Random Homoclinic Orbits

We introduce random homoclinic points and orbits for random dynamical systems with hyperbolic stationary orbits and investigate their meaning for irregular behaviour in form of a stochastic version of the Birkhoff-Smale Theorem. In particular we show that transversal random homoclinic points cause a conjugacy of the random dynamical system in a part of the phase space to a random shift system. Key words: random dynamical system, random difference equations, random subshift of finite type, homoclinic orbits, shadowing lemma, Birkhoff-Smale Theorem. 1991 Mathematics Subject Classification. Primary 58F03, 58F15; Secondary 60J10, 54H20. Introduction Random dynamical systems (RDS) generalize dynamical systems by allowing the dependence on an additional parameter evolving in time and describing a stochastic influence. The latter is modelled by an abstract dynamical system(\Omega ; F ; P; #) where(\Omega ; F ; P) is a probability space and # in case of discrete time is a measure-preservin..