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Dimension of contracting on average self-similar measures

Samuel Kittle, Constantin Kogler

TL;DR

The paper extends Hochman’s dimension formula for self-similar measures to contracting on average settings and under weakened separation conditions. It introduces the variance summation method as a replacement for entropy inverse theorems and develops a decomposition of stopped random walks to accumulate scale-wise variance, coupled with smoothings and Taylor bounds to control errors. Under contracting-on-average and irreducibility assumptions, it establishes that $\dim \nu = \min\{ d, h_{\mu}/|\chi_{\mu}| \}$, with a weak-separation variant that still recovers the formula under slower scale information via entropy-gap arguments. The work broadens the applicability to inhomogeneous self-similar measures across dimensions and provides a flexible framework linking entropy gaps, trace bounds, and Gaussian approximations to achieve full-dimensional results.

Abstract

We generalise Hochman's theorem on the dimension of self-similar measures to contracting on average measures and show that a weaker condition than exponential separation on all scales is sufficient. Our proof uses a technique we call the variance summation method, avoiding the use of inverse theorems for entropy.

Dimension of contracting on average self-similar measures

TL;DR

The paper extends Hochman’s dimension formula for self-similar measures to contracting on average settings and under weakened separation conditions. It introduces the variance summation method as a replacement for entropy inverse theorems and develops a decomposition of stopped random walks to accumulate scale-wise variance, coupled with smoothings and Taylor bounds to control errors. Under contracting-on-average and irreducibility assumptions, it establishes that , with a weak-separation variant that still recovers the formula under slower scale information via entropy-gap arguments. The work broadens the applicability to inhomogeneous self-similar measures across dimensions and provides a flexible framework linking entropy gaps, trace bounds, and Gaussian approximations to achieve full-dimensional results.

Abstract

We generalise Hochman's theorem on the dimension of self-similar measures to contracting on average measures and show that a weaker condition than exponential separation on all scales is sufficient. Our proof uses a technique we call the variance summation method, avoiding the use of inverse theorems for entropy.

Paper Structure

This paper contains 18 sections, 28 theorems, 167 equations.

Table of Contents

  1. Introduction
  2. Outline and Notation
  3. Outline of proofs
  4. Notation
  5. Preliminaries
  6. Smoothing functions on $\mathbb{R}^d$
  7. Gaussian Approximation and Full Dimension
  8. Taylor Expansion Bound
  9. Entropy and Trace Bounds
  10. Results from KittleKoglerEntropy
  11. Entropy Gap and Trace Bound for Theorem \ref{['GeneralHochman']}
  12. Entropy Gap and Trace Bound for Theorem \ref{['MainWeakHochman']}
  13. Decomposition of Stopped Random Walks and Variance Summation
  14. Proper Decompositions
  15. Preliminary results
  16. ...and 3 more sections

Key Result

Theorem 1.2

(Generalisation of Hochman's theorem) Let $\mu$ be a finitely supported, contracting on average and irreducible probability measure on $G$ without a common fixed point. Furthermore assume that there is $c > 0$ such that $M_n \geq e^{-cn}$ for infinitely many $n \geq 1$. Then

Theorems & Definitions (54)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Proposition 3.4
  • Lemma 3.5
  • ...and 44 more