The spectral gap for transfer operators of torus extensions over expanding maps

We study the spectral gap for transfer operators of the skew product $F: $\mathbb{T}$^d\times $\mathbb{T}$^\ell\to $\mathbb{T}$^d\times $\mathbb{T}$^\ell$ given by $F(x,y)=(Tx,y+\tau (x)(\mathrm{m}\mathrm{o}\mathrm{d}{\mathbb{Z}}^{\mathrm{\ell }}))$, where $T: $\mathbb{T}$^d\to $\mathbb{T}$^d$isa$C^\infty$ uniformly expanding endomorphism, and the fiber map $\tau :{\mathbb{T}}^d\to {\mathbb{R}}^{\mathrm{\ell }}$ is a $C^{\mathrm{\infty }}$ map. We construct a Hilbert space ${\mathcal{W}}^{-s}$ for any $s<0$, which contains all the H"older functions of H"older exponents $\mathrm{\mid }s\mathrm{\mid }$ on ${\mathbb{T}}^d\times {\mathbb{T}}^{\mathrm{\ell }}$. Applying the method of semiclassical analysis, we obtain the dichotomy: either the transfer operator has a spectral gap on ${\mathcal{W}}^{-s}$, or $\tau$ is an essential coboundary. In the former case, $F$ mixes exponentially fast for H"older observables with H"older exponents $\mathrm{\mid }s\mathrm{\mid }$; and in the latter case, either $F$ is not weak mixing and it is semiconjugate to a circle rotation, or $F$ is unstably mixing, i.e., it can be approximated by non-mixing skew products.