## Seminar 10.28.21 Sun

Title: Joint ergodicity conjecture for systems with commuting transformations

Speaker: Wenbo Sun – Virginia Tech

Abstract: It is well know by the Mean Ergodic Theorem that for any measure preserving system $(X,\mathcal{B},\mu,T)$ and $L^{\infty}$ function f, the time average of $T^{n}f$ converges to the integral of f if and only if T is ergodic. It is a natural question to ask when the average of products of polynomial iterates of  $L^{\infty}$ functions (known as multiple ergodic averages) converges to the product of the integrals of the functions. This question is called the Joint Ergodicity Problem. In this talk, I will introduce some recent advances in this problem. This talk is based on joint works with Sebasti\’an Donoso, Andreu Ferr\’e Moragues and Andreas Koutsogiannis.

Meeting ID: 916 3892 7725

Recorded Talk: https://osu.zoom.us/rec/play/dk–MjWHHK8ex0sdF0ILXOXT338U71LQ1awWnexrtUuyYiEtC-noT76YCLpX4bnCvTAT2mU-xxTPv9d1.OUHqT_H7CAgbLfFM?continueMode=true

## Seminar 10.21.21 Maass

Title: Spectral analysis of topological finite rank systems

Speaker: Alejandro Maass – University of Chile

Abstract: Finite topological rank Cantor minimal systems represent a broad class of sub shifts of zero entropy or odometers [Downarowiz-Maass], it contains well studied systems like substitution sub shifts or linearly recurrent systems. In this talk we will present the study of measure-theoretical and topological eigenvalues for such class of systems, given formulas characterizing them. This work is motivated by the seminal work of Bernard Host where it is proved that measure-theoretical and topological eigenvalues of substitutions systems coincide. This is a joint work with Fabien Durand and Alexander Frank.

Meeting ID: 916 3892 7725

Recorded Talk: https://osu.zoom.us/rec/share/6x1GYGHuPkjR4KFdpi4usJMP1ert17FX_RHSF_MVaxIkD6PrLXjLO83fXN_-CR8u.0rZGFYbboMqO4Ps7

## Seminar 10.08.21 Tanaka

Title: The Manhattan curve and rough similarity rigidity

Speaker: Ryokichi Tanaka – Kyoto University

Abstract: For every non-elementary hyperbolic group, we consider the Manhattan curve, which was originally introduced by M. Burger (1993),
associated with any pair of (say) word metrics. It is convex; we show that it is continuously differentiable and moreover is a straight line if and only if the corresponding two metrics are roughly similar, that is, they are within bounded distance after multiplying by a positive constant. I would like to explain how it is related to the central limit theorem for uniform counting measures on spheres, to ergodic theory of topological flows built on general hyperbolic groups, and to the multifractal structure of Patterson-Sullivan measures. Furthermore, I will present some explicit examples including a hyperbolic triangle group and compute the exact value of the mean distortion for a pair of word metrics by using automatic structures of the group.
Joint work with Stephen Cantrell (University of Chicago).