Classifying dynamical systems by their recurrence properties

Glasner, Eli

doi:10.12775/tmna.2004.018

Cited by 38 publications

(46 citation statements)

References 25 publications

(32 reference statements)

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“…Note that we use N to denote the set of positive integers. It turns out that many recurrence properties of TDS can be described using the return time sets N (U, V ) (see [1], [8], [14], [12], [13] and [10]). For example, for a TDS (X, T ) it is known that T is (topologically) strongly mixing iff N (U, V ) is cofinite, T is (topologically) weakly mixing iff N (U, V ) is thick [8], and T is (topologically) mildly mixing iff N (U, V ) is an (IP -IP ) * set [14], [12] for each pair of non-empty open subsets U and V .…”

Section: Rui Kuang and Xiangdong Ye (Hefei)mentioning

confidence: 99%

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Mixing via families for measure preserving transformations

Kuang

2008

Colloq. Math.

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Abstract. In topological dynamics a theory of recurrence properties via (Furstenberg) families was established in the recent years. In the current paper we aim to establish a corresponding theory of ergodicity via families in measurable dynamical systems (MDS). For a family F (of subsets of Z + ) and a MDS (X, B, µ, T ), several notions of ergodicity related to F are introduced, and characterized via the weak topology in the induced Hilbert space L 2 (µ). T is F -convergence ergodic of order k if for any A 0 , . . . , A k of positive measure, 0 = e 0 < · · · < e k and ε > 0, {n ∈ Z + : |µ(It is proved that the following statements are equivalent: (1) T is ∆ * -convergence ergodic of order 1; (2) T is strongly mixing; (3) T is ∆ * -convergence ergodic of order 2. Here ∆ * is the dual family of the family of difference sets.1. Introduction. By a topological dynamical system (TDS) (X, T ) we mean a compact metric space X together with a surjective continuous map T from X to itself. For a TDS (X, T ) and non-empty open subsets U and V of X let N (U, V ) = {n ∈ Z + : U ∩ T −n V = ∅}, where Z + is the set of nonnegative integers. Note that we use N to denote the set of positive integers. It turns out that many recurrence properties of TDS can be described using the return time sets N (U, V ) (see [1], [8], [14], [12], [13] and [10]). For example, for a TDS (X, T ) it is known that T is (topologically) strongly mixing iff N (U, V ) is cofinite, T is (topologically) weakly mixing iff N (U, V ) is thick [8], and T is (topologically) mildly mixing iff N (U, V ) is an (IP -IP ) * set [14], [12] for each pair of non-empty open subsets U and V . Recently, Huang and Ye [14] showed that a minimal system (X, T ) is weakly mixing iff the lower Banach density of N (U, V ) is 1, and (X, T ) is mildly mixing iff N (U, V ) is an IP * -set for each pair of non-empty open sets U and V .By a measurable dynamical system (MDS) we mean (X, B, µ, T ), where (X, B, µ) is a Lebesgue space and T : X → X is invertible and measure pre-

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Section: Rui Kuang and Xiangdong Ye (Hefei)mentioning

confidence: 99%

“…It was Furstenberg [8], [9] who first used subsets of Z + to describe dynamical properties in a systematic way. For the recent results, see [1], [12], [10], [13] and [14].…”

Section: Rui Kuang and Xiangdong Ye (Hefei)mentioning

confidence: 99%

Mixing via families for measure preserving transformations

Kuang

2008

Colloq. Math.

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“…Furstenberg started a systematic study of transitive dynamical systems in his paper on disjointness in topological dynamics and ergodic theory [11], and the theory was further developed in [13] and [12]. The main motivation for this paper comes from [39], [2], [24], [5], [14], [22], [38], [37] and recent papers [29], [34] and [31], which discusses a dynamical property called transitive compactness examined firstly for weakly mixing systems in [5]: transitive compactness is quite related to but different from the property of transitivity, it will be equivalent to weak mixing under some weak conditions, and it presents some kind of sensitivity of the system.…”

Section: Introductionmentioning

confidence: 99%

Dynamical compactness and sensitivity

Huang¹,

Khilko²,

Kolyada³

et al. 2016

Journal of Differential Equations

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Abstract. To link the Auslander point dynamics property with topological transitivity, in this paper we introduce dynamically compact systems as a new concept of a chaotic dynamical system (X, T ) given by a compact metric space X and a continuous surjective self-map T : X → X. Observe that each weakly mixing system is transitive compact, and we show that any transitive compact M-system is weakly mixing. Then we discuss the relationships among it and other several stronger forms of sensitivity. We prove that any transitive compact system is Li-Yorke sensitive and furthermore multi-sensitive if it is not proximal, and that any multi-sensitive system has positive topological sequence entropy. Moreover, we show that multi-sensitivity is equivalent to both thick sensitivity and thickly syndetic sensitivity for M-systems. We also give a quantitative analysis for multi-sensitivity of a dynamical system.

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“…Every E-system is topologically ergodic (see [60]), so (Y, T ) is topologically ergodic, and by corollary 4.2.2, T is topologically ergodic. 2…”

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confidence: 99%

“…As (Y, T ) is sensitive to initial conditions, it is not equicontinuous and by lemma 6.1 of [60], for all open cover U of Y , C(U, n) is unbounded. So the complexity function of X is unbounded.…”

mentioning

confidence: 99%

Strong mixing measures and invariant sets in linear dynamics

Arcila¹

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Classifying dynamical systems by their recurrence properties

Cited by 38 publications

References 25 publications

Mixing via families for measure preserving transformations

Mixing via families for measure preserving transformations

Dynamical compactness and sensitivity

Strong mixing measures and invariant sets in linear dynamics

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