[Paper Review] Intermediate dimensions of Bedford-McMullen carpets with applications to Lipschitz equivalence

Amlan Banaji, István Kolossváry|arXiv (Cornell University)|Nov 10, 2021

Mathematical Dynamics and Fractals4 citations

TL;DR

This paper establishes a precise formula for the intermediate dimensions of Bedford–McMullen carpets across the full spectrum $\theta \in [0,1]$, using a large deviations rate function. It reveals that intermediate dimensions are strictly increasing, analytic, and strictly concave between countably many phase transitions, and connects them to multifractal analysis by showing that two carpets have equal intermediate dimensions if and only if their Hausdorff multifractal spectra for uniform Bernoulli measures are equal, implying this is a necessary condition for bi-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 $\dim_{\, θ}Λ$ 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.

Motivation & Objective

• To determine the exact intermediate dimensions of Bedford–McMullen carpets for all $\theta \in [0,1]$.
• To analyze the geometric and analytic properties of the intermediate dimension function $\theta \mapsto \dim_\theta \Lambda$, including its monotonicity, analyticity, and concavity.
• To establish a deep connection between intermediate dimensions and multifractal analysis by relating equality of intermediate dimensions to equality of Hausdorff multifractal spectra.
• To apply this connection to the problem of bi-Lipschitz equivalence, showing that equal multifractal spectra are a necessary condition for two carpets to be bi-Lipschitz equivalent.

Proposed method

• Derives a formula for the intermediate dimension $\dim_\theta \Lambda$ using a large deviations rate function associated with the distribution of vertical fibers in the carpet.
• Applies the method of types from large deviations theory to analyze the asymptotic behavior of covering sets at different scales.
• Uses a recursive construction of approximate squares and their covering costs to bound the $s$-cost of covers under the $\theta$-intermediate dimension constraint.
• Relies on Legendre transforms of generating functions to relate the intermediate dimension to the rate function and to establish equivalence conditions.
• Employs a detailed analysis of the asymptotic growth of the number of cylinders at each level, using ordered counts $N_{\hat{\imath}}$ and weights $R_{\hat{\imath}}$ to model the distribution of vertical fibers.
• Proves equivalence of intermediate dimensions by showing that equality of the rate functions $I(t)$ and $I'(t - \gamma \log(M'/M))$ implies equality of the intermediate dimensions via a recursive argument on the sequence $t_\ell(s)$.

Experimental results

Research questions

• RQ1 What is the exact formula for the intermediate dimension $\dim_\theta \Lambda$ of a Bedford–McMullen carpet $\Lambda$ for all $\theta \in [0,1]$?
• RQ2 How does the function $\theta \mapsto \dim_\theta \Lambda$ behave—specifically, is it analytic, strictly increasing, and strictly concave between phase transitions?
• RQ3 What is the relationship between intermediate dimensions and the multifractal spectra of uniform Bernoulli measures on the carpet?
• RQ4 Under what conditions do two Bedford–McMullen carpets have equal intermediate dimensions?
• RQ5 Is the equality of Hausdorff multifractal spectra a necessary condition for two such carpets to be bi-Lipschitz equivalent?

Key findings

• The intermediate dimension $\dim_\theta \Lambda$ is strictly increasing and analytic on each interval between countably many phase transitions, with the function being strictly concave on each such interval.
• A precise formula for $\dim_\theta \Lambda$ is derived in terms of a large deviations rate function $I(t)$, which governs the asymptotic scaling of cylinder counts in the carpet.
• Two Bedford–McMullen carpets have equal intermediate dimensions if and only if the Hausdorff multifractal spectra of their uniform Bernoulli measures are equal.
• The equality of intermediate dimensions implies that the rate functions $I(t)$ and $I'(t - \gamma \log(M'/M))$ are related by a shift, which in turn forces a precise scaling relation between the fiber counts and weights of the carpets.
• The intermediate dimension function exhibits a novel structure with countably many phase transitions, a feature not observed in any previously studied fractal sets.
• As a consequence of the above, the equality of multifractal spectra is a necessary condition for bi-Lipschitz equivalence of carpets with non-uniform vertical fibers.