Pointwise Ergodic Theorems for Uniformly Behaved Sequences of Natural Numbers

Yunping Jiang,  CUNY
Pointwise Ergodic Theorems for Uniformly Behaved Sequences of Natural Numbers
Thursday, February 15 – 11 :00AM
Location: 016 Manchester Hall

Abstract: The colloquium commences with an exploration of the renowned Birkhoff ergodic theorem) delving into fundamental concepts in dynamical systems such as invariant measures, equal continuity, minimality, ergodicity, unique ergodicity, time-average and space average. The theorem asserts that in an ergodic measure-preserving dynamical system on a probability space, the time average of an L1-function over natural numbers matches the space average for almost all points. Additionally if the dynamical system is uniquely ergodic on a compact metric space, this equivalence extends to every point for continuous functions. Subsequently; attention turns to our current investigation into the
time averages of continuous functions along sequences of natural number , which holds significant implications for number theory. The concept of uniformly behaved sequences of natural numbers is introduced, accompanied by illustrative examples. Finally, a novel finding, jointly developed with Ph.D. student Jessica Liu, is presented: in a minimal uniquely ergodic, and a-mean Lyapunov table dynamical system on a compact metric space the time average of a continuous function along a uniformly behaved sequence a of natural numbers equals the space average for every point.

HOST: Dr. Miaohua Jiang [jiangm@wfu.edu]