Poincar\'e-Birkhoff theorems in random dynamics

TL;DR

This paper extends the classical Poincaré-Birkhoff Theorem to include area-preserving twist maps that are random and ergodic, broadening its applicability to stochastic dynamical systems.

Contribution

It introduces a generalized version of the Poincaré-Birkhoff Theorem for random area-preserving twist maps, encompassing the classical case as a special instance.

Findings

Established existence of periodic points in random twist maps

Unified deterministic and stochastic cases under a common framework

Extended classical results to ergodic random environments

Abstract

We propose a generalization of the Poincar\'e-Birkhoff Theorem on area-preserving twist maps to area-preserving twist maps that are random with respect to an ergodic probability measure. The classical theory is a particular instance of the random theory we propose.