Bedford [9], Keller discussed the relation between classical pressure and dimension for IFS [10]. He concluded that if {Si}i=1l is a conformal IFS satisfying the disjointness condition with local energy function ψ, then the pressure function has a unique zero root t0=dimHK. In 2000, using the definition of Carathe’ odory dimension characteristics, Barreira and Schmeling [11] introduced the notion of the u-dimension dimuZ for positive functions u, showing that dimuZ is the unique number t such that PZ(-tu)=0.On the progress of calculatingdimHK, [3–5] depend on the open set condition and separable condition. In fact, there are a lot of examples that do not satisfy this disjointness condition. Rao and  Wen once discussed a kind of self-similar fractal with overlap structure called λ-Cantor set [12].In order to study the Hausdorff dimension of an invariant measureμ for conformal and affine IFS with overlaps, Feng and Hu introduce a notion projection entropy (see [13]), which plays the similar role as the classical entropy of IFS satisfying the open set condition, and it becomes the classical entropy if the projection is finite to one.Bedford pointed out that the Bowen’s equation works if three elements are present: (i) conformal contractions, (ii) open set conditions, and (iii) subshift of finite-type (Markov) structure. Chen and Xiong [14] proved that subshift of finite-type (Markov) structure can be replaced by any subshift structure. In [15, 16], the authors defined projection pressure for two different types of IFS. In this paper, we consider projection pressure under asymptotically weak separation condition (AWSC) and check that Bowen’s equation still holds.


## 2.  The Projection Pressure for AWSC: Definition and Variational Principle

Let{Si}i=1l be an IFS on a closed set X⊂ℝd. Denote by K its attractor. Let Σ={1,…,l}ℕ associated with the left shift σ. Let Mσ(Σ) denote the space of σ-invariant measure on Σ, endowed with the weak-star topology, C(X) the space of real-valued continuous functions of X, and π:Σ→K be the canonical projection defined by(2.1){π(x)}=⋂n=1∞Sx1∘Sx2∘⋯∘Sxn(K),wherex={xi}i=1∞.A measureμ on K is called invariant (resp., ergodic) for the IFS if there is and invariant (resp., ergodic) measure ν on Σ such that μ=ν∘π-1.Let(Ω,ℱ,ν) be a probability space. For a sub-σ-algebra 𝒜 of ℱ and f∈L1(Ω,ℱ,ν), we denote by Eν(f|𝒜) the conditional expectation of f given 𝒜. For countable ℱ-measurable partition ξ of Ω. We denote by Iν(ξ|𝒜) the conditional information of ξ given 𝒜, which is given by the formular:(2.2)Iν(ξ∣A)=-∑A∈ξXAlog⁡Eν(XA∣A),
where 𝒳A denote the characteristic function on 𝒜.The conditional entropy ofξ given 𝒜, written Hν(ξ|𝒜) is defined by the formula Hν(ξ|𝒜)=∫Iν(ξ|𝒜)dν.The above information and entropy are unconditional when𝒜=𝒩, the trivial σ-algebra consisting of sets of measure zero and one, and in this case we write(2.3)Iν(ξ∣A)=Iν(ξ),Hν(ξ∣N)=Hν(ξ).Now we consider the space(Σ,B(Σ),m), where B(Σ) is the Borel σ-algebra on Σ and m∈Mσ(Σ). Let 𝒫 denote the Borel partition:(2.4)P={[j]:1≤j≤l}
of Σ, where [j]={(xi)i=1∞∈Σ,x1=j}. Let ℐ denotes the σ-algebra:(2.5)I={B∈B(Σ):σ-1B=B}.For convenience, we useγ to denote the Borel σ-algebra ℬ(ℝd) of ℝd. For f∈C(X), denote ∥f∥=supx∈Xf(x) and Snf(x)=∑i=0n-1f(σnx), for all x∈X. Let Σn={[b]:[b]=(x1,x2,…,xn),xi∈Σ,i=1,…,n}.Definition 2.1.
For anym∈Mσ(Σ), we call
(2.6)hπ(σ,m)=Hm(P∣σ-1π-1γ)-Hm(P∣π-1γ)
the projection entropy of m under π  w.r.t {Si}, and we call
(2.7)hπ(σ,m,x)=Em(f∣I)(x)
the local projection entropy of m at x under π w.r.t  {Si}i=1l, where f denote the function Im(𝒫|σ-1π-1γ)-Im(𝒫|σ-1γ).It is clear thathπ(σ,m)=∫hπ(σ,m,x)dm(x).The following Lemma2.2 gives the relation between the projection entropy and the classical entropy and the basic properties of the new entropy which are similar to the classical entropy’s. For more details we can see Theorem  2.2 in [13].Lemma 2.2.
