Absolute continuity of non-homogeneous self-similar measures
Abstract
We prove that self-similar measures on the real line are absolutely continuous for almost all parameters in the super-critical region, in particular confirming a conjecture of S-M. Ngai and Y. Wang. While recently there has been much progress in understanding absolute continuity for homogeneous self-similar measures, this is the first improvement over the classical transversality method in the general (non-homogeneous) case. In the course of the proof, we establish new results on the dimension and Fourier decay of a class of random self-similar measures.
Keywords
Cite
@article{arxiv.1709.05092,
title = {Absolute continuity of non-homogeneous self-similar measures},
author = {Santiago Saglietti and Pablo Shmerkin and Boris Solomyak},
journal= {arXiv preprint arXiv:1709.05092},
year = {2024}
}
Comments
v3: the statement of Theorem 1.3 was changed (the selection measure for the "model" of a random self-similar measure is assumed to be Bernoulli rather than an arbitrary ergodic shift-invariant measure; this was implicitly used in the proof. The original formulation is still correct; see the footnote on p.8 for details). The main result: Theorem 1.1 is unchanged