A note on quasi-weakly almost periodic point

Yan, Qi; Yin, Jian Dong; Wang, Tao

doi:10.1007/s10114-015-4202-z

Cited by 8 publications

(4 citation statements)

References 6 publications

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“…Equivalent statements of recurrence. Let us recall some equivalent statements of recurrence referring to [28,65,66,22] whose proofs are fundamental and standard.…”

Section: 3mentioning

confidence: 99%

Strongly distributional chaos of irregular orbits that are not uniformly hyperbolic

Hou¹,

Tian²

2021

Preprint

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In this article we prove that for a diffeomorphism on a compact Riemannian manifold, if there is a nontrival homoclinic class that is not uniformly hyperbolic or the diffeomorphism is a C 1+α and there is a hyperbolic ergodic measure whose support is not uniformly hyperbolic, then we find a type of strongly distributional chaos which is stronger than usual distributional chaos and Li-Yorke chaos in the set of irregular orbits that are not uniformly hyperbolic. Meanwhile, we prove that various fractal sets are strongly distributional chaotic, such as irregular sets, level sets, several recurrent level sets of points with different recurrent frequency, and some intersections of these fractal sets. In the process of proof, we give an abstract general mechanism to study strongly distributional chaos provided that the system has a sequence of nondecreasing invariant compact subsets such that every subsystem has exponential specification property, or has exponential shadowing property and transitivity. The advantage of this abstract framework is that it is not only applicable in systems with specification property including transitive Anosov diffeomorphisms, mixing expanding maps, mixing subshifts of finite type and mixing sofic subshifts but also applicable in systems without specification property including β-shifts, Katok map, generic systems in the space of robustly transitive diffeomorphisms and generic volume-preserving diffeomorphisms.

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“…Equivalent statements of recurrence. Let us recall some equivalent statements of recurrence referring to [28,65,66,22] whose proofs are fundamental and standard.…”

Section: 3mentioning

confidence: 99%

Strongly distributional chaos of irregular orbits that are not uniformly hyperbolic

Hou¹,

Tian²

2021

Preprint

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“…Let us recall some equivalent statements of recurrence referring to [33,74,75,35]. For a measure µ, define the support of µ by S µ := supp(µ) = {x ∈ X| µ(U ) > 0 for any neighborhood U of x}.…”

Section: Levels Of Recurrencementioning

confidence: 99%

“…Note that AP coincides with the set of all Banach lower recurrent points. From [33,74,75,72] W, QW, BR, Rec all have full measure for any invariant measure but AP maybe not. Note that…”

Section: Note That Ifmentioning

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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“…Note that AP coincides with the set of all Banach lower recurrent points. From [65,68,71] W, QW, BR, Rec all have full measure for any invariant measure but may not for AP. Note that AP ⊆ W ⊆ QW ⊆ BR ⊆ Rec.…”

Section: Introductionmentioning

confidence: 99%

Transitively-saturated property, Banach recurrence and Lyapunov regularity

Tian

Wang

2019

Nonlinearity

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The topological entropy of various gap-sets on periodic-like recurrence and Birkhoff regularity were considered in [69] but some Banach recurrence and Lyapunov regularity are not considered. In this paper we introduce five new levels on Banach recurrence and show they all carry full topological entropy, and simultaneously combine with Lyapunov regularity to get some refined theory on mixed multifractal analysis of [8,29].In this process, we strengthen Pfister and Sullivan's result of [58] from saturated property to transitively-saturated property (and from single-saturated property to transitively-convex-saturated property).

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A note on quasi-weakly almost periodic point

Cited by 8 publications

References 6 publications

Strongly distributional chaos of irregular orbits that are not uniformly hyperbolic

Strongly distributional chaos of irregular orbits that are not uniformly hyperbolic

Distributional chaos in multifractal analysis, recurrence and transitivity

Transitively-saturated property, Banach recurrence and Lyapunov regularity

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