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CHARACTERIZATIONS OF G‐LIKE CONTINUA CONTAINING
INDECOMPOSABLE SUBCONTINUA AND POSITIVETOPOLOGICAL ENTROPYHISAO KATOINSTITUTE OF MATHEMATICSUNIVERSITY OF TSUKUBA

1. INTRODUCTION

The theory of indecomposable continua is one of the most interesting branchesof continuum theory. Also, we know that many interesting connections between dy‐
namical systems and continuum theory have been studied by many mathematicians
and many indecomposable continua are frequently appeared in chaotic dynamicalsystems (see References). Such continua play important roles in order to investi‐
gate behaviors of the dynamics. We are interested in the following fact that chaotic
topological dynamics should imply the existence of complicated topological struc‐tures of underlying spaces. In many cases, such continua (=compact connected
metric spaces) are indecomposable continua. Also, in many cases, the composantsof such indecomposable continua are strongly related to stable or unstable (con‐nected) sets of the dynamics. For instance, in continuum theory and the theoryof dynamical systems, the Knaster continuum ( Smale’s horse shoe), the pseudo‐arc, solenoids and Wada’s lakes ( Plykin attractors) etc., are well‐known as such==

indecomposable continua. In the theory of indecomposable continua, the notion of“crookedness” has been essential and well‐known. In this article, we introduce a
new notion of “free tracing property by free \mathcal{P} ‐chains” and by use the notion we
study chaotic dynamics and indecomposability of continua.
2. PRELIMINARIESIn this article, we assume that all spaces are separable metric spaces and all maps
are continuous. Let \mathb {N} be the set of natural numbers, \mathbb{R} the real line, and I=[0,1]the unit interval. A graph is a compact connected 1‐dimensional polyhedron. A
graph T is a tree if T contains no simple closed curve. For a set A, |A| denotes

the cardinality of the set A . For a family \mathcal{A} of subsets of a space, \cup \mathcal{A} denotes theunion of all elements of \mathcal{A} , i.e.,\cup \mathcal{A}=\bigcup_{A\in A}A(=\cup\{A|A\in \mathcal{A}\}) .
For a subset A of a space X, \overline{A} denotes the closure of A in X . A subset E of X isan F_{\sigma}|‐set of X if E is a countable union of closed sets of X.

A continuum is a compact connected metric space. We say that a continuum isnondegenerate if it has more than one point. A continuum is indecomposable [24]if it is nondegenerate and it is not the union of two proper subcontinua. For any

continuum H , the set c(p) of all points of the continuum H , which can be joined

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with the point p by a proper subcontinuum of H , is said to be the composant of
the point p\in H , i.e.,c(p)=\cup { C|C is a proper subcontinuum of H containing the point p }.
Note that for an indecomposable continuum H , the following conditions are equiv‐
alent;(1) the two points p, q belong to same composant of H ;(2) c(p)\cap c(q)\neq\emptyset ;(3) c(p)=c(q) .
So, we know that if H is an indecomposable continuum, the family\{c(p)|p\in H\}
of all composants of H is a family of uncountable mutually disjoint sets c(p) which

are connected and dense F_{\sigma} ‐sets in H . Note that a (nondegenerate) continuum X

is indecomposable if and only if there are three distinct points of X such that any
subcontinuum of X containing any two points of the three points coincides with X,

i.e., X is irreducible between any two points of the three points [24].
Let H be an indecomposable continuum. We say that a subset Z of H is transver‐
sal for composants of H if no distinct two points of Z belong to the same com‐posant of H , i.e., if x, y are any distinct points of Z and E is any subcontinuum of
H containing x and y , then E=H . Note that a subset Z of H is transversal forcomposants of H if and only if Z is vertically embeded with respect to composantsof H (see [11]). In [27], Mazurukiewicz proved that if H is an indecomposable
continuum, then there is a Cantor set

Z in H which is transversal for composantsof H.

