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.

62 pages, revision based on referee's comments, results and proofs unchanged

References in corpus (5)

• On self-similar sets with overlaps and inverse theorems for entropy
• Ergodicity of group actions and spectral gap, applications to random walks and Markov shifts
• Random walks in Euclidean space
• Spectral gap in the group of affine transformations over prime fields
• A spectral gap theorem in simple Lie groups

Cited by in corpus (5)

• Fourier transform of self-affine measures
• Spectral gap in the group of affine transformations over prime fields
• A spectral gap theorem in simple Lie groups
• How long is the Chaos Game?
• Self-similar sets and measures on the line