Kaitlyn Loyd
University of Maryland
Title
Ergodic Averages along Ω(n)
Abstract
Let Ω(n) denote the total number of prime factors of n. Though the study of this sequence finds roots in analytic number theory, the following result of Bergelson and Richter introduced a dynamical perspective: in any uniquely ergodic system, Ω-ergodic averages (those with times sampled along Ω(n)) converge pointwise everywhere to the mean for every continuous function. In this talk, we will examine the behavior of Ω-ergodic averages for non-uniquely ergodic systems. We show that if a point x is quasi-generic for an ergodic measure μ, then μ is an accumulation point of the empirical measures formed from {TΩ(n)x}. Moreover, we study Ω-ergodic averages for non-compact horocycle flows to demonstrate that in general, a converse does not hold. This talk is based on joint work with Adam Kanigowski.