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

Abstract

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 $F(x,y)=(Tx,y+\tau (x)(\mathrm{mod}{\mathbb{Z}}^{\ell }))$, where $T:{\mathbb{T}}^d\to {\mathbb{T}}^d$ is a $C^{\infty }$ uniformly expanding endomorphism, and the fiber map $\tau :{\mathbb{T}}^d\to {\mathbb{R}}^{\ell }$ is a $C^{\infty }$ map. We construct a Hilbert space ${\mathcal{W}}^{-s}$ for any $s<0$, which contains all the Hölder functions of Hölder exponents $|s|$ on ${\mathbb{T}}^d\times {\mathbb{T}}^{\ell }$. 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 Hölder observables with Hölder exponents $|s|$; 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.

The second version has been seriously revised

References in corpus (1)

• Exponential Decay of Correlations for Finite Horizon Sinai Billiard Flows