Two dynamical perspectives on the randomness of the Mobius function

Abstract

Sarnak's approach to the Mobius randomness heuristic from the standpoint of dynamical systems is studied in two complementary settings. First, the Mobius function is realized in the context of symbolic dynamics, and we prove that its associated squarefree factor has a unique measure of maximal entropy. We then show that the Mobius function is linearly disjoint from all zero-entropy translations on homogeneous spaces of connected Lie groups, generalizing work of Vinogradov, Green-Tao, and Bourgain- Sarnak-Ziegler.