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The Assouad dimension of self-affine measures on sponges

Jonathan M. Fraser, István Kolossváry

Abstract

We derive upper and lower bounds for the Assouad and lower dimensions of self-affine measures in $\mathbb{R}^d$ generated by diagonal matrices and satisfying suitable separation conditions. The upper and lower bounds always coincide for $d=2,3$ yielding precise explicit formulae for the dimensions. Moreover, there are easy to check conditions guaranteeing that the bounds coincide for $d \geq 4$. An interesting consequence of our results is that there can be a `dimension gap' for such self-affine constructions, even in the plane. That is, we show that for some self-affine carpets of `Barański type' the Assouad dimension of all associated self-affine measures strictly exceeds the Assouad dimension of the carpet by some fixed $δ>0$ depending only on the carpet. We also provide examples of self-affine carpets of `Barański type' where there is no dimension gap and in fact the Assouad dimension of the carpet is equal to the Assouad dimension of a carefully chosen self-affine measure.

The Assouad dimension of self-affine measures on sponges

Abstract

We derive upper and lower bounds for the Assouad and lower dimensions of self-affine measures in generated by diagonal matrices and satisfying suitable separation conditions. The upper and lower bounds always coincide for yielding precise explicit formulae for the dimensions. Moreover, there are easy to check conditions guaranteeing that the bounds coincide for . An interesting consequence of our results is that there can be a `dimension gap' for such self-affine constructions, even in the plane. That is, we show that for some self-affine carpets of `Barański type' the Assouad dimension of all associated self-affine measures strictly exceeds the Assouad dimension of the carpet by some fixed depending only on the carpet. We also provide examples of self-affine carpets of `Barański type' where there is no dimension gap and in fact the Assouad dimension of the carpet is equal to the Assouad dimension of a carefully chosen self-affine measure.
Paper Structure (15 sections, 15 theorems, 97 equations, 1 figure)

This paper contains 15 sections, 15 theorems, 97 equations, 1 figure.

Table of Contents

  1. Introduction: dimensions of self-affine measures
  2. Main results: dimension bounds and dimension gaps
  3. Our model and assumptions
  4. Main result
  5. A dimension gap: examples and non-examples
  6. Comparing orderings of cubes and cylinders
  7. Examples
  8. Planar Barański carpets with different behaviour
  9. Characterisation of SPPC in dimension three
  10. Symbolic arguments
  11. Proof of Proposition \ref{['prop:ratioOfMeasures']}
  12. Proof of Theorem \ref{['thm:dimMeasure']}
  13. Transferring symbolic estimates to geometric estimates
  14. Lower bound for Assouad dimension
  15. Final claim for $\sigma$-ordered coordinate-wise natural measures

Key Result

Theorem 2.5

Let $\nu_{\mathbf{p}}$ be a self-affine measure fully supported on a self-affine sponge satisfying the very strong SPPC. Then and where In particular, for the $\sigma$-ordered coordinate-wise natural measure

Figures (1)

  • Figure 1: Defining maps for a Barański carpet with strictly positive dimension gap (left), and where the Assouad dimension of $F$ is attained for correctly chosen parameters (right).

Theorems & Definitions (32)

  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Theorem 2.5
  • Corollary 2.7
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 22 more