Scaled Entropy for Dynamical Systems

Yun, Zhao; Pesin, Yakov

doi:10.1007/s10955-014-1133-5

Cited by 13 publications

(7 citation statements)

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“…In this section, we point out that Theorem 2.6 in [4] is wrong. This theorem claimed that scaled topological entropy and lower and upper scaled topological entropies of a set Z coincide provided Z is invariant and compact.…”

Section: Scaled Topological Entropy: An Erratummentioning

confidence: 96%

“…This theorem claimed that scaled topological entropy and lower and upper scaled topological entropies of a set Z coincide provided Z is invariant and compact. In fact, Example 4.5 in [4] shows that there exists a non-degenerate dynamically coherent Bott system (L, φ H , f ) such that…”

Section: Scaled Topological Entropy: An Erratummentioning

confidence: 99%

“…The following theorem gives the relations between various versions of scaled metric entropies (see Theorem 3.3 in [4]). …”

Section: Scaled Metric Entropymentioning

confidence: 99%

“…In this section, we analyze the Example 4.7 in [4] to illustrate that: (1) the inequality in Theorem 3.1 can be strict and (2) the scaled metric entropy may not be invariant under an isomorphism of measurable spaces. Let T be a rotation by an irrational number θ on [0, 1).…”

Section: An Examplementioning

confidence: 99%

“…Let T be a rotation by an irrational number θ on [0, 1). Since T is an isometry, by Example 4.6 in [4] we know that the system (X, T ) has zero upper scaled topological entropy with respect to any scaled sequence. For any scaled sequence a = {a(n)} and T -invariant ergodic measure μ it follows from Theorem 4.1 that…”

Section: An Examplementioning

confidence: 99%

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Erratum to: Scaled Entropy for Dynamical Systems

Yun

Pesin

2016

J Stat Phys

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In [4] using the Carathéodory approach as described in [3], we introduced the notions of scaled topological and scaled metric entropies and we described some of their basic properties. Unfortunately, one of the results we stated is not true. In addition, we did not mention an important work in the area by Katok and Thouvenot [1] in which they introduced the notion of slow metric entropy based on an approach that utilizes the Hamming metrics. While our scaled metric entropy may not coincide with the slow metric entropy and they have different properties, the relations between them are instructive and help better understand the rescaling phenomena in dynamics. This note is intended to cover the gap and to correct a mistake in our paper. Scaled Topological EntropyWe begin by briefly recalling the definitions of scaled topological and scaled metric entropies. Consider a continuous transformation T : X → X of a compact metric space X equipped with metric d. A sequence of positive numbers a = {a(n)} n≥1 is a scaled sequence if it is monotonically increasing to infinity.The online version of the original article can be found under

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Section: Scaled Topological Entropy: An Erratummentioning

confidence: 96%

Section: Scaled Topological Entropy: An Erratummentioning

confidence: 99%

“…The following theorem gives the relations between various versions of scaled metric entropies (see Theorem 3.3 in [4]). …”

Section: Scaled Metric Entropymentioning

confidence: 99%

Section: An Examplementioning

confidence: 99%

Section: An Examplementioning

confidence: 99%