数理解析研究所講究録
第 1681 巻 2010 年 73-78

73

Chaotic continua of continuum-wise expansive
homeomorphisms
加藤久男 (Hisao Kato): 筑波大学 (University of Tsukuba)

1

Introduction.

In this note, we consider the following problem:

Problem 1.1. If : $Xarrow X$ is an expansive (or a continuum-wise expansive) homeomorphism of one-dimensional continuum $X$ , does $X$ contain an indecomposable subcontinuum? Moreover, what kinds of dynamical structures does such an indecomposable
continuum admit’;’ Is each chaotic continuum of indecomposable?
$f$

$a$

$f$

In this note, we will give some partial answers in the affirmative to the above problem.
It is well known that every continuum $X$ with $\dim X\geq 2$ contains an indecomposable
subcontimuum. Also there is an expansive homeomorphism on the 2-dimensional torus
such thatis the only chaotic continuum of and hence
is the decomposable
chaotic continuum of . In [5] and [6], we investigated chaotic continuum of homeomorphism. We proved the existence of (minimal) chaotic continuum of continuum-wise
expansive homeomorphism and we also investigated the indecomposability of chaotic continua and their composants. In fact, we proved that if $G$ is a finite graph and : $Xarrow X$of a G-like continuum $X$ is a continuum-wise expansive homeomorphism, then there isan indecomposable chaotic continuum of . In [9], Mouron proved the existence of anindecomposable subcontinuum of $X$ for the case that $X$ is a k-cyclic continuum $(k<\infty)$
and $X$ admits an expansive homeomorphism. In this note, we define the notion of closed
subset having uncountable handles and we show that if : $Xarrow X$ is a continumm-wise
expansive homeomorphism of a continuum $X$ and $Z$ is a ninimal chaotic continuum of, then for each proper closed subset of $Z$ with Int $zA\neq\phi,$
has uncountable handles
in $Z$ . As a corollary, we see that if : $Xarrow X$ is a continumm-wise expansive homeomorphism and $X$ does not contain any subcontinuum having uncountable handles, theneach minimal chaotic continuum of is indecomposable. This implies a stronger result
than the Mouron’s theorem above [9]. In fact, we obtain that if $X$ is a k-cyclic continuum
and $X$ admits a continuum-wise expansive homeomorphism , then each minimal chaotic
continuum of is indecomposable. The proof is different from the methods of the proofof Mouron [9].
$f$$T^{2}$$T^{2}$$T^{2}$

$f$

$f$

$f$

$f$

$f$

$A$

$f$

$A$

$f$

$f$

$f$

$f$

2

Expansive homeomorphism and continuum-wise expansive homeomorphism.

All spaces considered in this note are assumed to be separable metric spaces. By acompactum we mean a compact metric space. A continuum is connected, nondegeneratecompactum. A homeomorphism : $Xarrow X$ of a compactum $X$ with metric is called
expansive ([15]) if there is $c>0$ such that for any $x,$ $y\in X$ and $x\neq y$ , then there is aninteger
such that$f$$n\in \mathbb{Z}$$d$

74$d(f^{n}(x), f^{n}(y))>c$ .

A homeomorphism : $Xarrow X$ of a compactum $X$ is continuum-wise expansive (resp.
positively continuum-wise expansive) [4] if there is $c>0$ such that if is a nondegenerate
subcontinuum of $X$ , then there is an integer(resp. a positive integer) suchthat
$f$$A$$n\in \mathbb{Z}$$n\in \mathbb{N}$diam $f^{n}(A)>c$ ,where diam $B= \sup\{d(x, y)|x, y\in B\}$ for a set . Such a positive number is called
an expansive constant for . Note that each expansive homeomorphism is continuuunwise expansive, but the converse assertion is not true. There are many continuum-wise
expansive homeomorphisms which are not expansive (see [4]). These notions have been
extensively studied in the area of topological dynamics, ergodic theory and continuum
theory see $[1]-[3],[8],[12]-[15])$ .
The hyperspaceof $X$ is the set all nonempty closed subsets of $X$ with the Hausdorff
metric. Let$B$$c$$f$$($$2^{X}$$d_{H}$$C(X)=$ { $A\in 2^{X}|$ $A$ is connected}.

