Doubling property of self-affine measures on carpets of Bedford and McMullen

Li, Hui; Wei, Chun; Wen, Sheng You

doi:10.1512/iumj.2016.65.5826

Cited by 9 publications

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“…In stark contrast to the self-similar case, there exist self-affine sets satisfying the OSC such that all associated self-affine measures fail to be doubling. This was first established by Li, Wei and Wen [190]. In fact, Bedford Crucially, the approximate squares Π(Q(d, R)) and Π(Q(d , R)) are adjacent in that they share a common edge.…”

Section: Dimensions Of Self-affine Measuresmentioning

confidence: 90%

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: Dimensions Of Self-affine Measuresmentioning

confidence: 90%

Assouad Dimension and Fractal Geometry

Fraser

2020

111

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“…and it is called the uniform Bernoulli measure of E. (ii) A measure ν on a metric space X is said to be doubling if there is a constant C ≥ 1 such that 0 < ν(B(x, 2r)) ≤ Cν(B(x, r)) < ∞ for all balls B(x, r) ⊂ X with center x and radius r > 0. According to Li, Wei and Wen [12], µ E is doubling if and only if either a 0 a m−1 = 0, or a j a j+1 = 0 for all j = 0, . .…”

Section: Introductionmentioning

confidence: 99%

Lipschitz classification of Bedford-McMullen carpets with uniform horizontal fibers

Yang

Zhang

2020

Preprint

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Green’s Function and Pointwise Space-time Behaviors of the Vlasov-Poisson-Boltzmann System

Yang

Zhong

2019

Arch Rational Mech Anal

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The pointwise space-time behavior of the Green's function of the one-dimensional Vlasov-Maxwell-Boltzmann (VMB) system is studied in this paper. It is shown that the Green's function consists of the macroscopic diffusive waves and Huygens waves with the speed ± 5 3 at low-frequency, the hyperbolic waves with the speed ±1 at high-frequency, the singular kinetic and leading short waves, and the remaining term decaying exponentially in space and time. Note that these high-frequency hyperbolic waves are completely new and can not be observed for the Boltzmann equation and the Vlasov-Poisson-Boltzmann system. In addition, we establish the pointwise space-time estimate of the global solution to the nonlinear VMB system based on the Green's function. Compared to the Boltzmann equation and the Vlasov-Poisson-Boltzmann system, some new ideas are introduced to overcome the difficulties caused by the coupling effects of the transport of particles and the rotating of electro-magnetic fields, and investigate the new hyperbolic waves and singular leading short waves.

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Doubling property of self-affine measures on carpets of Bedford and McMullen

Cited by 9 publications

References 0 publications

Assouad Dimension and Fractal Geometry

Assouad Dimension and Fractal Geometry

Lipschitz classification of Bedford-McMullen carpets with uniform horizontal fibers

Green’s Function and Pointwise Space-time Behaviors of the Vlasov-Poisson-Boltzmann System

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