Topological entropy and IE-tuples of indecomposable continua
aa r X i v : . [ m a t h . D S ] M a r TOPOLOGICAL ENTROPY AND IE-TUPLES OF INDECOMPOSABLECONTINUA
HISAO KATOA
BSTRACT . In [3], by use of ergodic theory method, Blanchard, Glasner, Kolyada andMaass proved that if a map f : X → X of a compact metric space X has positive topolog-ical entropy, then there is an uncountable δ -scrambled subset of X for some δ > ( X , f ) is Li-Yorke chaotic. In [18], Kerr and Li developed local en-tropy theory and gave a new proof of this theorem. Moreover, by developing some deepcombinatorial tools, they proved that X contains a Cantor set Z which yields more chaoticbehaviors (see [18, Theorem 3.18]). In the paper [6], we proved that if G is any graphand a homeomorphism f on a G -like continuum X has positive topological entropy, then X has an indecomposable subcontinuum. Moreover, if G is a tree, there is a pair of twodistinct points x and y of X such that the pair ( x , y ) is an IE-pair of f and the irreduciblecontinuum between x and y in X is an indecomposable subcontinuum. In this paper, wedefine a new notion of ”freely tracing property by free chains” on G -like continua and weprove that a positive topological entropy homeomorphism on a G -like continuum admitsa Cantor set Z such that every tuple of finite points in Z is an IE -tuple of f and Z has thefreely tracing property by free chains. Also, by use of this notion, we prove the followingtheorem: If G is any graph and a homeomorphism f on a G -like continuum X has posi-tive topological entropy, then there is a Cantor set Z which is related to both the chaoticbehaviors of Kerr and Li [18] in dynamical systems and composants of indecomposablecontinua in topology. Our main result is Theorem 3.3 whose proof is also a new proofof [6]. Also, we study dynamical properties of continuum-wise expansive homeomor-phisms. In this case, we obtain more precise results concerning continuum-wise stablesets of chaotic continua and IE-tuples.
1. I
NTRODUCTION
During the last thirty years or so, many interesting connections between dynamical sys-tems and continuum theory have been studied by many authors(see [1,2,6,7,9-15,17,19,22-25,27,28]). We are interested in the following fact that chaotictopological dynamics should imply existence of complicated topological structures of un-derlying spaces. In many cases, such continua (=compact connected metric spaces) areindecomposable continua which are central subjects of continuum theory in topology. Weknow that many indecomposable continua often appear as chaotic attractors of dynami-cal systems. Also, in many cases, the composants of such indecomposable continua arestrongly related to stable or unstable (connected) sets of the dynamics. For instance, inthe theory of dynamical systems and continuum theory, the Knaster continuum (= Smale’shorse shoe), the pseudo-arc, solenoids and Plykin attractors (=Wada’s lakes) etc., are well-known as such indecomposable continua.
Mathematics Subject Classification.
Primary 37B45, 37B40, 54H20; Secondary 54F15.
Key words and phrases.
Topological entropy, Cantor set, IE-tuple, chaos, freely tracing property by freechains, indecomposable continuum, inverse limit, G -like continuum . In [3], by use of ergodic theory method, Blanchard, Glasner, Kolyada and Maass provedthat if a map f : X → X of a compact metric space X has positive topological entropy,then there is an uncountable δ -scrambled subset of X for some δ > ( X , f ) is Li-Yorke chaotic. In [18], Kerr and Li developed local entropy theoryand gave a new proof of this theorem. Moreover, they proved that X contains a Cantorset Z which yields more chaotic behaviors (see [18, Theorem 3.18]). In [2], Barge andDiamond showed that for piecewise monotone surjections of graphs, the conditions ofhaving positive entropy, containing a horse shoe and the inverse limit space containingan indecomposable subcontinuum are all equivalent. In [24], Mouron proved that if X is an arc-like continuum which admits a homeomorphism f with positive topologicalentropy, then X contains an indecomposable subcontinuum. In [6], as an extension ofthe Mouron’s theorem, we proved that if G is any graph and a homeomorphism f on a G -like continuum X has positive topological entropy, then X contains an indecomposablesubcontinuum. Moreover, if G is a tree, there is a pair of two distinct points x and y of X such that the pair ( x , y ) is an IE-pair of f and the irreducible continuum between x and y in X is an indecomposable subcontinuum.In this paper, for any graph G we define a new notion of ”freely tracing property by freechains” on G -like continua and by use of this notion, we prove that a positive topologicalentropy homeomorphism on a G -like continuum admits a Cantor set Z such that everytuple of finite points in Z is an IE -tuple of f and Z has the freely tracing property byfree chains. Also, we prove that the Cantor set Z is related to both the chaotic behaviorsof Kerr and Li [18] in dynamical systems and composants of indecomposable continua intopology. Our main result is Theorem 3.3 whose proof is also a new proof of [6]. Also, westudy dynamical properties of continuum-wise expansive homeomorphisms. In this case,we obtain more precise results concerning continuum-wise stable sets of chaotic continuaand IE-tuples. 2. D EFINITIONS AND NOTATIONS
In this paper, we assume that all spaces are separable metric spaces and all maps arecontinuous. Let N be the set of natural numbers and Z the set of integers.Let X be a compact metric space and U , V be two covers of X . Put U ∨ V = { U ∩ V | U ∈ U , V ∈ V } . The quantity N ( U ) denotes minimal cardinality of subcovers of U . Let f : X → X be amap and let U be an open cover of X . Put h ( f , U ) = lim n → ∞ log N ( U ∨ f − ( U ) ∨ . . . ∨ f − n + ( U )) n . The topological entropy of f , denoted by h ( f ) , is the supremum of h ( f , U ) for all opencovers U of X . The reader may refer to [3,4,5,6,8,18,22-25,27,28] for important factsconcerning topological entropy. Positive topological entropy of map is one of generallyaccepted definitions of chaos.We say that a set I ⊆ N has positive density iflim inf n → ∞ | I ∩ { , , ..., n }| n > . OPOLOGICAL ENTROPY AND IE-TUPLES OF INDECOMPOSABLE CONTINUA 3
Let X be a compact metric space and f : X → X a map. Let A be a collection of subsetsof X . We say that A has an independence set with positive density if there exists a set I ⊂ N with positive density such that for all finite sets J ⊆ I , and for all ( Y j ) ∈ ∏ j ∈ J A ,we have that \ j ∈ J f − j ( Y j ) = /0 . We observe a simple but important and and useful fact that if I is an independence setwith positive density for A then for all k ∈ Z , k + I is an independence set with positivedensity for A . For convenience, we may assume that I satisfies the condition ( kl ) ; for all ( Y j ) ∈ ∏ j ∈ J A and any Y ∈ A ( kl ) Y ∩ \ j ∈ J f − j ( Y j ) = /0 . We now recall the definition of IE-tuple. Let ( x , . . . , x n ) be a sequence of points in X . We say that ( x , . . . , x n ) is an IE-tuple for f if whenever A , . . . , A n are open setscontaining x , . . . , x n , respectively, we have that the collection A = { A , . . . , A n } has anindependence set with positive density. In the case that n =
2, we use the term IE-pair.We use IE k to denote the set of all IE-tuples of length k .Let f : X → X be a map of a compact metric space X with metric d and let δ >
