Generalizations of the Perron-Frobenius theorem for nonlinear maps

Let K-n = {x is an element of R-n \ x(i) greater than or equal to 0, 1 less than or equal to i less than or equal to n} and suppose that f : K-n --> K -n is nonexpansive with respect to the l(1)-norm, \\x\\(1) = Sigma(i=1)(n) x(i), and f(0) = 0. It is known (see [1]) that for every x is an element of K-n there exists a periodic point xi = xi(x) is an element of K-n (so f(p) (xi) = xi for some minimal positive integer p = p(xi)) and f(k)(x) approach es {f(j)(xi) \ 0 less than or equal to j < p} as k tends to infinity. In a previous paper [13] the set P-2(n) of positive integers p for which there e xists a map f as above and a periodic point xi is an element of K-n of mini mal period p was related to the idea of "admissible arrays" and a set Q(n) determined by certain arithmetical and combinatorial constraints. In a sequ el to this paper [14] it is proved that P-2(n) = Q(n) for all n, but the co mputation of Q(n) is highly nontrivial. Here we derive a variety of theorem s about admissible arrays and use these theorems to compute Q(n) explicitly for n < 50 and prove that P(n) = P-2(n) = Q(n) for n less than or equal to 50, where P(n) is a naturally occurring set defined below.