Random Iteration of Cylinder Maps and diffusive behavior away from resonances

Abstract

In this paper we propose a model of random compositions of cylinder maps, which in the simplified form is as follows: let $(\theta ,r)\in \mathbb{T}\times \mathbb{R}=\mathbb{A}$ and \[ f_{\pm 1}: \left(\begin{array}{c}θ\\r\end{array}\right) \longmapsto \left(\begin{array}{c}θ+r+\varepsilon u_{\pm 1}(θ,r) \\ r+\varepsilon v_{\pm 1}(θ,r) \end{array}\right), \] where $u_{\pm }$ and $v_{\pm }$ are smooth and $v_{\pm }$ are trigonometric polynomials in $\theta$ such that $\int v_{\pm }(\theta ,r)d\theta =0$ for each $r$. We study the random compositions \[ (θ_n,r_n)=f_{ω_{n-1}}\circ \dots \circ f_{ω_0}(θ_0,r_0), \] where ${\omega }_k=\pm 1$ with equal probability. We show that under non-degeneracy hypotheses and away from resonances for $n\sim {\epsilon }^{-2}$ the distributions of $r_n-r_0$ weakly converge to a stochastic diffusion process with explicitly computable drift and variance. In the case $u_{\pm }(\theta )=v_{\pm }(\theta )$ are trigonometric polynomials of zero average we prove a vertical central limit theorem, namely, for $n\sim {\epsilon }^{-2}$ the distributions of $r_n-r_0$ weakly converge to the normal distribution $\mathcal{N}(0,{\sigma }^2)$ with ${\sigma }^2=\frac{1}{4}\int (v_{+}(\theta )-v_{-}(\theta ){)}^{2}d\theta$.} The considered random model up to higher order terms in $\epsilon$ is conjugate to a restrictions to a Normally Hyperbolic Invariant Lamination of the generalized Arnold example. Combining the result of this paper with [8,23,28] we show formation of stochastic diffusive behaviour for the generalized Arnold example.

arXiv admin note: substantial text overlap with arXiv:1501.03319