Assouad dimension and fractal geometry

Assouad dimension and fractal geometry /

"This book provides a thorough treatment of the Assouad dimension, as well as its many variants, in the context of fractal geometry. The book is split into three parts. In the first part, the basic theory is set up including how the various dimensions relate to each other and how they behave un...

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Bibliographic Details
Main Author: Fraser, Jonathan M., 1987- (Author)
Format: Book
Language:English
Published: Cambridge, United Kingdom ; New York, NY : Cambridge University Press, 2021.
Series:Cambridge tracts in mathematics ; 222.
Subjects:
Fractals.
Dimension theory (Topology)
fractals.
Table of Contents:
  • Machine generated contents note: pt. 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?
  • pt. 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
  • pt. 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 [₂] 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.

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