A characterization of the periods of periodic points of 1-norm nonexpansive maps

Abstract

In this paper the set of minimal periods of periodic points of 1-norm nonexpansive maps $f:\mathbb{R}^n\rightarrow\mathbb{R}^n$ is studied. This set is denoted by R(n). The main goal is to present a characterization of R(n) by arithmetical and combinatorial constraints. More precisely, it is shown that $R(n)=Q'(2n)$, where $Q'(2n)$ denotes the set of periods of restricted admissible arrays on 2n symbols. The important point of this equality is that $Q'(2n)$is determined by arithmetical and combinatorial constraints only, and that it can be computed in finite time. By using this equality the set R(n) is computed for $1\leq n\leq 10$. Furthermore it is shown that the largest element $\psi (n)$of R(n) satisfies: $\log \psi (n) \sim \sqrt{2n\log n}.$