Note thatand $C(X)$ are compact metric spaces (e.g., see [7] or [11]). For a homeo$Xarrow
X$ , we define sets of stable and unstable nondegeneratemorphism :subcontinua
of $X$ as follows (see [6]):$2^{X}$$f$$V^{s}(=V_{f}^{s})=\{A|A$

is a nondegenerate subcontinuum of $X$ such thatdiam $f^{n}(A)=0\}$ ,$\lim_{narrow\infty}$$V^{u}(=V_{f}^{u})=\{A|A$

is a nondegenerate subcontinuum of $X$ such thatdiam $f^{-n}(A)=0\}$ .
$\lim_{narrow\infty}$

For each $0<\delta<\epsilon$ , put$V^{s}(\delta;\epsilon)=$$V^{u}(\delta;\epsilon)=${{ $A\in C(X)|$ diam$A\geq\delta$, and diam $f^{n}(A)\leq\epsilon$ for each $n\geq 0$ }$A\in C(X)|$ diam $A\geq\delta$ , and diam $f^{-n}(A)\leq\epsilon$ for each$n\geq 0$

Similarly, for each closed subset $Z$ of $X$ and $x\in Z$ , the continuum-wise$V^{\sigma}(x;Z)(\sigma=s, u)$ of
are defined as follows:$\sigma$}.

-stable sets$f$$V^{s}(x;Z)=${ $y\in Z|$ there is $A\in C(Z)$ such that$x,$ $y\in

A$and$V^{u}(x;Z)=${ $y\in Z|$ there is $A\in C(Z)$ such that$x,$ $y\inA$andA subcontinuum $Z$ of $X$ is called a -chaotic continuum of$\sigma$$f$$\lim_{narrow\infty}$$\lim_{narrow\infty}$

diam $f^{n}(A)=0$ },diam $f^{-n}(A)=0$ }.

(where $\sigma=s,$ ) if$u$

1. for each $x\in Z,$ $V^{\sigma}(x;Z)$ is dense in $Z$ , and2. there is $\tau>0$ such that for each $x\in Z$ and each neighborhood $U$ of
is $y\in U\cap Z$ such that$\lim\inf_{narrow\infty}d(f^{n}(x), f^{n}(y))\geq\tau$in case $\sigma=s$ , or$x$

in $X$ , there75$\lim\inf_{narrow\infty}d(f^{-n}(x), f^{-n}(y))\geq\tau$in case $\sigma=u$ .

A subcontinuum $Z$ of $X$ is called a minimal -chaotic continuum of (where $\sigma=s,$ ) if
$Z$ is a-chaotic continuum of and $Z$ does not contain any proper -chaotic continuumof . In this note, we often abbreviate -chaotic continuum to chaotic continuum. Notethatis closed in $C(X)$ . Also, note that if : $Xarrow X$ is a continuumwise expansive homeomorphism with an expansive constant $c>0$ , then (1) for each$0<\delta<\epsilon<c,$, andis an-set in $C(X)$ , and (2) $V^{u}(z;Z)$ is aconnected -set containing , because$f$$\sigma$$f$$\sigma$$u$$\sigma$$f$$\sigma$$f$$V^{\sigma}(\delta;\epsilon)(\sigma=u, s)$$V^{\sigma}$$V^{\sigma}(\delta;\epsilon)\subset V^{\sigma}$$F_{\sigma}$$z$$F_{\sigma}$$V^{u}(z;Z)= \bigcup_{n=0}^{\infty}(\cup\{A\in C(Z)|z\in A, diam f^{-i}(A)\leq\epsilon for i\geq n\})(see[4, (2.1)])$.

Similarly, $V^{s}(z;Z)$ is a connected -set containing . In [5], we showed that if : $Xarrow X$
is a continuum-wise expansive homeomorphism of a compactum $X$ with $\dim X>0$ , thenthere exists a minimal chaotic continuum ofsee 5, 3.6 $)])$ . In this case, if $Z$ is achaotic continuum of , then the decomposition $\{V^{\sigma}(z;Z)|z\in Z\}$ of $Z$ is an uncountablefamily of mutually disjoint, dense connected -sets in $Z$ .
A continuum $X$ is decomposable if there are two proper subcontinua
and $B$ of $X$
such that $A\cup B=X$ . A continuum $X$ is indecomposable if it is not decomposable. Let$X$ be a continuum and let $p\in X$ . Then the set
$f$

