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

Journal of the European Mathematical Society (2016)

- Volume: 018, Issue: 2, page 327-351
- ISSN: 1435-9855

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topKempton, Tom. "Sets of $\beta $-expansions and the Hausdorff measure of slices through fractals." Journal of the European Mathematical Society 018.2 (2016): 327-351. <http://eudml.org/doc/277254>.

@article{Kempton2016,

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.},

author = {Kempton, Tom},

journal = {Journal of the European Mathematical Society},

keywords = {Bernoulli convolution; $\beta $ expansion; slicing fractals; conditional measures; Bernoulli convolution; beta expansion; slicing fractals; conditional measures},

language = {eng},

number = {2},

pages = {327-351},

publisher = {European Mathematical Society Publishing House},

title = {Sets of $\beta $-expansions and the Hausdorff measure of slices through fractals},

url = {http://eudml.org/doc/277254},

volume = {018},

year = {2016},

}

TY - JOUR

AU - Kempton, Tom

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

JO - Journal of the European Mathematical Society

PY - 2016

PB - European Mathematical Society Publishing House

VL - 018

IS - 2

SP - 327

EP - 351

AB - 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.

LA - eng

KW - Bernoulli convolution; $\beta $ expansion; slicing fractals; conditional measures; Bernoulli convolution; beta expansion; slicing fractals; conditional measures

UR - http://eudml.org/doc/277254

ER -

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