Ergodic Theory and Dynamical Systems

Research Article

An ergodic theorem for iterated maps

John H. Eltona1

a1 School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA

Abstract

Consider a Markov process on a locally compact metric space arising from iteratively applying maps chosen randomly from a finite set of Lipschitz maps which, on the average, contract between any two points (no map need be a global contraction). The distribution of the maps is allowed to depend on current position, with mild restrictions. Such processes have unique stationary initial distribution [BE], [BDEG].

We show that, starting at any point, time averages along trajectories of the process converge almost surely to a constant independent of the starting point. This has applications to computer graphics.

(Received April 10 1986)

(Revised August 11 1986)