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Entropy rates in the dimension theory of self-similar measures

Péter P. Varjú

Abstract

This is a survey paper on the dimension theory of self-similar measures on the real line focusing on the role of entropy rates.

Entropy rates in the dimension theory of self-similar measures

Abstract

This is a survey paper on the dimension theory of self-similar measures on the real line focusing on the role of entropy rates.

Paper Structure

This paper contains 8 sections, 5 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Algebraic contractions
  3. Bernoulli convolutions
  4. Exceptional parameters in $[1/2,1)$
  5. Continuity properties of entropy rates
  6. A word about the proof of Theorem \ref{['th:Bernoulli']}
  7. Generalizations
  8. Two transcendental parameters

Key Result

Theorem 5

Let $\Phi$ be an IFS that satisfies the weak exponential separation condition, let $p$ be a probability vector, and let $\mu$ be the corresponding self-similar measure. Then

Theorems & Definitions (10)

  • Definition 1: Exact overlaps
  • Definition 2: Entropy rate
  • Conjecture 3
  • Definition 4
  • Theorem 5: Hochman
  • Theorem 6: Rapaport
  • Theorem 7: Hochman, Varjú
  • Definition 10: Entropy rate for subsets in parameter space
  • Theorem 12: Streck
  • Theorem 13