© 1998 by London Mathematical Society
© The London Mathematical Society
Admissible Arrays and a Nonlinear Generalization of Perron–Frobenius Theory
Mathematics Department, Rutgers University New Brunswick, New Jersey 08903, USA
Fachbereich Mathematik, Technische Universität Berlin Berlin, Germany
Received 16 May 1995.
Let Kn={x
Rn:xi
0 for 1
in} and suppose that f:Kn
Kn is nonexpansive with respect to the L1-norm and f(0)=0. It is known that for every xKn there exists a periodic point 
=xKn (so fp()= for some minimal positive integer p=p) and fk(x) approaches {fj():0j<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 Kn of f of minimal period p? If f is linear (so that f is a nonnegative, column stochastic matrix) and 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)
P*(n)Q(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 1n50, and it is proved that P(n)=P*(n)=Q(n) for n50, although in general P(n)
Q(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.