Measure complexity and M\"obius disjointness

In this paper, the notion of measure complexity is introduced for a topological dynamical system and it is shown that Sarnak’s M"{o}bius disjointness conjecture holds for any system for which every invariant Borel probability measure has sub-polynomial measure complexity. Moreover, it is proved that the following classes of topological dynamical systems $(X,T)$ meet this condition and hence satisfy Sarnak’s conjecture: (1) Each invariant Borel probability measure of $T$ has discrete spectrum. (2) $T$ is a homotopically trivial $C^{\mathrm{\infty }}$ skew product system on ${\mathbb{T}}^2$ over an irrational rotation of the circle. Combining this with the previous results it implies that the M"{o}bius disjointness conjecture holds for any $C^{\mathrm{\infty }}$ skew product system on ${\mathbb{T}}^2$. (3) $T$ is a continuous skew product map of the form $(ag,y+h(g))$ on $G\times {\mathbb{T}}^1$ over a minimal rotation of the compact metric abelian group $G$ and $T$ preserves a measurable section. (4) $T$ is a tame system.