Dynamical entropy of probability measures on infinite product spaces

TL;DR

This paper introduces a new concept called infinite-product entropy for probability measures on infinite product spaces, exploring its properties, relation to classical entropy, and applications in dynamical systems.

Contribution

It defines and analyzes the properties of infinite-product entropy, connecting it to existing entropy concepts and providing foundational results for infinite-dimensional dynamical systems.

Findings

Infinite-product entropy is shift invariant and convex.

For translation-invariant measures, it equals classical shift entropy.

The paper establishes a variational inequality linking infinite-product and topological entropy.

Abstract

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.