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Intermediate dimensions of Bedford-McMullen carpets with applications to Lipschitz equivalence

Amlan Banaji, István Kolossváry

Abstract

Intermediate dimensions were recently introduced to provide a spectrum of dimensions interpolating between Hausdorff and box-counting dimensions for fractals where these differ. In particular, the self-affine Bedford-McMullen carpets are a natural case for investigation, but until now only very rough bounds for their intermediate dimensions have been found. In this paper, we determine a precise formula for the intermediate dimensions $\dim_{\, θ}Λ$ of any Bedford-McMullen carpet $Λ$ for the whole spectrum of $θ\in [0,1]$, in terms of a certain large deviations rate function. The intermediate dimensions exist and are strictly increasing in $θ$, and the function $θ\mapsto \dim_{\, θ}Λ$ exhibits interesting features not witnessed on any previous example, such as having countably many phase transitions, between which it is analytic and strictly concave. We make an unexpected connection to multifractal analysis by showing that two carpets with non-uniform vertical fibres have equal intermediate dimensions if and only if the Hausdorff multifractal spectra of the uniform Bernoulli measures on the two carpets are equal. Since intermediate dimensions are bi-Lipschitz invariant, this shows that the equality of these multifractal spectra is a necessary condition for two such carpets to be Lipschitz equivalent.

Intermediate dimensions of Bedford-McMullen carpets with applications to Lipschitz equivalence

Abstract

Intermediate dimensions were recently introduced to provide a spectrum of dimensions interpolating between Hausdorff and box-counting dimensions for fractals where these differ. In particular, the self-affine Bedford-McMullen carpets are a natural case for investigation, but until now only very rough bounds for their intermediate dimensions have been found. In this paper, we determine a precise formula for the intermediate dimensions of any Bedford-McMullen carpet for the whole spectrum of , in terms of a certain large deviations rate function. The intermediate dimensions exist and are strictly increasing in , and the function exhibits interesting features not witnessed on any previous example, such as having countably many phase transitions, between which it is analytic and strictly concave. We make an unexpected connection to multifractal analysis by showing that two carpets with non-uniform vertical fibres have equal intermediate dimensions if and only if the Hausdorff multifractal spectra of the uniform Bernoulli measures on the two carpets are equal. Since intermediate dimensions are bi-Lipschitz invariant, this shows that the equality of these multifractal spectra is a necessary condition for two such carpets to be Lipschitz equivalent.

Paper Structure

This paper contains 28 sections, 27 theorems, 262 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Bedford--McMullen carpets
  3. Summary of results
  4. Complete statements and examples
  5. Main result: formula for intermediate dimensions
  6. Form of the intermediate dimensions
  7. Connection to multifractal analysis and bi-Lipschitz equivalence
  8. Equivalent forms of the rate function
  9. Illustrative examples
  10. Proof of Proposition \ref{['prop:1']}
  11. Preliminaries
  12. Proof of Step 1
  13. Proof of Step 2
  14. Proof of Step 3
  15. Proof of Step 4
  16. ...and 13 more sections

Key Result

Theorem 2.1

Let $\Lambda$ be a Bedford--McMullen carpet with non-uniform vertical fibres. For all $\theta \in (0,1)$, $\dim_{\, \theta} \Lambda$ exists and is given in the following way. For fixed $\theta \in (0,1)$ let $L = L(\theta) \coloneqq 1 + \lfloor \frac{-\log \theta}{\log \gamma} \rfloor$, so $\gamma^{ and $s(\theta) = \dim_{\, \theta} \Lambda$.

Figures (5)

  • Figure 1: Intermediate dimensions give finer geometric information about the fractal set. Left: a Bedford--McMullen carpet $\Lambda$ in red and the shaded images of $[0,1]^2$ under the iterated function system generating $\Lambda$. Right: certain bounds for the intermediate dimensions $\dim_{\, \theta}\Lambda$ in dashed orange, and the true value in blue obtained in this paper.
  • Figure 2: Parameters $n=100$ and $\mathbf{N}=(51,50,50,50,50,50)$ are the same in each example; only $m$ varies from $30$ on the left to $50$ on the right.
  • Figure 3: Left: plot of $\dim_{\, \theta}\Lambda$ (blue) and $\dim_{\, \theta}\Lambda'$ (orange) from Example \ref{['ex:biLip']}. Right: ratio of $\dim_{\, \theta}\Lambda'/\dim_{\, \theta}\Lambda$ for $\theta\geq \gamma^{-35}$.
  • Figure 4: Plots of the intermediate dimensions of carpets in Example \ref{['ex:Nice']}.
  • Figure 5: Visualising the cover in \ref{['eq:definerealcover']} for $L \geq 3$. Here, $l$ denotes an arbitrary number in $\{1,2,\dotsc,L-2\}$. The indices of the symbolic representation and the lengths of the different parts are in black. Above the scales explicitly written out are the sets (in blue) which make up the part of the cover consisting of approximate squares of the corresponding level. The 'critical' thresholds $t_i$ for the averages of the different parts of the symbolic representation are in red. Recall that the $t_i$ depend on $s$, and the sets that make up the cover depend on $s$ and $\theta$.

Theorems & Definitions (69)

  • Definition 1.1
  • Theorem 2.1
  • Corollary 2.2
  • Corollary 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Remark 2.6
  • Corollary 2.10
  • Proposition 2.11
  • Remark 2.12
  • ...and 59 more