Sarkovskiĭ’s theorem for hereditarily decomposable chainable continua

Minc, Piotr; Transue, William

doi:10.1090/s0002-9947-1989-0965302-9

Cited by 11 publications

(4 citation statements)

References 9 publications

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“…This arises from the fact that Sharkovskii's theorem does not hold in graphs. However, by Sharkovskii's theorem on the interval and by the Minc-Transue's generalization of Sharkovskii's theorem on arc-like continua [19], we have that if f has a periodic point not a power of 2, then for some n ∈ N, f n has a periodic point of every period.) We conjecture that the above equivalent conditions diagram holds for all X hereditarily decomposable and G-like for some graph G. If this conjecture holds, then our main results in this paper can be deduced from the existence of a horseshoe.…”

Section: Final Remarks and Problemsmentioning

confidence: 99%

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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Section: Final Remarks and Problemsmentioning

confidence: 99%

Chaos and indecomposability

Darji

Kato

2017

Advances in Mathematics

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“…The topological entropy of f , denoted by h( f ), is the supremum of h( f , U ) for all open covers U of X . The reader may refer to [3,4,5,6,8,18,[22][23][24][25]27,28] for important facts concerning topological entropy. Positive topological entropy of map is one of generally accepted definitions of chaos.…”

Section: Definitions and Notationsmentioning

confidence: 99%

“…During the last thirty years or so, many interesting connections between dynamical systems and continuum theory have been studied by many authors (see [1,2,6,7,[9][10][11][12][13][14][15]17,19,[22][23][24][25]27,28]). We are interested in the following fact that chaotic topological dynamics should imply existence of complicated topological structures of underlying spaces.…”

Section: Introductionmentioning

confidence: 99%

Topological entropy and IE-tuples of indecomposable continua

Kato

2019

Fund. Math.

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In [3], by use of ergodic theory method, Blanchard, Glasner, Kolyada and Maass proved that 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 > 0 and hence the dynamics (X, f ) is Li-Yorke chaotic. In [18], Kerr and Li developed local entropy theory and gave a new proof of this theorem. Moreover, by developing some deep combinatorial tools, they proved that X contains a Cantor set Z which yields more chaotic behaviors (see [18, Theorem 3.18]). In the paper [6], we proved that if G is any graph and 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 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, we define a new notion of "freely tracing property by free chains" on G-like continua and we prove that a positive topological entropy homeomorphism on a G-like continuum admits a Cantor set Z such that every tuple of finite points in Z is an IE-tuple of f and Z has the freely tracing property by free chains. Also, by use of this notion, we prove the following theorem: If G is any graph and a homeomorphism f on a G-like continuum X has positive topological entropy, then there is a Cantor set Z which is related to both the chaotic behaviors of Kerr and Li [18] in dynamical systems and composants of indecomposable continua in topology. Our main result is Theorem 3.3 whose proof is also a new proof of [6]. Also, we study dynamical properties of continuum-wise expansive homeomorphisms. In this case, we obtain more precise results concerning continuum-wise stable sets of chaotic continua and IE-tuples.

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“…We give two arguments for this result. One proof uses the recent result of Piotr Mine and W. R. R. Transue [4] that Sarkovskii's Theorem holds for hereditarily decomposable chainable continua. The other proof uses an extension (see Theorem 5 of section 3) of a result of the author [2,Theorem 3].…”

Section: Introductionmentioning

confidence: 99%