Chaos and indecomposability of continua . . Chaos and indecomposability of continua . . . . . Hisao Kato University of Tsukuba May 25-29, 2015 . . . . . . Hisao Kato (University of Tsukuba) Chaos and indecomposability of continua May 25-29, 2015 1 / 31
Chaos and indecomposability of continua This is a joint work with U. Darji in Tsukuba, 2014. Recently, we know that C. Mouron have same results as our main results, independently. . . . . . . Hisao Kato (University of Tsukuba) Chaos and indecomposability of continua May 25-29, 2015 2 / 31
Chaos and indecomposability of continua 1 Abstract 2 Introduction 3 Main Theorems 4 Indecomposable continua 5 Inverse limits and shift maps 6 Topological entropy 7 Independence sets and positive density 8 IE-tuple 9 Z-pairs 10 Lemmas and Propositions . . . . . . Hisao Kato (University of Tsukuba) Chaos and indecomposability of continua May 25-29, 2015 3 / 31
Chaos and indecomposability of continua Abstract Abstract Abstract : We use recent developments in local entropy theory to prove that positive topological entropy implies the existence of chaos in dynamical systems and complicated structures in the underlying spaces. In 1994, Barge and Diamond proved that if G is a finite graph and f : G → G is any map with positive topological entropy, then the inverse limit space lim − ( X , f ) contains an indecomposable continuum. ← In 2011, Mouron proved that if X is a chainable continuum which admits a homeomorphism f with positive topological entropy, then X contains an indecomposable subcontinuum. . . . . . . Hisao Kato (University of Tsukuba) Chaos and indecomposability of continua May 25-29, 2015 4 / 31
Chaos and indecomposability of continua Abstract In this talk, we generalize the results of Barge, Diamond and Mouron. We show that if X is a G -like continuum for some finite graph G and f : X → X is a any homeomorphism with positive topological entropy, then X contains an indecomposable continuum. Moreover we obtain that if X is a G -like continuum for some finite tree G and f : X → X is a any monotone map with positive topological entropy, then X contains an indecomposable continuum. This answers some questions raised by Mouron and generalizes the theorem of Barge and Diamond. . . . . . . Hisao Kato (University of Tsukuba) Chaos and indecomposability of continua May 25-29, 2015 5 / 31
Chaos and indecomposability of continua Introduction Introduction During the last thirty years or so, many interesting connections between topological dynamics and continuum theory have been established. One of the underlying themes is that somehow complicated dynamics should imply existence of complicated continua. More precisely, if X is a continuum (i.e., compact connected metric space) and h : X → X is a homeomorphism of X such that ( X , h ) is chaotic in some sense, then X should contain a complicated subcontinuum. In the case that h happens to be simply a continuous surjection, then the inverse limit space lim − ( X , h ) should ← contain a complicated continuum. . . . . . . Hisao Kato (University of Tsukuba) Chaos and indecomposability of continua May 25-29, 2015 6 / 31
Chaos and indecomposability of continua Introduction In 1985, Barge and Martin showed that if f is a map of the the interval I = [0 , 1] which has a periodic point not a power of 2, then lim − ( I , f ) contains an indecomposable continuum. By Misiurewicz’s ← theorem, we have that a map of the interval has a periodic point not a power of 2 if and only if the map has positive topological entropy. Hence, we have that if f : I → I has positive topological entropy, then lim − ( I , f ) contains an indecomposable continuum. ← . . . . . . Hisao Kato (University of Tsukuba) Chaos and indecomposability of continua May 25-29, 2015 7 / 31
Chaos and indecomposability of continua Introduction In 1989, Ingram showed that if X is an arc-like continuum and h is a homeomorphism of X with a periodic point not a power of two, then X contains an indecomposable continuum. In 2011, Mouron proved that if h is a homeomorphism of an arc-like continuum and h has positive entropy, then X contains an indecomposable continuum. He raised the questions if the same result holds for the monotone open maps of arc-like continua or, in general, for monotone maps of arc-like continua. . . . . . . Hisao Kato (University of Tsukuba) Chaos and indecomposability of continua May 25-29, 2015 8 / 31
