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.

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

References in corpus (6)

• Absolute continuity of self-similar measures, their projections and convolutions
• Absolute continuity of Bernoulli convolutions for algebraic parameters
• On self-similar sets with overlaps and inverse theorems for entropy in ${\mathbb{R}}^d$
• $L^q$ dimensions and projections of random measures
• The Hoelder Property for the Spectrum of Translation Flows in Genus Two
• Singularity versus exact overlaps for self-similar measures

Cited by in corpus (6)

• Fourier transform of self-affine measures
• Polynomial Fourier decay for fractal measures and their pushforwards
• Families of infinite parabolic IFS with overlaps: the approximating method
• Self-similar sets and measures on the line
• Disintegration results for fractal measures and applications to Diophantine approximation
• Exceptions in the domain of generic absolute continuity of non-homogeneous self-similar measures