Random iteration of analytic maps

Iteration of analytic self - maps of the disk : an overview

In this expository article we discuss the theory of iteration for functions analytic in the unit disk, and bounded by one in modulus. This is a classical subject that started in the eighteen-hundreds and continues today. We will review the historical background, and then conclude with more recent work.

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 ∈ {−...

Continuous iteration of dynamical maps

Backward-iteration Sequences with Bounded Hyperbolic Steps for Analytic Self-maps of the Disk

A lot is known about the forward iterates of an analytic function which is bounded by 1 in modulus on the unit disk D. The Denjoy-Wolff Theorem describes their convergence properties and several authors, from the 1880’s to the 1980’s, have provided conjugations which yield very precise descriptions of the dynamics. Backward-iteration sequences are of a different nature because a point could hav...

Coupled Analytic Maps

We consider a lattice of weakly coupled expanding circle maps. We construct, via a cluster expansion of the Perron-Frobenius operator, an invariant measure for these innnite dimensional dynamical systems which exhibits space-time-chaos.