Let{Si}i=1l be an IFS. Then(i)
For anym∈Mσ(Σ), one has 0≤hπ(σ,m)≤h(σ,m), where h(σ,m) denotes the classical measure-theoretic entropy of m associated with σ.(ii)
The mapm↦hπ(σ,m) is affine on Mσ(Σ). Furthermore if m=∫νdℙ(ν) is the ergodic decomposition of m, one has
(2.8)hπ(σ,m)=∫hπ(σ,ν)dP(ν).(iii)
For anym∈Mσ(Σ), one has
(2.9)lim⁡n→∞1nIm(P0n-1∣π-1γ)(x)=h(σ,m,x)-hπ(σ,m,x),
for m-a.e. x∈Σ, where h(σ,m,x) denotes the local entropy of m at x, that is, h(σ,m,x)=Im(𝒫∣σ-1ℬ(Σ))(x).Definition 2.3.
Letk∈ℕ and ν∈Mσk(Σ). Define
(2.10)hπ(σk,ν)=Hν(P0k-1∣σ-kπ-1γ)-Hν(P0k-1∣π-1γ).The termhπ(σk,ν) can be viewed as the projection measure-theoretic entropy of ν w.r.t. the IFS {Si1∘⋯∘Sik:1≤ij≤lfor1≤j≤k}. The following lemma exploits the connection between hπ(σk,ν) and hπ(σ,m), where m=(1/k)∑i=0k-1ν∘σ-i.Lemma 2.4.
Letk∈ℕ and ν∈MσkΣ. Set m=(1/k)∑i=0k-1ν∘σ-i. Then m is σ-invariant, and hπ(σ,ν)=(1/k)hπ(σk,ν)=(1/k)hπ(σk,m).Proof.
See Proposition  4.3 in [13].Definition 2.5.
An IFS{Si}i=1l on a compact set X⊂ℝd is said to satisfy the asymptotically weak separation condition (AWSC), if
(2.11)lim⁡n→∞1nlog⁡tn=0,
where tn is given by
(2.12)tn=sup⁡x∈Rd#{Su:u∈{1,…,l}n,x∈Su(K)},
here K is the attractor of {Si}i=1l.Lemma 2.6.
Let{Si}i=1l be an IFS with attractor K. Suppose that Ω is a subset of {1,…,l} such that there is a map g:  {1,…,l}→Ω so that
(2.13)Si=Sg(i)(i=1,…,l).
Let (Ωℕ,σ̃) denote the one-side full shift over Ω. Define G: Σ→Ωℕ by (xj)j=1∞↦(g(xj))j=1∞. Then(i)
K is also the attractor of {Si}i∈Ω. Moreover, if one lets π̃:Ωℕ→K denote the canonical projection w.r.t. {Si}i∈Ω, then one has  π=π̃∘G.(ii)
Letm∈Mσ(Σ). Then ν=m∘G-1∈Mσ̃(Ωℕ). Furthermore, hπ(σ,m)=hπ̃(σ̃,ν).Proof.
See Lemma  4.23 in [13].Lemma 2.7.
Let{Si}i=1l be an IFS with attractorK⊂ℝd. Assume that
(2.14)#{1≤i≤l:x∈Si(K)}≤k
for some k∈ℕ and each x∈ℝd. Then hπ(σ,m)≥h(σ,m)-logk for any m∈Mσ(Σ).Proof.
See Lemma 4.21 in [13].Lemma 2.8.
Leta1,a2,…,ak be given real numbers. If  pi≥0 and ∑i=0kpi=1, then ∑i=0kpi(ai-logpi)≤log(∑i=0keai) and equality holds iff pi=eai/∑j=1keaj.Proof.
See Lemma  9.9 in [17].For convenience, forn∈ℕ, write Σn={1,…,l}n. According to Lemma 2.6 there is a set Ωn⊂Σn and a map g:Σn→Ωn such that Su=Sg(u) for u∈Σn. Let (Ωnℕ,T) denote the one-sided full shift space over the alphabet Ωnℕ and ξn denote the natural generator. Let G:Σ→Ωnℕ be defined by(2.15)(xi)i=1∞⟼(g(xjn+1⋯x(j+1)n))j=0∞.Theorem 2.9.
Suppose an IFS{Si}i=1l satisfies the AWSC with attractor K and f:Σ→ℝ is continuous. Then
(2.16)lim⁡n→∞1n(log⁡∑[u]∈ξnsup⁡x∈G-1[u]eSnf(x))=sup⁡{hπ(σ,m)+∫fdm:m∈Mσ(Σ)}.Proof.
We divided the proof into two steps.
Step 1.
(2.17)liminf⁡n→∞1n(log⁡∑[u]∈ξnsup⁡x∈G-1[u]eSnf(x))≥sup⁡{hπ(σ,m)+∫fdm:m∈Mσ(Σ)}.
