Shadowing is generic---a continuous map case

Kościelniak, Piotr; Mazur, Marcin; Oprocha, Piotr; Pilarczyk, Paweł

doi:10.3934/dcds.2014.34.3591

Cited by 26 publications

(29 citation statements)

References 37 publications

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“…For any m, n ∈ N, V m can be covered by |I By our assumptions, when n → ∞ then also |I(n)| → ∞. Thus we see that for any t > 0, C(Λ; t, σ, f ) := lim Step 4: On one hand it is known (see [23] and [21,22], respectively) that shadowing is a generic property in C(M ) and in H(M ). On the other hand, [22,45] show that for generic maps in C(M ) (resp.…”

Section: Proof For Theorem 17mentioning

confidence: 68%

See 1 more Smart Citation

On the irregular points for systems with the shadowing property

Dong¹,

Oprocha

Tian³

2017

Ergod. Th. Dynam. Sys.

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We prove that when f is a continuous self-map acting on a compact metric space (X, d) which satisfies the shadowing property, then the set of irregular points (i.e. points with divergent Birkhoff averages) has full entropy.Using this fact we prove that in the class of C 0 -generic maps on manifolds, we can only observe (in the sense of Lebesgue measure) points with convergent Birkhoff averages. In particular, the time average of atomic measures along orbit of such points converges to some SRB-like measure in the weak * topology. Moreover, such points carry zero entropy. In contrast, irregular points are nonobservable but carry infinite entropy.

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Section: Proof For Theorem 17mentioning

confidence: 68%

“…A property P is called generic in the space X if there is a residual subset R of X such that any f ∈ R satisfies P. Recent results show that in the class of manifolds with a decomposition shadowing property is generic (e.g. see [37,23]).…”

Section: Introductionmentioning

confidence: 99%

On the irregular points for systems with the shadowing property

Dong¹,

Oprocha

Tian³

2017

Ergod. Th. Dynam. Sys.

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“…The analogous result was established by Mizera for continuous maps on [0, 1] and the unit circle, [9]. Recently this type of result was also established for compact manifolds by Mazur and Oprocha [6], and also for surjections on manifolds that admit a decomposition by Kościelniak, Mazur, Oprocha, and Pilarczyk [8]. Using different techniques, Bernardes and Darji [2] established that shadowing is generic for homeomorphisms of the Cantor space.…”

mentioning

confidence: 61%

Shadowing is generic on dendrites

Brian¹,

Meddaugh²,

Raines³

2019

Discrete &Amp; Continuous Dynamical Systems - S

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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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Shadowing is generic---a continuous map case

Cited by 26 publications

References 37 publications

On the irregular points for systems with the shadowing property

On the irregular points for systems with the shadowing property

Shadowing is generic on dendrites

Distributional chaos in multifractal analysis, recurrence and transitivity

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