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 dim θ Λ of any Bedford-McMullen carpet Λ for the whole spectrum of θ ∈ [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 → 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.

Assouad type dimensions of infinitely generated self-conformal sets

“…As the main result of [3], we proved that the upper intermediate dimensions dim θ also satisfy the naïve formula, that is, dim θ F = max{dim H F, dim θ {S i (x)}} for all θ ∈ [0, 1]. The intermediate dimensions are a family of dimensions (introduced in [10] and studied further in [2,4,5,6]) which are parameterised by θ ∈ [0, 1] and interpolate between the Hausdorff and box dimensions. We provided some (often counterintuitive) applications to the dimensions of projections, fractional Brownian images, and general Hölder images.…”

Attainable forms of intermediate dimensions

“…Moreover, intermediate dimensions can distinguish bi-Lipschitz equivalence even when other notions of dimension give no information [3]. In fact, quantitative information about Hölder exponents can be obtained for any bi-Hölder map between sets with distinct intermediate dimensions [5,Thm.…”

“…On the other hand, intermediate dimensions have been computed for some specific families of sets. For example, Bedford-McMullen carpets are a recent example of a natural family of sets for which the intermediate dimensions exhibit interesting properties [3]. Other sets which have been studied in the literature include infinitely generated self-conformal sets [2] and elliptical polynomial spirals [6].…”

Attainable forms of intermediate dimensions

“…Moreover, intermediate dimensions can distinguish bi-Lipschitz equivalence even when other notions of dimension give no information [3]. In fact, quantitative information about Hölder exponents can be obtained for any bi-Hölder map between sets with distinct intermediate dimensions [5,Thm.…”

“…On the other hand, intermediate dimensions have been computed for some specific families of sets. For example, Bedford-McMullen carpets are a recent example of a natural family of sets for which the intermediate dimensions exhibit interesting properties [3]. Other sets which have been studied in the literature include infinitely generated self-conformal sets [2] and elliptical polynomial spirals [6].…”