Asymptotic estimates for the periods of periodic points of non-expansive maps

For each positive integer n we use the concept of ‘admissible arrays on n symbols’ to define a set of positive integers Q(n) which is determined solely by number theoretical and combinatorial constraints and whose computation reduces to a finite problem. In earlier joint work with M. Scheutzow, it was shown that the set Q(n) is intimately connected to the set of periods of periodic points of classes of nonexpansive nonlinear maps defined on the positive cone in R. In this paper we continue the characterization of Q(n) and present precise asymptotic estimates for the largest element of Q(n). For example, if γ (n) denotes the largest element of Q(n), then we show that limn→∞(n logn)−1/2 log γ (n) = 1. We also discuss why understanding further details about the fine structure ofQ(n) involves some delicate number theoretical issues.