In 2010 Sarnak formulated the following conjecture on so called Möbius
disjointness: for each zero entropy topological system $(T,X)$, each
$f\in C(X)$ and $x\in X$, we have
$$
\frac1N \sum_{n\leq N}f(T^nx)\mu(n)\to 0 as N\to\infty,
$$
where $\mu:\N\to\{-1,0,1\}$ stands for the arithmetic Möbius function.
Although, since 2010, the conjecture has been proved in numerous cases,
it still remains open. One of the difficulties is that it is unknowm
whether if it holds
in a uniquely ergodic system $(T,X,\nu)$ then the conjecture holds also
in any other
uniquely ergodic system $(T',X',\nu')$ such that $(T,\mu)$ and
$(T',\mu')$ are
measure-theoretically isomorphic.
The aim of my talk will be to show that Sarnak's conjecture holds in all
uniquely ergodic
models of totally ergodic rotations. The talk is based on my joint work
with H. Abdalaoui and T. de la Rue.