For arbitrary n∈ℕ,m∈Mσ(Σ), then m∈Mσn(Σ). By Lemma 2.8, we have
(2.18)log⁡∑[u]∈ξnsup⁡x∈G-1[u]eSnf(x)≥∑[u]∈ξnm∘G-1([u])(sup⁡x∈G-1[u]Snf(x)-log⁡m∘G-1([u]))=Hm∘G-1(ξn)+∑[u]∈ξnm∘G-1([u])∘sup⁡x∈G-1[u]Snf(x)≥Hm∘G-1(ξn)+∫Snf(x)dm=h(T,m∘G-1)+n∫fdm.
By Lemma 2.2(i) and Lemma 2.6(ii), divided by n yields
(2.19)1nlog⁡∑[u]∈ξnsup⁡x∈G-1[u]eSnf(x)≥h(T,m∘G-1)n+∫fdm≥hπ̃(T,m∘G-1)n+∫fdm=hπ(σn,m)n+∫fdm=hπ(σ,m)+∫fdm.
By the arbitrariness of m and n, we have Step 1.
Step 2.
(2.20)sup⁡{hπ(σ,m)+∫fdm:m∈Mσ(Σ)}≥limsup⁡n→∞1n(log⁡∑[u]∈ξnsup⁡x∈G-1([u])eSnf(α)).
By the continuity of f, for arbitrary ϵ>0, there exists N∈ℕ such that for arbitrary aN∈ΣN and any x,y∈aN, we have
(2.21)|f(x)-f(y)|<ϵ.
Now, for any n∈ℕ and an+N∈Σn+N(2.22)|Sn+Nf(x)-Sn+Nf(y)|≤nϵ+2N‖f‖,∀x,y∈an+N.
Define a Bernoulli measureν on Ωn+Nℕ by
(2.23)ν([u])=sup⁡x∈G-1[u]⁡e(Sn+Nf)(x)∑[v]∈ξnsup⁡y∈G-1[v]⁡e(Sn+Nf)(y),ν([w1,…,wk])=∏i=1kν([wi]),wi∈ξnk∈N.
Then ν can be viewed as a σn+N-invariant measure on Σ (by viewing Ωnℕ as a subset of Σ). By Lemma 2.6, we have hπ(σn+N,ν)=hπ̃(T,ν). Define μ=(1/(n+N))∑i=0n+N-1ν∘σ-i∈Mσ(Σ). We have
(2.24)hπ(σ,μ)+∫fdμ=hπ((σn+N,ν))n+N+∫Sn+Nfdνn+N=1n+N(hπ̃(T,ν)+∫Sn+Nfdν)≥1n+N(h(T,ν)-log⁡tn+N+∫Sn+Nfdν)=1n+N(Hν(ξn+N)-log⁡tn+N+∫Sn+Nfdν)≥1n+N(∑[u]∈ξn+N(-ν([u])log⁡ν([u])+ν([u])inf⁡x∈G-1[u]Sn+Nf(x)))-log⁡tn+Nn+N≥1n+N(∑[u]∈ξn+N(-ν([u])log⁡ν([u])+ν([u])(sup⁡x∈G-1[u]Sn+Nf(x)-nϵ-2N‖f‖)))-log⁡tn+Nn+N=1n+Nlog⁡∑[u]∈ξn+Nsup⁡x∈G-1[u]e(Sn+Nf)(x)-nϵ+2N‖f‖+log⁡tn+Nn+N.
Let k=n+N and let n→∞, then k→∞. We have
(2.25)sup⁡{hπ(σ,μ)+∫fdμ,m∈Mσ(Σ)}≥limsup⁡k→∞1klog⁡∑[u]∈ξn+Nsup⁡x∈G-1[u]eSkf(x)-ϵ.
Since ϵ is arbitrary, we finish the proof of Step 2.Definition 2.10.
If an IFS{Si}i=1l satisfies AWSC with attractor K and f∈C(Σ). We call
(2.26)Pπ(f)=lim⁡n→∞1n(log⁡∑[u]∈ξnsup⁡x∈G-1[u]eSnfπ(x))
the projection pressure of f under π w.r.t. {Si}i=1l.It is clearly that, iff=0, we have the same result of Lemma  9.1 in [13].Corollary 2.11.
limn→∞(log#{Su:u∈Σn}/n)=sup{hπ(σ,m):m∈Mσ(Σ)}.


## 3.  Application for Projection Pressure

Definition 3.1.
