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

TL;DR

This paper investigates the spectral gap of transfer operators for torus extensions over expanding maps, establishing conditions for exponential mixing or structural properties of the system using semiclassical analysis.

Contribution

It constructs a new Hilbert space framework and applies semiclassical methods to characterize when such systems exhibit a spectral gap or are essentially coboundaries.

Findings

Spectral gap exists or the system is an essential coboundary.

Exponential mixing occurs when a spectral gap is present.

System can be non-weakly mixing or unstably mixing depending on coboundary conditions.

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 $F(x,y)=(Tx, y+\tau(x) \pmod{ \mathbb{Z}^\ell})$, 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 H\"older functions of H\"older 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 H\"older observables with H\"older 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.