Ergodic Theory Motivated by Conjectures in Number Theory

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:2024-06-03 10:30





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. 


蒋云平,美国纽约城市大学杰出教授(Distinguished Professor)。1982年毕业于北京大学数学系,1990年毕业于美国纽约城市大学,获博士学位。蒋云平教授是中科院人才计划入选者,并多次在美国获得国家和学校的科学研究奖励。蒋云平教授曾任美国数学学会的著名杂志Memoirs of the AMS Transactions of the AMS编委。蒋云平教授的研究方向是动力系统,研究兴趣主要集中在一维动力系统,复动力系统,在CMP, Adv. Math., Trans. AMS, ETDS, Chaos等权威学术期刊发表论文80多篇。