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Topological entropy of nonautonomous dynamical systems (1911.07993v1)
Published 18 Nov 2019 in math.DS
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.