Absolutely continuous invariant measures for non-autonomous dynamical systems

We consider the non-autonomous dynamical system {tau(n),}, where tau(n) is a continuous map X -> X, and X is a compact metric space. We assume that {tau(n)} converges uniformly to tau. The inheritance of chaotic properties as well as topological entropy by tau from the sequence {tau(n)} has been studied in [4,5,10,13,17]. In [16] the generalization of SRB measures to non-autonomous systems has been considered. In this paper we study absolutely continuous invariant measures (acim) for non-autonomous systems. After generalizing the Krylov-Bogoliubov Theorem [7] and Straube's Theorem [14] to the non-autonomous setting, we prove that under certain conditions the limit map T of a non-autonomous sequence of maps {tau(n)} with acims has an acim. (C) 2018 Elsevier Inc. All rights reserved.