Dynamics of metrics in measure spaces and their asymptotic invariants

TL;DR

This paper introduces scaling entropy as a more powerful metric invariant for measure-preserving actions, replacing traditional partition techniques with metric iteration, and proposes a new geometric framework for understanding entropy and dynamical systems.

Contribution

It defines scaling entropy, develops a metric-based approach to ergodic theory, and proposes a research program on asymptotic metric dynamics in measure spaces.

Findings

Scaling entropy is a more powerful invariant than traditional entropy.

A new metric iteration approach replaces measurable partitions.

The framework links entropy to asymptotic Hausdorff dimension.

Abstract

We discuss the Kolmogorov's entropy and Sinai's definition of it; and then define a deformation of the entropy, called {\it scaling entropy}; this is also a metric invariant of the measure preserving actions of the group, which is more powerful than the ordinary entropy. To define it, we involve the notion of the $\epsilon$-entropy of a metric in a measure space, also suggested by A. N. Kolmogorov slightly earlier. We suggest to replace the techniques of measurable partitions, conventional in entropy theory, by that of iterations of metrics or semi-metrics. This leads us to the key idea of this paper which as we hope is the answer on the old question: what is the natural context in which one should consider the entropy of measure-preserving actions of groups? the same question about its generalizations--scaling entropy, and more general problems of ergodic theory. Namely, we propose a certain research program, called {\it asymptotic dynamics of metrics in a measure space}, in which, for instance, the generalized entropy is understood as {\it the asymptotic Hausdorff dimension of a sequence of metric spaces associated with dynamical system.} As may be supposed, the metric isomorphism problem for dynamical systems as a whole also gets a new geometric interpretation.