0. Asubset S of X is a δ - scrambled set of f if | S | ≥ x , y ∈ S with x = y , then onehas lim inf n → ∞ d ( f n ( x ) , f n ( y )) = n → ∞ d ( f n ( x ) , f n ( y )) ≥ δ . We say that f : X → X is Li-Yorke chaotic if there is an uncountable subset S of X suchthat for any x , y ∈ S with x = y , then one haslim inf n → ∞ d ( f n ( x ) , f n ( y )) = n → ∞ d ( f n ( x ) , f n ( y )) > . Also, f has sensitive dependence on initial conditions if there is a positive number c > x ∈ X and any neighborhood U of x , one can find y ∈ U and n ∈ N suchthat d ( f n ( x ) , f n ( y )) ≥ c .Let X i ( i ∈ N ) be a sequence of compact metric spaces and let f i , i + : X i + → X i be amap for each i ∈ N . The inverse limit of the inverse sequence { X i , f i , i + } ∞ i = is the spacelim ←−{ X i , f i , i + } = { ( x i ) ∞ i = | x i = f i , i + ( x i + ) for each i ∈ N } ⊂ ∞ ∏ i = X i which has the topology inherited as a subspace of the product space ∏ ∞ i = X i .If f : X → X is a map, then we use lim ←− ( X , f ) to denote the inverse limit of X with f asthe bonding maps , i.e.,lim ←− ( X , f ) = n ( x i ) ∞ i = ∈ X N | f ( x i + ) = x i ( i ∈ N ) o . Let σ f : lim ←− ( X , f ) → lim ←− ( X , f ) be the shift homeomorphism defined by σ f ( x , x , x , ...., ) = ( x , x , ...., ) . A continuum is a compact connected metric space. We say that a continuum is nonde-generate if it has more than one point. A continuum is indecomposable (see [19,20,23,26]) KATO if it is nondegenerate and it is not the union of two proper subcontinua. For any contin-uum H , the set c ( p ) of all points of the continuum H , which can be joined with the point p by a proper subcontinuum of H , is said to be the composant of the point p ∈ H (see [20,p.208]). Note that for an indecomposable continuum H , the following are equivalent;(1) the two points p , q belong to same composant of H ;(2) c ( p ) ∩ c ( q ) = /0;(3) c ( p ) = c ( q ) .So, we know that if H is an indecomposable continuum, the family { c ( p ) | p ∈ H } of all composants of H is a family of uncountable mutually disjoint sets c ( p ) which areconnected and dense F σ -sets in H (see [20, p.212, Theorem 6]). Note that a (nondegen-erate) 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 coincideswith X , i.e., X is irreducible between any two points of the three points.Let H be an indecomposable continuum. We say that a subset Z of H is verticallyembedded to composants of H if no two of points of Z belong to the same composant 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 .A map g from X onto G is an ε -map ( ε > ) if for every y ∈ G , the diameter of g − ( y ) is less than ε . A continuum X is G-like if for every ε > ε -map from X onto G . For any finite polyhedron G , X is G -like if and only if X is homeomorphic to aninverse limit of an inverse sequence of G . Arc-like continua are those which are G -like for G = [ , ] . Our focus in this article is on G -like continua where G is a graph (= connected1-dimensional compact polyhedron). A graph G is a tree if G contains no simple closedcurve. A continuum X is tree-like if for any ε > G ε and an ε -mapfrom X onto G ε . In this case, G ε depends on ε . If G is a collection of subsets of X , thenthe nerve N ( G ) of G is the polyhedron whose vertices are elements of G and there is asimplex < g , g , ..., g k > with distinct vertices g , g , ..., g k if \ i g i = /0 . In this paper, we consider the only case that nerves are graphs.If { C , . . . , C n } is a subcollection of G we call it a chain if C i ∩ C i + = /0 for 1 ≤ i < n and C i ∩ C j = /0 implies that | i − j | ≤
1. We say that { C , . . . , C n } is a free chain in G ifit is a chain and, moreover, for all 1 < i < n we have that C ∈ G with C ∩ C i = /0 impliesthat C = C i , C = C i − or C = C i + . By the mesh of a finite collection G of sets, wemeans the largest of diameters of elements of G . Note that for a graph G , a continuum X is a G -like if and only if for any ε >
0, there is a finite open cover G of X such that N ( G ) = G (which means that N ( G ) and G are homeomorphic) and the mesh of G is lessthan ε . The Knaster continuum (= Smale’s horse shoe) and the pseudo-arc are arc-likecontinua, solenoids are circle-like continua and Plykin attractors are ( S ∨ S ∨ · · · ∨ S m ) -like continua, where S ∨ S ∨ · · · ∨ S m ( m ≥ ) denotes the one point union of m circles S i . Such spaces are typical indecomposable continua. The reader may refer to [20] and[26] for standard facts concerning continuum theory. OPOLOGICAL ENTROPY AND IE-TUPLES OF INDECOMPOSABLE CONTINUA 5
Let X be a continuum and m ∈ N . Suppose that A i ( ≤ i ≤ m ) are m (nonempty)open sets in X and x i ( ≤ i ≤ m ) are m distinct points of X . We identify the order A → A → · · · → A m and the converse order A m → A m − → · · · → A . Then we considerthe equivalence class [ A → A → · · · → A m ] = { A → A → · · · → A m ; A m → A m − → · · · → A } . Suppose that G is a finite open cover of X . We say that a chain { C , · · · , C n } ⊆ G followsfrom the pattern [ A → A → · · · → A m ] if there exist1 ≤ k < k < · · · < k m ≤ n o r ≤ k m < k m − < · · · < k ≤ n such that C k i ⊂ A i for each i = , , ..., m . In this case, more precisely we say that the chain [ C k → C k → · · · → C k m ] follows from the pattern [ A → A → · · · → A m ] . Similarly, wesay that a chain { C , . . . , C n } ⊆ G follows from the pattern [ x → x → · · · → x m ] if thereexist 1 ≤ k < k < · · · < k m ≤ n o r ≤ k m < k m − < · · · < k ≤ n such that x i ∈ C k i for each i = , , ..., m , where [ x → x → · · · → x m ] = { x → x → · · · → x m ; x m → x m − → · · · → x } . More precisely, we say that the chain [ C k → C k → · · · → C k m ] follows from the pattern [ x → x → · · · → x m ] . Let Z be a subset of a G -like continuum X . We say that Z has the freely tracing propertyby (resp. free) chains if for any ε >
0, any m ∈ N and any order x → x → · · · → x m ofany m distinct points x i ( i = , , ..., m ) of Z , there is an open cover U of X such that themesh of U is less than ε , the nerve N ( U ) of U is G and there is a (resp. free) chain in U which follows from the pattern [ x → x → · · · → x m ] .Example 1. (1) Let X = [ , ] be the unit interval and D a subset of X . If | D | ≥ D does not have the freely tracing property by chains.(2) Let X = S be the unit circle and D a subset of X . If | D | ≤
3, then D has the freelytracing property by free chains. If | D | ≥
4, then D does not have the freely tracing prop-erty by chains.For the case that X is a tree-like, we obtain the following proposition. Proposition 2.1.
Let X be a tree-like continuum and let D be a subset of X with | D | ≥ .Then the following are equivalent. (1) For any order x → x → x of three distinct points x i ( i = , , ) of D and any ε > , there is an open cover U of X such that the mesh of U is less than ε ,the nerve N ( U ) of U is a tree and there is a chain in U which follows from thepattern [ x → x → x ] . (2) D has the freely tracing property by chains; for any ε > , any m ∈ N and anyorder x → x → · · · → x m of any m distinct points x i ( i = , , ..., m ) of D, there isan open cover U of X such that the mesh of U is less than ε , the nerve N ( U ) of U is a tree and there is a chain in U which follows from the pattern [ x → x →· · · → x m ] . KATO (3)
The minimal continuum H in X containing D is indecomposable and no two ofpoints of D belong to the same composant of H, i.e., D is vertically embedded tocomposants of H.Proof.
