Irregular sets, the 𝛽-transformation and the almost specification property

Thompson, Daniel

doi:10.1090/S0002-9947-2012-05540-1

Irregular sets, the $\beta$-transformation and the almost specification property
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Trans. Amer. Math. Soc. 364 (2012), 5395-5414 Request permission

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 $\varphi : X \mapsto \mathbb {R}$ be a continuous function. We show that the set of points for which the Birkhoff average of $\varphi$ 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.

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