Rate distortion theory, metric mean dimension and measure theoretic entropy

TL;DR

This paper establishes a variational principle linking metric mean dimension and measure-theoretic entropy, offering a more computation-friendly formulation and exploring relations with rate distortion functions.

Contribution

It introduces a new variational principle for metric mean dimension using a measure-theoretic entropy-like function, simplifying calculations and connecting to existing rate distortion concepts.

Findings

Proved a variational principle for metric mean dimension.

Established relations between rate distortion functions and a modified entropy function.

Reproved key results from previous studies using new methods.

Abstract

We prove a variational principle for the metric mean dimension analog to the one in [LT]. Instead of using the rate distortion function we use the function $h_\mu(\epsilon,T,\delta)$ that is closely related to the entropy $h_\mu(T)$ of $\mu$. Our formulation has the advantage of being, in the authors opinion, more natural when doing computations. As a corollary we obtain a proof of the standard variational principle. We also obtain some relations between the rate distortion function with our function $\tilde{h}_\mu(\epsilon,T,\delta)$, a modification of $h_\mu(\epsilon,T,\delta)$ when replacing the dynamical metrics with the average dynamical metrics. Using our methods we also reprove the main result in [LT]. We will explain how to construct homeomorphisms on closed manifolds with maximal metric mean dimension. We end this paper with some questions that naturally arise from this work.