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

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.