Let X_{i}(i\in \mathbb{N}) be a sequence of compact metric spaces and let f_{i,i+1} : X_{i+1}arrow X_{i}be a map for each i\in \mathbb{N} . The inverse limit of the inverse sequence \{X_{i}, f_{i,i+1}\}_{i=1}^{\infty}
is the space\frac{\lim}{\backslash }\{X_{i}, f_{i,i+1}\}= { (x_{i})_{i=1}^{\infty}|x_{i}=f_{i,i+1}(x_{i+1}) for each i\in \mathbb{N} }\subset\prod_{i=1}^{\infty}X_{i}which has the topology inherited as a subspace of the product space \prod_{i=1}^{\infty}X_{i} . Fora map f : Xarrow X , put\lim_{arrow}(X, f)= { (x_{i})_{i=1}^{\infty}|x_{i}=f(x_{i+1}) for each i\in \mathbb{N} }.
A map g from X onto G is an \epsilon ‐map (\epsilon>0) if for every y\in G , the diameterof g^{-1}(y) is less than \epsilon . For any collection \mathcal{P} of graphs, X is \mathcal{P} ‐like if for any\epsilon>0 there exist an element G\in \mathcal{P} and an \epsilon ‐map from X onto G . A continuumX
is G‐like if X is \mathcal{P} ‐like, where \mathcal{P}=\{G\} . Note that X is G‐like if only if X is
homeomorphic to the inverse limit of an inverse sequence of G , i.e.,

X=1\not\simeq\underline{m}\{G_{i}, f_{i,i+1}\},
where G_{i}=G and f_{i,i+1} : G_{i+1}arrow G_{i} is an onto map for each i\in \mathbb{N} . Arc‐like

continua ( chainable continua) are those which are G ‐like for G=I , and circle‐like=

continua are those which are S‐like, where S is a simple closed curve. Our focus inthis article is on G‐like continua where G is any graph.
Let \mathcal{U} be a collection of subsets of X . The nerve N(\mathcal{U}) of \mathcal{U} is the polyhedron
whose vertices are elements of \mathcal{U} and there is a simplex

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G ‐LIKE CONTINUA<U_{1}, U_{2},U_{k}> with distinct vertices U_{1}, U_{2},U_{k}\in \mathcal{U} if\bigcap_{i=1}^{k}U_{i}\neq\emptyset.

In this paper, we consider the only case that nerves are graphs.
If \{C_{1}, C_{n}\} is a subcollection of \mathcal{U} , we call it a chain if C_{i}\cap C_{i+1}\neq\emptyset for
1\leq i<n and \overline{C_{i} \cap\overline{C_{j} \neq\emptyset implies that |i-j|\leq 1 . We say that \{C_{1}, C_{n}\}is a free chain in \mathcal{U} if it is a chain and, moreover, for all 1<i<n we have that
C\in \mathcal{U} with \overline{C}\cap\overline{C_{i} \neq\emptyset implies that C=C_{i}, C=C_{i-1} or C=C_{i+1} . By themesh of a finite collection \mathcal{U} of sets, we means the largest of diameters of elementsof \mathcal{U} . Note that for a graph G , a continuum X is G ‐like if and only if for any \epsilon>0,
there is a finite open cover \mathcal{U} of X such that N(\mathcal{U}) is homeomorphic to G and themesh of \mathcal{U} is less than \epsilon . The Knaster continuum [21] ( =Smale ’s horse shoe) and thepseudo‐arc ( hereditarily indecomposable arc‐like continuum) are arc‐like continua,solenoids are circle‐like continua and the Wada’ lake [35] ( Plykin attractor [32])is a(S_{1}\vee S_{2}\vee S_{3}) ‐like continuum, where S_{1}\vee S_{2}\vee S_{3} denotes the one point union==of 3 circles. Such spaces are typical indecomposable continua which often appear

in continuum theory and chaotic dynamical systems. The reader may refer to [24]and [31] for standard facts concerning continuum theory.
3. FREE TRACING PROPERTY BY FREE G ‐CHAINSLet X be a continuum and m\in \mathbb{N} . Suppose that A_{i}(1\leq i\leq m) are nonempty
open sets in X and x_{i}(1\leq i\leq m) are m distinct points of X . We identify theorder A_{1}arrow A_{2}arrow
arrow A_{1}.
arrow A_{m} and the converse order A_{m}arrow A_{m-1}arrow
Then we consider the equivalence classm[A_{1}arrow A_{2}arrow arrow A_{m}]=\{A_{1}arrow A_{2}arrow arrow A_{m};A_{m} arrow A_{m-1}arrow arrow A_{1}\}.
Suppose that \mathcal{U} is a finite open cover of X . We say that a chain \{C_{1}, , C_{n}\}\subseteq \mathcal{U}follows from the pattern [A_{1}arrow A_{2}arrow arrow A_{m}][11] if there exist
1\leq k_{1}<k_{2}<<k_{m}\leq n or 1\leq k_{m}<k_{m-1}<<k_{1}\leq nsuch that C_{k_{i}}\subset A_{i} for each i=1,2,. In this case, more precisely we say thatarrow
the chain [C_{k_{1}}arrow C_{k_{2}}arrow arrow C_{k_{m}}] follows from the pattern [A_{1}arrow A_{2}arrow
A_{m}] . Similarly, we say that a chain \{C_{1}, C_{n}\}\subseteq \mathcal{U} follows from the patternm[x_{1}arrow x_{2}arrow arrow x_{m}][11] if there exist
1\leq k_{1}<k_{2}<<k_{m}\leq n or 1\leq k_{m}<k_{m-1}<

