Some Novel Types of Fractal Geometry [Hardcover]

This book deals with fractal geometries that have features similar to ones of ordinary Euclidean spaces, while at the same time being quite different from Euclidean spaces.. A basic example of this feature considered is the presence of Sobolev or Poincar? inequalities, concerning the relationship between the average behavior of a function and the average behavior of its small-scale oscillations. Remarkable results in the last few years through Bourdon-Pajot and Laakso have shown that there is much more in the way of geometries like this than have been realized, only examples related to nilpotent Lie groups and Carnot metrics were known previously. On the other had, 'typical' fractals that might be seen in pictures do not have these same kinds of features. This text examines these topics in detail and will interest graduate students as well as researchers in mathematics and various aspects of geometry and analysis.

1. Introduction
2. Some background material
3. A few basic topics
4. Deformations
5. Mapping between spaces
6. Some more general topics
7. A class of constructions to consider
8. Geometric structures and some topological configurations
Appendix. Some side comments
References
Index

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 dirlS+