Strong orthogonality between the Möbius function and skew products

Abstract

For $\tau >3$ and $\alpha \in \mathbb {R}$, let T be a skew product map of the form $T(x_1,x_2)=(x_1+\alpha ,x_2+h(x_1))$ on $\mathbb {T}^2$ over a rotation of the circle, for which h is of zero topological degree and of class $C^{\tau }$. We prove that for a measure-theoretically generic set of $\alpha$, such a $C^{\tau }$ skew product map T is strongly orthogonal to the Möbius function. Moreover, we establish the strong orthogonality between the Möbius function and some irregular $C^{\tau }$ skew product map on $\mathbb {T}^2$. The key tools used are Fourier analysis and exponential sums estimate concerning the Möbius function.