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 $(X,{f_n}_{n=1}^{+infty})$ 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 Bowens inequality, we construct a nonautonomous dynamical system whose topological entropy is not preserved under a finite-to-one extension.