The irregular set for maps with almost weak specification property has full metric mean dimension

TL;DR

This paper proves that for certain dynamical systems with the almost weak specification property, the set of points with non-converging Birkhoff averages either doesn't exist or has maximal metric mean dimension.

Contribution

It establishes that the irregular set for continuous functions in systems with almost weak specification has full metric mean dimension or is empty.

Findings

Irregular set is either empty or has full metric mean dimension.

Full metric mean dimension applies to systems with almost weak specification.

Provides a dichotomy for the size of irregular sets in these systems.

Abstract

Let $(X, d)$ be a compact metric space, $f: X \to X$ be a continuous transformation with the almost weak specification property and $\varphi: X \to \mathbb{R}$ be a continuous function. We consider the set (called the irregular set for $\varphi$) of points for which the Birkhoff average of $\varphi$ does not exist and show that this set is either empty or carries full Bowen upper and lower metric mean dimension.