L^q dimensions of self-similar measures, and applications: a survey

2019/07/13 by Shmerkin, Pablo
#28A80 #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics #Primary: 28A75

paper · doi:10.48550/arxiv.1907.07121

Abstract

We present a self-contained proof of a formula for the L^q dimensions of self-similar measures on the real line under exponential separation (up to the proof of an inverse theorem for the L^q 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 L^q 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.

Related

• Dynamical self-similarity, L^q-dimensions and Furstenberg slicing in ℝ^d
• Self-similar sets and measures on the line
• Local dimensions of overlapping self-similar measures
• The L^q-spectrum for a class of self-similar measures with overlap
• On multifractal formalism for self-similar measures with overlaps