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An ergodic theorem for iterated maps. (English) Zbl 0621.60039
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.
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.

60J05 Discrete-time Markov processes on general state spaces
37H15 Random dynamical systems aspects of multiplicative ergodic theory, Lyapunov exponents
28D05 Measure-preserving transformations
47A35 Ergodic theory of linear operators
Full Text: DOI
[1] Stout, Almost Sure Convergence (1974) · Zbl 0321.60022
[2] Karlin, Pacific J. of Math. 3 pp 725– (1953) · Zbl 0051.10603 · doi:10.2140/pjm.1953.3.725
[3] DOI: 10.1512/iumj.1981.30.30055 · Zbl 0598.28011 · doi:10.1512/iumj.1981.30.30055
[4] Breiman, Probability (1968)
[5] DOI: 10.1214/aoms/1177699364 · Zbl 0147.16404 · doi:10.1214/aoms/1177699364
[6] Doob, Stochastic Processes (1953)
[7] Barnsley, Prod. R. Soc. Lond. 399 pp 243– (1985)
[8] DOI: 10.1007/BF02760620 · Zbl 0528.60028 · doi:10.1007/BF02760620
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