2017
DOI: 10.1016/j.aim.2016.09.012
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Chaos and indecomposability

Abstract: 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 indecompos… Show more

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Cited by 7 publications
(12 citation statements)
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References 22 publications
(33 reference statements)
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“…Also, we prove that the Cantor set Z 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.…”
Section: Introductionmentioning
confidence: 91%
See 2 more Smart Citations
“…Also, we prove that the Cantor set Z 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.…”
Section: Introductionmentioning
confidence: 91%
“…In [24], 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. In [6], as an extension of the 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 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.…”
Section: Introductionmentioning
confidence: 95%
See 1 more Smart Citation
“…Theorem 2.1. ( [5]) Let G be any graph. If a homeomorphism f on a G-like continuum X has positive topological entropy, i.e., h(f ) > 0, then X contains an indecomposable subcontinuum.…”
Section: Monotone Maps Of Continua and Topological Entropymentioning
confidence: 99%
“…During the last thirty years or so, many interesting connections between topological dynamics and continuum theory have been studied by many authors. In particular, we are interested in the fact that several complicated dynamics should imply existence of complicated continua and in many cases, such continua are indecomposable continua which are central subjects in continuum theory (see the references [1,2,[5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][23][24][25][26]28,29]).…”
Section: Introductionmentioning
confidence: 99%