Seminar 12.6.18 Donoso

Title: Quantitative multiple recurrence in ergodic theory

SpeakerSebastian Donoso (Universidad de O’Higgins)

Abstract: In this talk I will survey recent development of the multiple recurrence problem in ergodic theory. For a probability space (X,,μ) and measure preserving transformations T1,,Td, the problem is to study the largeness of the set of n such that

μ(ATa1(n)1ATad(n)dA)>F(μ(A))

where a1,,ad take integer values on the integers and F is a suitable function. I will mention key results and comment on the problem for commuting transformations, linear and polynomial functions ai. I plan to provide some proofs that rely on combinatorial constructions.