数理解析研究所講究録
1042 巻 1998 年 78-83

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Dynamical systems of Certain
maps$\mathrm{n}\mathrm{o}\mathrm{n}-\mathrm{h}\mathrm{o}1_{\mathrm{o}\mathrm{m}_{\check{i}}\mathrm{o}}\mathrm{r}\mathrm{P}\mathrm{h}\mathrm{i}\mathrm{c}$東海大理学部内村桂輔 (Keisuke Uchimura)

Introduction

1Complex dynamical systems given by quadratic polynomials$f_{c}(z)=z2+C$

have many well-known important analytic and geometric properties. When $c=-2$ , $f_{-2}(Z)$has relation to a Chebyshev polynomial. The Julia set of $f_{-2}(Z)$ is the interval [-2, 2].
For any in [-2, 2], we set,$z$$z=\psi(\theta)=e^{\pi\theta i}+e^{-}\pi\theta i$

where$\psi$

is afunction from$\mathrm{R}$to [-2, 2]. Clearly$\psi(\theta)=\psi(\theta+2)=\psi(2-\theta)$

Then we introduce an equivalence
we have an induced map.

by$\mathrm{r}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\sim \mathrm{g}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{d}$$\psi$:$\mathrm{R}/\simarrow[-2,2]$$\theta\sim\theta+2$,$\theta\sim 2-\theta$. Hence.

\sim deflned by $t(\theta)=2\theta$ . Then two dynamical systems
be a transformation on$f-2\}$ andis aare topological conjugate and the induced map
is equivalent to the dynamical system given by a tent
topological conjugacy.
map on $[0,1]$ .
Two-dimensinal extension of these dynamical systems is considered. LetLet$t$$\mathrm{R}/$$\{\mathrm{R}/\sim, t\}$$\mathrm{t}1^{-2,2}],$$\psi$$\{\mathrm{R}/\sim, t\}$$z=\Psi(\sigma, \tau)=e+\pi\sigma e^{-}+e2i2\pi\tau i2\pi(\mathcal{T}-\sigma)i$

Then$\Psi$

maps$\mathrm{R}^{2}$

to a subset$\mathrm{S}$in the complex plane C. Clearly$\Psi(\sigma, \tau)=\Psi(\sigma+1, \tau)=\Psi(\sigma, \tau+1)$

Then$\Psi$.

.

may be considered as a map from two-dimensional tours$\Psi(\sigma, \tau)=\Psi(\sigma, \sigma-\mathcal{T})=$$T^{2}$

to$S$

申 $(-\sigma+\tau, \tau)=\Psi(1-\mathcal{T}, 1-\sigma)$ .

. Further

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Then we introduce an equivalence

defined by$\mathrm{r}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\sim \mathrm{o}\mathrm{n}T^{2}$.

(1.1)$F_{2}(z)$is a two-dimension$(\sigma, \tau)\sim(\sigma, \sigma-\tau)\sim(-\sigma+\tau, \tau)\sim(1-\tau, 1-\sigma)$Hence we have an induced map$\Psi$: $T^{2}/\simarrow S$ .

An extension of $f_{-2}(Z)$ is a map :$F_{2}(z)=z^{2}-2\overline{Z}$.

Then$F_{2}(\Psi(\sigma, \tau))=\Psi(2\sigma, 2\tau)$Let$d$be adouble angle map of.

itself. That is,$T^{2}/\sim \mathrm{o}\mathrm{n}\mathrm{t}\mathrm{o}$$d(\sigma, \mathcal{T})=(2\sigma, 2\mathcal{T})$.

and $\{T^{2}/\sim, d\}$ are topological conjugate. HenceThenis chaotic.
extension of $f_{-2}(Z)$ . Uchimura [1996] shows that$\{S, F_{2}\}$$\{S, F_{2}\}$2

The dynamics of$F_{2}$

We study the dynamical system given by.

$F_{2}(z)=$ $z2-2\overline{Z}$.

(2.1)

partition the complex plane into two sets. One is a closed
We shall show thatand the other is its complement. The dynamics of
on is chastic and thein ofof .
complement of is theas a function of two real
is not holomorphic, we regard the function
Sincevariables. That is,$F_{2}((x, y))=(x^{2}-y^{2}-2X,2_{X}y+2y)$ .
(2.2)$\mathrm{C}$$F_{2}$$\mathrm{d}_{\mathrm{o}\mathrm{n}\mathrm{l}\mathrm{a}}\mathrm{i}\mathrm{n}S$$\mathrm{S}$$\mathrm{A}I7_{2}$$S$$\mathrm{b}\mathrm{a}s$$\infty$$F_{2}$$F_{2}(z)$$F_{2}$

is $4x^{2}+4y^{2}-4$ . So the set of critical
Here $z=x+iy$ . The Jacobian determinant ofvalues is an algebraic curve of the fourth degree which is known as Steiner’s hypocycloid.
It takes the form$(x^{2}+y^{2}+9)^{2}+8(-x^{3}+3xy^{2})-108=0$ .
(2.3)$F_{2}$

Let

be a closed region bounded by Steiner’s hypocycloid (2.3). See [Koornwinder, 1974].
restricted to the set is a finiteto-one factor
We shall show that the function$S$$F_{2}(z)$$S$of the algebraic endomorphism of the torus given by the matrix$z=\exp(2\pi\sigma i)+\exp(-2\pi\tau i)+\exp(2\pi(\tau-\sigma)i)$

.