First, we will show that (1) implies (3). Consider the family K of all subcontinuaof X containing D . Since X is a tree-like continuum, the intersection H = \ { K ∈ K } is the unique minimal subcontinuum containing D . Suppose, on the contrary, that H isdecomposable. Then there are proper subcontinua H , H of H with H = H ∪ H . Since H is the minimal continuum containing D , we can choose x , y ∈ D such that x ∈ H − H , y ∈ H − H . Also let z be a point of D with z = x , z = y . We may assume that z ∈ H . Choose ε > ε < d ( y , H ) . By (1), we can choose an open cover U of X such that themesh of U is less than ε , the nerve N ( U ) of U is a tree and there is a chain in U whichfollows from the pattern [ x → y → z ] . Since H is connected, the family { U ∈ U | U ∩ H = /0 } contains a chain from x to z . Then we have a circular chain in U . Since N ( U ) is a tree,this is a contradiction. Hence H is indecomposable.Suppose, on the contrary, that there are two distinct points x , y ∈ D which are containedin the same composant of H . Choose a proper subcontinuum J of H containing x , y . Let z be any point of D with z = x and z = y . Let ε > U of X such that the mesh of U is less than ε , the nerve N ( U ) of U is a tree and there is a chain in U which follows from the pattern [ x → z → y ] .Since N ( U ) is a tree and x , y ∈ J , we see that d ( z , J ) < ε . Since ε is arbitrary small, thenwe see that z ∈ J . Hence D ⊂ J . Since H is the minimal continuum containing D , this is acontradiction. Consequently, no two of points of D belong to the same composant of H .Next we will show that (3) implies (1). We assume that the minimal continuum H in X containing D is indecomposable and no two of points of D belong to the same composantof H . Let a , b , c be any three distinct points of D . We consider the order a → b → c . Let ε > ε < min { d ( a , b ) , d ( b , c ) , d ( c , a ) } . Since X is tree-like,we have a finite open cover V of X such that the mesh of V is less than ε , the nerve N ( V ) of V is a tree. For any x ∈ X , let V x be an element of V containing the point x .We consider the following cases.Case(i): V b separates the vertices V a from V c in the nerve N ( V ) .In this case, we can easily see that there is a chain in V which follows from the pattern [ a → b → c ] .Case(ii): V b does not separate the vertices V a from V c in the nerve N ( V ) .Since N ( V ) is a tree, we can choose subfamilies V ′ and V ” of V such that V = V ′ ∪ V ”, V ′ ∩ V ” = { V b } , the nerves N ( V ′ ) and N ( V ” ) are connected (tree), and N ( V ′ ) contains V a and V c . Put Y = X − [ V ” , where [ V ” = [ { V | V ∈ V ” } . Consider the component Y a containing a in Y and the component Y c containing c in Y .Then we see that Y a ∩ Y c = /0. Suppose, on the contrary, that Y a ∩ Y c = /0 and hence Y a = Y c . OPOLOGICAL ENTROPY AND IE-TUPLES OF INDECOMPOSABLE CONTINUA 7
Since X is a tree-like continuum, we see that H ∩ Y a is a proper subcontinuum of H containing a and c . This implies that a and c belong to the same composant of H . Thisis a contradiction. Hence Y a ∩ Y c = /0. Since Y a is a component of Y , we can choose asufficiently small closed and open neighborhood Y of Y a in Y such that Y ∩ Y c = /0. Put Y = Y − Y . Consider the following families V and V of open sets of Y defined by V = V ′ | Y , V = V ′ | Y , where V ′ | Y = { V ∩ Y | V ∈ V ′ } . Put U ′ = V ∪ V ∪ V ” . By use of the cover U ′ of X , we can easily construct the desired open cover U of X suchthat the mesh of U is less than ε , the nerve N ( U ) of U is a tree and there is a chain in U which follows from the pattern [ a → b → c ] .Note that it is trivial that (2) implies (1). Finally, we prove that (1) implies (2). By theinduction on m , we prove the implication. In the statement of ( ) , the case m = ( ) . We assume that the statement of ( ) is true for the case m ≥
3. Let x → x → x → · · · → x m → x m + be any order of distinct m + x i ( i = , , ..., m + ) of D and ε >
0. By induction, there is a finite open cover V of X such that the mesh of V is less than ε , the nerve N ( V ) of V is a tree and there exists a chain in V which followsfrom the pattern [ x → x → x → · · · → x m ] . As above, we consider the following cases.Case(i): V x m separates the vertices V x i ( i = , , ..., m − ) from V x m + in the nerve N ( V ) .In this case, we can easily see that there is a chain in V which follows from the pattern [ x → x → x → · · · → x m → x m + ] .Case(ii): V x m does not separate the vertices V x i ( i = , , ..., m − ) from V x m + in the nerve N ( V ) .Since N ( V ) is a tree, we can choose subfamilies V ′ and V ” of V such that V = V ′ ∪ V ”, V ′ ∩ V ” = { V x m } , the nerves N ( V ′ ) and N ( V ” ) are connected (tree) and N ( V ′ ) containsthe vertices V x i ( i = , , ..., m − ) and V x m + . As above, we put Y = X − [ V ” . Note that (1) and (3) are equivalent and hence we can use the conditions of (3). Bythe arguments as above, we can choose a closed and open set Y ′ of Y containing x m + such that Y ′ does not contain any x i ( i = , , ..., m − ) . Put Y ” = Y − Y ′ . Consider thefollowing families V and V of open sets of X defined by V = V ′ | Y ′ , V = V ′ | Y ” . Put U ′ = V ∪ V ∪ V ” . By use of the cover U ′ of X , we have the desired open cover U of X such that the mesh of U is less than ε , the nerve N ( U ) of U is a tree and there is a chain in U which followsfrom the pattern [ x → x → x → · · · → x m → x m + ] . This completes the proof. (cid:3)
3. T
OPOLOGICAL ENTROPY ON G - LIKE CONTINUA AND C ANTOR SETS WHICH HAVETHE FREELY TRACING PROPERTY BY FREE CHAINS
KATO
In [3], by use of ergodic theory method, Blanchard, Glasner, Kolyada and Maass provedthat if a map f : X → X of a compact metric space X has positive topological entropy,then there is an uncountable δ -scrambled set of f for some δ > ( X , f ) is Li-Yorke chaotic. In [8], Huang and Ye studied local entropy theory and theygave a characterization of positive topological entropy by use of entropy tuples. Moreover,in [18], by use of local entropy theory (IE-tuples), Kerr and Li proved the following moreprecise theorem. Theorem 3.1. ([18, Theorem 3.18])
Suppose that f : X → X is a positive topologicalentropy map on a compact metric space X , and x , x , ..., x m ( m ≥ ) are finite distinctpoints of X such that the tuple ( x , x , ..., x m ) is an IE-tuple of f . If A i ( i = , , ..., m ) is any neighborhood of x i , then there are Cantor sets Z i ⊂ A i such that the followingconditions hold; ( ) every tuple of finite points in the Cantor set Z = ∪ i Z i is an IE-tuple; ( ) for all k ∈ N , k distinct points y , y , ..., y k ∈ Z and any points z , z , ..., z k ∈ Z, one has lim inf n → ∞ max { d ( f n ( y i ) , z i ) | ≤ i ≤ k } = . In particular, Z is a δ -scrambled set of f for some δ > . In [6], by use of local entropy theory (IE-tuples), we proved the following theorem.
Theorem 3.2. ([6])
Suppose that G is any graph and f : X → X is a homeomorphism ona G-like continuum X with positive topological entropy. Then X contains an indecompos-able subcontinuum. Moreover, if G is a tree, there is a pair of two distinct points x and yof X such that the pair ( x , y ) is an IE-pair of f and the irreducible continuum between xand y in X is an indecomposable subcontinuum. The next theorem is a structure theorem for positive topological entropy homeomor-phisms on G -like continua. The result is the main theorem in this paper which impliesthat for any graph G , a positive topological entropy homeomorphism on a G -like contin-uum X admits Cantor set Z which yields both some complicated structures in topologyand the chaotic behaviors of Kerr and Li [18] in dynamical systems. Especially, the Cantorset Z has the freely tracing property by free chains. Theorem 3.3.
