Geometry of self-similar measures on intervals with overlaps and applications to sub-Gaussian heat kernel estimates

Abstract

We study the geometric properties of self-similar measures on intervals generated by iterated function systems (IFS's) that do not satisfy the open set condition (OSC) and have overlaps. The examples studied in this paper are the infinite Bernoulli convolution associated with the golden ratio, and a family of convolutions of Cantor-type measures. We make use of Strichartz second-order identities defined by auxiliary IFS's to compute measures of cells on different levels. These auxiliary IFS's do satisfy the OSC and are used to define new metrics. As an application, we obtain sub-Gaussian heat kernel estimates of the time changed Brownian motions with respect to these measures. The walk dimensions obtained under these new metrics are strictly greater than $ 2 $ and are closely related to the spectral dimension of fractal Laplacians.