Entropy dimensions and a class of constructive examples

Ferenczi, Sébastien; Park, Kyewon Koh

doi:10.3934/dcds.2007.17.133

Cited by 48 publications

(37 citation statements)

References 3 publications

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“…Like topological entropy, the calculation of entropy dimension is also a tough work. Readers can refer to[6, Proposition 5] for the system with the existence of entropy dimension and [6, Proposition 4] for the system whose upper and lower entropy dimension are not the same. More examples whose upper entropy dimension have been computed are available in [3].…”

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confidence: 99%

Zero sequence entropy and entropy dimension

Qiao¹,

Zhou²

2017

Discrete &Amp; Continuous Dynamical Systems - A

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Let (X, T) be a topological dynamical system and M (X) the set of all Borel probability measures on X endowed with the weak *-topology. In this paper, it is shown that for a given sequence S, a homeomorphism T of X has zero topological sequence entropy if and only if so does the induced homeomorphism T of M (X). This extends the result of Glasner and Weiss [9, Theorem A] for topological entropy and also the result of Kerr and Li [15, Theorem 5.10] for null systems. Moreover, it turns out that the upper entropy dimension of (X, T) is equal to that of (M (X), T). We also obtain the version of ergodic measure-preserving systems related to the sequence entropy and the upper entropy dimension.

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confidence: 99%

Zero sequence entropy and entropy dimension

Qiao¹,

Zhou²

2017

Discrete &Amp; Continuous Dynamical Systems - A

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“…Although systems with positive entropy are much more complicated than those with zero entropy, zero entropy systems own various levels of complexity, and recently have been discussed in [3,4,6,7,9,13,18,22]. Those authors adopted various methods to classify zero entropy dynamical systems.…”

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confidence: 99%

Relative entropy dimensions for amenable group actions

Xiao¹,

Yin²

2022

Preprint

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We study the topological complexities of relative entropy zero extensions acted by countableinfinite amenable groups. Firstly, for a given Følner sequence {F n } +∞ n=0 , we define respectively the relative entropy dimensions and the dimensions of the relative entropy generating sets to characterize the sub-exponential growth of the relative topological complexity. Meanwhile, we investigate the relations among them. Secondly, we introduce the notion of a relative dimension set. Moreover, using it, we discuss the disjointness between the relative entropy zero extensions which generalizes the results of Dou, Huang and Park[Trans. Amer. Math. Soc. 363(2) (2011), 659-680].

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“…a factor map). Since zero entropy systems make up a dense G δ subset of all homeomorphisms, there are several kinds of works about conjugate invariants of zero entropy systems, for example, sequence entropy [19,15], maximal pattern entropy [13], entropy dimension [2,6,3] and Slow entropy [17]. But there is no relative conjugate invariants for zero entropy factor maps.…”

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confidence: 99%

“…Entropy dimension for a topological dynamical system first introduced by M. De Carvalho in [2] measures the superpolynomial, but subexponential growth rate of the number of open sets that cover the space out of the sequence of iterated open covers. S. Ferenczi and K. K. Park have introduced the entropy dimension in [6] to measure the complexity of entropy zero measurable dynamics. It measures the growth rate of H( n−1 i=0 T −i P ).…”

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confidence: 99%

Relative entropy dimension of topological dynamical systems

Zhou¹

2019

Discrete &Amp; Continuous Dynamical Systems - A

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We introduce the notion of relative topological entropy dimension to classify the different intermediate levels of relative complexity for factor maps. By considering the dimension or "density" of special class of sequences along which the entropy is encountered, we provide equivalent definitions of relative entropy dimension. As applications, we investigate the corresponding localization theory and obtain a disjointness theorem involving relative entropy dimension.2010 Mathematics Subject Classification. Primary: 37B99, 54H20. Key words and phrases. Relative entropy dimension, relative dimension set. (*) This research is supported by NSFC(11801193). 6631 6632 XIAOMIN ZHOUfind an infinite subset W ⊂ Z + in the case of positive entropy such that along the sequence W , the symbolic names are "independent" and there exists c > 0 with lim inf n→∞ |W ∩[1,n]| n

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Entropy dimensions and a class of constructive examples

Cited by 48 publications

References 3 publications

Zero sequence entropy and entropy dimension

Zero sequence entropy and entropy dimension

Relative entropy dimensions for amenable group actions

Relative entropy dimension of topological dynamical systems

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