such that x_{i}\in C_{k_{i}} for each i=1,2,m<k_{1}\leq n, where[x_{1}arrow x_{2}arrow arrow x_{m}]=\{x_{1}arrow x_{2}arrow arrow x_{m};x_{m} arrow x_{m-1}arrow arrow x_{1}\}.
More precisely, we say that the chain [C_{k_{1}}arrow C_{k_{2}}arrow arrow C_{k_{m}}] follows from thepattern[x_{1}arrow x_{2}arrow arrow x_{m}].

Let \mathcal{P} be a collection of graphs and let

X. We say thatZ

be a subset of a \mathcal{P} ‐like continuum

Z has the free tracing property by(resp. free) \mathcal{P} ‐chains if for any\epsilon>0 ,arrow x_{m} of any m distinct points
any m\in \mathbb{N} and any order x_{1}arrow x_{2}arrowx_{i}(i=1,2, \ldots, m) of Z , there is an open cover \mathcal{U} of X such that the mesh of \mathcal{U} isless than \epsilon , the nerve N(\mathcal{U}) of \mathcal{U} is homeomorphic to an element of \mathcal{P} and there isa(resp. free) chain in \mathcal{U} which follows from the pattern [x_{1}arrow x_{2}arrow arrow x_{m}].

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Especially, for a

G ‐like continuum X ,we say that a subset

G ‐chains if Z

tracing property by (resp. free)(resp. free) \mathcal{P} ‐chains, where \mathcal{P}=\{G\}.

Z

of

X

has the free

has the free tracing property by

In the special the case that X itself is a graph G , for points x_{i}(i=1,2, \ldots, m) of

G , we can similarly define that an edge of G follows from the patternarrow x_{m}].

[x_{1}arrow x_{2}arrow

4. CHARACTERIZATIONS OF INDECOMPOSABLE CONTINUA AND FREE TRACINGPROPERTY

A continuum X is tree‐like if X is T‐like, where T is the collection of all trees.
For the case that X is a tree‐like continuum, we have the following characterization
theorem.

Theorem 4.1. ([12]) Let T be the collection of all trees and let X be a T ‐like
continuum, i. e., X is tree‐like. Suppose that D is a subset of X with |D|\geq 3 . Thenthe following conditions are equivalent.

(1) For any order x_{1}arrow x_{2}arrow x_{3} of three distinct points x_{i}(i=1,2,3) of D and
any \epsilon>0 , there is an open cover \mathcal{U} of X such that the mesh of \mathcal{U} is less than \epsilon,
the nerve N(\mathcal{U}) of \mathcal{U} is homeomorphic to an element of T and there is a chain in\mathcal{U} which follows from the pattern [x_{1}arrow x_{2}arrow x_{3}].

(2) D has the free tracing property by T ‐chains.
(3) The minimal continuum H in X containingD

is indecomposable and Z istransversal for composants of H.
For the special case of arc‐like continua, we have the following characterization
theorem.

Theorem 4.2. ([12]) Let X be an arc‐like continuum. Suppose that Z is a subsetof X with |Z|\geq 3 . Then the following conditions are equivalent.
(1) X is indecomposable and Z is transversal for composants of X.
(2) Z has the free tracing property by free I ‐chains and X is the minimal continuum
containing

Z.

Next result is the main theorem in this section.

Theorem 4.3. ([12]) Suppose that X is any G ‐like continuum for a graph G and
H is a subcontinuum of X.
Then the following conditions (1), (2) and (3) areequivalent.

(1) H is indecomposable.
(2) There is a Cantor set Z in H such that Z has the free tracing property by free
G ‐chains and H is the unique minimal continuum containing Z. In particular,
is transversal for composants of H.

