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.
