Decay of correlation for expanding toral endomorphisms

Abstract

Let $A$ be an expanding endomorphism on the torus ${\mathbb{T}}^d={\mathbb{R}}^d/{\mathbb{Z}}^d$ with its smallest eigenvalue $\lambda >1$. Consider the ergodic system $({\mathbb{T}}^d,A,\mu )$ where $\mu$ is Haar measure. We prove that the correlation ${\rho }_{f,g}(n)$ of a pair of functions $f,g\in L^2(\mu )$ is controlled by the modulus of $L^2$-continuity ${\Omega }_{f,2}({\lambda }^{-n})$ and that the estimate is to some extent optimal. We also prove the central limit theorem for the stationary process $f(A^{n}x)$ defined by a function $f$ satisfying ${\Sigma }_n{\Omega }_{f,2}({\lambda }^{-n})<\infty$. An application is given to the Ulam-von Neumann system.