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. 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.
Misiurewicz[19]introducedtheconceptofpseudo-entropyandproved this quantity coincides with topological entropy. Richeson et al. [21] obtained the lower bounded of topological entropy by means of the definition of pseudo-entropy. This paper aims to generalize the main results obtained by Misiurewicz and Rich-eson et al. to free semigroup actions. Firstly, the pseudo-entropy is introduced for free semigroup actions, and it is shown that the pseudo-entropy coincides with the topological entropy defined by Bufetov [9]. Secondly, these concepts of the chain recurrence, the chain mixing, the chain re…
The aim of this manuscript is to study some local properties of the topological entropy of a free semigroup action. In order to do that we focus on the set of entropy points of a free semigroup action, show that this set carries the full entropy of the system (which, with respect to the chaocity of the system, gives a fundamental relevance to such set) and obtain many interesting properties of such set. Our results are inspired by the ones presented in \cite{YZ}.
We consider the existence of the topological entropy of shift spaces on a finitely generated semigroup whose Cayley graph is a tree. The considered semigroups include free groups. On the other hand, the notion of stem entropy is introduced. For shift spaces on a strict free semigroup, the stem entropy coincides with the topological entropy. We reveal a sufficient condition for the existence of the stem entropy of shift spaces on a semigroup. Furthermore, we demonstrate that the topological entropy exists in many cases and is identical to the stem entropy.
We study two notions of topological entropy of correspondences introduced by Friedland and Dinh-Sibony. Upper bounds are known for both. We identify a class of holomorphic correspondences whose entropy in the sense of Dinh-Sibony equals the known upper bound. This provides an exact computation of the entropy for rational semigroups. We also explore a connection between these two notions of entropy.
This paper's major purpose is to continue the work of Zhu and Ma[1]. To begin, the $\mathbf{g}$-almost product property, more general irregular and regular sets, and some new notions of the Banach upper density recurrent points and transitive points of free semigroup actions are introduced. Furthermore, under the $\mathbf{g}$-almost product property and other conditions, we coordinate the Banach upper recurrence, transitivity with (ir)regularity, and obtain lots of generalized multifractal analyses for general observable functions of free semigroup actions. Finally, statistical $ω$-limit sets …
Let $P_1,P_2,\dots, P_k$ be complex polynomials of degree at least two that are not simultaneously conjugate to monomials or to Chebyshev polynomials, and $S$ the semigroup under composition generated by $P_1,P_2,\dots, P_k$. We show that all elements of $S$ share a measure of maximal entropy if and only if the intersection of principal right ideals $SP_1\cap SP_2\cap \dots \cap SP_k$ is non-empty.
In this manuscript we show that the metric mean dimension of a free semigroup action satisfies three variational principles: (a) the first one is based on a definition of Shapira's entropy, introduced in \cite{SH} for a singles dynamics and extended for a semigroup action in this note; (b) the second one treats about a definition of Katok's entropy for a free semigroup action introduced in \cite{CRV-IV}; (c) lastly we consider the local entropy function for a free semigroup action and show that the metric mean dimension satisfies a variational principle in terms of such function. Our results a…
Motivated by recent applications to entropy theory in dynamical systems, we generalise notions introduced by Matthews and define weakly weighted and componentwisely weakly weighted (generalised) quasi-metrics. We then systematise and extend to full generality the correspondences between these objects and other structures arising in theoretical computer science and dynamics. In particular, we study the correspondences with weak partial metrics, and, if the underlying space is a semilattice, with invariant (generalised) quasi-metrics satisfying the descending path condition, and with strictly mo…
The first aims of this work are to endorse the advent of finitely additive set functions as equilibrium states and the possibility to replace the metric entropy by an upper semi-continuous map associated to a general variational principle. More precisely, using methods from Convex Analysis, we construct for each generalized convex pressure function an upper semi-continuous entropy-like map (which, in the context of continuous transformations acting on a compact metric space and the topological pressure, turns out to be the upper semi-continuous envelope of the Kolmogorov-Sinai metric entropy),…
