Modul: MAT760 Ergodic theory and dynamical systems seminar

## Some new results regarding convergence under xqxp-invariant measures on the circle

Talk by Prof. Dr. Sophie Grivaux

**Date:** 27.11.23 **Time:** 15.00 - 16.00 **Room:** Y27H25

For each integer n \geq 1, denote by T_n the map x \mapsto nx mod 1 from the circle group T=R/Z into itself. Let p,q \geq 2 be two multiplicatively independent integers. Using Baire Category arguments, we will show that generically, a continuous T_p-invariant probability measure \mu on T is such that (T_q^{n}\mu)_{n\geq 0} does not weak-star converge to the Lebesgue measure on T. This disproves Conjecture (C3) from a 1988 paper by R. Lyons, which is a stronger version of Furstenberg's rigidity conjecture on xp and xq invariant measures on T, and complements previous results by Johnson and Rudolph. If time permits, I will also present some generalizations of this result concerning convergence to the Lebesgue measure of sequences of the form (T_{c_{n}}\mu)_{n\geq 0}, as well as some extensions to the multidimensional setting. The talk will be based on a joint work with Catalin Badea (Lille). ''