{Si:X→X}i=1l is called a C1 IFS on a compact set X⊂ℝd if each Si extends to a contracting C1-diffeomorphism Si:U→Si(U)⊂U on an open set U⊃X.For anyd×d real matrix M, we use ∥M∥ to denote the usual norm of M and ⫾M⫾ the smallest singular value of M, that is,(3.1)‖M‖=max⁡{|Mv|:v∈Rd,|v|=1},⫾M⫾=min⁡{|Mv|:v∈Rd,|v|=1}.Definition 3.2.
The IFS{Si}i=1l is conformal if for every i∈{1,2,…,l}, (1) Si:U→Si(U) is C1, (2) ∥Si′(x)∥≠0 for all x∈U, and (3) |Si′(x)y|=∥Si′(x)∥|y| for all x∈U, y∈ℝd.Definition 3.3.
Let{Si}i=1l be a C1 IFS. For x=(xj)j=1∞∈Σ, the upper and lower  Lyapunov exponents of  {Si}i=1l  at  x are defined, respectively, by
(3.2)λ¯(x)=-liminf⁡n→∞1nlog⁡⫾Sx1,…,xn′(πσnx)⫾,λ̲(x)=-limsup⁡n→∞1nlog⁡‖Sx1,…,xn′(πσnx)‖,
where Sx1,…,xn′(πσnx) denote the differential of Sx1,…,xn:=Sx1∘Sx2∘⋯∘Sxn at πσnx. When λ¯(x)=λ̲(x), the common value, denoted as λ(x), is called the Lyapunov exponents of   {Si}i=1l at x.
It is easy to check that bothλ¯ and λ̲ are positive-valued σ-invariant functions on Σ (i.e., λ¯=λ¯∘σ and λ̲=λ̲∘σ).Definition 3.4.
AC1 IFS {Si}i=1l is said to be weakly conformal if
(3.3)1n(log⁡⫾Sx1,…,xn′(πσnx)⫾-log⁡‖Sx1,…,xn′(πσnx)‖)
converges to 0 uniformly on Σ as n tends to ∞.If IFS{Si}i=1l is weakly conformal, by Birkhoff’s ergodic theorem, we can conclude ∫λ(x)dm=-∫log∥Sx1′(πσx)∥dm=-∫log⫾Sx1′(πσx)⫾dm.Lemma 3.5.
Let K  be the attractor of a weakly conformal IFS{Si}i=1l. Then we have
(3.4)dim⁡HK=dim⁡BK(3.5)=sup⁡{dim⁡Hμ:μ=m∘π-1,m∈Mσ(Σ),mis ergodic}(3.6)=max⁡⁡{dim⁡Hμ:μ=m∘π-1,m∈Mσ(Σ)}(3.7)=sup⁡{hπ(σ,m)∫λdm:m∈Mσ(Σ)}.Proof.
See Theorem  2.13 in [13].Theorem 3.6.
Let{Si(x)}i=1l be a weakly conformal IFS satsifying AWSC. Let ψ(x)=log∥Sx1′πσ(x)∥:Σ→ℝ and π:Σ→K be the canonical projection. Then dimHK is the unique root of Pπ(tψ)=0.Proof.
According to Theorem2.9, we have
(3.8)Pπ(tψ)=sup⁡{hπ(σ,m)+∫tψdm,m∈Mσ(Σ)}.
Let t0=sup{hπ(σ,m)/∫λdm:m∈Mσ(Σ)}, according to (3.7) t0=dimHK.
First we showPπ(tψ) is decreased with respect to t. If 0≤t1≤t2, then for any m∈Mσ(Σ), we have hπ(σ,m)+∫t1ψdm≥hπ(σ,m)+∫t2ψdm. Hence according to variational principle, we have Pπ(t1ψ)≥Pπ(t2ψ).
Ast0≥hπ(σ,m)/∫λdm for all m∈Mσ(Σ), hπ(σ,m)+t0∫ψdm≤0 for all m∈Mσ(Σ), whence Pπ(t0ψ)≤0.
However,Pπ(0)>0, the existence of a positive zero t1 for t↦Pπ(tψ) follows from the intermediate value theorem, that is, sup{hπ(σ,m)+∫t1ψdm,m∈Mσ(Σ)}=0.
For allϵ>0 there is a m∈Mσ(Σ) such that hπ(σ,m)+∫t1ψdm≥-ϵ. Thus, t1≤hπ(σ,m)/∫-ψdm+ϵ/∫-ψdm≤sup{hπ(σ,m)/∫-ψdm,m∈Mσ(Σ)}+ϵ/∫-ψdm, let ϵ→0 we have t1≤t0. And t0≤t1 as Pπ(t1ψ)=0 implies hπ(σ,m)+t1∫ψdμ≤0 for all μ∈Mσ(Σ). So any root of Pπ(tψ)=0 is equal to dimHK.



---

*Source: 807405-2012-06-20.xml*