Locally Contractive Iterated Function Systems

• Published in: The Annals of probability Vol. 27; no. 4; pp. 1952 - 1979
• Main Author: Steinsaltz, David
• Format: Journal Article
• Language: English
• Published: Hayward, CA Institute of Mathematical Statistics 01-10-1999; The Institute of Mathematical Statistics
• Subjects: 26A18; 58F08; 60J05; Ergodic theory; Exact sciences and technology; Global analysis, analysis on manifolds; Iteration; Iterations of functions; logistic maps; Logistics; Lyapunov functions; Markov chains; Markov processes; Mathematical analysis; Mathematical functions; Mathematical monotonicity; Mathematics; Perceptron convergence procedure; Probability and statistics; Probability theory and stochastic processes; random attractors; Random variables; Real functions; Real numbers; Sciences and techniques of general use; single-server queues; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds; Zeno’s walk; Markov process; Almost sure convergence; Random attractor; Random walk; Iteration; Zeno walk; Stochastic process; Single server queue; Stationary system; Contractivity; Markov operator; Logistic map; Limit distribution; Convergence rate; Lyapunov function
• ISSN: 0091-1798; 2168-894X
• DOI: 10.1214/aop/1022874823

An iterated function system on$\mathscr{X} \subset \mathbb{R}^d$is defined by successively applying an i.i.d. sequence of random Lipschitz functions from X to X. This paper shows how Fn= f1⚬ ⋯ ⚬ fnmay converge even in the absence of the strong contraction conditions, for instance, Lipschitz constant smaller than 1 on average, which earlier work has required. Instead, it is posited that there be a region of contraction which compensates for the noncontractive or even expansive part of the functions. Applications to queues, to self-modifying random walks and to random logistic maps are given.