Some results on the entropy of non-autonomous dynamical systems

ABSTRACT

In this paper, we advance the entropy theory of discrete non-autonomous 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 non-stationary 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.

Keywords:

• Non-autonomous dynamical system
• topological entropy
• metric entropy
• non-stationary subshift of finite type

AMS Subject Classification (2010):

• 37B55
• 37B40
• 37B10
• 37A35

Acknowledgments

Yuri Latushkin sincerely thanks Lai-Sang Young for the opportunity to spend the academic year 2014/15 at the Courant Institute of Mathematical Sciences and for many illuminating discussions. Christoph Kawan thanks Tomasz Downarowicz and Sergiy Kolyada for very helpful comments.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the NSF [grant DMS-1067929], by the Research Board and Research Council of the University of Missouri, and by the Simons Foundation. Partly supported by DFG fellowship [KA 3893/1-1].