Some results on entropy dimension for non-autonomous systems
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Some results on entropy dimension for non-autonomous systems YANG Yan-juan
WANG Lin∗
WANG Wei
Abstract. In this paper, the preimage branch t-entropy and entropy dimension for nonautonomous systems are studied and some systems with preimage branch t-entropy zero are introduced. Moreover, formulas calculating the s-topological entropy of a sequence of equi-continuous monotone maps on the unit circle are given. Finally, examples to show that the entropy dimension of non-autonomous systems can be attained by any positive number s are constructed.
§1
Introduction
In the study of the autonomous dynamical systems which are induced by the iterations of a single transformation, entropies are important invariants to describe the complexity of dynamical systems. In 1958, with the notion of entropy in information theory, Kolmogorov [16] defined the measure-theoretic entropy for a measure-preserving transformation on a probability space, which describes the maximal information we can get from a system. In 1965, Adler, Konheim and McAndrew [1] introduced the topological entropy for continuous maps on a compact topological space using open covers, which measures the exporential growth rate of the number of orbits of a dynamical system. In 1971, Bowen gave the same definition by spanning sets and separated sets respectively for uniformly continuous maps on a metric space. Since then the notion of entropy including measure-theoretic entropy and topological entropy has been playing an important role in the study of dynamical systems. These two entropies are connected by the variational principle, that is, the topological entropy is equal to the supremum of the measure-theoretic entropies with respect to all invariant probability measures for a given dynamical system [11]. We also refer to [10, 19, 22] for more information about entropies for dynamical systems. Received: 2017-03-11. Revised: 2019-06-07. MR Subject Classification:37D30, 37B40, 37C50. Keywords: entropy dimension, homemorphism, finite graphs, non-autonomous systems. Digital Object Identifier(DOI): https://doi.org/10.1007/s11766-020-3537-0. Lin Wang is supported by the National Natural Science Foundation of China (No.11801336,11771118), the Science and Technology Innovation Project of Shanxi Higher Education (No.2019L0475) and the Applied Basic Research Program of Shanxi Province(No:201901D211417). ∗ Corresponding author.
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Appl. Math. J. Chinese Univ.
Vol. 35, No. 3
In recent years, more and more people keep a watchful eye on the non-autonomous dynamical systems. In order to investigate the complexity of them from different points of view, various of entropy-type invariants are introduced and investigated. Y. Zhu and J. Zhang [25, 29] discuss the topological entropy of non-autonomous systems, for which the lower bound is given. In [15], Kawan and Latushkin obtained some useful results about entropy for non-autonomous systems. In [21, 22], the authors give a description of chaos in term of topological entropy for nonautonomous systems, particularly, some tools
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