Abstract: We study the spectral gaps of transfer operators and mixing property for the skew product given

by $F(x,y)=(fx, y+\tau(x))$,  wherethe base map  $f$ is a $C^\infty$ uniformly expanding endomorphism over

a $d$-dimensional torus, and the fiber map is a rotation by $\tau(x)$.  We construct a Hilbert space Hilbert space that contains H\"older functions.

We apply the semiclassical analysis approach to get the dichotomy: either the transfer operator has a spectral

gap  on Hilbert space, or $\tau$ is an essential coboundary.  In the former case, $F$ mixes exponentially fast

for H\"older observables; 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.

This is a jiont work with Jianyu Chen.