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E-raamat: Assouad Dimension and Fractal Geometry

(University of St Andrews, Scotland)
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Assouad Dimension and Fractal GeometrySuurem pilt
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The Assouad dimension is a notion of dimension in fractal geometry that has been the subject of much interest in recent years. This book, written by a world expert on the topic, is the first thorough account of the Assouad dimension and its many variants and applications in fractal geometry and beyond. It places the theory of the Assouad dimension in context among up-to-date treatments of many key advances in fractal geometry, while also emphasising its diverse connections with areas of mathematics including number theory, dynamical systems, harmonic analysis, and probability theory. A final chapter detailing open problems and future directions for research brings readers to the cutting edge of this exciting field. This book will be an indispensable part of the modern fractal geometer's library and a valuable resource for pure mathematicians interested in the beauty and many applications of the Assouad dimension.

This book is the first thorough treatment of the Assouad dimension in fractal geometry. Aimed at researchers and graduate students, it will have broad appeal among pure mathematicians due to its discussion of the Assouad dimension's many applications to number theory, dynamical systems, harmonic analysis, and probability theory.

Arvustused

'The book is very well written and illustrated. The reader gets to know an almost complete spectrum of resent results and historical developments concerning Assouad dimension.' Jörg Neunhäuserer, European Mathematical Society

Muu info

The first thorough treatment of the Assouad dimension in fractal geometry, with applications to many fields within pure mathematics.
Preface xi
Acknowledgements xv
PART ONE THEORY
1(78)
1 Fractal Geometry and Dimension Theory
3(7)
1.1 The Emergence of Fractal Geometry
3(2)
1.2 Dimension Theory
5(5)
2 The Assouad Dimension
10(12)
2.1 The Assouad Dimension and a Simple Example
10(3)
2.2 A Word or Two on the Definition
13(2)
2.3 Some History
15(2)
2.4 Basic Properties: The Greatest of All Dimensions
17(5)
3 Some Variations on the Assouad Dimension
22(34)
3.1 The Lower Dimension
22(2)
3.2 The Quasi-Assouad Dimension
24(1)
3.3 The Assouad Spectrum
25(12)
3.4 Basic Properties: Revisited
37(19)
4 Dimensions of Measures
56(8)
4.1 Assouad and Lower Dimensions of Measures
56(4)
4.2 Assouad Spectrum and Box Dimensions of Measures
60(4)
5 Weak Tangents and Microsets
64(15)
5.1 Weak Tangents and the Assouad Dimension
64(8)
5.2 Weak Tangents for the Lower Dimension?
72(1)
5.3 Weak Tangents for Spectra?
73(2)
5.4 Weak Tangents for Measures?
75(4)
PART TWO EXAMPLES
79(118)
6 Iterated Function Systems
81(12)
6.1 IFS Attractors and Symbolic Representation
81(3)
6.2 Invariant Measures
84(2)
6.3 Dimensions of IFS Attractors
86(2)
6.4 Ahlfors Regularity and Quasi-Self-Similarity
88(5)
7 Self-Similar Sets
93(17)
7.1 Self-Similar Sets and the Hutchinson-Moran Formula
93(2)
7.2 The Assouad Dimension of Self-Similar Sets
95(7)
7.3 The Assouad Spectrum of Self-Similar Sets
102(3)
7.4 Dimensions of Self-Similar Measures
105(5)
8 Self-Affine Sets
110(27)
8.1 Self-Affine Sets and Two Strands of Research
110(1)
8.2 Falconer's Formula and the Affinity Dimension
111(3)
8.3 Self-Affine Carpets
114(10)
8.4 Self-Affine Sets with a Comb Structure
124(3)
8.5 A Family of Worked Examples
127(2)
8.6 Dimensions of Self-Affine Measures
129(8)
9 Further Examples: Attractors and Limit Sets
137(23)
9.1 Self-Conformal Sets
137(3)
9.2 Invariant Sets for Parabolic Interval Maps
140(5)
9.3 Limit Sets of Kleinian Groups
145(9)
9.4 Mandelbrot Percolation
154(6)
10 Geometric Constructions
160(21)
10.1 Products
160(6)
10.2 Orthogonal Projections
166(12)
10.3 Slices and Intersections
178(3)
11 Two Famous Problems in Geometric Measure Theory
181(9)
11.1 Distance Sets
181(6)
11.2 Kakeya Sets
187(3)
12 Conformal Dimension
190(7)
12.1 Lowering the Assouad Dimension by Quasi-Symmetry
190(7)
PART THREE APPLICATIONS
197(53)
13 Applications in Embedding Theory
199(16)
13.1 Assouad's Embedding Theorem
200(3)
13.2 The Spiral Winding Problem
203(9)
13.3 Almost Bi-Lipschitz Embeddings
212(3)
14 Applications in Number Theory
215(11)
14.1 Arithmetic Progressions
215(4)
14.2 Diophantine Approximation
219(5)
14.3 Definability of the Integers
224(2)
15 Applications in Probability Theory
226(4)
15.1 Uniform Dimension Results for Fractional Brownian Motion
226(3)
15.2 Dimensions of Random Graphs
229(1)
16 Applications in Functional Analysis
230(7)
16.1 Hardy Inequalities
230(2)
16.2 Lp → Lq Bounds for Spherical Maximal Operators
232(2)
16.3 Connection with Lp-Norms
234(3)
17 Future Directions
237(13)
17.1 Finite Stability of Modified Lower Dimension
237(1)
17.2 Dimensions of Measures
237(1)
17.3 Weak Tangents
238(1)
17.4 Further Questions of Measurability
239(1)
17.5 IFS Attractors
240(2)
17.6 Random Sets
242(1)
17.7 General Behaviour of the Assouad Spectrum
243(2)
17.8 Projections
245(1)
17.9 Distance Sets
246(1)
17.10 The Holder Mapping Problem and Dimension
247(1)
17.11 Dimensions of Graphs
248(2)
References 250(14)
List of Notation 264(3)
Index 267
Jonathan M. Fraser is a Reader in Mathematics at the University of St Andrews. He works in fractal geometry and related areas.
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