Abstract: The Birkhoff ergodic theorem, a fundamental result in ergodic theory, asserts that, in an ergodic measure-preserving dynamical system, the time average equates to the space average for almost every point. In the realm of uniquely ergodic systems on compact metric spaces, this equivalence extends to every point, extending the theorem's applicability. Our current research, inspired by Sarnak's conjecture and Chowla’s conjecture in number theory, delves into the study of weighted time averages and time averages along a sequence of natural numbers for continuous functions within certain dynamical systems on compact metric spaces. We aim to exploit the oscillatory nature of weights and the uniform behavior of sequences of natural numbers as tools for categorizing zero entropy dynamical systems. Two arithmetic functions, the Möbius function as a weight and the big prime omega function as a sequence of natural numbers, exhibit these properties, respectively. We will introduce additional weights and sequences of natural numbers with similar properties. This presentation will offer an overview of recent developments in this field. Sarnak's conjecture, intimately linked with Chowla's conjecture in number theory, provides a crucial motivation for our research. Further investigation into this connection reveals intriguing relationships between invariant measures, particularly in Möbius and square-free flows. I will also discuss recent advancements in this area.
Bio: Yunping Jiang is a distinguished professor at the City University of New York (CUNY), encompassing both Queens College and the Graduate Center. He specializes in dynamical systems and related fields. His academic journey began at Peking University for his undergraduate studies, followed by obtaining his master's degree under the mentorship of Professor Shantao Liao. He furthered his education by pursuing a Ph.D. at the CUNY Graduate Center, working under the guidance of Professor Dennis Sullivan. Beyond his scholarly endeavors, Jiang has contributed to the field, serving as an editor for AMS Transactions and Memoirs, and holding a directorial role at the National Science Foundation (NSF) Analysis Program.
