Assouad Dimension And The Open Set Condition
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Assouad Dimension and the Open Set Condition
The Assouad dimension is a measure of the complexity of a fractal set similar to the box counting dimension, but with an additional scaling requirement. In this thesis, we generalize Moran's open set condition and introduce a notion called grid like which allows us to compute upper bounds and exact values for the Assouad dimension of certain fractal sets that arise as the attractors of self-similar iterated function systems. Then for an arbitrary fractal set [Special characters omitted.] , we explore the question of whether the Assouad dimension of the set of differences [Special characters omitted.] obeys any bound related to the Assouad dimension of [Special characters omitted.] . This question is of interest, as infinite dimensional dynamical systems with attractors possessing sets of differences of finite Assouad dimension allow embeddings into finite dimensional spaces without losing the original dynamics. We find that even in very simple, natural examples, such a bound does not generally hold. This result demonstrates how a natural phenomenon with a simple underlying structure has the potential to be difficult to measure.
Assouad Dimension and Fractal Geometry
Author: Jonathan M. Fraser
language: en
Publisher: Cambridge University Press
Release Date: 2020-10-29
The first thorough treatment of the Assouad dimension in fractal geometry, with applications to many fields within pure mathematics.
Conformal Dimension
Author: John M. Mackay
language: en
Publisher: American Mathematical Soc.
Release Date: 2010
Conformal dimension measures the extent to which the Hausdorff dimension of a metric space can be lowered by quasisymmetric deformations. Introduced by Pansu in 1989, this concept has proved extremely fruitful in a diverse range of areas, including geometric function theory, conformal dynamics, and geometric group theory. This survey leads the reader from the definitions and basic theory through to active research applications in geometric function theory, Gromov hyperbolic geometry, and the dynamics of rational maps, amongst other areas. It reviews the theory of dimension in metric spaces and of deformations of metric spaces. It summarizes the basic tools for estimating conformal dimension and illustrates their application to concrete problems of independent interest. Numerous examples and proofs are provided. Working from basic definitions through to current research areas, this book can be used as a guide for graduate students interested in this field, or as a helpful survey for experts. Background needed for a potential reader of the book consists of a working knowledge of real and complex analysis on the level of first- and second-year graduate courses.