Irregular sets, the -transformation and the almost specification property

By Daniel J. Thompson. 2012

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Let (X, d) be a compact metric space, f : X → 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 φ : X → R be a continuous function. We show that the set of points for which the Birkhoff average of φ does not exist (which we call the irregular set) is either empty or has full topological entropy. Every β-shift satisfies almost specification and we show that the irregular set for any β-shift or β-transformation is either empty or has full topological entropy and Hausdorff dimension.

Subfields: Cell Biology, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0497339bde09bb686d3a1003cb118d9e0569db32