# Volume Doubling Measures and Heat Kernel Estimates on Self-Similar Sets

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*Jun Kigami*

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?

These three problems turn out to be closely related. The author
introduces 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 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 Li-Yau type
sub-Gaussian diagonal estimate of a heat kernel.