Decay of correlation for expanding toral endomorphisms

Abstract

Let $A$ be an expanding endomorphism on the torus ${\Bbb T}^d = {\Bbb R}^d /{\Bbb Z}^d$ with its smallest eigenvalue $\lambda >1$. Consider the ergodic system $({\Bbb T}^d, A, \mu)$ 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.