Exponential Mixing of Torus Extensions over Expanding Maps

We study the mixing property for the skew product F : TdˆTl Ñ Td ˆ Tl given by F px, yq “ pTx, y ` τpxqq, where T : Td Ñ Td is a C8 uniformly expanding endomorphism, and the fiber map τ : Td Ñ Tl is a C8 map. We apply the semiclassical analytic approach to get the dichotomy: either F mixes exponentially fast or τ is an essential coboundary. In the former case, the Koopman operator p F of F has spectral gap in some Hilbert space that contains all L2pTd ˆ Tlq functions, and in the latter case the system is semiconjugate to an expanding endomorphism crossing a torus rotation. 0. Introduction. In this paper we study the mixing properties for torus extension of expanding maps. The systems F we consider are of the form of skew products with expanding T : T Ñ T on the base and torus rotations with rotation vectors τpxq, x P T, on the fibers T. (See (1.2) for the maps.) We obtain a dichotomy: either such a system has exponential decay of correlations with respect to the smooth invariant measure, or the rotation function τpxq over T is an essential coboundary. The latter implies that the system is semiconjugate to an expanding endomorphism crossing a torus rotation, or simply semiconjugate to a circle rotation, and therefore cannot be weak mixing (Theorem 3 (iii)) or stably ergodic. The methods we use to get exponential mixing is the semiclassical analytic approach. Instead of the Ruelle-Perron-Frobenius transfer operators acting on some Hölder function space, we study the dual operator, Koopman operator p F , given by p Fφ “ φ ̋F , acting on certain distribution space. By Fourier transform along T, the fiber direction, the operator can be decompose to a family of operators t p FνuνPZl , where ν is the frequency. Such operators can be regarded as Fourier integral operators. Using semiclassical analysis theory we can obtain that the spectral radius of p Fν is strictly less than 1 for all ν ­“ 0, and uniformly less than 1 for all ν with |ν| large whenever τ is not an essential coboundary, while 1 is the only eigenvalue of p F0 on the unit circle and it is simple. Hence the operator p F has a spectral gap, and the system has exponential decay of correlations. Dolgopyat established exponential mixing property for compact group extensions of expanding maps under a generic condition called infinitesimally completely nonintegrability (see [3]). Faure used semiclassical analysis in [5] to obtain exponential mixing for a simpler but intuitive model a circle extension of an expanding map of T under a so-called partially captive condition. For the low dimensional case, the dichotomy similar to that in Theorem 1 was obtained by Butterley and Eslami recently in [1] and discontinuities are allowed at a finite set for maps T and τ there. 1 2 JIANYU CHEN AND HUYI HU Naud showed in [9] that such skew product cannot mix super-exponentially, not even for analytic observables. Similar results were also obtained in the context of suspension semiflows over linear expanding maps. Pollicott [12] used Dolgopyat’s estimates [2] to show that the generic suspension semiflows is exponentially mixing. Tsujii [15] constructed an anisotropic Sobolev space on which the transfer operator has spectral gap. This paper is organized as the following. The setting and statements of results are given in Section 1. In Section 2 we introduce some notions and results from classical and semiclassical analysis, including Fourier transform, Sobolev spaces, Pseudo-differential operators, Fourier Integral Operators, Egorov’s Theorem, and L-continuity theorems. This section is not necessary for the reader who is familar with the theory. We prove the theorems of the paper in Section 3 based on Proposition 3.1 and 3.2, which give the spectral radius of the Koopman operator, the dual operator of the transfer operator. The propositions are proved in Section 4, using classical and semiclassical analysis. A key estimates in the proof, stated in Lemma 5.1, is postponed in Section 5. 1. Statement of results. Let T “ R{Z, and let T : T Ñ T be a C8 uniformly expanding map such that (1.1) γ :“ inf px,vqPSTd |DxT pvq| ą 1, where ST is the unit tangent bundle over T. It is well known that T has a unique smooth invariant probability measure dμpxq “ hpxqdx, where the density function h P C8pTd,R`q. Further, T is mixing with respect to μ. Given a function τ P C8pTd,Tlq, we define the skew product F : T ˆ T Ñ T ˆ T by (1.2) F ˆ