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On absolute continuity of inhomogeneous and contracting on average self-similar measures

Samuel Kittle, Constantin Kogler

Abstract

We give a condition for absolute continuity of self-similar measures in arbitrary dimensions. This allows us to construct the first explicit absolutely continuous examples of inhomogeneous self-similar measures in dimension one and two. In fact, for $d\geq 1$ and any given rotations in $O(d)$ acting irreducibly on $\mathbb{R}^d$ as well as any distinct translations, all having algebraic coefficients, we construct absolutely continuous self-similar measures with the given rotations and translations. We furthermore strengthen Varjú's result for Bernoulli convolutions, treat complex Bernoulli convolutions and in dimension $\geq 3$ improve the condition on absolute continuity by Lindenstrauss-Varjú. Moreover, self-similar measures of contracting on average measures are studied, which may include expanding similarities in their support.

On absolute continuity of inhomogeneous and contracting on average self-similar measures

Abstract

We give a condition for absolute continuity of self-similar measures in arbitrary dimensions. This allows us to construct the first explicit absolutely continuous examples of inhomogeneous self-similar measures in dimension one and two. In fact, for and any given rotations in acting irreducibly on as well as any distinct translations, all having algebraic coefficients, we construct absolutely continuous self-similar measures with the given rotations and translations. We furthermore strengthen Varjú's result for Bernoulli convolutions, treat complex Bernoulli convolutions and in dimension improve the condition on absolute continuity by Lindenstrauss-Varjú. Moreover, self-similar measures of contracting on average measures are studied, which may include expanding similarities in their support.
Paper Structure (41 sections, 87 theorems, 437 equations)

This paper contains 41 sections, 87 theorems, 437 equations.

Table of Contents

  1. Introduction
  2. Main Result and Outline
  3. Main Result
  4. Outline
  5. Notation
  6. Organisation
  7. Preliminaries
  8. Derivative Bounds
  9. Basic Properties
  10. Taylor Expansion Bound
  11. Large Deviation Principle
  12. $\mathcal{I}_k$ is open
  13. Order $k$ Detail
  14. Definitions
  15. Bounding Detail
  16. ...and 26 more sections

Key Result

Theorem 1.2

(KittleKoglerDimension*Theorem 1.2 and Theorem 1.3) Let $\mu$ be a finitely supported, contracting on average and irreducible probability measure on $G$ without a common fixed point. Assume that either of the following two properties holds: Then

Theorems & Definitions (164)

  • Definition 1.1
  • Theorem 1.2
  • Conjecture 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Corollary 1.9
  • Corollary 1.10
  • ...and 154 more