Topological entropy of nonautonomous dynamical systems

Let M (X) be the space of all Borel probability measures on a compact metric space X endowed with the weak*-topology. In this paper, we prove that if the topological entropy of a nonautonomous dynamical system (X, {f(n)}(n=1)(+infinity)) vanishes, then so does that of its induced system (M(X), {f(n)}(n=1)(+infinity)) moreover, once the topological entropy of (X, {f(n)}(n=1)(+infinity)) is positive, that of its induced system (M(X), {f(n)}(n=1)(+infinity)) 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. (C) 2019 Elsevier Inc. All rights reserved.