Volume doubling measures and heat kernel estimates on self-similar sets

Abstract

This paper studies the following three problems. 1. When does a measure on a self-similar set have the volume doubling property with respect to a given distance? 2. Is there any distance on a self-similar set under which the contraction mappings have the prescribed values of contractions ratios? 3. When does a heat kernel on a self-similar set associated with a self-similar Dirichlet form satisfy the Li-Yau type sub-Gaussian diagonal estimate? Those three problems turns out to be closely related. We introduce a new class of self-similar set, called rationally ramified self-similar sets containing both the Sierpinski gasket and the (higher dimensional) Sierpinski carpet and give complete solutions of the above three problems for this class. In particular, the volume doubling property is shown to be equivalent to the upper Li-Yau type sub-Gaussian diagonal estimate of a heat kernel. Received by the editor May 26, 2005; and in revised form September 18, 2006. 2000 Mathematics Subject Classification. Primary 28A80, 60J35; Secondary 31C25, 60J45.