Sets of beta-expansions and the Hausdorff Measure of Slices through\n Fractals

2013/07/08 by Tom Kempton, Kempton, Tom · 1 citation
Computer Science · Mathematics · #Advanced Topology and Set Theory #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #Rough Sets and Fuzzy Logic

paper · doi:10.48550/arxiv.1307.2091

openalex publication_date 2013/07/08 · openalex created_date 2019/07/30 · openalex updated_date 2026/07/28

Abstract

We study natural measures on sets of beta-expansions and on slices through\nself similar sets. In the setting of beta-expansions, these allow us to better\nunderstand the measure of maximal entropy for the random beta-transformation\nand to reinterpret a result of Lindenstrauss, Peres and Schlag in terms of\nequidistribution. Each of these applications is relevant to the study of\nBernoulli convolutions. In the fractal setting this allows us to understand how\nto disintegrate Hausdorff measure by slicing, leading to conditions under which\nalmost every slice through a self similar set has positive Hausdorff measure,\ngeneralising long known results about almost everywhere values of the Hausdorff\ndimension.\n

Citations

• Representations for real numbers and their ergodic properties

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