(2.4)

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Then the point$(\sigma, \tau)$

is regarded as a point in the torus

a torus endomorphism

$l$on

$T$

$T\simeq \mathrm{R}^{2}/\mathrm{Z}^{2}$. From Then, we havegiven byis equivalent to the points $(-\sigma+\tau, \tau),$
we have to consider an equivalence relation given by$(\sigma, \sigma-\tau)$$(\sigma, \tau)$$(\sigma, \tau)\sim(-\sigma+\tau, \mathcal{T}),$$(\sigma, \tau)\sim(\sigma, \sigma-\tau),$and $(1-\tau, 1-\sigma)$ . ,hat is to say,$\mathrm{T}$$(\sigma, \tau)\sim(1-.\mathcal{T}, 1-\sigma).\cdot.$.

(2.5)

Let $R=T/\sim$ .
is regarded as a closed region bounded by a regular triangle $OAB$ with
The set, in the-plane, where$R$$O=(\mathrm{O}, 0),$$A=( \frac{1}{2}, \frac{-1}{2\sqrt{3}}),$$B=( \frac{1}{2}, \frac{1}{2\sqrt{3}})$$(\mathrm{s},\mathrm{t})$$s=(\sigma+\tau)/2,$ $t=\sqrt{3}(\sigma-\tau)/2$.

Now we consider the symbolic dynamics. For any point $x\in R$ , we define the. itinerary$h(x)$ by the rule$h(x)=(s_{0}s_{12}s\ldots)$

where $s_{j}=k$ if and only if $d^{j}(x)\in\triangle(k),$ $(k=0,1,2,3)$ . Here $\triangle(0)=\triangle DEF,$$\triangle(2)=\triangle ADF,$$\triangle(3)=\triangle BED$ , where $D,$ $E$ and $F$ are the midpoints of
AB, $OB$ and $OA$ , respectively. Set$\triangle(1)=$$\triangle OEF,$$\Sigma_{4}=${ $(s0s1s2\cdots)|s_{j}=0,1,2$ or 3}.

We introduce an equivalence$\mathrm{r}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\sim \mathrm{o}\mathrm{n}\Sigma_{4}$$(w_{1}0w_{2})\sim(w_{1}2w_{2})$which is defined by, for $w_{1}\in\{0,1,2,3\}*,$ $(w_{2})\in h(OA)$ ,, for $w_{1}\in\{0,1,2,3\}*,$ $(w_{2})\in h(OB)$ ,$(w_{12}0w)\sim(w_{1}1w_{2})$ , for$(w_{2})\in h(AB)$ .

($w_{1,-}\mathrm{o}_{w_{2})}\sim(w_{1}3w_{2})$$!.\mathrm{t}$$w_{1}\in \mathrm{t}0,1,2,3\}^{*},$

Let$\Sigma’$denote the quotient$\Sigma_{4}/\sim$. Note that$\sigma$

is naturally defined on$\sigma[(s_{0^{S_{1}\ldots s}}n\cdots)]=[(S_{1}\ldots s_{l},\ldots)]$

Clearly$\sigma$$\Sigma’$. That is,.

is well-defined.

Theorem 2.1. The mapping
diagram commutative:
$d$

$R$

$R$

$\tilde{h}\}\Sigma$$\Sigma\downarrow\tilde{h,}$’$\sigma$$\tilde{h}$:$Rarrow\Sigma’$is a bijection and it makes the following81
Let be a Sierpinsky gasket
seen that$\Gamma$$\mathrm{w}\backslash$hose outermost triangle is $OAB$ . Then it can be easily$d(\Gamma)=\Gamma$

Let$\Sigma"=$

The shift map$\sigma${.

$[(s0S1\cdots Sn\cdots)]$ $|s_{i}\in\{1,2,3\}$is naturally defined onCorrollary 2.1. The mapping
diagram commutative:$\tilde{h}$:$\Sigma$, for all }.
$i$”. Hence we have the following corrollary.

is a bijection and it makes the following$\Gammaarrow\Sigma$ ”$d$$\Gamma$$\Gamma$$\tilde{h}\}\Sigma$$\Sigma \mathrm{I}^{\tilde{h}},$,”$\sigma$

Theorem 2.2.

