Admissible arrays and a nonlinear generalization of Perron-Frobenius theory

Roger D. Nussbaum, Michael Scheutzow

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Let Kn = {x∈ℝn:xi ≥ 0 for 1 ≤ i ≤ n} and suppose that f:Kn→Kn is nonexpansive with respect to the ℓ1-norm and f(0) = 0. It is known that for every x∈Kn there exists a periodic point ξ = ξx∈Kn (so fp(ξ) = ξ for some minimal positive integer p = pξ) and fk(x) approaches {fi(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) ⊂ 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.

Original languageEnglish (US)
Pages (from-to)526-544
Number of pages19
JournalJournal of the London Mathematical Society
Issue number3
StatePublished - Dec 1998

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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