Admissible Arrays and a Nonlinear Generalization of Perron–Frobenius Theory

Let K n ={ x ∈ℝ n : x i ⩾0 for 1⩽ i ⩽ n } and suppose that f : K n → K n is nonexpansive with respect to the L 1 ‐norm and f (0)=0. It is known that for every x ∈ K n there exists a periodic point ξ=ξ x ∈ K n (so f p (ξ)=ξ for some minimal positive integer p = p ξ ) and f k ( x ) approaches { 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∈ K n of f of minimal period p ? If f is linear (so that f is a nonnegative, column stochastic matrix) and ξ∈ K n 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 1⩽ n ⩽50, and it is proved that P ( n )= P *( n )= Q ( n ) for n ⩽50, 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.