$L^q$ dimensions of self-similar measures, and applications: a survey
Dynamical Systems
2019-07-17 v1 Classical Analysis and ODEs
Abstract
We present a self-contained proof of a formula for the dimensions of self-similar measures on the real line under exponential separation (up to the proof of an inverse theorem for the norm of convolutions). This is a special case of a more general result of the author from [Shmerkin, Pablo. On Furstenberg's intersection conjecture, self-similar measures, and the norms of convolutions. Ann. of Math., 2019], and one of the goals of this survey is to present the ideas in a simpler, but important, setting. We also review some applications of the main result to the study of Bernoulli convolutions and intersections of self-similar Cantor sets.
Keywords
Cite
@article{arxiv.1907.07121,
title = {$L^q$ dimensions of self-similar measures, and applications: a survey},
author = {Pablo Shmerkin},
journal= {arXiv preprint arXiv:1907.07121},
year = {2019}
}
Comments
33 pages, no figures. arXiv admin note: text overlap with arXiv:1609.07802