Z(3) There is a dense F_{\sigma} ‐set Z_{\infty} of H such that

Z_{\infty}= \bigcup_{i\in \mathb {N} Z_{i}is the countable union of Cantor sets Z_{i} in H, Z_{\infty} has the free tracing property
by free G ‐chains and H is the unique minimal continuum containing Z_{i} for eachi\in \mathbb{N} . In particular, Z_{\infty} is transversal for composants of H.

Proposition 4.4. ([12]) Suppose that X is a G ‐like continuum for a graph G andZ

is a Cantor set inXsuch thatZ

has the free tracing property by free chains

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G ‐LIKE CONTINUA

and H is the unique minimal continuum H in X containing Z. Let z\in Z andlet c(z, H) be the composant of z in the indecomposable continuum H. Then any
subcontinuum A inc(z, H) is an arc‐like continuum.

For hereditarily indecomposable continua, we have the following.

Corollary 4.5. ([12]) Suppose that X is any G ‐like continuum for a graph G and
H is a subcontinuum of X. Then the following (1) and (2) are equivalent.
(1) H is hereditarily indecomposable.
(2) For any subcontinuum K of H , there is a Cantor set Z_{K} in K such that Z_{K} has
the free tracing property by free G ‐chains and K is the unique minimal continuum
containing Z_{K} . In particular, Z_{K} is transversal for composants of K.
The following is a characterization of pseudo‐arc.

Corollary 4.6. ([12]) Suppose that X is an arc‐like continuum and H is a subcon‐
tinuum of X. Then the following (1) and (2) are equivalent.
(1) H is the pseudo‐arc.
(2) For any subcontinuum K of H , there is a Cantor set Z_{K} in K such that Z_{K} has
the free tracing property by free I ‐chains and K is the unique minimal continuum
containing Z_{K} . In particular, Z_{K} is transversal for composants of K.

In [23], Kuykendall studied irreducibility and indecomposability in inverse limitsof continua. Also, we have the following.

Corollary 4.7. ([12]) Let G be a graph and let X=1\dot{{\imath}}m\{G_{i}, f_{i,i+1}\}arrow be an inverse
limit with onto bonding maps f_{i,i+1} , where G_{i}=G for each i\in \mathbb{N} . Then thefollowings holds.

(1) There is an indecomposable subcontinuum in X if and only if there is a Cantor

arrow z^{m} of any finite points
such that for any order z^{1}arrow z^{2}arrowZn\in\mathbb{N}, there is k\geq n and an edge ofz^{j}=(z_{i}^{j})_{i=1}^{\infty}(j=1,2, \ldots, m) of and any
G_{k} which follows from the pattern

set

Z

in

X

[z_{k}^{1}arrow z_{k}^{2}arrow arrow z_{k}^{m}].
(2) Moreover, if G is a tree, there is an indecomposable subcontinuum in X if andonly if there is a three points set Z in X such that for any order z^{1}arrow z^{2}arrow z^{3} of
Z and any n\in \mathbb{N} , there is k\geq n and an edge of G_{k} which follows from the pattern[z_{k}^{1}arrow z_{k}^{2}arrow z_{k}^{3}].

5. POSITIVE TOPOLOGICAL ENTROPYLet

X be a compact metric space and \mathcal{U}, \mathcal{V} be two covers of X .

Put\mathcal{U}\vee \mathcal{V}=\{U\cap V|U\in \mathcal{U}, V\in \mathcal{V}\}.
The quantity N(\mathcal{U}) denotes minimal cardinality of subcovers of \mathcal{U} . Let f : Xarrow X
be a map and let \mathcal{U} be an open cover of X . Puth(f, \mathcal{U})=nar ow\infty 1\dot{ \imath} m\frac{\log N(\mathcal{U}Vf^{-1}( \mathcal{U})\ve \ldots Vf^{-n+1}(\mathcal{U}) }{n}.
The topological entropy of f , denoted by h(f) , is the supremum of h(f, \mathcal{U}) for all
open covers \mathcal{U} of X . Positive topological entropy of map is one of generally accepted

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definitions of chaos. We say that a set I\subseteq \mathbb{N} has positive density if

