Conformal dimension via p-resistance: Sierpinski carpet

Kwapisz, Jarosław

doi:10.5186/aasfm.2020.4515

Cited by 4 publications

(4 citation statements)

References 22 publications

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“…The fact that Cdim A F < dim A F is due to Keith and Laakso [164, Corollary 1.0.5], see also [195,Example 6.2.3]. For other more recent developments on this problem, see [169,176]. Rossi and Suomala [243] investigated the conformal Hausdorff dimension of the Mandelbrot percolation sets, recall Section 9.4.…”

Section: Lowering the Assouad Dimension By Quasi-symmetrymentioning

confidence: 99%

Assouad Dimension and Fractal Geometry

Fraser

2020

111

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Dimension theory of orthogonal projections vi Contents 10.2.2 The Assouad dimension of orthogonal projections 10.2.3 An application to the box dimensions of projections 10.2.4 The lower dimension and projections 10.3 Slices and intersections 11 Two famous problems in geometric measure theory 11.1 Distance sets 11.2 Kakeya sets 12 Conformal dimension 12.1 Lowering the Assouad dimension by quasi-symmetry PART THREE APPLICATIONS 13 Applications in embedding theory 13.1 Assouad's embedding theorem 13.1.1 Doubling and uniformly perfect metric spaces 13.1.2 Assouad's embedding theorem 13.2 The spiral winding problem 13.3 Almost bi-Lipschitz embeddings 14 Applications in number theory 14.1 Arithmetic progressions 14.1.1 Discrete Kakeya sets 14.2 Diophantine approximation 14.3 Definability of the integers 15 Applications in probability theory 15.1 Uniform dimension results 15.2 Dimensions of random graphs 16 Applications in functional analysis 16.1 Hardy inequalities 16.2 L p → L q bounds for spherical maximal operators 16.3 Connection with L p -norms 17 Future directions 17.1 Finite stability of modified lower dimension 17.2 Dimensions of measures 17.3 Weak tangents 17.4 Further questions of measurability 17.5 IFS attractors Contents vii 17.6 Random sets 17.7 General behaviour of the Assouad spectrum 17.8 Projections 17.9 Distance sets 17.10 The Hölder mapping problem and dimension 17.11 Dimensions of graphs References List of notation Index PART ONE THEORY 1 Fractal geometry and dimension theoryIn this introductory chapter we briefly discuss the history and development of fractal geometry and dimension theory. We introduce and motivate some important concepts such as Hausdorff and box dimension. As part of this discussion we encounter covers and packings, which are central notions in dimension theory, and introduce the dimension theory of measures. The emergence of fractal geometryA fractal can be described as an object which exhibits interesting features on a large range of scales, see Figure 1.1. In pure mathematics, the Sierpiński triangle, the middle third Cantor set, the boundary of the Mandelbrot set, and the von Koch snowflake are archetypal examples and, in 'real life', examples include the surface of a lung, the horizon of a forest, and the distribution of stars in the galaxy. The fractal story began in the nineteenth century with the appearance of a multitude of strange examples exhibiting what we now understand as fractal phenomena. These included Weierstrass' example of a continuous nowhere differentiable function, Cantor's construction of an uncountable set with zero length, and Brown's observations on the path taken by a piece of pollen suspended in water (Brownian motion). During the first half of the twentieth century the mathematical foundations for fractal geometry were laid down by, for example, Besicovitch, Bouligand, Hausdorff, Julia, Marstrand and Sierpiński, and the theory was unified and popularised by the extensive writings of Mandelbrot in the 1970s, for example [197]. It was Mandelbrot who co...

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Section: Lowering the Assouad Dimension By Quasi-symmetrymentioning

confidence: 99%

Assouad Dimension and Fractal Geometry

Fraser

2020

111

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“…As an example, it is known that cdim AR (S) is strictly less than the Hausdorff dimension of the standard Sierpiński carpet S [35], but not whether the infimum in (1.1) is achieved by some Ahlfors regular space Y when X = S. See [11,42] for additional details. (Finding the exact value of cdim AR (S) is also a well-known open problem; see, e.g., [38] for recent progress. )…”

Section: Conformal Dimensionmentioning

confidence: 99%

Infinitesimal splitting for spaces with thick curve families and Euclidean embeddings

David,

Eriksson-Bique

2024

Annales De l'Institut Fourier

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We study metric measure spaces that admit "thick" families of rectifiable curves or curve fragments, in the form of Alberti representations or curve families of positive modulus. We show that such spaces cannot be bi-Lipschitz embedded into any Euclidean space unless they admit some "infinitesimal splitting": their tangent spaces are bi-Lipschitz equivalent to product spaces of the form Z×R k for some k ⩾ 1. We also provide applications to conformal dimension and give new proofs of some previously known non-embedding results.Résumé. -On étudie des espaces métriques mesurés qui possèdent des familles "épaisses" de courbes rectifiables ou de fragments de courbes, sous la forme de représentations d'Alberti ou de familles de courbes de module strictement positif. On montre que de tels espaces ne possèdent pas de plongement bi-lipschitzien dans un espace euclidien, sauf s'ils admettent une "décomposition infinitésimale": leurs espaces tangents sont bi-lipschitz équivalents à des produits d'espaces de la forme Z × R k pour un certain k ⩾ 1. On donne aussi des applications à la dimension conforme et de nouvelles preuves de certains résultats de non plongement déjà connus.

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“…We conjecture that lim n→+∞ CD(X n , d E ) = 2. Observe that it is not even known the value CD(X 1 , d E ), the conformal dimension of the standard Sierpinski carpet: see [Kwa20] for the most recent results about this topic.…”

Section: The Following Two Questions Remain Openmentioning

confidence: 99%

Ahlfors regular conformal dimension and Gromov-Hausdorff convergence

Cavallucci¹

2022

Preprint

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We prove that the conformal dimension is upper semicontinuous with respect to Gromov-Hausdorff convergence when restricted to the class of uniformly perfect, uniformly quasi-selfsimilar metric spaces. A corollary is the upper semicontinuity of the conformal dimension of limit sets of discrete, quasiconvex-cocompact group of isometries of uniformly bounded codiameter of δ-hyperbolic metric spaces under equivariant pointed Gromov-Hausdorff convergence of the spaces.

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Conformal dimension via p-resistance: Sierpinski carpet

Cited by 4 publications

References 22 publications

Assouad Dimension and Fractal Geometry

Assouad Dimension and Fractal Geometry

Infinitesimal splitting for spaces with thick curve families and Euclidean embeddings

Ahlfors regular conformal dimension and Gromov-Hausdorff convergence

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