Mina Shahrestani | Searcharxiv

arXiv2024

Dynamical entropy of probability measures on infinite product spaces

The aim of this note is to introduce a notion of dynamical entropy, which we call infinite-product entropy, for probability measures on (countable) infinite cartesian product of any measurable space with itself. The idea behind the definition is that any infinite product space may be considered as a type of dynamical object. We have considered in a previous note a similar idea in topological dynamics to define a notion of dynamical entropy for arbitrary subsets of infinite products of compact topological spaces. We consider some basic properties of infinite-product entropy, e.g. shift invariance, convexity, subadditivity with respect to product of probability measures, the behavior with respect to dilation and restriction. We show that for a translation invariant probability measure the infinite-product entropy coincides with the usual entropy of a shift transformation. We consider some basic examples and computations. We also consider a variational inequality related to infinite-product entropy and topological entropy of subsets of infinite product spaces.

math.PR↗

arXiv2020

Topological Entropy for Arbitrary Subsets of Infinite Product Spaces

In this note a notion of generalized topological entropy for arbitrary subsets of the space of all sequences in a compact topological space is introduced. It is shown that for a continuous map on a compact space the generalized topological entropy of the set of all orbits of the map coincides with the classical topological entropy of the map. Some basic properties of this new notion of entropy are considered; among them are: the behavior of the entropy with respect to disjoint union, cartesian product, component restriction and dilation, shift mapping, and some continuity properties with respect to Vietoris topology. As an example, it is shown that any self-similar structure of a fractal given by a finite family of contractions gives rise to a notion of intrinsic topological entropy for subsets of the fractal. A generalized notion of Bowen's entropy associated to any increasing sequence of compatible semimetrics on a topological space is introduced and some of its basic properties are considered. As a special case for $1\leq p\leq\infty$ the Bowen $p$-entropy of sets of sequences of any metric space is introduced. It is shown that the notions of generalized topological entropy and Bowen $\infty$-entropy for compact metric spaces coincide.

math.DS↗