Self-affine sets with non-compactly supported random perturbations

Jordan, Thomas; Jurga, Natalia

doi:10.5186/aasfm.2014.3948

Cited by 5 publications

(5 citation statements)

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“…In fact, the initial proof in [65] required that the singular values be strictly less than 1/3 but this was relaxed to 1/2 by Solomyak [256], who also noted that 1/2 is the optimal constant, based on an example of Przytycki and Urbański [237]. See [150] for a similar result where measures other than Lebesgue measure L md are used in determining the 'generic result'.…”

Section: Falconer's Formula and The Affinity Dimensionmentioning

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: Falconer's Formula and The Affinity Dimensionmentioning

confidence: 99%

Assouad Dimension and Fractal Geometry

Fraser

2020

111

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“…The affinity dimension can be considered the 'best guess' for the Hausdorff, packing and box-counting dimension of self-affine sets and it is of major current interest to establish exactly when these notions do, or do not, coincide. Jordan, Pollicott and Simon [23], and Jordan and Jurga [22] studied limit sets with random perturbations of the translation ('noise') at every level of the construction, and found the same almost sure coincidence with the affinity dimension. Fraser and Shmerkin [14] considered a Bedford-McMullen construction with random translation vectors that keep the column structure intact.…”

Section: Introductionmentioning

confidence: 76%

The box-counting dimension of random box-like self-affine sets

Troscheit¹

2018

Indiana Univ. Math. J.

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In this paper we study two random analogues of the box-like selfaffine attractors introduced by Fraser, which themselves were generalisations of Sierpiński carpets. We determine the almost sure box-counting dimension for the homogeneous random case (1-variable random), and give a sufficient condition for the almost sure box-counting dimension to be the expectation of the box-counting dimensions of the corresponding deterministic attractors. Furthermore, we find the almost sure box-counting dimension of the random recursive model (∞-variable), which includes affine fractal percolation.Restricting the functions f i : R d → R d to contracting similitudes, one obtains the class of self-similar sets. Much is known about these and we refer the reader to Falconer [6] and the many references contained therein. Here we are specifically concerned with the class of self-affine carpets. These are the attractors of finite IFSs

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“…Therefore there are numerous explicit and nonexplicit situations where Corollary 2.2 provides a precise result, and an affirmative solution to (1.1) in the self-affine setting. For example, a well-known result by Falconer [5] states that s = dim B F ∅ = dim H F ∅ almost surely if one randomises the translation vectors associated with the affine maps, provided the linear parts all have norm strictly bounded above by 1/2, see also [15]. Falconer proved in a subsequent paper that if F ∅ ⊂ R 2 satisfies some separation conditions and contains a connected component not contained in a straight line, then s = dim B F ∅ holds, see [7,Corollary 5].…”

Section: Resultsmentioning

confidence: 99%

“…Establishing precise conditions for the affinity dimension to coincide with dim B F ∅ is a major open problem in fractal geometry and has been the focus of considerable amounts of work, for example [2,5,7,9,11,12,14,15]. Therefore there are numerous explicit and nonexplicit situations where Corollary 2.2 provides a precise result, and an affirmative solution to (1.1) in the self-affine setting.…”

Section: Resultsmentioning

confidence: 99%

The dimensions of inhomogeneous self-affine sets

Burrell¹,

Fraser²

2020

Ann. Acad. Sci. Fenn. Math.

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We prove that the upper box dimension of an inhomogeneous self-affine set is bounded above by the maximum of the affinity dimension and the dimension of the condensation set. In addition, we determine sufficient conditions for this upper bound to be attained, which, in part, constitutes an exploration of the capacity for the condensation set to mitigate dimension drop between the affinity dimension and the corresponding homogeneous attractor. Our work improves and unifies previous results on general inhomogeneous attractors, low-dimensional affine systems, and inhomogeneous self-affine carpets, while providing inhomogeneous analogues of Falconer's seminal results on homogeneous self-affine sets.Mathematics Subject Classification 2010: primary: 28A80.

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Self-affine sets with non-compactly supported random perturbations

Cited by 5 publications

References 10 publications

Assouad Dimension and Fractal Geometry

Assouad Dimension and Fractal Geometry

The box-counting dimension of random box-like self-affine sets

The dimensions of inhomogeneous self-affine sets

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