Periodic points of nonexpansive maps and nonlinear generalizations of the Perron-Frobenius theory
The paper studies the dynamical behavior of iterates of (nonlinear) nonexpansive mappings of the closed positive cone \(K^n= \{x\in\mathbb{R}^n\mid x_i\geq 0,\;i=1,2,\dots, n\}\) in the Euclidean \(n\)-space \(\mathbb{R}^n\) endowed with the usual \(\ell^1\)-norm, \(\| x\|_1= \sum^n_{i=1}| x_i|\) \((x\in\mathbb{R}^n)\). The main result of this highly interesting paper may be summarized in the following statement: The set \(P(n)\) of positive integers \(p\) such that there is a nonexpansive \(h: K^n\to K^n\) with \(h(0)= 0\) having a periodic point \(x\in K^n\) with (minimal) period \(p\) is equal to the set \(Q(n)\) of positive integers \(p\) such that there exists an admissible array of \(n\) symbols of period \(p\) (this concept is defined in the paper). The set \(Q(n)\) admits a purely theoretical and combinatorial description and its computation actually reduces to a finite number of steps. The authors provide the list of \(Q(n)\) for \(1\leq n\leq 50\). The main result is to be considered as a precise generalization of an aspect of the classical Perron-Frobenius theory for matrices with nonnegative entries. Some consequences and open questions are collected in the last part of the paper.
scientific article; zbMATH DE number 1192406