The dynamical system$\{S, F_{2}\}$is chaotic. That is,(1) it is transitive;(2) it is sensitive to intial conditions;is dense in .
(3) the set of periodic orbits of, we have the following theorem.
For any complex number$S$$F_{2}$$z\not\in\grave{S}$Theorem 23For any complex number$z\not\in S$$(F_{2})^{n}(z)arrow\infty$

3

Attracting basins of,as$narrow\infty$.

$F_{4,3}$In this section we consider perturbations of the mapping
namical systems given by$F_{c}(Z)=z-2C\overline{Z}$ .

Such a map is one of the simplest
a simple property
also$\mathrm{q}.\mathrm{c}$. We study the dy-. mappings except for affine mappings. The mapping$\mathrm{S}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{S}\mathrm{p}_{\mathrm{e}\mathrm{s}}$$\frac{\partial^{2}}{\partial z\partial\overline{z}}F_{c}^{2}=-4Cz$

for Laplace operator.

$F_{2}(z)$,82
points of prime period
First we note that tllere are four fixed points and.
2 which are listed inwhen. The value
is a special value in
We study the dynamics ofthe sense that all attractive fixed points and attractive periodic points of p.eriod 2 lie ona critical set. We will study attractive basins of .
$12_{\mathrm{P}^{\mathrm{e}\mathrm{r}}}.\mathrm{i}\mathrm{o}\mathrm{d}\mathrm{i}\mathrm{C}$$\mathrm{U}\mathrm{c}\mathrm{h}\mathrm{i}\mathrm{m}\mathrm{u}\mathrm{r}\mathrm{a}[1977]$$F_{c}(z)$$c= \frac{4}{3}$$c= \frac{4}{3}$$F_{\frac{4}{3}}$When$c \cdot=\frac{4}{3}$, two attracting fixed points

and$P_{\langle 1)}^{2}$

are written as$P_{(1)}^{3}$$P_{11}^{2}=)- \frac{1}{6}+i\sqrt{\frac{5}{12}}$$F_{(1)}^{3}=\overline{P_{(}21)}$.

There are four attracting periodic points of period 2 which are written as$P^{2}=\omega P^{2}P^{3}=\omega(2)(1)’(2)(1)2P2$$P_{(2)}^{4}=\omega P_{(}^{3}P^{5}=1)’(2)\omega P_{()}231$

are the zeros of Jacobian determinent of . Then critical set ofis equal to a circle $\{z;2|z|=\frac{4}{3}\}$ . Note that the attracting periodic points,
lie on the circle. Set,and.

Critical points of$F_{\frac{4}{3}}(z)$.

$F_{\frac{4}{3}}(z)$$F_{\frac{4}{3}}$$P_{(1)}^{2}$$P_{(1)’(2}^{3}P^{2})’ P_{(2)’(2}^{3}P^{4})’ P_{(2)}^{5}$$A=( \frac{4}{3},0)$$‘ f \mathit{3}=(-\frac{2}{3}, \frac{2\sqrt{3}}{3})$$C=(- \frac{2}{3}, -\frac{2\sqrt{3}}{3})$Theorem 3.1. The interiors of the triangular regions
andmediate
ins of Pxed points
and. The interiors ofimmediate
ins of periodic points
and. The interiors of
are immediate basin of periodic points
and.
$\triangle OBD$$\mathrm{b}\mathrm{a}s$$P_{(1)}^{2}$$\triangle OBE$

$P_{(1)}^{3}$$\mathrm{b}\mathrm{a}s$$P_{(2)}^{4}$$P_{(2)}^{5}$$P_{(2)}^{2}$

are the imandare. and$\triangle OCF$$\triangle OCE$$\triangle OAD$$\triangle OAF$

$P_{(2)}^{3}$

By the symmetry we know that to prove Theorem 3.1 it suffices to prove the following
proposition.
Proposition 3.1. Let$z$

be any interior point of$\triangle OBD$. Then.
$(F_{\frac{4}{3}})^{n}(z)$This proposition is proved in$arrow P_{(1)}^{2}$$\mathrm{U}\mathrm{c}\mathrm{h}\mathrm{i}\mathrm{m}\mathrm{u}\mathrm{r}\mathrm{a}[1997]$.

$(narrow\infty)$.

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References
Koornwinder, T. H. [1974] ” Orthogonal polynimials in two variables
eigenfunctions of two algebraically independent partial differential operators III IV”,Indag. Math. 36, 357-381.
$\mathrm{w}\mathrm{h}\mathrm{i}\mathrm{c}\mathrm{h}_{t}|\mathrm{a}\mathrm{r}\mathrm{e}$Uchimura, K. [1996] ” The dynamical systems associated with Chebyshev polynomials

in two variables”, Int. J.

Bifurcation and Chaos, 6,$12\mathrm{B},$$26l$

1-2618.

Uchimura, K. [1997] ”Attracting basins of certain non-holomorphic map”,Res. Rep. Dept. Math. Tokai Univ., 13, 1-13.