Fractal geometry of Bedford-McMullen carpets

Fraser, Jonathan M.

doi:10.48550/arxiv.2008.10555

Cited by 4 publications

(6 citation statements)

References 24 publications

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“…where O (i,j) is an element of the collection of rotations of 0 • , 180 • around the point (1/2, 1/2) and flips along the lines x = 1/2 or y = 1/2. When O (i,j) = id for all (i, j) ∈ D, the attractor associated with the IFS {ψ (i,j) : (i, j) ∈ D} is called a Barański carpet (please see [1]); if we further have a p ≡ n −1 and b q ≡ m −1 then the attractor is called a Bedford-McMullen carpet (please see [13]). In this case, it is easy to see that the intersection of every pair of level-k rectangles is their largest common face and our arguments work in this setting.…”

Section: Further Remarksmentioning

confidence: 99%

An intersection problem of graph-directed attractors and an application

Ruan¹,

Xiao²

2022

Preprint

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Let (K1, . . . , Kn) be a Cantor type graph-directed attractors in R d . By creating an auxiliary graph, we provide an effective criterion of whether Ki ∩ Kj is empty for every pair of 1 ≤ i, j ≤ n. Moreover, the emptiness can be checked by examining only a finite number of the attractor's geometric approximations. The method is also applicable for more general graph-directed systems. As an application, we are able to determine the connectedness of all d-dimensional generalized Sierpiński sponges of which the corresponding IFSs are allowed to have rotational and reflectional components.

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Section: Further Remarksmentioning

confidence: 99%

An intersection problem of graph-directed attractors and an application

Ruan¹,

Xiao²

2022

Preprint

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“…There are explicit formulas for the box, packing, Hausdorff, Assouad and lower dimensions of Bedford-McMullen sets, while the box dimension and the packing dimension always coincide. For example, see theorem 2.1 in [11] for these formulas. However, in general, it is not easy to obtain the intermediate dimension of Bedford-McMullen sets (see [1]).…”

Section: Application To Lipschitz Equivalencementioning

confidence: 99%

“…Given a Bedford-McMullen set E, we say E has uniform fibres if all non-empty rows contain the same number of rectangles. Notice that there is a dichotomy (see [11]), in the uniform fibres case,…”

Section: Application To Lipschitz Equivalencementioning

confidence: 99%

Gap sequences and Topological properties of Bedford–McMullen sets*

2022

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In this paper, we study the topological properties and the gap sequences of Bedford–McMullen sets. First, we introduce a topological condition, the component separation condition (CSC), and a geometric condition, the exponential rate condition (ERC). Then we prove that the CSC implies the ERC, and that both of them are sufficient conditions for obtaining the asymptotic estimate of gap sequences. We also explore topological properties of Bedford–McMullen sets and prove that all normal Bedford–McMullen sets with infinitely many connected components satisfy the CSC, from which we obtain the asymptotic estimate of the gap sequences of Bedford–McMullen sets without any restrictions. Finally, we apply our result to Lipschitz equivalence.

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“…• A well-studied class of fractals are the self-affine carpets of Bedford and McMullen; for the definition and a survey of the dimension theory of such carpets, see [8]. If F is any Bedford-McMullen carpet with non-uniform fibres then 0 Section 4] and [19], but a precise formula remains elusive.…”

Section: The Intermediate Dimensionsmentioning

confidence: 99%

“…For some (but not all) Bedford-McMullen carpets, Proposition 3.10 gives the best lower bound for the intermediate dimensions for θ close to 1 that is known at the time of writing. Indeed, using the notation in [8,19] We have already seen that for sets such as F with 0 < dim B F < dim A F , Proposition 3.10 improves both general lower bounds [6, Proposition 2.4] and [5, (2.6)], and indeed in this case the gradient at θ = 1 of the former is dim A F − dim B F ≈ 0.148, and the gradient of the latter is dim B F ≈ 1.852. This example suggests that, at least for some carpets such as this one, Kolossváry's upper bound [19,Theorem 1.2] is closer to capturing the true behaviour of the intermediate dimensions than his lower bound.…”

Section: The Intermediate Dimensionsmentioning

confidence: 99%

Generalised intermediate dimensions

Amlan¹

2020

Preprint

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We introduce a family of dimensions, which we call the Φ-intermediate dimensions, that lie between the Hausdorff and box dimensions and generalise the intermediate dimensions introduced by Falconer, Fraser and Kempton. We do this by restricting the allowable covers in the definition of Hausdorff dimension, but in a wider variety of ways than in the definition of the intermediate dimensions. We also extend the theory from Euclidean space to a wider class of metric spaces. We investigate relationships between the Φ-intermediate dimensions and other notions of dimension, and study many analytic and geometric properties of the dimensions. We prove continuity-like results which improve similar results for the intermediate dimensions and give a sharp general lower bound for the intermediate dimensions that is positive for all θ ∈ (0, 1] for sets with positive box dimension. We prove Hölder distortion estimates which imply bi-Lipschitz stability for the Φ-intermediate dimensions. We prove a mass distribution principle and Frostman type lemma, and use these to study dimensions of product sets, and to show that the lower versions of the dimensions, unlike the the upper versions, are not finitely stable. We show that for any compact subset of an appropriate space, these dimensions can be used to 'recover the interpolation' between the Hausdorff and box dimensions of sets for which the intermediate dimensions are discontinuous at θ = 0, thus providing more refined geometric information about such sets.

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Fractal geometry of Bedford-McMullen carpets

Cited by 4 publications

References 24 publications

An intersection problem of graph-directed attractors and an application

An intersection problem of graph-directed attractors and an application

Gap sequences and Topological properties of Bedford–McMullen sets*

Generalised intermediate dimensions

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