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 Poincare 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: 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.