$z$

$F_{\sigma}$$f$$($$($$[$$\sigma-$$f$$F_{\sigma}$$A$$c(p)=${ $x\in X|$ there is a proper subcontinuum$A$of $X$ containing$p$and$x$}

is called the composant of $X$ containing . Note that if $X$ is an indecomposable continuum,
then $\{c(p)|p\in X\}$ is an uncountable family of mutually disjoint, dense connected
sets in $X$ . See [7] for some fundamental properties of indecomposable continua andcomposants. A closed subset $A$ of $X$ has uncountable handles if there is a familyi.e.,offorof mutually disjoint nondegenerate subcontiua$X$ such that eachhas at least two components and is an uncountable set.
, there is a fimite open coverof $X$ such that
A continuum $X$ is k-cyclic if for anymeshand the nerveof is a one-dimensional polyhedron which has at mostdistinct simple closed curves.
$p$$F_{\sigma^{-}}$$\{H_{\alpha}|\alpha\in$$H_{\alpha}$$\Lambda\}$$A\cap H_{\alpha}(\neq\phi)$$N(\mathcal{U})$$H_{\alpha}\cap H_{\beta}=\phi$$\alpha\neq\beta)$$\Lambda$$\epsilon>0$$(\mathcal{U})<\epsilon$$($$\mathcal{U}$$\mathcal{U}$$k$

3Results.

Proposition 3.1. If a continuum $X$ is k-cyclic for some $k<\infty_{2}$ then $X$ contains nosubcontinuum having uncountable handles.
$H$

Remark. The converse assertion of the above proposition is not true. Hawaiian earring
contains no subcontinuum having uncountable handles and $H$ is not k-cyclic for any$k<\infty$.

: $Xarrow X$ be a continuum-wise expansive homeoLemma 3.2. (see [5, (3.2)]) Let
morphism of a compactum $X$ with an expansive constant $c>0$ , and let $0<\epsilon<c/2$ .
Then there is $\epsilon>\delta>0$ such that if $A$ is a subcontinuum of $X$ with diam $A\leq\delta$ and, then one of the following two conditions holds:
diam $f^{m}(A)\geq\epsilon$ for some
$f$$m\in \mathbb{Z}$76
1. If $m\geq 0$ , for each $n\geq m$ and $x\in f^{n}(A)$ , there is a subcontinuum$x\in f^{n}(B)$ , diam
for $0\leq j\leq n$ and diam $f^{n}(B)=\delta$ .

$B$of

$A$

such that$f^{j}(B)\leq\epsilon$

2. If $m<0$ , for each $n\geq-m$ and $x\in f^{-n}(A)$ , there is a subcontinuum $B$ ofthat $x\in f^{-n}(B)$ , diam
for $0\leq j\leq n$ , and diam $f^{-n}(B)=\delta$ .

$A$

such

$f^{-j}(B)\leq\epsilon$

Lemma 3.3. [5, (3.3) and (3.4)] Let $f:Xarrow X$ be a continuum-wise expansive homeomorphism of a compactum $X$ with $\dim X>0$ . Then the following are true.
$)$

$($

1. $V^{u}\neq\phi$ or$V^{s}\neq\phi$.

2. If $\delta>0$ is as in the above lemma, then for each $\gamma>0$ there is a natural number, then either diam $f^{n}(A)\geq\delta$
such that if is a subcontinuum of $X$ with diam$n\geq
N(\gamma)$$f^{-n}(A)\geq\delta$$n\geqN(\gamma)$
holds.
or diamfor eachfor each$N(\gamma)$$A$$A\geq\gamma$

Theorem 3.4. If : $Xarrow X$ is a continuum-wise expansive homeomorphism of a continuum $X$ andis a minimal chaotic continuum of , then for any proper closed subset
of with $Int_{Z}A\neq\phi,$ has uncountable handles. Moreover, is decomposable if
and only if there exists a proper subcontinuum $C$ of with $In.t_{Z}C\neq\phi$ such that $C$ hasuncountable handles in $Z$ .
$f$$Z$$A$$f$

$Z$

$A$

$Z$

$Z$

Corollary 3.5. Suppose that a continuum $X$ contains no subcontinuum having uncountable handles. If : $Xarrow X$ is a continuum-wise expansive homeomorphism of $X$ and $Z$
is a minimal chaotic continuum of , then $Z$ is indecomposable.
$f$

$f$

Corollary 3.6. If $X$ is a k-cyclic continuum for some $k<\infty$ and $X$ admits a continuumwise expansive homeomorphism f) then each minimal chaotic continuum of is indecomposable.
$f$

Next, we consider the following problem.

