Shift spaces and distributional chaos

Oprocha, Piotr; Wilczyński, Paweł

doi:10.1016/j.chaos.2005.09.069

Cited by 34 publications

(22 citation statements)

References 8 publications

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“…A very important generalization of Li-Yorke chaos is that proposed by Schweizer and Smítal in [10], mainly because it is equivalent to positive topological entropy and some other concepts of chaos when restricted to the compact interval case [10] or hyperbolic symbolic spaces [8]. It is also remarkable that this equivalence does not transfer to higher dimensions, e.g., positive topological entropy does not imply distributional chaos (DC1) in the case of triangular maps of the unit square [12] (the same happens when the dimension is zero [9]).…”

Section: Introductionmentioning

confidence: 99%

Maximal distributional chaos of weighted shift operators on Köthe sequence spaces

2014

Czech Math J

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Section: Introductionmentioning

confidence: 99%

Maximal distributional chaos of weighted shift operators on Köthe sequence spaces

2014

Czech Math J

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“…The example similar to that presented below was first discovered by R. Piku la in [47]. As a general method for such examples, a construction was independently obtained in [46]. Example 9.…”

Section: Strengthened Definitionsmentioning

confidence: 99%

Distributional chaos revisited

Oprocha

2009

Trans. Amer. Math. Soc.

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Abstract. In their famous paper, Schweizer and Smítal introduced the definition of a distributionally chaotic pair and proved that the existence of such a pair implies positive topological entropy for continuous mappings of a compact interval. Further, their approach was extended to the general compact metric space case.In this article we provide an example which shows that the definition of distributional chaos (and as a result Li-Yorke chaos) may be fulfilled by a dynamical system with (intuitively) regular dynamics embedded in R 3 . Next, we state strengthened versions of distributional chaos which, as we show, are present in systems commonly considered to have complex dynamics.We also prove that any interval map with positive topological entropy contains two invariant subsets X, Y ⊂ I such that f | X has positive topological entropy and f | Y displays distributional chaos of type 1, but not conversely.

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“…In 2007 a sufficient condition for a shift to be DC was given in [8]. The asymptotic orbits of substitution systems were investigated in [9].…”

Section: Introductionmentioning

confidence: 99%