Statistical properties of skew-endomorphisms with Bernoulli base

Abstract

We analyse numerically the statistical properties of a class of mappings of the 2-torus T 2 onto itself, whose investigation is suggested by some models of modulated diffusion. These transformations can be written as a skew-product of the endomorphism B p ( x) = px mod 1, p ϵ Z ⧹ {−1, 0, 1} , on the 1-torus T 1 := [0, 1] and a translation on T 1. Under suitable assumptions the skew-product can be proven to be mixing w.r.t. the Lebesgue measure. Central Limit (CL) and Functional Central Limit (FCL) properties are numerically checked for analytic observables. The result is remarkable because the mappings show no hyperbolic or quasi-hyperbolic structure, crucial for the proof of Central Limit Theorem and Donsker's Invariance Principle in all of the dynamical systems where these properties have been established up to now. Moreover, CL and FCL behaviours seem to hold also in the case of purely ergodic endomorphisms and even for observables whose correlations do not decay at infinity.