Problem 3.7. Suppose that $f:Xarrow X$ is a continuum-wise expansive homeomorphismof one-dimensional continuum $X$ and is an indecomposable -chaotic continuum of. Does the composant $c(z)$ of $Z$ containing coincide withfor each $z\in Z^{t}$?
$Z$

$a$

$f$

$\sigma$$V^{\sigma}(z;Z)$$z$

We give a partial answer in the affirmative to the problem. A subcontinuum of $X$ hasof mutually disjoint nondegenerate
uncountable handlebars if there is a family
and is anfor eachof $X$ such that $H_{\alpha}-A\neq\phi,$subcontiua$X$handles,then hasuncountable set. Note that if a subcontinuum of has uncountable, there is aif for any
uncountable handlebars. Acontinuum $X$ is k-branched$X$
is a one-dimensional
and the nerve
finite open cover of such that meshpolyhedron which has at most distinct branch points. Note that Hawaiian earring $H$ isa l-branched continuum.
$A$$\{H_{\alpha}|\alpha\in\Lambda\}$$\alpha\in\Lambda$$A\cap H_{\alpha}(\neq\phi)$$H_{\alpha}$

$A$

$A$

$\epsilon>0$$(k\in \mathbb{N})$$(\mathcal{U})<\epsilon$$\mathcal{U}$$\Lambda$$N(\mathcal{U})$$k$

Proposition 3.8. If a continuum $X$ is k-branched for some $k<\infty$ , then $X$ contains nosubcontinuum having uncountable handlebars.

We need the following lemma.

77Lemma 3.9. (Sum theorem of dimension) Ifand
metric space $X$ such that $\dim$ $X$ .
$\leq n$$X_{i}(i\in \mathbb{N})$$X= \bigcup_{i\in \mathbb{N}}X_{i}$are closed subsets of a separable, then $\dim X\leq n$ .

Theorem 3.10. Suppose that a continuum $X$ contains no subcontinuum having uncountable handlebars. If $f:Xarrow X\iota s$ a continuum-wise expansive homeomorphism, then there
is a -chaotic continuum $Z$ of such that $Z$ is an indecomposable continuum and for each$z\in Z$ , the composant $c(z)$of $Z$ containing coincides with $V^{\sigma}(z;Z)$ .
$f$$\sigma$$z$

Corollary 3.11. If $f:Xarrow X$ is a continuum-wise expansive homeomorphism of a kbranched continuum $X(k<\infty)$ , then there is a -chaotic continuum $Z$ of $f(\sigma=s$ oru such that $Z$ is an indecomposable continuum such that for each $z\in Z$ , the composant$c(z)$ of $Z$ containing
coincides with $V^{\sigma}(z;Z)$ .
$\sigma$$)$$z$

In [10], Mouron proved that if : $Xarrow X$ is an expansive homeomorphism, then $X$
is not tree-like. We will give a more precise result than Mouron’s result. We need thefollowing simple lemmas.
$f$Lemma 3.12. Let (X, d) be a metric space and let $\delta>0$ . Then for each positive integer, there is a positive number $\eta=\eta(\delta, n)>0$ such that if
is any connected subset $M$$X$ with diam$y_{i}(i=1,2,\ldots, n)$ in $M$ such thatthen there are distinct pointsof$d(y_{i}, y_{j})\geq\eta$for $i\neq j$ .
$A$

$n$$(M)\geq\delta_{f}$Lemma 3.13. Let : $Xarrow X$ be an expansive homeomorphism of a compactum $X$ withan expansive constant $c>0$ . For each $\eta>0$ , there is a positive integer $n=n(\eta)$ suchthat if $x,$ $y\in X$ with $d(x, y)\geq\eta$ , then $\max\{d(f^{i}(x), f^{i}(y))|-n\leq i\leq n\}\geq c$ .
$f$Theorem 3.14. Let : $Xarrow X$ be an expansive homeomorphism of a continuum X. Ifa subcontinuum $Y$ of $X$ satisfies the condition
for some $y\in Y$ , then $Y$ is not atree-like continuum. In particular, evew chaotic continuum of is not tree-like.
$f$$P_{\sigma}(y;Y)$$f$

Corollary 3.15. Suppose that a continuum $X$ contains no subcontinuum having uncountable handles. If $f$ : $Xarrow X$ is an expansive homeomorphism of $X$ and $Z$ is a minimal
then $Z$ is an indecomposable continuum which is not tree-like.
chaotic continuum of$f_{f}$

Corollary 3.16. Suppose that a continuum $X$ contains no subcontinuum having uncountable handlebars. If $f:Xarrow X$ is an expansive homeomorphism, then there is a -chaotic
continuum $Z$ of such that $Z$ is not tree-like, $Z$ is indecomposable and for each $z\in Z$ ,the composant $c(z)$ of $Z$ containing coincides with $V^{\sigma}(z;Z)$ .
$\sigma$$f$$z$Remark. There are many tree-like chaotic continua of continuum-wise expansive homeomorphisms.

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