Devaney chaos plus shadowing implies distributional chaos

Li, Jian; Li, Jie; Tu, Siming

doi:10.1063/1.4962131

Cited by 16 publications

(18 citation statements)

References 37 publications

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“…Now we show that if the transitive system with shadowing property has positive topological entropy then, it is mean sensitive. This result is inspired by [22].…”

Section: Conditions That Imply Mean Sensitivitymentioning

confidence: 73%

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When is a dynamical system mean sensitive?

García-Ramos

Zhang³

2017

Ergod. Th. Dynam. Sys.

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This article is devoted to study which conditions imply that a topological dynamical system is mean sensitive and which do not. Among other things we show that every uniquely ergodic, mixing system with positive entropy is mean sensitive. On the other hand we provide an example of a transitive system which is cofinitely sensitive or Devaney chaotic with positive entropy but fails to be mean sensitive.As applications of our theory and examples, we negatively answer an open question regarding equicontinuity/sensitivity dichotomies raised by Tu, we introduce and present results of locally mean equicontinuous systems and we show that mean sensitivity of the induced hyperspace does not imply that of the phase space.2010 Mathematics Subject Classification. 54H20, 37B10, 54B20, 37B05.

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“…Now we show that if the transitive system with shadowing property has positive topological entropy then, it is mean sensitive. This result is inspired by [22].…”

Section: Conditions That Imply Mean Sensitivitymentioning

confidence: 73%

“…Proof. Following from the proof of Lemma 3.3 in [22] we know that there exists a sensitive and distal pair (v 1 , v 2 ) in X 2 . By the definition of distal pair, there are ε > 0 and m ∈ N such that…”

Section: Conditions That Imply Mean Sensitivitymentioning

confidence: 99%

When is a dynamical system mean sensitive?

García-Ramos

Zhang³

2017

Ergod. Th. Dynam. Sys.

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“…Proof. We apply an argument in [7]. As in the proof of Lemma 1.1, for δ > 0, express DC1 δ (X, f ) as U δ ∩ V where…”

Section: It Follows That For Anymentioning

confidence: 99%

“…Shadowing is a feature of topologically hyperbolic dynamical systems, and its implications for chaos is a subject of ongoing research [2,5,6,7,10] (see [1,12] for a general background). One of the definitions of chaos is the generic chaos proposed by Lasota (see [13]).…”

Section: Introductionmentioning

confidence: 99%

Generic and dense distributional chaos with shadowing

Kawaguchi¹

2021

Preprint

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For continuous self-maps of compact metric spaces, we consider the notions of generic and dense chaos introduced by Lasota and Snoha and their variations for the distributional chaos, under the assumption of shadowing. We give some equivalent properties to generic uniform (distributional) chaos in terms of chains and prove their equivalence to generic (distributional) chaos under the shadowing. Several examples illustrating the main results are also given.

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“…Shadowing is a natural candidate for such an assumption. Recently in [8], Li et al proved that for any f ∈ C(X) with the shadowing property, f exhibits DC1 if one of the following properties holds: (1) f is non-periodic transitive and has a periodic point, or (2) f is non-trivial weakly mixing. Here, note that we have h top (f ) > 0 in both cases.…”

Section: Introductionmentioning

confidence: 99%