The present book deals with fractal geometries which have features similar to ones of ordinary Euclidean spaces, while at the same time being quite different from Euclidean spaces in other ways. A basic type of feature being considered is the presence of Sobolev or Poincare inequalities, concerning the relationship between the average behaviour of a function and the average behaviour of its small-scale oscillations. Remarkable results in the last few years of Bourdon-Pajot and Laakso have shown that there is much more in the way of geometries like this than has been realized. Examples related to nilpotent Lie groups and Carnot metrics were known previously. On the other hand, 'typical' fractals that might be seen in pictures do not have these same kinds of features. 'Some Novel Types of Fractal Geometry' will be of interest to graduate students and researchers in mathematics, working in various aspects of geometry and analysis.
1: Introduction 2: Some background material 3: A few basic topics 4: Deformations 5: Mappings between spaces 6: Some more general topics 7: A class of constructions to consider 8: Geometric structures and some topological configurations Appendix A. A few side comments References Index
Professor Stephen Semmes, Mathematics Department, Rice University, Houston.
Reviews for Some Novel Types of Fractal Geometry
The purpose of the book under review is to present perspectives for the development of the theory of spaces that have 'decent calculus' like the spaces supporting Poincar inequalities. . . .The book is written in a very informal style. There are almost no theorems or proofs, just questions and elaborated comments. One of the features of most of the mathematical books and papers is that the reader is forced to spend hours on checking painful details in order to follow the text. However, a consequence of the informal style of Semmes is that there is no such need here. On the contrary one can enjoy reading the whole book in a couple of evenings without any danger of being exhausted! The purpose of the book is to suggest possible directions of research in the fascinating area of analysis on metric spaces. Everybody doing research in this area should read the book. -- MathematicalReviews [A] rich overview of this recent and fascinating subject at the crossroad between analysis and geometry. The great quality of this book may be found in the many open-ended directions of research that it suggests and explores. --Bulletin of the London Mathematical Society