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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

REFERENCES

[B]Breiman, L.. Probability. Addison-Wesley (Reading, Massachusetts, 1968).Google Scholar
[BD]Barnsley, M. F. & Demko, S.. Iterated function systems and the global construction of fractals. Prod. R. Soc. Lond. A 399 (1985), 243–275.Google Scholar
[BDEG]Barnsley, M. F., Demko, S., Elton, J. & Geronimo, J.. Markov processes arising from function iteration with place-dependent probabilities. Preprint, 12 1985.Google Scholar
[BE]Barnsley, M. F. & Elton, J.. Stationary attractive measures for a class of Markov chains arising from function iteration. Preprint, 08 1985.Google Scholar
[D]Doob, J. L.. Stochastic Processes. Wiley (New York, 1953).Google Scholar
[DF]Dubins, L. & Freedman, D.. Invariant probabilities for certain Markov processes. Ann. Math. Stat. 32 (1966), 837–848.CrossRefGoogle Scholar
[DS]Diaconis, P. & Shashahani, M.. Products of random matrices and computer image generation. Stanford U. preprint (1984).Google Scholar
[FK]Furstenberg, H. & Kifer, Y.. Random matrix products and measures on projective spaces. Israel J. Math. 46 (1983), 12–32.Google Scholar
[H]Hutchinson, J.. Fractals and self-similarity. Indiana U. Journal of Math. 30 (1981), 713–747.CrossRefGoogle Scholar
[K]Karlin, S.. Some random walks arising in learning models. Pacific J. of Math. 3 (1953), 725–756.CrossRefGoogle Scholar
[S]Stout, W. F.. Almost Sure Convergence. Academic Press (New York, 1974).Google Scholar