Let G be any graph, X a G-like continuum and f : X → X a homeo-morphism on X with positive topological entropy. Suppose that x , x , ..., x m ( m ≥ ) are finite distinct points of X such that the tuple ( x , x , ..., x m ) is an IE-tuple of f andA i ( i = , , ..., m ) is any neighborhood of x i . Then there are Cantor sets Z i ⊂ A i and anindecomposable subcontinuum H of X such that the following conditions hold; ( ) the Cantor set Z = ∪ mi = Z i is vertically embedded to composants of H; i.e., if x , y aredistinct points of Z, then the irreducible continuum Ir ( x , y ; H ) between x and y in H is H, ( ) Z has the freely tracing property by free chains; for any m ∈ N and any orderx → x → · · · → x m of m distinct points x i ( i = , , ..., m ) of Z and any ε > , thereis an open cover U of X such that the mesh of U is less than ε , the nerve N ( U ) of U isG and there is a free chain in U which follows from the pattern [ x → x → · · · → x m ] , ( ) every tuple of finite points in the Cantor set Z is an IE-tuple of f , and ( ) for all k ∈ N , any distinct k points y , y , ..., y k ∈ Z and any points z , z , ..., z k ∈ Z, the
OPOLOGICAL ENTROPY AND IE-TUPLES OF INDECOMPOSABLE CONTINUA 9 following condition holds lim inf n → ∞ max { d ( f n ( y i ) , z i ) | ≤ i ≤ k } = . In particular, Z is a δ -scrambled set of f for some δ > . In the statement of Theorem 3.3, we need the condition that X is a G -like continuumfor a graph G .Example 2. Let g : Z → Z be a homeomorphism on a Cantor set Z which has positivetopological entropy. Let X = Cone ( Z ) be the cone of Z and let f : X → X be a homeomor-phism which is the natural extension of g . Then h ( f ) > X is tree-like, but X is not G -like for any graph G . Note that X contains no indecomposable subcontinuum. Also, if D is a subset with | D | ≥
3, then D does not have the freely tracing property by chains.We will freely use the following facts from the local entropy theory. Proposition 3.4. ([18, Propositions, 3.8, 3.9])
Let X be a compact metric space and letf : X → X be a map. (1)
Let ( A , . . . , A k ) be a tuple of closed subsets of X which has an independent set ofpositive density. Then, there is an IE-tuple ( x , . . . , x k ) with x i ∈ A i for ≤ i ≤ k. (2) h ( f ) > if and only if f has an IE-pair ( x , x ) with x = x . (3) IE k is closed and f × . . . × f invariant subset of X k . (4) If ( A , . . . , A k ) has an independence set with positive density and, for ≤ i ≤ k, A i is a finite collection of sets such that A i ⊆ ∪ A i , then there is A ′ i ∈ A i such that ( A ′ , . . . , A ′ k ) has an independence set with positive density. To prove Theorem 3.3, we need the following results.
Proposition 3.5. ([6, Proposition 3.1])
Let I ⊆ N be a set with positive density and n ∈ N .Then, there is a finite set F ⊆ I with | F | = n and a positive density set B such that F + B ⊆ I. Proposition 3.6. ([6, Proposition 3.2])
Let X be a compact metric space and let f : X → Xbe a map. Let A be a collection which has an independence set with positive density andn ∈ N . Then, there is a finite set F with | F | = n such that A F = { \ i ∈ F f − i ( Y i ) : Y i ∈ A } has an is an independence set with positive density. Let m ≥ { , , ..., m } n be the set of all functions from { , , ..., n } to { , , ..., m } .For σ ∈ { , , ..., m } n ( m ≥ ) , we write σ = ( σ ( ) , σ ( ) , ... σ ( n )) , where σ ( i ) ∈ { , , ..., m } .Note that |{ , , ..., m } n | = m n . Proposition 3.7. (cf. [6, Proposition 3.3])
Let m , n ∈ N , and σ , . . . , σ [( m − ) n + ][( m − ) n + ] be any sequence of distinct elements of { , ..., m } n . Then there are ≤ i ≤ n and ≤ k < k < k < . . . < k m ≤ [( m − ) n + ][( m − ) n + ] such that σ k j ( i ) = j for j = , ..., m. Proof.
First, we prove the following claim ( ∗ ) : If B is a subset of { , , ..., m } n with | B | = ( m − ) n +
1, then there is 1 ≤ i ≤ n such that B ( i ) = { , , ..., m } , where B ( i ) = { σ ( i ) | σ ∈ B } .Suppose, on the contrary that for each 1 ≤ j ≤ n , | B ( j ) | ≤ m − , where B ( j ) = { σ ( j ) | σ ∈ B } . Then we may consider that the set B is a subset of B ( ) × B ( ) × · · · × B ( n ) whose cardinality is ≤ ( m − ) n . This is a contradiction.Next, we will prove this proposition. We divide the given sequence σ , . . . , σ [( m − ) n + ][( m − ) n + ] into ( m − ) n + B = { σ , . . . , σ ( m − ) n + } , B = { σ ( m − ) n + , . . . , σ [( m − ) n + ] } , · · · B ( m − ) n + = { σ [( m − ) n ][( m − ) n + ]+ , . . . , σ [( m − ) n + ][( m − ) n + ] } . Since | B s | = ( m − ) n + ( s = , , .., ( m − ) n + ) , the above claim ( ∗ ) implies that foreach B s , there is 1 ≤ i s ≤ n such that B s ( i s ) = { σ ( i s ) | σ ∈ B s } = { , .., m } . Define afunction F : B = { B s | ≤ s ≤ ( m − ) n + } → { , , .., n } by F ( B s ) = i s . Since | B | =( m − ) n +
1, we can find 1 ≤ i ≤ n such that | F − ( i ) | ≥ m . Then we can choose 1 ≤ s < s < · · · < s m ≤ ( m − ) n + B s j ( i ) = { , , .., m } ( j = , , ..., m ) . By use ofthis fact, we can choose σ k j ∈ B s j such that σ k j ( i ) = j for j = , ..., m . Then1 ≤ k < k < . . . < k m ≤ [( m − ) n + ][( m − ) n + ] and σ k j ( i ) = j for j = , ..., m . (cid:3) To check the chaotic behaviors of Kerr and Li ([18, Theorem 3.18]), we need the fol-lowing lemma.
Lemma 3.8.
Let f : X → X be a map of a compact metric space X . Suppose that ( A , . . . , A k ) is a tuple of closed subsets of X which has an independent set of positive den-sity. Then there is a tuple ( A ′ , . . . , A ′ k ) of closed subsets of X which has an independent setwith positive density such that A ′ j ⊂ A j ( j = , , ..., k ) , and if h : { , , ..., k } → { , , ..., k } is any function, then there is n h ∈ N such that f n h ( A ′ j ) ⊂ A h ( j ) for each j = , , ..., k.Proof. Suppose that ( A , . . . , A k ) has an independent set I ⊂ N with positive density sat-isfying the above condition ( kl ) . Let K be the set { , , · · · , k } k of all functions on { , , ..., k } . Since | K | = k k (= p ) , we can put K = { h , h , · · · , h p } . By Proposition 3.5,there is a finite set F ⊂ I with | F | = p and a positive density set B such that F + B ⊂ I .Let F = { i , i , · · · , i p } . For each 1 ≤ j ≤ k , we put A ′ j = A j ∩ \ i s ∈ F f − i s ( A h s ( j ) ) OPOLOGICAL ENTROPY AND IE-TUPLES OF INDECOMPOSABLE CONTINUA 11
Then A ′ = { A ′ j | ≤ j ≤ k } is a desired family. In fact, f i s ( A ′ j ) ⊂ A h s ( j ) for each j = , , ..., k . (cid:3) Proposition 3.9.
