Journal of the European Mathematical Society

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Volume 18, Issue 2, 2016, pp. 327–351
DOI: 10.4171/JEMS/591

Published online: 2016-02-08

Sets of $\beta$-expansions and the Hausdorff measure of slices through fractals

Tom Kempton[1]

(1) University of Utrecht, Netherlands

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

Keywords: Bernoulli convolution, $\beta$ expansion, slicing fractals, conditional measures

Kempton Tom: Sets of $\beta$-expansions and the Hausdorff measure of slices through fractals. J. Eur. Math. Soc. 18 (2016), 327-351. doi: 10.4171/JEMS/591