Some results on the entropy of nonautonomous dynamical systems

TL;DR

This paper advances the entropy theory for discrete nonautonomous dynamical systems, providing conditions for invariant measure sequences and applying the theory to nonstationary subshifts of finite type.

Contribution

It develops a measure-theoretic entropy framework for nonautonomous systems and establishes conditions for the variational principle in nonstationary subshifts.

Findings

Conditions for invariant measure sequences with fine-scale entropy capture

Sufficient conditions for the variational principle in nonstationary subshifts

Extension of entropy theory to nonautonomous dynamical systems

Abstract

In this paper we advance the entropy theory of discrete nonautonomous dynamical systems that was initiated by Kolyada and Snoha in 1996. The first part of the paper is devoted to the measure-theoretic entropy theory of general topological systems. We derive several conditions guaranteeing that an initial probability measure, when pushed forward by the system, produces an invariant measure sequence whose entropy captures the dynamics on arbitrarily fine scales. In the second part of the paper, we apply the general theory to the nonstationary subshifts of finite type, introduced by Fisher and Arnoux. In particular, we give sufficient conditions for the variational principle, relating the topological and measure-theoretic entropy, to hold.