Assouad Dimension and Fractal Geometry

Copyright
Contents
Preface
Acknowledgements
PART ONE THEORY
	1 Fractal Geometry and Dimension Theory
		1.1 The Emergence of Fractal Geometry
		1.2 Dimension Theory
	2 The Assouad Dimension
		2.1 The Assouad Dimension and a Simple Example
		2.2 A Word or Two on the Definition
		2.3 Some History
		2.4 Basic Properties: The Greatest of All Dimensions
	3 Some Variations on the Assouad Dimension
		3.1 The Lower Dimension
		3.2 The Quasi-Assouad Dimension
		3.3 The Assouad Spectrum
		3.4 Basic Properties: Revisited
	4 Dimensions of Measures
		4.1 Assouad and Lower Dimensions of Measures
		4.2 Assouad Spectrum and Box Dimensions of Measures
	5 Weak Tangents and Microsets
		5.1 Weak Tangents and the Assouad Dimension
		5.2 Weak Tangents for the Lower Dimension?
		5.3 Weak Tangents for Spectra?
		5.4 Weak Tangents for Measures?
PART TWO EXAMPLES
	6 Iterated Function Systems
		6.1 IFS Attractors and Symbolic Representation
		6.2 Invariant Measures
		6.3 Dimensions of IFS Attractors
		6.4 Ahlfors Regularity and Quasi-Self-Similarity
	7 Self-Similar Sets
		7.1 Self-Similar Sets and the Hutchinson–Moran Formula
		7.2 The Assouad Dimension of Self-Similar Sets
		7.3 The Assouad Spectrum of Self-Similar Sets
		7.4 Dimensions of Self-Similar Measures
	8 Self-Affine Sets
		8.1 Self-Affine Sets and Two Strands of Research
		8.2 Falconer’s Formula and the Affinity Dimension
		8.3 Self-Affine Carpets
		8.4 Self-Affine Sets with a Comb Structure
		8.5 A Family of Worked Examples
		8.6 Dimensions of Self-Affine Measures
	9 Further Examples: Attractors and Limit Sets
		9.1 Self-Conformal Sets
		9.2 Invariant Sets for Parabolic Interval Maps
		9.3 Limit Sets of Kleinian Groups
		9.4 Mandelbrot Percolation
	10 Geometric Constructions
		10.1 Products
		10.2 Orthogonal 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
		13.2 The Spiral Winding Problem
		13.3 Almost Bi-Lipschitz Embeddings
	14 Applications in Number Theory
		14.1 Arithmetic Progressions
		14.2 Diophantine Approximation
		14.3 Definability of the Integers
	15 Applications in Probability Theory
		15.1 Uniform Dimension Results for Fractional Brownian Motion
		15.2 Dimensions of Random Graphs
	16 Applications in Functional Analysis
		16.1 Hardy Inequalities
		16.2 Lp → Lq Bounds for Spherical Maximal Operators
		16.3 Connection with Lp-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
		17.6 Random Sets
		17.7 General Behaviour of the Assouad Spectrum
		17.8 Projections
		17.9 Distance Sets
		17.10 The Holder Mapping Problem and Dimension
		17.11 Dimensions of Graphs
References
List of Notation
Index