Topological conditional entropy for amenable group actions

Abstract

We introduce the topological conditional entropy for countable discrete amenable group actions and establish a variational principle for it.

Tail variational principle and asymptotic $h$-expansiveness for amenable group actions

“…We resolve the subtleties by providing a topological analog of the Pinsker formula, known for measure-preserving actions. Other than that, our proof relies on two already existing results on the interplay between topological and measure-theoretic dynamics of countable amenable group actions: the variational principle for topological relative entropy [5,Theorem 13.3] (another proof can be found in [21,Theorem 5.1]) and a characterization of the topological tail entropy in terms of selfjoinings of the action [22,Theorem 3.1]. Eventually, we were able to achieve our initial goal and characterize asymptotically h-expansive G-actions.…”

“…They are straightforward adaptations of the Adler-Conheim-McAndrew notion of topological entropy and Misiurewicz' topological conditional entropy [18], respectively (the term tail entropy was introduced later). We remark that topological tail entropy for actions of countable amenable groups has been addressed in [4,22] and also several equivalent definitions are given in [23].…”

“…But since we intend to use it in the derivation of the tail variational principle, we need an independent proof. The shortest proof of Proposition 6.1 that we were able to come up with relies heavily on the already mentioned variational principle for topological relative entropy ( [5,Theorem 13.3] or [21,Theorem 5.1]) and, in the most crucial place, on a result from [22]. We use these results to jump several times between topological and measure-theoretic notions.…”

Entropy for semigroup actions on general topological spaces

“…From invariance entropy for control systems, in 2013, Colonius, Fukuoka, and Santana [10] introduced the notion of invariance entropy for semigroup actions on metric spaces. In 2014, Zhou, Zhang, and Chen [39], inspired by the topological conditional entropy defined by Misiurewicz, presented the topological conditional entropy for countable discrete amenable group actions on compact metric spaces. Since in topological dynamics the phase space need not be metrizable, the development of the theory requires notions of entropy which do not depend on a distance on the space.…”

On the tail pressure

“…Recently, there appeared some works which study the tail entropy of dynamics of group actions (e.g. see [4], [15], [16]).…”