Random Iteration of Maps on a Cylinder and diffusive behavior

In this paper we propose a model of random compositions of maps of a cylinder, which in the simplified form is as follows: (θ, r) ∈ T×R = A and f±1 : ( θ r ) 7−→ ( θ + r + εu±1(θ, r). r + εv±1(θ, r). ) , where u± and v± are smooth and v± are trigonometric polynomials in θ such that ∫ v±(θ, r) dθ = 0 for each r. We study the random compositions (θn, rn) = fωn−1 ◦ · · · ◦ fω0(θ0, r0) with ωk ∈ {−1, 1} with equal probabilities. We show that under natural non-degeneracy hypothesis for n ∼ ε−2 the distributions of rn − r0 weakly converge to a diffusion process with explicitly computable drift and variance. In the case of random iteration of the standard maps f±1 : ( θ r ) 7−→ ( θ + r + εv±1(θ). r + εv±1(θ) ) , where v± are trigonometric polynomials such that ∫ v±(θ) dθ = 0 we prove a vertical central limit theorem. Namely, for n ∼ ε−2 the distributions of rn − r0 weakly converge to a normal distribution N (0, σ2) for σ2 = 1 4 ∫ (v+(θ)− v−(θ)) dθ. Such random models arise as a restrictions to a Normally Hyperbolic Invariant Lamination for a Hamiltonian flow of the generalized example of Arnold. We hope that this mechanism of stochasticity sheds some light on formation of diffusive behaviour at resonances of nearly integrable Hamiltonian systems. ∗Universitat Politècnica de Catalunya, [email protected] †University of Maryland at College Park, [email protected]