Let X be a G-like continuum for a graph G and let T be a Cantor set inX . Suppose that T has the freely tracing property by chains. Then any minimal continuumH in X containing T is indecomposable and there is s ∈ N such that for any composantc of H, | c ∩ T | ≤ s. In particular, no infinite points of T belong to the same composant ofH. Also, there is a subset Z of T such that Z is a Cantor set and Z is vertically embeddedto composants of H.Proof. For the graph G , we can find sufficiently large s ∈ N such that G does not contain s simple closed curves.Consider the family K of all subcontinua of X containing T . By the the Zorn’s lemma,there is a minimal element H of K . We will show that H is indecomposable. Suppose,on the contrary, that H is decomposable. There are two proper subcontinua A and B of H such that H = A ∪ B . Since H is a minimal continuum (=irreducible continuum)containing T , there are two points x , y ∈ T with x ∈ A − B , y ∈ B − A . Since T is perfect, T ∩ ( A − B ) and T ∩ ( B − A ) are infinite sets. Then there are distinct points a i ∈ T ∩ ( A − B ) ( i = , , , ..., s ) , b i ∈ T ∩ ( B − A ) ( i = , , ..., s ) . Let ε > d ( A , { b i | i = , , ..., s } ) > ε . Since T has the the freely tracing property by chains,there is an open cover U of X such that the mesh of U is less than ε , the nerve N ( U ) of U is G and there is a chain in U which follows from the pattern [ a → b → a → b → a → b → · · · → a s − → b s → a s ] . Since A is connected, for any two points z , z ′ ∈ A there a chain { C , C f rm − e , ..., C n } in U from z to z ′ such that C j ∩ A = /0. By use of these facts, we can easily find distinct s simple closed curves in N ( U ) = G . This is a contradiction. By the similar arguments,we see that for any composant c of H , | c ∩ T | ≤ s . To find a desired Cantor set Z ⊂ T , weconsider the following subset of T : R = { ( x , y ) ∈ T | there is a proper subcontinuum F of H with x , y ∈ F } Let { U i | i ∈ N } be an open base of H and let R i = { ( x , y ) ∈ T | there is a subcontinuum F in H − U i with x , y ∈ F } . Note that R i is a closed set of T and R = [ { R i | i ∈ N } . Since each composant of H contains no infinite points of T , we see that R is a nowheredense F σ -set in T (see [21, p. 71, Application 2]). By [21, p. 70, Corollary 3], thereis a Cantor set Z in T such that Z is independent in R , i.e., if x , y ∈ Z and x = y , then ( x , y ) / ∈ R . Then we see that Z is vertically embedded to composants of H . This completesthe proof. (cid:3) The following lemma is the key lemma to prove the main theorem.
Lemma 3.10.
Let G be a graph and let f be a homeomorphism on a G-like continuum Xwith positive topological entropy. Suppose that A is a finite open collection of X whichhas an independence set of f with positive density, any distinct elements of A are disjoint,and | A | = m ≥ . Then for any ε > and any order A → A → · · · → A m of all elements of A , there exists a finite open cover V of X satisfying the following conditions; ( ) the mesh of V is less than ε , ( ) the nerve N ( V ) of V is G, ( ) for each A ∈ A there is a shrink s ( A ) ∈ V with s ( A ) ⊂ A such thats ( A ) = { s ( A ) | A ∈ A } has an independence set with positive density, and ( ) there is a free chain [ s ( A ) → s ( A ) → · · · → s ( A m )] from s ( A ) to s ( A m ) in V whichfollows from the pattern [ A → A → · · · → A m ] .Proof. We put A = { A , A , · · · , A m } . For each A ∈ A , we can choose an open set A ′ ⊂ A ′ ⊂ A so that A ′ = { A ′ | A ∈ A } has an independence set I with positive density (see Proposition 3.4). We choose a suffi-ciently small positive number ε ′ < ε so that d ( A ′ , X − A ) > ε ′ for each A ∈ A . Let E ( G ) be the set of all edges of the graph G and let | E ( G ) | be the cardinality of the set E ( G ) . Wecan choose a sufficiently large natural number n ∈ N such that m n > | E ( G ) | · [( m − ) n + ][( m − ) n + ] . By Proposition 3.5, we have F with | F | = n and B satisfying the condition of Proposition3.5. Put F = { j , j , ..., j n } . Recall A ′ F = { \ j ∈ F f − j ( Y j ) | Y j ∈ A ′ } = { n \ i = f − j i ( A σ ( i ) ) | σ ∈ { , , ..., m } n } . Note that | A ′ F | = m n and A ′ F has the independence set B with positive density. Since F is a finite set, we can choose a sufficiently small positive number δ > A ′ F is at least δ apart and if U is a subset of X whose diameteris less than δ , then the diameter of f i ( U ) ( i ∈ F ) is less than ε ′ . Since X is G -like, wecan choose an open cover U of X such that N ( U ) is G and the mesh of U is less than δ .Since δ is so small, we see that each element of U intersects at most one element of A ′ F .By Proposition 3.4 (4), we obtain a subcollection U ′ of U such that each element of A ′ F intersects with only one element of U ′ , | U ′ | = m n and the family U ′ has an independentset of positive density. Then we can choose a free chain C in U such that C contains atleast m n / | E ( G ) | ≥ [( m − ) n + ][( m − ) n + ] many elements of U ′ . Put C = { C , C , · · · , C p } . Note that each U ′ ∈ U ′ determines theelement σ ∈ { , , ..., m } n . By Proposition 3.7, we can choose i ∈ F such that there is asequence 1 ≤ k < k < k < . . . < k m ≤ [( m − ) n + ][( m − ) n + ] such that C k j ∈ U ′ and f i ( C k j ) ∩ A ′ j = /0for each j = , , ..., m . By the choice of ε ′ , f i ( C k j ) ⊂ A j for each j = , , ..., m . Then thefree chain [ f i ( C k ) → f i ( C k ) → · · · → f i ( C k m )] OPOLOGICAL ENTROPY AND IE-TUPLES OF INDECOMPOSABLE CONTINUA 13 in f i ( U ) follows from the pattern [ A → A → · · · → A m ] . Put s ( A j ) = f i ( C k j ) and V = f i ( U ) . Then s ( A ) = { s ( A ) | A ∈ A } is the desired family. This completes the proof. (cid:3) Now, we will prove Theorem 3.3.
Proof.
Let A = { A i | i = , , ..., m } . Then we may assume that A has an independenceset with positive density and the closures of any distinct elements of A are disjoint. Also,we may assume | A | = m (= m ) ≥ A is less than ε ∈ ( , / ) . ByLemma 3.8, we have a collection A ′ = { A ′ , A ′ , · · · , A ′ m } of open sets which has an independent set with positive density and satisfies the followingcondition (KL) for A ;(KL) A ′ i ⊂ A i ( i = , , ..., m ) , and if h : { , , ..., m } → { , , ..., m } is any function,then there is n h ∈ N such that f n h ( A ′ i ) ⊂ A h ( i ) for each i = , , ..., m .Consider the set A ′ ( m ) of orders (=permutations) of all elements of A ′ . Note that thecardinality of A ′ ( m ) is m ! = m · ( m − ) · · · · ·
1. We consider the set
Ord A ′ ( m ) ofequivalence classes of elements of A ′ ( m ) , i.e., Ord A ′ ( m ) = { [ A i → A i → · · · → A im ] | i = , , ..., q } , where q = m ! /
2. By Lemma 3.10, there exists a finite open cover U of X such that themesh of U is less than ε and U satisfies the following conditions; ( ) the nerve N ( U ) of U is homeomorphic to G , ( ) for each A ∈ A there is s ( A ) ∈ U such that s ( A ) ⊂ A ′ ⊂ A , the family A ( ) = { s ( A ) | A ∈ A } has an independence set with positive density, and we have a free chain [ s ( A ) → s ( A ) → · · · → s ( A m )] to from s ( A ) to s ( A m ) in U which follows from the pattern [ A → A → · · · → A m ] . This is the case i =
1. If we continue this procedure by induction on i = , , ..., q , weobtain a sequence U , U , · · · , U q of finite open covers of X and s i ( A ) ∈ U i ( A ∈ A , i = , , ..., q ) such that the following conditions hold; ( ) the nerve N ( U i ) of U i is homeomorphic to G , ( ) U i + is a refinement of U i , ( ) for each A ∈ A , s i ( A ) ∈ U i ( i = , , ..., q ) satisfies that A ⊃ A ′ ⊃ s i ( A ) ⊃ s i + ( A ) and the family A ( i ) = { s i ( A ) | A ∈ A } ( i = , , ..., q ) is an independence set with positive density, and there is a free chain [ s i ( A i ) → s i ( A i ) → · · · → s i ( A im )] from s i ( A i ) to s i ( A im ) in U i follows from the pattern [ s i − ( A i ) → s i − ( A i ) → · · · → s i − ( A im )] ( i = , , ..., q ) , where s ( A j ) = A j , etc.By Proposition 3.6, for each A ∈ A ( q ) , we can choose nonempty open sets s q ( A ) + and s q ( A ) − in s q ( A ) such that s q ( A ) + ∩ s q ( A ) − = /0 and the collection A = { s q ( A ) + , s q ( A ) − | A ∈ A ( q ) } has an independence set with positive density.Let | A | = m (= m ) and 0 < ε ≤ · ε . By Lemma 3.8, for A we can choose acollection A ′ such that the mesh of A ′ is less than ε and A ′ satisfies the condition (KL)for A as above. Also, we consider the set A ′ ( m ) of permutations of all elements of A ′ and the set Ord A ′ ( m ) as above.By repeated use of Lemma 3.10, we obtain desired families A ( i ) = { s i ( A ) | A ∈ A } ( i = , , ..., q ) as above, where q = m ! /
2. By use of A ( q ) , we obtain A as above. Note that | A | = m (= · m ) .If we continue this procedure, we have a sequence ε i ( i ∈ N ) of positive numbers andsequences of of families A i and A ′ i of open sets of X satisfying the following conditions; ( ) ε i > ε i + ( i ∈ N ) and lim i → ∞ ε i = ( ) the closures of any distinct elements of A i are disjoint, each A ∈ A i contains theclosures of two elements of A i + and the mesh of A i is less than ε i , ( ) A i and A ′ i have independence sets with positive density, ( ) A ′ i satisfies the condition (KL) for A i , and ( ) for any order E → E → · · · → E m i of all elements of A i , there is an open cover U of X such that the mesh of U is less than ε i , N ( U ) is G and there is a free chain in U which follows from [ E → E → · · · → E m i ] . For each i ∈ N , we put T i = [ A i ( i ∈ N ) and T = \ i ∈ N T i . Then T is a Cantor set. By the above constructions, we see that for any k ∈ N and anyorder x → x → · · · → x k of k distinct points x i ( i = , , ..., k ) of T and any ε >
0, there is an open cover U of X such that the mesh of U is less than ε , the nerve N ( U ) of U is G and there is a free chainin U which follows from the pattern [ x → x → · · · → x k ] . By Proposition 3.9, any minimal continuum H in X containing T is indecomposable andno infinite points of T belong to the same composant of H . Also, by Proposition 3.9, wecan choose a subset Z of T such that Z is a Cantor set and Z is vertically embedded tocomposants of H . Also, by the constructions, we see that T satisfies the conditions (2), OPOLOGICAL ENTROPY AND IE-TUPLES OF INDECOMPOSABLE CONTINUA 15 (3) and (4) of Theorem 3.3. Note that any subset of T satisfies the conditions. Hence theCantor set Z satisfies the conditions of Theorem 3.3. This completes the proof. (cid:3) Corollary 3.11.
