Random iteration of isometries

Abstract: This thesis consists of four papers, all concerning random iteration of isometries. The papers are:I. Ambroladze A, Ådahl M, Random iteration of isometries in unbounded metric spaces. Nonlinearity 16 (2003) 1107-1117.II. Ådahl M, Random iteration of isometries controlled by a Markov chain. Manuscript.III. Ådahl M, Melbourne I, Nicol M, Random iteration of Euclidean isometries. Nonlinearity 16 (2003) 977-987.IV. Johansson A, Ådahl M, Recurrence of a perturbed random walk and an iterated function system depending on a parameter. Manuscript.In the first paper we consider an iterated function system consisting of isometries on an unbounded metric space. Under suitable conditions it is proved that the random orbit {Zn} ∞n=0, of the iterations corresponding to an initial point Z0, “escapes to infinity" in the sense that P(Zn Є K) → 0, as n → ∞ for every bounded set K. As an application we prove the corresponding result in the Euclidean and hyperbolic spaces under the condition that the isometries do not have a common fixed point.In the second paper we let a Markov chain control the random orbit of an iterated function system of isometries on an unbounded metric space. We prove under necessary conditions that the random orbit escapes to infinity" and we also give a simple geometric description of these conditions in the Euclidean and hyperbolic spaces. The results generalises the results of Paper I.In the third paper we consider the statistical behaviour of the reversed random orbit corresponding to an iterated function system consisting of a finite number of Euclidean isometries of Rn. We give a new proof of the central limit theorem and weak invariance principles, and we obtain the law of the iterated logarithm. Our results generalise immediately to Markov chains. Our proofs are based on dynamical systems theory rather than a purely probabilistic approach.In the fourth paper we obtain a suficient condition for the recurrence of a perturbed (one-sided) random walk on the real line. We apply this result to the study of an iterated function system depending on a parameter and defined on the open unit disk in the complex plane.