Topological and Measure-Theoretical Entropies of Nonautonomous Dynamical Systems

We consider the dynamics of a nonautonomous dynamical system determined by a sequence of continuous self-maps where defined on a compact metric space X. Applying the theory of the Carath,odory structures, elaborated by Pesin (Dimension Theory in Dynamical Systems. Chicago Lectures in Mathematics. The University of Chicago Press, Chicago, 1997), we construct a Carath,odory structure whose capacity coincides with the topological entropy of the considered system. Generalizing the notion of local measure entropy, introduced by Brin and Katok (in: Palis (ed) Geometric Dynamics, Lecture Notes in Mathematics. Springer, Berlin 1983) for a single map, to a nonautonomous dynamical system we provide a lower and upper estimations of the topological entropy by local measure entropies. The theorems of the paper generalize results of Kawan (Nonautonomous Stoch Dyn Syst 1:26-52, 2013) and of Feng and Huang (J Funct Anal 263:2228-2254, 2012). Also, we construct a new entropy-like invariant such the entropy of a sequence of Lipschitz continuous maps with the same Lipschitz constant restricted to a subset is upper bounded by Hausdorff dimension of Y multiplied by the logarithm of the Lipschitz constant L. This gives a generalizations of results of Dai et al. (Sci China Ser A 41:1068-1075, 1998) and Misiurewicz (Discret Contin Dyn Syst 10:827-833, 2004).