Chaos and indecomposability of continua Introduction Also, Barge and Diamond in 1994 showed that for piecewise monotone surjections of graphs, the conditions of having positive entropy, containing a horse shoe and the inverse limit space containing an indecomposable subcontinuum are all equivalent. However, one cannot hope to generalize their result to arbitrary maps. A classical example of Henderson shows that there is a map of the interval with no chaotic behavior, in particular with topological entropy zero, whose inverse limit space is the hereditarily indecomposable. . . . . . . Hisao Kato (University of Tsukuba) Chaos and indecomposability of continua May 25-29, 2015 9 / 31
Chaos and indecomposability of continua Introduction Hence the best one can hope for is to obtain results where positive entropy implies indecomposability in the corresponding inverse limit space. On the other hand, in 2011 Mouron also showed that there is a homeomorphism of the Cantor fan (which is hereditarily decomposable) such that the homeomorphism has positive topological entropy. . . . . . . Hisao Kato (University of Tsukuba) Chaos and indecomposability of continua May 25-29, 2015 10 / 31
Chaos and indecomposability of continua Main Theorems Main Theorems We have the following main theorems. . Theorem 1 . . . Let G be a finite graph, X be a G-like continuum and h : X → X a homeomorphism with positive entropy. Then X contains an indecomposable continuum. . . . . . . Corollary 2 . . . Let G be a finite graph, X be a G-like continuum and f : X → X any map with positive entropy. Then lim − ( X , f ) contains an ← indecomposable continuum. . . . . . . . . . . . Hisao Kato (University of Tsukuba) Chaos and indecomposability of continua May 25-29, 2015 11 / 31
Chaos and indecomposability of continua Main Theorems Moreover, we obtain . Corollary 3 . . . Let G be a finite tree, X be a G-like continuum and f : X → X a monotone map with positive entropy. Then X contains an indecomposable continuum. . . . . . For the case of uniform positive entropy, we have the following. . Theorem 4 . . . Let G be a finite graph, X be a G-like continuum and f : X → X a homeomorphism with uniform positive entropy. Then X is itself indecomposable. . . . . . . . . . . . Hisao Kato (University of Tsukuba) Chaos and indecomposability of continua May 25-29, 2015 12 / 31
Chaos and indecomposability of continua Indecomposable continua Indecomposable continua A continuum is a compacted connected metric space. We say that a continuum is nondegenerate if it has more than one point. A continuum is indecomposable if it has more than one point and is not the union of two proper subcontinua. Let G be a compact metric space. A continuous mapping g from X onto G is an ϵ -mapping if for every x ∈ X , the diameter of g − 1 ( x ) is less than ϵ . A continuum X is G -like if for every ϵ > 0 there is a ϵ -mapping from X onto G . Our focus in this article is on G -like continua where G is a finite graph. . . . . . . Hisao Kato (University of Tsukuba) Chaos and indecomposability of continua May 25-29, 2015 13 / 31
Chaos and indecomposability of continua Inverse limits and shift maps Inverse limits and shift maps If f : X → X is a map, then we use lim − ( X , f ) to denote the inverse ← limit of X with f as the bonding maps, i.e., ( x i ) ∈ X N : f ( x i +1 ) = x i { } lim − ( X , f ) = . ← Define the shift map ˜ f : lim − ( X , f ) → : lim − ( X , f ) by ← ← ˜ f ( x 1 , x 2 , x 3 , ..... ) = ( f ( x 1 ) , f ( x 2 ) , f ( x 3 ) , ... ) = ( f ( x 1 ) , x 1 , x 2 , , ..... ) . . . . . . . Hisao Kato (University of Tsukuba) Chaos and indecomposability of continua May 25-29, 2015 14 / 31
Chaos and indecomposability of continua Topological entropy Topological entropy Let X be a compact metric space and U , V be two covers of X . Then, U ∨ V = { U ∩ V : U ∈ U , V ∈ V} . The quantity N ( U ) denote minimal cardinality of subcover of U . Let f : X → X be continuous and U be an open cover of X . Then, ln [ N ( U ∨ f − 1 ( U ) ∨ . . . ∨ f − n +1 ( U ))] h top ( U , f ) = lim sup . n n →∞ The topological entropy of f , denoted by h top ( f ), is simply the supremum of h top ( f , U ) over all open covers U of X . . . . . . . Hisao Kato (University of Tsukuba) Chaos and indecomposability of continua May 25-29, 2015 15 / 31
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