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arXiv · 1911.07993

Topological entropy of nonautonomous dynamical systems

Kairan Liu·Yixiao Qiao·Leiye Xu
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Abstract

Let $\mathcal{M}(X)$ be the space of Borel probability measures on a compact metric space $X$ endowed with the weak$^\ast$-topology. In this paper, we prove that if the topological entropy of a nonautonomous dynamical system $(X,\{f_n\}_{n=1}^{+\infty})$ vanishes, then so does that of its induced system $(\mathcal{M}(X),\{f_n\}_{n=1}^{+\infty})$; moreover, once the topological entropy of $(X,\{f_n\}_{n=1}^{+\infty})$ is positive, that of its induced system $(\mathcal{M}(X),\{f_n\}_{n=1}^{+\infty})$ jumps to infinity. In contrast to Bowen's inequality, we construct a nonautonomous dynamical system whose topological entropy is not preserved under a finite-to-one extension.

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Kairan Liu, Yixiao Qiao, Leiye Xu. 2019-11-18. Topological entropy of nonautonomous dynamical systems. https://arxiv.org/abs/1911.07993

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