Analysis Seminar

A topological dynamical system $(X,T)$ is said to be Möbius disjointnes if \[\tag{$*$} \lim_{N\to\infty}\frac1N\sum_{n\leq N}f(T^nx)\mu(n)=0\] for all $f\in C(X)$ and $x\in X$ ($\mu$ stands for the classical Möbius function).Sarnak's conjecture from 2010 states that all zero entropy systems are Möbius disjoint. The celebrated Chowla conjecture on autocorrelations of Möbius function implies Sarnak's conjecture. However, also positive entropy systems can be Möbius 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.