Topological entropy of nonautonomous dynamical systems

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 $(X,\{f_n{\}}_{n=1}^{+\mathrm{\infty }})$ vanishes, then so does that of its induced system $(\mathcal{M}(X),\{f_n{\}}_{n=1}^{+\mathrm{\infty }})$; moreover, once the topological entropy of $(X,\{f_n{\}}_{n=1}^{+\mathrm{\infty }})$ is positive, that of its induced system $(\mathcal{M}(X),\{f_n{\}}_{n=1}^{+\mathrm{\infty }})$ 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.