Absolute Continuity of Self Similar Measures

Abstract

We prove that a self similar measure is absolutely continuous providing that it satisfies a condition depending on its Garsia entropy, contraction ratio, and the separation between different points in approximations of the self similar measure. In the special case of Bernoulli convolutions this paper gives a sufficient condition for the Bernoulli convolution with parameter lambda to be absolutely continuous in terms of lambda and its Mahler measure.

This version of the paper has been accepted by Annales scientifiques de l'École normale supérieure

Cited by in corpus (5)

• Fourier transform of self-affine measures
• Self-similar sets and measures on the line
• Bernoulli convolutions with Garsia parameters in $(1,\sqrt{2}]$ have continuous density functions
• Towards Absolutely Continuous Bernoulli Convolutions
• Thermodynamic formalism for invariant measures in iterated function systems with overlaps