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Pointwise Assouad dimension for measures

Roope Anttila

Abstract

We introduce a pointwise variant of the Assouad dimension for measures on metric spaces, and study its properties in relation to the global Assouad dimension. We show that, in general, the value of the pointwise Assouad dimension differs from the global counterpart, but in many classical cases, it exhibits similar exact dimensionality properties as the classical local dimension, namely it equals the global Assouad dimension at almost every point. We also compute the Assouad dimension of invariant measures with place dependent probabilities supported on strongly separated self-conformal sets.

Pointwise Assouad dimension for measures

Abstract

We introduce a pointwise variant of the Assouad dimension for measures on metric spaces, and study its properties in relation to the global Assouad dimension. We show that, in general, the value of the pointwise Assouad dimension differs from the global counterpart, but in many classical cases, it exhibits similar exact dimensionality properties as the classical local dimension, namely it equals the global Assouad dimension at almost every point. We also compute the Assouad dimension of invariant measures with place dependent probabilities supported on strongly separated self-conformal sets.
Paper Structure (16 sections, 16 theorems, 104 equations)

This paper contains 16 sections, 16 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. Main results and the structure of the paper
  3. Preliminaries
  4. Assouad dimension of sets and connection to weak tangents
  5. Assouad dimension of measures and the doubling property
  6. Pointwise Assouad dimension
  7. Relationships to other dimensions
  8. Quasi-Bernoulli measures on self-conformal sets
  9. Self-conformal sets
  10. Quasi-Bernoulli measures
  11. Measures with place dependent probabilities
  12. Place dependent invariant measures
  13. Self-similar and self-affine measures
  14. Self-similar measures and the open set condition
  15. Self-affine measures on Bedford-McMullen sponges
  16. ...and 1 more sections

Key Result

Proposition 2.1

If $X\subset \mathbb{R}^d$ is compact, then $\dim_{\mathrm{A}} X\geq \dim_{\mathrm{A}}F$, for all $F\in\mathrm{Tan}(X)$.

Theorems & Definitions (34)

  • Proposition 2.1
  • Proposition 3.1
  • proof
  • Remark 3.2
  • Example 3.3
  • Example 3.4
  • Example 3.5
  • Theorem 4.1
  • Remark 4.2
  • Lemma 4.3
  • ...and 24 more