JournalsjemsVol. 18, No. 2pp. 327–351

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

  • Tom Kempton

    University of Utrecht, Netherlands
Sets of $\beta$-expansions and the Hausdorff measure of slices through fractals cover
Download PDF

Abstract

We study natural measures on sets of β\beta-expansions and on slices through self similar sets. In the setting of β\beta-expansions, these allow us to better understand the measure of maximal entropy for the random β\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.

Cite this article

Tom Kempton, Sets of β\beta-expansions and the Hausdorff measure of slices through fractals. J. Eur. Math. Soc. 18 (2016), no. 2, pp. 327–351

DOI 10.4171/JEMS/591