Admissible Arrays and a Nonlinear Generalization of Perron-frobenius Theory

Let Kn ̄2x `2n :x i & 0 for 1% i% n ́ and suppose that f :Kn MNKn is nonexpansive with respect to the F " -norm and f(0) ̄ 0. It is known that for every x `Kn there exists a periodic point ξ ̄ ξ x `Kn (so f p(ξ ) ̄ ξ for some minimal positive integer p ̄ pξ) and f k(x) approaches 2 f j(ξ ) :0% j! p ́ as k approaches infinity. What can be said about P*(n), the set of positive integers p for which there exists a map f as above and a periodic point ξ `Kn of f of minimal period p? If f is linear (so that f is a nonnegative, column stochastic matrix) and ξ `Kn is a periodic point of f of minimal period p, then, by using the Perron–Frobenius theory of nonnegative matrices, one can prove that p is the least common multiple of a set S of positive integers the sum of which equals n. Thus the paper considers a nonlinear generalization of Perron–Frobenius theory. It lays the groundwork for a precise description of the set P*(n). The idea of admissible arrays on n symbols is introduced, and these arrays are used to define, for each positive integer n, a set of positive integers Q(n) determined solely by arithmetical and combinatorial constraints. The paper also defines by induction a natural sequence of sets P(n), and it is proved that P(n)ZP*(n)ZQ(n). The computation of Q(n) is highly nontrivial in general, but in a sequel to the paper Q(n) and P(n) are explicitly computed for 1% n% 50, and it is proved that P(n) ̄P*(n) ̄Q(n) for n% 50, although in general P(n) 1Q(n). A further sequel to the paper (with Sjoerd Verduyn Lunel) proves that P*(n) ̄Q(n) for all n. The results in the paper generalize earlier work by Nussbaum and Scheutzow and place it in a coherent framework.