Metric mean dimension via subshifts of compact type

TL;DR

This paper explores the metric mean dimension of subshifts of compact type, extending existing formulas and applying results to maps like the Gauss map and Manneville-Pomeau family, linking dimension and pressure parameters.

Contribution

It generalizes Bowen's entropy formula to metric mean dimension for continuous and discontinuous maps using subshifts of compact type.

Findings

Metric mean dimensions of a map and its inverse limit coincide.

Extension of metric mean dimension to discontinuous maps via subshifts.

Equality of metric mean dimension and box dimension for specific maps.

Abstract

We investigate the metric mean dimension of subshifts of compact type. We prove that the metric mean dimensions of a continuous map and its inverse limit coincide, generalizing Bowen's entropy formula. Building upon this result, we extend the notion of metric mean dimension to discontinuous maps in terms of suitable subshifts. As an application, we show that the metric mean dimension of the Gauss map and that of induced maps of the Manneville-Pomeau family is equal to the box dimension of the corresponding set of discontinuity points, which also coincides with a critical parameter of the pressure operator associated to the geometric potential.