Positive entropy homeomorphisms of chainable continua and indecomposable subcontinua

Mouron, Christopher

doi:10.1090/s0002-9939-2010-10783-9

Cited by 17 publications

(18 citation statements)

References 5 publications

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“…This theorem generalizes that of Mouron [20]. It also generalizes the above mentioned theorem of Barge and Diamond [2].…”

Section: Introductionsupporting

confidence: 80%

“…It also generalizes the above mentioned theorem of Barge and Diamond [2]. Moreover, using some easily proved lemmas and our main result, we can answer some questions left open in [20].…”

Section: Introductionsupporting

confidence: 70%

“…Thus, 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, there are restrictions on the domain space also as observed by Mouron [20]. There is a homeomorphism of the Cantor fan which has positive topological entropy.…”

Section: Introductionmentioning

confidence: 86%

“…Ye in the same paper and also in [17] proved that for certain special types of homeomorphisms of the arc-like continua, positive entropy implies that the domain contains an indecomposable continuum. In 2011, Mouron [20] settled the more general problem by proving that if h is a homeomorphism of an arc-like continuum and h has positive entropy, then X contains an indecomposable continua. 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.…”

Section: Introductionmentioning

confidence: 99%

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Chaos and indecomposability

Darji

Kato

2017

Advances in Mathematics

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We use recent developments in local entropy theory to prove that chaos in dynamical systems implies the existence of complicated structure in the underlying space. Earlier Mouron proved that if X is an arc-like continuum which admits a homeomorphism f with positive topological entropy, then X contains an indecomposable subcontinuum. 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 ← − (G, f ) contains an indecomposable continuum. In this paper we show that if X is a G-like continuum for some finite graph G and f : X → X is any map with positive topological entropy, then lim ← − (X, f ) contains an indecomposable continuum. As a corollary, we obtain that in the case that f is a homeomorphism, X contains an indecomposable continuum. Moreover, if f has uniformly positive upper entropy, then X is an indecomposable continuum. Our results answer some questions raised by Mouron and generalize the above mentioned work of Mouron and also that of Barge and Diamond. We also introduce a new concept called zigzag pair which attempts to capture the complexity of a dynamical systems from the continuum theoretic perspective and facilitates the proof of the main result.

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“…This theorem generalizes that of Mouron [20]. It also generalizes the above mentioned theorem of Barge and Diamond [2].…”

Section: Introductionsupporting

confidence: 80%

“…It also generalizes the above mentioned theorem of Barge and Diamond [2]. Moreover, using some easily proved lemmas and our main result, we can answer some questions left open in [20].…”

Section: Introductionsupporting

confidence: 70%

Section: Introductionmentioning

confidence: 86%

Section: Introductionmentioning

confidence: 99%

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Chaos and indecomposability

Darji

Kato

2017

Advances in Mathematics

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“…The special case of Barge's conjecture was proved by Ye in 1995 [30] for homeomorphisms induced by square commuting diagrams on inverse limits of arcs. Barge's conjecture has been recently proved by Mouron [26], and consequently, hereditarily decomposable arc-like continua admit only zero entropy homeomorphisms. Barge's conjecture has been recently proved by Mouron [26], and consequently, hereditarily decomposable arc-like continua admit only zero entropy homeomorphisms.…”

Section: Theoremmentioning

confidence: 95%

On indecomposability in chaotic attractors

Boroński¹,

Oprocha

2015

Proc. Amer. Math. Soc.

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We exhibit a Li-Yorke chaotic interval map F such that the inverse limit X F = lim ← − {F, [0, 1]} does not contain an indecomposable subcontinuum. Our result contrasts with the known property of interval maps: if ϕ has positive entropy then X ϕ contains an indecomposable subcontinuum. Each subcontinuum of X F is homeomorphic to one of the following: an arc, or X F , or a topological ray limiting to X F . Through our research, we found that it follows that X F is a chaotic attractor of a planar homeomorphism. In addition, F can be modified to give a cofrontier that is a chaotic attractor of a planar homeomorphism but contains no indecomposable subcontinuum. Finally, F can be modified, without removing or introducing new periods, to give a chaotic zero entropy interval map, such that the corresponding inverse limit contains the pseudoarc.

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Chaotic Continua in Chaotic Dynamical Systems

Kato

2021

Springer Proceedings in Mathematics &Amp; Statistics

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Positive entropy homeomorphisms of chainable continua and indecomposable subcontinua

Abstract: Abstract. It is shown that if X is a chainable continuum and h : X −→ X is a homeomorphism such that the topological entropy of h is greater than 0, then X must contain an indecomposable subcontinuum. This answers a question of Barge.

Cited by 17 publications

References 5 publications

Chaos and indecomposability

Chaos and indecomposability

On indecomposability in chaotic attractors

Chaotic Continua in Chaotic Dynamical Systems

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