Fractal Geometry of Bedford-McMullen Carpets

Fraser, Jonathan M.

doi:10.1007/978-3-030-74863-0_15

Cited by 10 publications

(9 citation statements)

References 51 publications

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“…noting that dim θ F log = dim B F log = dim A F log = 1 for θ ∈ (0, 1] using (4.5) and Propositions 3.3 and 3.4. [22] independently, see also [23], and these carpets have been widely studied and generalised, see [5,14] and references therein.…”

Section: Examples Of Different Behavioursmentioning

confidence: 99%

Intermediate dimensions

2019

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We introduce a continuum of dimensions which are 'intermediate' between the familiar Hausdorff and box dimensions. This is done by restricting the families of allowable covers in the definition of Hausdorff dimension by insisting that |U | ≤ |V | θ for all sets U, V used in a particular cover, where θ ∈ [0, 1] is a parameter. Thus, when θ = 1 only covers using sets of the same size are allowable, and we recover the box dimensions, and when θ = 0 there are no restrictions, and we recover Hausdorff dimension.We investigate many properties of the intermediate dimension (as a function of θ), including proving that it is continuous on (0, 1] but not necessarily continuous at 0, as well as establishing appropriate analogues of the mass distribution principle, Frostman's lemma, and the dimension formulae for products. We also compute, or estimate, the intermediate dimensions of some familiar sets, including sequences formed by negative powers of integers, and Bedford-McMullen carpets.Mathematics Subject Classification 2010 : primary: 28A80; secondary: 37C45.

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Section: Examples Of Different Behavioursmentioning

confidence: 99%

Intermediate dimensions

2019

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“…Secondly, it is a natural question (see e.g. [10,Question 8.3]) if Theorem 1.2 may be upgraded to a Marstrand-like result of the form…”

Section: Background and Main Resultsmentioning

confidence: 99%

“…However, the best possible upper bounds remain unknown (see e.g. Fraser's question [9,Question 8.3]).…”

Section: Introductionmentioning

confidence: 99%

Improved Versions of Some Furstenberg Type Slicing Theorems for Self-Affine Carpets

2021

International Mathematics Research Notices

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Let $F$ be a Bedford–McMullen carpet defined by independent integer exponents. We prove that for every line $\ell \subseteq \mathbb{R}^2$ not parallel to the major axes, $$\begin{align*} & \dim_H (\ell \cap F) \leq \max \left\lbrace 0,\, \frac{\dim_H F}{\dim^* F} \cdot (\dim^* F-1) \right\rbrace\end{align*}$$and $$\begin{align*} & \dim_P (\ell \cap F) \leq \max \left\lbrace 0,\, \frac{\dim_P F}{\dim^* F} \cdot (\dim^* F-1) \right\rbrace,\end{align*}$$where $\dim ^*$ is Furstenberg’s star dimension (maximal dimension of microsets). This improves the state-of-the-art results on Furstenberg type slicing Theorems for affine invariant carpets.

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“…Bedford-McMullen carpets are selfaffine sets on the plane introduced independently by Bedford [8] and McMullen [37]. There is an abundant amount of literature on them, we refer to the recent survey [19] and references therein, thanks to the simplicity of their construction and at the same time exhibiting many interesting features. Here we show another such interesting feature regarding the optimisation problem min…”

Section: Application To Bedford-mcmullen Carpetsmentioning

confidence: 99%

On the Convergence Rate of the Chaos Game

Bárány

Jurga

Kolossváry

2022

International Mathematics Research Notices

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This paper studies how long it takes the orbit of the chaos game to reach a certain density inside the attractor of a strictly contracting IFS of which we only assume that its lower dimension is positive. We show that the rate of growth of this cover time is determined by the Minkowski dimension of the push-forward of the shift invariant measure with exponential decay of correlations driving the chaos game. Moreover, we bound the expected value of the cover time from above and below with multiplicative logarithmic correction terms. As an application, for Bedford–McMullen carpets, we completely characterise the family of probability vectors that minimise the Minkowski dimension of Bernoulli measures. Interestingly, these vectors have not appeared in any other aspect of Bedford–McMullen carpets before.

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Fractal Geometry of Bedford-McMullen Carpets

Cited by 10 publications

References 51 publications

Intermediate dimensions

Intermediate dimensions

Improved Versions of Some Furstenberg Type Slicing Theorems for Self-Affine Carpets

On the Convergence Rate of the Chaos Game

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