Conformal Dimension: Theory and Application
About this Title
John M. Mackay, University of Illinois at Urbana-Champaign, Urbana, IL and Jeremy T. Tyson, University of Illinois at Urbana-Champaign, Urbana, IL
Publication: University Lecture Series
Publication Year 2010: Volume 54
ISBNs: 978-0-8218-5229-3 (print); 978-1-4704-1649-2 (online)
MathSciNet review: MR2662522
MSC: Primary 30L10; Secondary 28A78, 28A80, 37F35
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.
Graduate students and research mathematicians interested in geometric function theory.
Table of Contents
- Chapter 1. Background material
- Chapter 2. Conformal gauges and conformal dimension
- Chapter 3. Gromov hyperbolic groups and spaces and their boundaries
- Chapter 4. Lower bounds for conformal dimension
- Chapter 5. Sets and spaces of conformal dimension zero
- Chapter 6. Gromov–Hausdorff tangent spaces and conformal dimension
- Chapter 7. Ahlfors regular conformal dimension
- Chapter 8. Global quasiconformal dimension