Transitively-saturated property, Banach recurrence and Lyapunov regularity

Yu, Huan; Tian, Xueting; Wang, Xiao Yi

doi:10.1088/1361-6544/ab090c

Cited by 19 publications

(41 citation statements)

References 72 publications

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“…Many people pay attention to these recurrent points and measure them [28,66]. In [61,30] the authors considered various recurrence and showed many different recurrent levels carry strong dynamical complexity from the perspective of topological entropy. Now we state our second result as follows.…”

Section: Rec Lowmentioning

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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Section: Rec Lowmentioning

confidence: 99%

Strongly distributional chaos of irregular orbits that are not uniformly hyperbolic

Hou¹,

Tian²

2021

Preprint

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“…The property on entropy estimate was firstly established by Pfister and Sullivan in [36], provided that the system has g-product property (which is weaker than specification property) and uniform separation property (which is weaker than expansiveness). Recently, in [21,Theorem 1.4] the authors proved that if further there is an invariant measure with full support, then…”

Section: Saturated Setsmentioning

confidence: 99%

Entropy of irregular points that are not uniformly hyperbolic

Hou¹,

Tian²

2021

Preprint

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In this article we prove that for a C 1+α diffeomorphism on a compact Riemannian manifold, if there is a hyperbolic ergodic measure whose support is not uniformly hyperbolic, then the topological entropy of the set of irregular points that are not uniformly hyperbolic is larger than or equal to the metric entropy of the hyperbolic ergodic measure. In the process of proof, we give an abstract general mechanism to study topological entropy of irregular points provided that the system has a sequence of nondecreasing invariant compact subsets such that every subsystem has shadowing property and is transitive.Theorem 1.1. [16, Corollary B] Let M be a compact Riemannian manifold of dimension at least 2 and f be a C 1+α diffeomorphism. Then one has h top (f, IR(f )) ≥ sup{h top (f, Λ) : Λ is a transitive locally maximal hyperbolic set} = sup{h µ (f ) : µ is a hyperbolic ergodic measure}.

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“…Further, many people pay attention to more refinements of recurrent points according to the 'recurrent frequency' such as almost periodic points(which naturally exists in any dynamical system since it is equivalent that it belongs to a minimal set), weakly almost periodic points and quasi-weakly almost periodic points and measure them [19,38]. In [33,21,11] the authors considered various Banach recurrence and showed many different recurrent levels carry strong dynamical complexity from the perspective of topological entropy and distributional chaos. In present paper we aim to study the existence and complexity of points that are recurrent but not Banach recurrent.…”

Section: Introductionmentioning

confidence: 99%

“…All of the gap-sets are invariant sets with totally zero measure. The complexity of (W \ R) and (QW \ W ) were described in [33,11], the complexity of (BR \ QW ) was described in [21,11] and the complexity of (Ω \ Rec) ) was described in [16] for a large class of systems including all transitive Anosov diffeomorphisms. However, the complexity characterization of (Rec \ BR) is still unknown.…”

Section: Introductionmentioning

confidence: 99%

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Existence and Distributional Chaos of Points that are Recurrent but Not Banach Recurrent

Jiang¹,

Tian²

2019

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Recurrence is a classical and basic concept in the study of dynamical systems. According to the 'recurrent frequency' (i.e., the probability of finding the orbit of an initial point entering in a neighborhood), many different levels of recurrent points are found such as periodic points, almost periodic points, weakly almost periodic points, quasi-weakly almost periodic points and Banach recurrent points [19,38,22]. In this paper we are mainly to search the existence on six new levels of recurrent points between Banach recurrence and general recurrence. Moreover, from the perspective of distributional chaos, we will show that five levels of the new recurrent points contain uncountable DC1-scrambled subsets.

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Transitively-saturated property, Banach recurrence and Lyapunov regularity

Cited by 19 publications

References 72 publications

Strongly distributional chaos of irregular orbits that are not uniformly hyperbolic

Strongly distributional chaos of irregular orbits that are not uniformly hyperbolic

Entropy of irregular points that are not uniformly hyperbolic

Existence and Distributional Chaos of Points that are Recurrent but Not Banach Recurrent

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