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.

Zoom link: https://osu.zoom.us/j/93885989739?pwd=bUNWdjgzMS93NHRUcmVZRkljTDBHZz09

Meeting ID: 938 8598 9739

Password: Mixing

Recorded Talk: https://osu.zoom.us/rec/share/Gf98gFbI9Itd1STAukYTGjTHeePNXMHIsdoCITVDNs0cCpKQbNDEjUaYfEEVHbms.BBHTyrGjdrrvmvPr

Title: Marked boundary rigidity and Anosov extension

Speaker: Dong Chen – Penn State University

Abstract: In this talk we will show how a sufficiently small geodesic ball in any Riemannian manifold can be embedded into an Anosov manifold with the same dimension. Furthermore, such embedding exists for a larger family of domains even with hyperbolic trapped sets. We will also present some applications to boundary rigidity and related open questions. This is a joint work with Alena Erchenko and Andrey Gogolev.

Title: Convergence of polynomial multiple ergodic averages for totally ergodic systems

Speaker: Andreas Koutsogiannis – Aristotle University of Thessaloniki (Greece)

Abstract: A collection of integer sequences is jointly ergodic if for every ergodic measure preserving system the multiple ergodic averages, with iterates given by this collection of sequences, converge to “the expected limit” in the mean, i.e., the product of the integrals. Exploiting a recent approach of Frantzikinakis, which allows one to avoid deep tools from ergodic theory that were previously used to establish similar results, we study joint ergodicity in totally ergodic systems for integer parts of real polynomial iterates. More specifically, our main results in this direction are a sufficient condition for k terms, and a characterization in the k=2 case. Joint work with Wenbo Sun.