Ergodic Theory Motivated by Sarnak’s and Chowla’s Conjectures in Number Theory
Yunping Jiang (CUNY - Queens Coll. and Grad. Center, EUA)

The Birkho ergodic theorem, a fundamental concept in ergodic theory, asserts that, in an ergodic measure-preserving dynamical system, the time average converges to the space average for almost every point. This convergence holds for every point, extending the theorem’s applicability when dealing with a uniquely ergodic system on a compact metric space. Our current research, inspired by Sarnak’s conjecture, delves into the study of weighted time averages for continuous functions within zero topological entropy dynamical systems on compact metric spaces. We aim to leverage the oscillating property of weight as a tool for classifying such systems. This talk will provide an overview of recent developments in this area. Sarnak’s conjecture, closely tied to Chowla’s conjecture in number theory, serves as a key motivator for our work. Further exploration of this connection unveils intriguing relationships between invariant measures, particularly in the Möbius and square-free ows. I will also touch upon recent advancements on this topic.