Abstract
We study the receptive metric entropy for semigroup actions on probability spaces, inspired by a similar notion of topological entropy introduced by Hofmann and Stoyanov (Adv Math 115:54–98, 1995). We analyze its basic properties and its relation with the classical metric entropy. In the case of semigroup actions on compact metric spaces we compare the receptive metric entropy with the receptive topological entropy looking for a Variational Principle. With this aim we propose several characterizations of the receptive topological entropy. Finally we introduce a receptive local metric entropy inspired by a notion by Bowen generalized in the classical setting of amenable group actions by Zheng and Chen, and we prove partial versions of the Brin–Katok Formula and the local Variational Principle.
Highlights
In [29] a general notion of receptive topological entropy was introduced and studied for a uniformly continous action T : G × X → X of a locally compact semigroup G on a metric space (X, d)
In the present paper we study a natural similar definition of a receptive metric entropy hμ(T, ) of a measure-preserving action T : G × X → X of a discrete semigroup G on a probability space (X, μ) with respect to a regular system (Nn)n∈N in G
It is well-known that the Variational Principle holds for continuous actions of amenable groups or of countable cancellative semigroups on compact metric spaces, for the classical topological and metric entropy defined by means of a Følner sequence
Summary
In [29] a general notion of receptive topological entropy (the term “receptive” was coined later on) was introduced and studied for a uniformly continous action T : G × X → X of a locally compact semigroup G on a metric space (X , d). The classical Variational Principle due to Goodwyn [27] and Goodman [24] states that, in case G = Z+ and f = T (1, −) : X → X is a continuous selfmap of the compact metric space X , h( f ) = sup{hμ( f ) : μ f -invariant Borel probability measure} It is well-known that the Variational Principle holds for continuous actions of amenable groups or of countable cancellative semigroups on compact metric spaces, for the classical topological and metric entropy defined by means of a Følner sequence (see [38,39] and [36,46] respectively). We warmly thank the referees for their careful reading and useful comments and suggestions
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
