Exponential mixing of torus extensions over expanding maps

We study the mixing property for the skew product $F: \TT^d\times \TT^\ell\to \TT^d\times \TT^\ell$ given by $F(x,y)=(Tx, y+\tau(x) \pmod{\ZZ^\ell})$, where $T: \TT^d\to \TT^d$ is a $C^\infty$ uniformly expanding endomorphism, and the fiber map $\tau: \TT^d\to \RR^\ell$ is a $C^\infty$ map. We apply the semiclassical approach to get the dichotomy: either $F$ mixes exponentially fast or $\tau$ is an essential coboundary. In the former case, the Koopman operator $\wF$ of $F$ has spectral gap in some Hilbert space $\CW^s$, $s<0$, which contains all $(-s)$-H\"older continuous functions on $\TT^d\times \TT^\ell$; and in the latter case, either $F$ is not weak mixing, or it can be approximated by non-mixing skew products that are semiconjugate to circle rotations.