Irregular sets, the $\beta$-transformation and the almost specification property

TL;DR

This paper investigates the irregular sets in dynamical systems with the almost specification property, showing they are either empty or have full entropy and dimension, especially in $eta$-shifts and transformations.

Contribution

It introduces the almost specification property and demonstrates that irregular sets in such systems have full entropy and dimension, extending previous results.

Findings

Irregular sets are either empty or have full topological entropy.

$eta$-shifts satisfy almost specification.

Irregular sets in $eta$-systems have full Hausdorff dimension.

Abstract

Let $(X,d)$ be a compact metric space, $f:X \mapsto X$ be a continuous map satisfying a property we call almost specification (which is slightly weaker than the $g$-almost product property of Pfister and Sullivan), and $\phi$ be a continuous function on $X$. We show that the set of points for which the Birkhoff average of $\phi$ does not exist (which we call the irregular set) is either empty or has full topological entropy. Every $\beta$-shift satisfies almost specification and we show that the irregular set for any $\beta$-shift or $\beta$-transformation is either empty or has full topological entropy and Hausdorff dimension.