Let G be any graph. If f : G → G is a positive entropy map on G, thenthere exist an indecomposable subcontinuum H of X = lim ←− ( G , f ) and a Cantor set Z in Hsatisfies the following conditions; ( ) Z is vertically embedded to composants of H, ( ) Z has the freely tracing property by free chains, ( ) every tuple of finite points in the Cantor set Z is an IE-tuple of the shift map σ f and ( ) for all k ∈ N , any distinct k points y , y , ..., y k ∈ Z and any points z , z , ..., z k ∈ Z, thefollowing condition holds lim inf n → ∞ max { d ( σ nf ( y i ) , z i ) | ≤ i ≤ k } = . In particular, Z is a δ -scrambled set of σ f for some δ > .Proof. Note that h ( f ) = h ( σ f ) > σ f is a homeomorphism on the G -like continuum X = lim ←− ( G , f ) . This result follows from Theorem 3.3. (cid:3) For a special case, we have the following.
Corollary 3.12.
Let X be one of the Knaster continuum, solenoids or Plykin attractors.If f is any positive topological entropy homeomorphism on X , then there is a Cantor setZ in X such that the Cantor set Z satisfies the following conditions; ( ) Z is vertically embedded to composants of X , ( ) Z has the freely tracing property by free chains, ( ) every tuple of finite points in the Cantor set Z is an IE-tuple of f , and ( ) for all k ∈ N , any distinct k points y , y , ..., y k ∈ Z and any points z , z , ..., z k ∈ Z, thefollowing condition holds lim inf n → ∞ max { d ( f n ( y i ) , z i ) | ≤ i ≤ k } = . In particular, Z is a δ -scrambled set of f for some δ > .Proof. By Theorem 3.3, there is an indecomposable subcontinuum H in X . Note thatany proper subcontinuum of X is not indecomposable and hence H = X . This corollaryfollows from Theorem 3.3. (cid:3) An onto map f : X → Y of continua is monotone if for any y ∈ Y , f − ( y ) is connected.In [16], we proved that if G is a graph and f : X → X is a monotone map on a G -likecontinuum X which has positive topological entropy, then X contains an indecomposablesubcontinuum. Here we give the following more precise result. Theorem 3.13.
Suppose that G is a graph and X is a G-like continuum. If f : X → X is amonotone map on X with positive topological entropy, then there exist an indecomposablesubcontinuum H of X and a Cantor set Z in H such that the Cantor set Z satisfies thefollowing conditions; ( ) Z is vertically embedded to composants of H, ( ) every tuple of finite points in the Cantor set Z is an IE-tuple of f , ( ) for all k ∈ N , any distinct k points y , y , ..., y k ∈ Z and any points z , z , ..., z k ∈ Z, thefollowing condition holds lim inf n → ∞ max { d ( f n ( y i ) , z i ) | ≤ i ≤ k } = . In particular, Z is a δ -scrambled set of f for some δ > . A continuum E is an n - od ( ≤ n < ∞ ) if E contains a subcontinuum A such that thecomplement of A in E is the union n nonempty mutually separated sets, i.e., E − A = [ { E i | i = , , ..., n } for some subsets E i satisfying the condition: E i ∩ E j = /0 ( i = j ) . For any continuum X , let T ( X ) = sup { n | there is an n -od in X } . Note that if X is a G -like continuum for a graph G , then T ( X ) < ∞ .To prove Theorem 3.13, we need the following lemma. Lemma 3.14. (cf. [16, Lemma 2.3])
Let X and Y be continua with T ( X ) < ∞ . Supposethat f : X → Y is an (onto) monotone map, H ′ is an indecomposable subcontinuum of Xand Z ′ is a Cantor set which is vertically embedded to composants of H ′ . If H = f ( H ′ ) isnondegenerate, then H is an indecomposable subcontinuum of Y and there is a subset Zof f ( Z ′ ) such that Z is a Cantor set and Z is vertically embedded to composants of H.Proof. Note that if f : X → Y is monotone, then f − ( C ) is connected for any subcontin-uum C in Y . By use of this fact, we can see that T ( Y ) ≤ T ( X ) < ∞ . For each x ∈ Z ′ , let c ( x ) denote the composant of H ′ containing x ∈ Z ′ . LetComp ( Z ′ ; H ′ ) = { c ( x ) | x ∈ Z ′ } . Since Z ′ is vertically embedded to composants of H ′ , Comp ( Z ′ ; H ′ ) is a family of mutuallydisjoint dense connected subsets c ( x ) of H ′ . For each x , y ∈ Z ′ , we define x ∼ f y providedthat f ( c ( x )) ∩ f ( c ( y )) = /0. Also, we define x ∼ y provided that there is a finite sequence x = x , x , ..., x s = y of x i ∈ Z ′ such that x i ∼ f x i + for each i = , , .., s −
1. Then therelation ∼ is an equivalence relation on Z ′ . Note that f ( c ( x )) ∩ f ( c ( y )) = /0 if and only ifthere is a point z ∈ Y with f − ( z ) ∩ c ( x ) = /0 = f − ( z ) ∩ c ( y ) . Let [ x ] denote the equivalence class containing x ∈ Z ′ , i.e., [ x ] = { y ∈ Z ′ | x ∼ y } . Since f − ( z ) is a subcontinuum of X for each z ∈ Y , we can conclude that | [ x ] | ≤ T ( X ) .In particular, f | Z ′ : Z ′ → f ( Z ′ ) is a finite-one map and hence f ( Z ′ ) is a perfect set, i.e., f ( Z ′ ) has no isolated point.Since Z ′ is an uncountable set, we we can choose an uncountable subset Z ” of Z ′ suchthat the family { f ( c ( x )) | x ∈ Z ” } is a family of mutually disjoint subsets of H = f ( H ′ ) . OPOLOGICAL ENTROPY AND IE-TUPLES OF INDECOMPOSABLE CONTINUA 17
We will prove that H = f ( H ′ ) is indecomposable. Suppose, on the contrary, that H isdecomposable. There is a proper subcontinuum A of H = f ( H ′ ) with Int H ( A ) = /0 . Since each composant of H ′ is dense in H ′ and hence f ( c ( x )) is dense in H for any x ∈ Z ”, f ( c ( x )) ∩ A = /0. This implies that | T ( Y ) | = ∞ . This is a contradiction. Hence H = f ( H ′ ) is indecomposable.We show that for each composant c of H , | c ∩ f ( Z ′ ) | ≤ T ( X ) . Suppose, on the contrary,that there is a proper subcontinuum C of H such that | C ∩ f ( Z ′ ) | ≥ T ( X ) +
1. Then f − ( C ) is a continuum which intersects T ( X ) + H ′ . This is a contradiction.Since f ( Z ′ ) is perfect, by the proof of Proposition 3.9, we can find a Cantor set Z in f ( Z ′ ) such that Z is vertically embedded to composants of H . (cid:3) We will give the proof of Theorem 3.13.