1 i\cdot\inf_{narrow\infty}\frac{|I\cap\{1,2,\ldots,n\}|}{n}>0.
Let
f : Xarrow X a map. Let \mathcal{A} be a collection ofX
I\subset\mathbb{N}subsets of . We say that a set
is an independence set for \mathcal{A} if for all finite
J\subseteq
I
sets, and for all (Y_{j}) \in\prod_{j\in J}\mathcal{A} , we have that
X be a compact metric space and\bigcap_{j\in J}f^{-j}(Y_{j})\neq\emptyset.
We now recall the definition of IE‐tuple which is related to independence set in \mathb {N}and (topological) entropy (see [20]). Let (x_{1}, \ldots, x_{n}) be a sequence of points in X.
A_{n} are open sets
We say that (x_{1}, \ldots, x_{n}) is an IE‐tuple of f if whenever A_{1},x_{n} , respectively, we have that the collection \mathcal{A}=\{A_{1}, A_{n}\}containing x_{1},has an independence set with positive density. In the case that n=2 , we use theterm IE‐pair. We use IE_{k} to denote the set of all IE‐tuples of length k.
Let f : Xarrow X be a map of a compact metric space X with metric d and let\delta>0 . A subset S of X is a \delta ‐scrambled set of f if |S|\geq 2 and for any x, y\in S
with x\neq y , then one has\lim_{narrow}\inf_{\infty}d(f^{n}(x), f^{n}(y))=0 and 1 \dot{ \imath} m\sup_{narrow\infty}d(f^{n}(x), f^{n}(y) \geq\delta.
We say that f : Xarrow X is Li‐Yorke chaotic if there is an uncountable subset S of
X such that for any x, y\in S with x\neq y , then one has\lim\dot{{\imath}}nfd(f^{n}(x), f^{n}(y))narrow\infty=0 and 1 \dot{ \imath} m\sup_{narrow\infty}d(f^{n}(x), f^{n}(y) >0.
In [3], by use of ergodic theory method, Blanchard, Glasner, Kolyada and Maass
proved the following theorem.

Theorem 5.1. ([3]) If a map f : Xarrow X of a compact metric space X has positive
topological entropy, then there is an uncountable \delta ‐scrambled subset of X for some\delta>0 and hence the dynamics (X, f)

is Li‐Yorke chaotic.

In [20], by use of local entropy theory (IE‐tuples), Kerr and Li proved the fol‐lowing theorem.

Theorem 5.2. ([20, Theorem 3.18]) Suppose that f : Xarrow X is a positive topo‐
logical entropy map of a compact metric space X , and x_{1}, x_{2},x_{m}(m\geq 2) are
distinct points of X such that the tuple (x_{1}, x_{2}, \ldots, x_{m}) is an IE ‐tuple of f . If
A_{i}(i=1,2, \ldots, m) is any neighborhood of x_{i} , then there are Cantor sets Z_{i}\subset A_{i}such that the following conditions hold;(1) any sequence (z_{1}, z,\cdots, z_{n}) of points in the Cantor set Z= \bigcup_{i}Z_{i} is an IE ‐tupleof f , and(2) for all k\in \mathbb{N}, k distinct points y_{1}, y_{2},y_{k}\in Z and any points z_{1}, z_{2},z_{k}\in Z,

one has\lim\dot{{\imath}}nf\max\{d(f^{n}(y_{i}), z_{i})narrow\infty|1\leq i\leq k\}= 0.
In particular,

Z is a \delta ‐scrambled set off for some \delta>0.

In [11] we have the following structure theorem for homeomorphisms on G‐like
continua.

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G ‐LIKE CONTINUA

Theorem 5.3. ([11]) In the setting of Theorem 5.2 assume additionally that X isa G ‐like continuum for a graph G and f : Xarrow X is a homeomorphism. Then the
Cantor sets Z_{i}\subset A_{i}(i=1,2, \ldots, m) can be chosen so as to satisfy, in addition to

the above conditions (1) and (2), also the following two ones;(3) Z= \bigcup_{i=1}^{m}Z_{i} has the free tracing property by free G ‐chains, and(4) the unique minimal subcontinuum H of X containing Z is indecomposable and
Z is transversal for composants of H.

An onto map f :Xarrow Y of continua ismonotone if for any y\in Y, f^{-1}(y) isconnected.

Theorem 5.4. ([11]) Let X be a G ‐like continuum, where G is a graph. If f : XarrowX

is a monotone map with positive topological entropy, then there exists a Cantor

set Z in X satisfying conditions (1) and (2) of Theorem 5.2 and embedded verticallyto the composants of a certain indecomposable subcontinuum H of X. Moreover,H can be taken to be the unique minimal subcontinnum of X containing Z.
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Institute of Mathematics

University of Tsukuba
Tsukuba‐shi, Ibaraki, JapanE‐mail: hkato@math.tsukuba.ac.jp