Laboratoire de Mathématiques Raphaël Salem

UMR 6085 CNRS - University of Rouen Normandy

In this talk I will provide some counter-examples for quantitative multiple recurrence problems for systems with more than one transformation.  For instance, I will show that there exists an ergodic system $(X,\mathcal{X},\mu,T_1,T_2)$ with two commuting transformations such that for every $\ell < 4$ there exists $A\in\mathcal{X}$ such that  \[ \mu(A\cap T_1^n A\cap T_2^n A) <\mu(A)^{\ell} \]  for every $n \in \mathbb{N}$.  The construction of such a system is based on the study of “big” subsets of $\mathbb{N}^2$ and $\mathbb{N}^3$  satisfying combinatorial properties.  

This a joint work with Wenbo Sun.