On Dimension and Absolute Continuity of Self-Similar Measures

A similarity is a map from Rd to itself that uniformly scales distances. When one repeatedly applies randomly chosen contracting similarities, the resulting Markov chain converges to a limiting stationary distribution known as a self-similar measure. The central problems in the study of self-similar measures are to determine their dimension and to understand when they are absolutely continuous, meaning that they admit a density with respect to the Lebesgue measure. 

I will present joint work with Samuel Kittle addressing these questions. Indeed, we prove Hochman’s result on the dimension of self-similar measures under a weakened Diophantine assumption. Moreover, we establish numerous novel explicit examples of absolutely continuous self-similar measures. In fact, we give the first inhomogeneous examples in dimension 1 and 2 and construct examples for essentially any given rotations and translations, provided they have algebraic coefficients. Moreover, we strengthen Varju’s result for Bernoulli convolutions and Lindenstrauss-Varju’s result in dimension greater than or equal to 3