Random Iteration of Cylinder Maps and diffusive behavior away from resonances

TL;DR

This paper models random compositions of cylinder maps and demonstrates that, away from resonances, the distribution of the radial component converges to a diffusion process, with a central limit theorem established for specific polynomial cases.

Contribution

It introduces a stochastic model for cylinder map compositions and proves convergence to diffusion and a central limit theorem under certain conditions.

Findings

Radial component converges to a diffusion process.

Established a central limit theorem for polynomial cases.

Connected the model to generalized Arnold examples.

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 \[ f_{\pm 1}: \left(\begin{array}{c}\theta\\r\end{array}\right) \longmapsto \left(\begin{array}{c}\theta+r+\varepsilon u_{\pm 1}(\theta,r) \\ r+\varepsilon v_{\pm 1}(\theta,r) \end{array}\right), \] 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 \[ (\theta_n,r_n)=f_{\omega_{n-1}}\circ \dots \circ f_{\omega_0}(\theta_0,r_0), \] where $\omega_k =\pm 1$ with equal probability. We show that under non-degeneracy hypotheses and away from resonances for $n\sim \varepsilon^{-2}$ 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 $n\sim \varepsilon^{-2}$ the distributions of $r_n-r_0$ weakly converge to the normal distribution $\mathcal N(0,\sigma^2)$ with $\sigma^2=\frac14\int (v_+(\theta)-v_-(\theta))^2\,d\theta$.} The considered random model up to higher order terms in $\varepsilon$ 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.