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Absolute continuity of non-homogeneous self-similar measures

Santiago Saglietti, Pablo Shmerkin, Boris Solomyak

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.

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.

Paper Structure

This paper contains 17 sections, 32 theorems, 245 equations.

Table of Contents

  1. Introduction and main results
  2. Non-homogeneous self-similar measures
  3. Outline of proof
  4. A class of random self-similar measures
  5. Notation
  6. Exact dimensionality and dimension conservation
  7. Dimension, entropy and super-exponential concentration
  8. Preliminaries on entropy and entropy dimension
  9. A theorem on entropy growth
  10. Component measures and the inverse theorem
  11. Uniform entropy dimension of $\eta^{(\omega)}$
  12. Proof of Theorem \ref{['teo2']}
  13. Proof of Theorem \ref{['thm:sup-exp-conc']}
  14. Fourier transform estimates
  15. Probabilistic estimates
  16. ...and 2 more sections

Key Result

Theorem 1.1

Fix $k\ge 2$, distinct translations $t_1,\ldots,t_k\in\mathbb{R}$ and a probability vector $\mathbf{p}=(p_1,\ldots,p_k)$. There exists a set of zero Lebesgue measure such that the following holds: for any $\lambda\in \mathcal{R}_{\mathbf{p}}\setminus E_{\mathbf{p}}$, the self-similar measure $\nu_{\lambda,t}^\mathbf{p}$ is absolutely continuous.

Theorems & Definitions (55)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 1.4
  • Theorem 1.5
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Remark 3.3
  • ...and 45 more