Sets of \(\beta\)-expansions and the Hausdorff measure of slices through fractals
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Publication:252674
DOI10.4171/JEMS/591zbMath1339.28006arXiv1307.2091OpenAlexW2963922978MaRDI QIDQ252674
Publication date: 3 March 2016
Published in: Journal of the European Mathematical Society (JEMS) (Search for Journal in Brave)
Abstract: 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.
Full work available at URL: https://arxiv.org/abs/1307.2091
Fractals (28A80) Convolution, factorization for one variable harmonic analysis (42A85) Radix representation; digital problems (11A63) Hausdorff and packing measures (28A78)
Related Items (7)
On the Hausdorff and packing measures of slices of dynamically defined sets ⋮ Local dimensions for the random \(\beta \)-transformation ⋮ On the packing measure of slices of self-similar sets ⋮ Hausdorff dimension of the level sets of self-affine functions ⋮ On the complexity of the set of codings for self-similar sets and a variation on the construction of Champernowne ⋮ Unnamed Item ⋮ Hausdorff dimension of multiple expansions
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