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An ergodic theorem for iterated maps

Published online by Cambridge University Press:  19 September 2008

John H. Elton
John H. Elton
Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA
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Abstract

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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.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 7 , Issue 4 , December 1987 , pp. 481 - 488
Copyright
Copyright © Cambridge University Press 1987

References

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