A Self Similar Measure is Absolutely Continuous

Absolute Continuity of Self Similar Measures

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 selfsimilar 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.