Proof.
We consider the inverse ˜ f : lim ←− ( X , f ) → lim ←− ( X , f ) of the shift map σ f , i.e.,˜ f ( x , x , x , · · · ) = ( f ( x ) , x , x , · · · ) . Note that h ( f ) = h ( ˜ f ) >
0. By Theorem 3.3, we can find an indecomposable subcontin-uum H ′ and a Cantor set Z ′ in lim ←− ( X , f ) as in Theorem 3.3. Since f is a monotone map,we see that the projection p n : lim ←− ( X , f ) → X n = X to the n -th coordinate X n is also mono-tone. If we choose sufficiently large natural number n , then H = p n ( H ′ ) is nondegenerate.By the above lemma, H is indecomposable and there is a Cantor set Z ⊂ p n ( Z ′ ) such that Z is vertically embedded to composants of H . Note that the projection p n preserves theproperties of IE -tuples and (4) of Theorem 3.3. Then we see that H and Z are desiredspaces. (cid:3)
4. C
HAOTIC CONTINUA OF CONTINUUM - WISE EXPANSIVE HOMEOMORPHISMS AND
IE-
TUPLES
In this section, we study dynamical behaviors of continuum-wise expansive home-omorphisms related to IE-tuples and chaotic continua in topology. Any continuum-wiseexpansive homeomorphism f on a continuum X has positive topological entropy andhence f has IE-tuples (see Theorem 4.1 below). Also, X contains a chaotic continuumand chaotic continuum has uncountable mutually disjoint (unstable or) stable dense con-nected F σ -sets (see Theorem 4.1). In this section, we study some precise results of IE-tuples related to (unstable) stable connected sets of chaotic continua and composants ofindecomposable continua.A homeomorphism f : X → X of a compact metric space X with metric d is called expansive ([5,13]) if there is c > x , y ∈ X and x = y , then there is aninteger n ∈ Z such that d ( f n ( x ) , f n ( y )) > c .A homeomorphism f : X → X of a compact metric space X is continuum-wise expansive (resp. positively continuum-wise expansive ) [15] if there is c > A is anondegenerate subcontinuum of X , then there is an integer n ∈ Z (resp. a positive integer n ∈ N ) such that diam f n ( A ) > c , where diam B = sup { d ( x , y ) | x , y ∈ B } for a set B . Such a positive number c is calledan expansive constant for f . Note that each expansive homeomorphism is continuum-wise expansive, but the converse assertion is not true. There are many continuum-wiseexpansive homeomorphisms which are not expansive (see [15]). These notions have beenextensively studied in the area of topological dynamics, ergodic theory and continuumtheory (see [5,10-15,27]).The hyperspace 2 X of X is the set of all nonempty closed subsets of X with the Haus-dorff metric d H . Let C ( X ) = { A ∈ X | A is connected } .Note that 2 X and C ( X ) are compact metric spaces (e.g., see [20] and [26]). For a home-omorphism f : X → X and for each closed subset H of X and x ∈ H , the continuum-wise σ -stable sets V σ ( x ; H ) ( σ = s , u ) of f are defined as follows: V s ( x ; H ) = { y ∈ H | there is A ∈ C ( H ) such that x , y ∈ A and lim n → ∞ diam f n ( A ) = } , V u ( x ; H ) = { y ∈ H | there is A ∈ C ( H ) such that x , y ∈ A and lim n → ∞ diam f − n ( A ) = } .Note that V s ( x ; H ) ⊂ W s ( x ) = { y ∈ X | lim n → ∞ d ( f n ( y ) , f n ( x )) = } , V u ( x ; H ) ⊂ W u ( x ) = { y ∈ X | lim n → ∞ d ( f − n ( y ) , f − n ( x )) = } . A subcontinuum H of X is called a σ -chaotic continuum (see [13]) of f (where σ = s , u )if (1) for each x ∈ H , V σ ( x ; H ) is dense in H , and(2) there is τ > x ∈ H and each neighborhood U of x in X , thereis y ∈ U ∩ H such thatlim inf n → ∞ d ( f n ( x ) , f n ( y )) ≥ τ in case σ = s , orlim inf n → ∞ d ( f − n ( x ) , f − n ( y )) ≥ τ in case σ = u .We know that if f : X → X is a continuum-wise expansive homeomorphism, then V σ ( z ; H ) is a connected F σ -set containing z . If H is a σ -chaotic continuum of f , thenthe decomposition { V σ ( z ; H ) | z ∈ H } of H is an uncountable family of mutually disjoint,dense connected F σ -sets in H . Note that σ -chaotic continua of f have very similar struc-tures of composants of indecomposable continua. In fact, for the case of 1-dimensionalcontinua, σ -chaotic continua may be indecomposable (see [10]).Example 3. Let f : T → T be an Anosov diffeomorphism on the 2-dimensional torus T , say (cid:20) (cid:21) Then f is expansive and T itself is a σ -chaotic continuum of f for σ = u , s . Note that T contains no indecomposable σ -chaotic subcontinuum.For continuum-wise expansive homeomorphisms, we have obtained the following re-sults (see [11,13,15]). Theorem 4.1.
Let f : X → X be a continuum-wise expansive homeomorphism on a con-tinuum X . Then the followings hold.
OPOLOGICAL ENTROPY AND IE-TUPLES OF INDECOMPOSABLE CONTINUA 19 (1) ([15, Theorem 4.1]) f has positive topological entropy and hence there are IE-tuples. (2) ([13, Theorem 3.6 and Theorem 4.1])
There is a σ -chaotic continuum H of f .Moreover, if H is a u-chaotic continuum (resp. s-chaotic continuum), then thereexists a Cantor set Z in H satisfying the conditions; ( i ) no two of points of Z belong to the same V u ( x ; H ) ( x ∈ X ) (resp. V s ( x ; H ) ( x ∈ X ) ), i.e., Z is vertically embedded to V σ ( x , H ) ( x ∈ H ) , ( ii ) Z is a δ -scrambled set of f − for some δ > (resp. f ). (3) ([11, Theorem 2.4]) Moreover, if f : X → X is a positively continuum-wise expan-sive homeomorphism, then X contains a u-chaotic continuum H such that H isindecomposable and the set of composants of H coincides to { V u ( x ; H ) | x ∈ H } .Also, there exists a Cantor set Z in H satisfying the conditions; ( i ) Z is vertically embedded to composants V u ( x , H ) ( x ∈ H ) , ( ii ) Z is a δ -scrambled set of f − for some δ > . (4) ( d ) ([11, Corollary 2.7]) Moreover, if G is any graph and X is a G-like continuum,then X contains a σ -chaotic continuum H such that H is indecomposable and theset of composants of H coincides to { V σ ( x ; H ) | x ∈ H } . Moreover if σ = u (resp.s), then there exists a Cantor set Z in H satisfying the conditions; ( i ) Z is vertically embedded to V σ ( x , H ) ( x ∈ H ) , ( ii ) Z is a δ -scrambled set of f − for some δ > (resp. f ). (5) ([11, Theorem 2.6]) Moreover, if X is a continuum in the plane R , then X con-tains a σ -chaotic continuum H of f such that H is indecomposable and the set ofcomposants of H coincides to { V σ ( x , H ) | x ∈ H } . Moreover if σ = u (resp. s),then there exists a Cantor set Z in H satisfying the conditions; ( i ) Z is vertically embedded to V σ ( x , H ) ( x ∈ H ) , ( ii ) Z is a δ -scrambled set of f − for some δ > (resp. f ). We consider the case that σ -chaotic continua are periodic. By combining Theorem 3.1and Theorem 4.1, we have the following results. Corollary 4.2.
