A multiplicative ergodic theorem for lipschitz maps

John H. Elton

Stochastic Processes and their Applications, 1990, vol. 34, issue 1, 39-47

Abstract: If (Fn, n [greater-or-equal, slanted] 0) is a stationary (ergodic) sequence of Lipschitz maps of a locally compact Polish space X into itself having a.s. negative Lyapunov exponent function, the composition process Fn...F1x converges in distribution to a stationary (ergodic) process in X (independent of x). For every x, the empirical distribution of a trajectory converges with probability one, and for every [var epsilon]>0, almost every trajectory is eventually within [var epsilon] of the support. We use the fact that the Lyapunov exponent of a process "run backwards" is the same as forwards. A set invariance condition is given for the case when (Fn) is a Markov chain. The result has applications to computer graphics and stability in control theory.

Keywords: stationary; sequence; Lipschitz; map; Lyapunov; exponent; ergodic; theorem (search for similar items in EconPapers)
Date: 1990
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Citations: View citations in EconPapers (18)

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