On the genericity of chaos

Kościelniak, Piotr

doi:10.1016/j.topol.2007.01.014

Cited by 11 publications

(11 citation statements)

References 10 publications

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“…Let C(M )be the set of continuous maps on a compact manifold M and H(M ) the set of homeomorphisms on M . Recall that C 0 generic f ∈ H(M ) (or f ∈ C(M )) has the shadowing property and infinite topological entropy (see [41] and [39,40], respectively). Thus Theorem 6.4 applies in C 0 generic dynamical systems.…”

Section: Minimal Pointsmentioning

confidence: 99%

Distributional chaos in multifractal analysis, recurrence and transitivity

Tian

2019

Ergod. Th. Dynam. Sys.

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There are lots of results to study dynamical complexity on irregular sets and level sets of ergodic average from the perspective of density in base space, Hausdorff dimension, Lebesgue positive measure, positive or full topological entropy (and topological pressure) etc.. However, it is unknown from the viewpoint of chaos. There are lots of results on the relationship of positive topological entropy and various chaos but it is known that positive topological entropy does not imply a strong version of chaos called DC1 so that it is non-trivial to study DC1 on irregular sets and level sets. In this paper we will show that for dynamical system with specification property, there exist uncountable DC1-scrambled subsets in irregular sets and level sets. On the other hand, we also prove that several recurrent levels of points with different recurrent frequency all have uncountable DC1-scrambled subsets. The main technique established to prove above results is that there exists uncountable DC1-scrambled subset in saturated sets.

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Section: Minimal Pointsmentioning

confidence: 99%

Distributional chaos in multifractal analysis, recurrence and transitivity

Tian

2019

Ergod. Th. Dynam. Sys.

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“…Devaney chaos implies Li-Yorke chaos [19,21,23] and positive topological entropy implies Li-Yorke chaos [5]. If X is the unit interval we have that positive topological entropy is equivalent to Devaney chaos on a subsystem.…”

Section: Li-yorke Chaosmentioning

confidence: 99%

Chaotic behaviour of the map x ↦ ω(x, f)

D’Aniello

Steele

2014

Open Mathematics

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Let K(2 N ) be the class of compact subsets of the Cantor space 2 N , furnished with the Hausdorff metric. Let ∈ C (2 N ). We study the map ω : 2 N → K(2 N ) defined as ω ( ) = ω( ), the ω-limit set of under . Unlike the case of -dimensional manifolds, ≥ 1, we show that ω is continuous for the generic self-map of the Cantor space, even though the set of functions for which ω is everywhere discontinuous on a subsystem is dense in C (2 N ). The relationships between the continuity of ω and some forms of chaos are investigated. MSC:37B99, 54H20, 37B10

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“…(The proof presented there did not admit 2-dimensional manifolds with boundary. In [13,14] Kościelniak gave an independent argumentation that could be applied for this case, too.) In this paper we characterize such a chaotic behavior for systems with the shadowing property, in terms of the existence of unstable chain recurrent points.…”

Section: Introductionmentioning

confidence: 99%

Chaos and the shadowing property

Kościelniak¹,

Mazur²

2007

Topology and its Applications

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We present, as a simpler alternative for the results of [P. Kościelniak, On genericity of shadowing and periodic shadowing property, J. Math. Anal. Appl. 310 (2005) 188-196; P. Kościelniak, M. Mazur, On C 0 genericity of various shadowing properties, Discrete Contin. Dynam. Syst. 12 (2005) 523-530], an elementary proof of C 0 genericity of the periodic shadowing property. We also characterize chaotic behavior (in the sense of being semiconjugated to a shift map) of shadowing systems.

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On the genericity of chaos

Cited by 11 publications

References 10 publications

Distributional chaos in multifractal analysis, recurrence and transitivity

Distributional chaos in multifractal analysis, recurrence and transitivity

Chaotic behaviour of the map x ↦ ω(x, f)

Chaos and the shadowing property

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