Let f : X → X be a continuum-wise expansive homeomorphism on a con-tinuum X . Suppose that X contains a periodic σ -chaotic continuum H of f . Then thereexists a Cantor set Z in H such that if σ = u (resp. σ = s ) , then the following conditionshold; ( ) Z is vertically embedded to V σ ( x , H ) ( x ∈ H ) , ( ) every tuple of finite points in the Cantor set Z is an IE-tuple of f − (resp. f ), and ( ) for all k ∈ N , any distinct k points y , y , ..., y k ∈ Z and any points z , z , ..., z k ∈ Z, thefollowing condition holds lim inf n → ∞ max { d ( f − n ( y i ) , z i ) | ≤ i ≤ k } = ( resp . lim inf n → ∞ max { d ( f n ( y i ) , z i ) | ≤ i ≤ k } = ) . Proof.
We may assume σ = s . Since the chaotic continuum H is periodic, there is i ∈ N such that f i ( H ) = H . Then f i | H : H → H is continuum-wise expansive and hence itstopological entropy is positive. By Theorem 3.1, there is a Cantor set Z in H as in Theorem Z is a δ -scrambled set of f for some δ > Z is vertically embedded to V s ( x , H ) ( x ∈ H ) . (cid:3) Similarly, we have the following result.
Corollary 4.3.
Suppose that f : X → X is a positively continuum-wise expansive home-omorphism on a continuum X such that X has a periodic u-chaotic continuum H whichis indecomposable and the set of composants of H coincides to { V u ( x ; H ) | x ∈ H } . Thenthere exists a Cantor set Z in H which is vertically embedded to composants of H andsatisfies the conditions; ( ) if x , y belong to the same composant of H, then lim n → ∞ d ( f − n ( x ) , f − n ( y )) = , ( ) every tuple of finite points in the Cantor set Z is an IE-tuple of f − , and ( ) for all k ∈ N , any distinct k points y , y , ..., y k ∈ Z and any points z , z , ..., z k ∈ Z, thefollowing condition holds lim inf n → ∞ max { d ( f − n ( y i ) , z i ) | ≤ i ≤ k } = . For special cases, we have the following.
Corollary 4.4.
Suppose that X is one of the Knaster continuum, Plykin attractors orsolenoids. If f : X → X is a continuum-wise expansive homeomorphism on X , then f orf − is positively continuum-wise expansive. In particular, if f is positively continuum-wise expansive, then there exists a Cantor set Z in X such that the Cantor set Z is verticallyembedded to composants of X and satisfies the conditions; ( ) if x , y belong to the same composant of X , then lim n → ∞ d ( f − n ( x ) , f − n ( y )) = , ( ) every tuple of finite points in the Cantor set Z is an IE-tuple of f − , ( ) Z has the freely tracing property by free chains, and ( ) for all k ∈ N , any distinct k points y , y , ..., y k ∈ Z and any points z , z , ..., z k ∈ Z, thefollowing condition holds lim inf n → ∞ max { d ( f − n ( y i ) , z i ) | ≤ i ≤ k } = . .Proof. Note that X is a G -like continuum for some graph G . Recall that X is indecom-posable and each proper subcontinuum of X is not indecomposable. Hence a σ -chaoticcontinuum of f coincides with X . Then we can easily see that f or f − is positivelycontinuum-wise expansive. The corollary follows from Theorems 3.3 and 4.1. (cid:3) For the case of the shift map σ f : lim ←− ( G , f ) → lim ←− ( G , f ) of a map f : G → G on a graph G which has sensitive dependence on initial conditions, we can find a periodic indecom-posable s -chaotic continuum in lim ←− ( G , f ) . Hence we have the following corollary. Corollary 4.5.
Suppose that f : G → G is a map on a graph G which has sensitive de-pendence on initial conditions and σ f : X = lim ←− ( G , f ) → X is the shift map of f . Thenthere exists an indecomposable s-chaotic continuum H in X such that σ nf ( H ) = H forsome n ∈ N and the set of composants of H coincide to { V s ( x ; H ) | x ∈ H } . Hence there isa Cantor set Z in H such that Z is vertically embedded to composants of H and satisfiesthe conditions; ( ) if x , y belong to the same composant of H, then lim n → ∞ d ( σ nf ( x ) , ( σ nf ( y )) = , ( ) every tuple of finite points in the Cantor set Z is an IE-tuple of σ f , OPOLOGICAL ENTROPY AND IE-TUPLES OF INDECOMPOSABLE CONTINUA 21 ( ) Z has the freely tracing property by free chains, and ( ) for all k ∈ N , any distinct k points y , y , ..., y k ∈ Z and any points z , z , ..., z k ∈ Z, thefollowing condition holds lim inf n → ∞ max { d ( σ nf ( y i ) , z i ) | ≤ i ≤ k } = . Proof.
Note that ( σ f ) − = ˜ f . By [11, Corollary 2.8], there is a s -chaotic indecomposablecontinuum H of σ f and a natural number n such that σ nf ( H ) = H and the set of composantsof H coincides to { V s ( x , H ) | x ∈ H } . By use of Theorem 3.3, we can find a desired Cantorset Z in H . (cid:3) Example 4. Let f : I = [ , ] → I be the map defined by f ( t ) = t ( − t ) ( t ∈ I ) .Note that f has sensitive dependence on initial conditions and lim ←− ( X , f ) is the Knastercontinuum. Then lim ←− ( X , f ) is the s -chaotic continuum of the shift homeomorphism σ f :lim ←− ( X , f ) → lim ←− ( X , f ) satisfying the conditions of Corollary 4.5, where H = lim ←− ( X , f ) .For the general case that any σ -chaotic continua of a continuum-wise expansive home-omorphism are not periodic, we do not know if the statements of Corollaries 4.2 and 4.3hold. In fact, the following problems remain open. Question 4.6.
Let f : X → X be a continuum-wise expansive homeomorphism on a con-tinuum X . Is it true that there exist a σ -chaotic continuum H of f and a Cantor set Zin H such that Z is vertically embedded to V σ ( x , H ) ( x ∈ H ) and satisfies the followingconditions? :If σ = u (resp. σ = s ) , then ( ) every tuple of finite points in the Cantor set Z is an IE-tuple of f − (resp. f ), and ( ) for all k ∈ N , any distinct k points y , y , ..., y k ∈ Z and any points z , z , ..., z k ∈ Z, thefollowing condition holds lim inf n → ∞ max { d ( f − n ( y i ) , z i ) | ≤ i ≤ k } = (resp. lim inf n → ∞ max { d ( f n ( y i ) , z i ) | ≤ i ≤ k } = ) . Question 4.7.
Let f : X → X be a positively continuum-wise expansive homeomorphismon a continuum X . Is it true that there exist an indecomposable continuum H and a Cantorset Z in H satisfying the following conditions? : ( ) Z is vertically embedded to composants of H. ( ) If x , y belong to the same composant of H, then lim n → ∞ d ( f − n ( x ) , ( f − n ( y )) = . ( ) Every tuple of finite points in the Cantor set Z is an IE-tuple of f − . ( ) For all k ∈ N , any distinct k points y , y , ..., y k ∈ Z and any points z , z , ..., z k ∈ Z, thefollowing condition holds lim inf n → ∞ max { d ( f − n ( y i ) , z i ) | ≤ i ≤ k } = . Acknowledgments . The author would like to thank Dr. Masatoshi Hiraki for usefuldiscussions on Proposition 3.7. R EFERENCES [1] M. Barge and J. Martin,
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