Random walks in the group of Euclidean isometries and self-similar measures

Abstract: We study products of random isometries acting on Euclidean space. Building on previous work of the second author, we prove a local limit theorem for balls of shrinking radius with exponential speed under the assumption that a Markov operator associated to the rotation component of the isometries has spectral gap. We also prove that certain self-similar measures are absolutely continuous with smooth densities. These families of self-similar measures give higher dimensional analogues of Bernoulli convolutions on which absolute continuity can be established for contraction ratios in an open set.