Seminar 04.07.22 Tsinas

Title:Multiple ergodic theorems for sequences of polynomial growth

Speaker:  Konstantinos Tsinas – University of Crete (Greece)

Abstract: Following the classical results of Host-Kra and Leibman on the polynomial ergodic theorem, it is natural to ask whether we can establish mean convergence of multiple ergodic averages along several other sequences, which arise from functions that have polynomial growth. In 1994, Boshernitzan proved that for a function f, which belongs to a large class of smooth functions (called a Hardy field) and which has polynomial growth, its “distance” from rational polynomials is crucial in determining whether or not the sequence of the fractional parts of f(n) is equidistributed on [0,1]. This, also, implies a corresponding mean convergence theorem in the case of single ergodic averages along the sequence ⌊f(n)⌋ of integer parts. In the case of multiple averages, it was conjectured by Frantzikinakis that a similar condition on the linear combinations of the involved functions should imply mean convergence. We verify this conjecture and show that in all ergodic systems we have convergence to the “expected limit”, namely, the product of the integrals. We rely mainly on the recent joint ergodicity results of Frantzikinakis, as well as some seminorm estimates for functions belonging to a Hardy field. We will also briefly discuss the “non-independent” case, where the L^2-limit of the averages exists but is not equal to the product of the integrals.

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Meeting ID: 938 8598 9739

Password: Mixing

Recorded Talk: