Porosities of Mandelbrot Percolation

Berlinkov, Artemi; Järvenpää, Esa

doi:10.1007/s10959-019-00895-z

Cited by 3 publications

(3 citation statements)

References 34 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The Assouad dimension behaves rather differently, as seen in the following result of Fraser, Miao and Troscheit [104]. This result can also be derived from earlier work of Berlinkov and Järvenpää on porosity [29]. An immediate and somewhat counter-intuitive consequence of this result Proof Since we condition on M being non-empty we may assume that, for all levels k 1, there exists at least one kept cube Q k ⊆ M k .…”

Section: Mandelbrot Percolationsupporting

confidence: 56%

Assouad Dimension and Fractal Geometry

Fraser

2020

111

View full text Add to dashboard Cite

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...

show abstract

Section: Mandelbrot Percolationsupporting

confidence: 56%

Assouad Dimension and Fractal Geometry

Fraser

2020

111

View full text Add to dashboard Cite

show abstract

“…Many properties of this simple model have been studied, including e.g. its connectivity [10,5,6], its visibility (or behaviour under projections) [1,27,26], its porosity [3,12], path properties [11,7] and very recently its (un-)rectifiability [8].…”

Section: Introductionmentioning

confidence: 99%

Almost sure convergence and second moments of geometric functionals of fractal percolation

Klatt,

Winter

2023

Adv. Appl. Probab.

View full text Add to dashboard Cite

We determine almost sure limits of rescaled intrinsic volumes of the construction steps of fractal percolation in ${\mathbb R}^d$ for any dimension $d\geq 1$ . We observe a factorization of these limit variables which allows one, in particular, to determine their expectations and covariance structure. We also show the convergence of the rescaled expectations and variances of the intrinsic volumes of the construction steps to the expectations and variances of the limit variables, and we give rates for this convergence in some cases. These results significantly extend our previous work, which addressed only limits of expectations of intrinsic volumes.

show abstract

“…Further, Troscheit [Tro15] proved that the Hausdorff, box-counting, and packing dimensions all coincide for these sets, while the Assouad dimension is 'maximal' in some sense. For example, the limit set of Mandelbrot percolation of the d-dimensional unit cube for supercritical probabilities has Assouad dimension d, conditioned on non-extinction, see also Berlinkov and Järvenpää [BJ16]; and Fraser, Miao, and Troscheit [FMT14] for earlier results. The notion of Assouad spectrum, dim θ A for θ ∈ (0, 1), was applied to Mandelbrot percolation by Fraser and Yu [FY16a,FY16b] to obtain more information about its scaling.…”

Section: Introductionmentioning

confidence: 99%

The quasi-Assouad dimension of stochastically self-similar sets

Troscheit¹

2019

Proceedings of the Royal Society of Edinburgh: Section A Mathem

View full text Add to dashboard Cite

The class of stochastically self-similar sets contains many famous examples of random sets, e.g. Mandelbrot percolation and general fractal percolation. Under the assumption of the uniform open set condition and some mild assumptions on the iterated function systems used, we show that the quasi-Assouad dimension of self-similar random recursive sets is almost surely equal to the almost sure Hausdorff dimension of the set. We further comment on random homogeneous and V -variable sets and the removal of overlap conditions.

show abstract

Porosities of Mandelbrot Percolation

Cited by 3 publications

References 34 publications

Assouad Dimension and Fractal Geometry

Assouad Dimension and Fractal Geometry

Almost sure convergence and second moments of geometric functionals of fractal percolation

The quasi-Assouad dimension of stochastically self-similar sets

Contact Info

Product

Resources

About