Periodic points of nonexpansive maps and nonlinear generalizations of the Perron-Frobenius theory

Abstract.

Let $K^n = \{x \in {\Bbb R}^n \mid x_i \ge 0,\ 1 \le i \le n \}$ and suppose that f : K ⁿ→K ⁿ is nonexpansive with respect to the l ₁-norm, $\|x\|_1 = \sum_{i=1}^n |x_i|$, and satisfies f (0) = 0. Let P ₃(n) denote the (finite) set of positive integers p such that there exists f as above and a periodic point $\xi \in K^n$ of f of minimal period p. For each n≥ 1 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 a separate paper the sets Q(n) have been explicitly determined for 1 ≤n≤ 50, and we provide this information in an appendix. In our main theorem (Theorem 3.1) we prove that P ₃(n) = Q(n) for all n≥ 1. We also prove that the set Q(n) and the concept of admissible arrays are intimately connected to the set of periodic points of other classes of nonlinear maps, in particular to periodic points of maps g : D _g→D _g, where $D{_g} \subset {\Bbb R}^n$ is a lattice (or lower semilattice) and g is a lattice (or lower semilattice) homomorphism.