Walks with optimal reward on metric spaces

Abstract

Let (M, d) be a complete metric space and suppose that there are given finitely many contractions Γρ:M→ M and Lipschitz maps (ρ = 1, …, r).We consider ‘walks’ of length m with a given starting point x0 in M. They are defined as follows: one chooses a sequence (ρμ)μ=1,…, m of length m in {1, …, r}, and this choice induces the ‘walk’ Associated with x1, …, xm is the ‘reward’ We denote by the maximal possible reward.The aim of this paper is to investigate the behaviour of the sequence for large m. It will be shown that the growth is nearly linear: there is a constant γ (which does not depend on x0) such that tends to γ. However, an explicit calculation of γ might be hard. The complexity depends on the fractal dimension of the smallest nonempty compact subset of M which is invariant with respect to all Γρ.In the case of finite M one can say much more. Then—after a suitable rescaling—the sequence is periodic where the length of the period can be described in terms of the length of certain cycles of a graph associated with M.The motivation to study this problem came from a variant of Parrondo's paradox from probability theory: what is the optimal choice of games if a great number of players is involved?