SELF-SIMILAR MEASURES AND HARMONIC ANALYSIS

Abstract

This is a survey of some recent work on the spectral properties and the Fourier asymptotics of self-similar measures defined on R. Given an IFS {F1,...,Fm} and a set of probability weights p1,...,pm, then there is a unique self-similar probability µ which satisfies
µ =
m j=1
pjµ◦F−1 j .
It is known that µ is either purely singular or absolutely continuous. We will explain how this question is closely related to the asymptotic properties of its Fourier transform. We also explore the existence of the orthonormal bases of exponential functions in the L2(µ) space and its relation to tiling. Some open questions are listed.