Decay of correlation for expanding toral endomorphisms

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Let A be an expanding endomorphism on the torus Tᵈ = Rᵈ / Zᵈ with its smallest eigenvalue λ>1. Consider the ergodic system (Tᵈ, A, μ) where μ is Haar measure. We prove that the correlation ρf, g(n) of a pair of functions f, g ∈ L²(μ) is controlled by the modulus of L²-continuity Ωf, ₂(λ⁻ⁿ) and that the estimate is to some extent optimal. We also prove the central limit theorem for the stationary process f(Aⁿ x) defined by a function f satisfying Σₙ Ωf,₂(λ⁻ⁿ) <∞. An application is given to the Ulam-von Neumann system.