The M\"obius disjointness conjecture: uniform convergence and entropy (Mariusz Lemanczyk, Nicolaus Copernicus University in Toruń)

12.04.2018 16:15 – 17:15

(*) \lim_{N\to\infty}\frac1N\sum_{n\leq N}f(T^nx)\mu(n)=0 for all f\in C(X) and x\in X (\mu stands for the classical M\"obius function).
Sarnak's conjecture from 2010 states that all zero entropy systems are M\"obius disjoint. The celebrated Chowla conjecture on autocorrelations of M\"obius function implies Sarnak's conjecture, and, by a recent theorem of Tao, the logarithmic version of the two conjectures are equivalent. However, also positive entropy systems can be M\"obius disjoint. We will be discussing uniform (in x\in X) convergence in (*), show that the seemingly stronger requirement of uniform convergence in (*) for zero entropy systems is equivalent to Sarnak's conjecture, and show (under Chowla conjecture) to which extent uniformity fails in positive entropy systems.

PS. The Colloquium will be followed by an aperitif

Lieu

Acacias, Room 17, Colloque

entrée libre