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

Kempton, 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 -