ابعاد هموار: نظریه و کاربرد (سری سخنرانی‌های دانشگاهی)

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. 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 John M. Mackay, Jeremy T. Tyson. Includes Bibliographical References (p. 131-138) And Index.