This paper studies the following three problems. 1. When does a measure on a selfsimilar set have the volume doubling property with respect to a given distance? 2. Is there any distance on a selfsimilar set under which the contraction mappings have the prescribed values of contractions ratios? 3. When does a heat kernel on a selfsimilar set associated with a selfsimilar Dirichlet form satisfy the LiYau type subGaussian diagonal estimate? These three problems turn out to be closely related. The author introduces a new class of selfsimilar set, called rationally ramified selfsimilar sets containing both the Sierpinski gasket and the (higher dimensional) Sierpinski carpet and gives complete solutions of the above three problems for this class. In particular, the volume doubling property is shown to be equivalent to the upper LiYau type subGaussian diagonal estimate of a heat kernel. Table of Contents  Prologue
 Scales and volume doubling property of measures
 Construction of distances
 Heat kernel and volume doubling property of measures
 Appendix
 Bibliography
