Periodic points of nonexpansive maps and nonlinear generalizations of the Perron-Frobenius theory

• R.D. Nussbaum ^[1] ; M. Scheutzow ^[2] ; S.M. Verduyn Lunel ^[3]
  1. [1] Rutgers University

     Rutgers University

     City of New Brunswick, Estados Unidos

     
  2. [2] Technical University of Berlin

     Technical University of Berlin

     Berlin, Stadt, Alemania

     
  3. [3] VU University Amsterdam

     VU University Amsterdam

     Países Bajos

     

  Mostrar afiliaciones +
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 4, Nº. 1, 1998, págs. 141-181
• Idioma: inglés
• DOI: 10.1007/s000290050028
• Enlaces
  • Texto completo (pdf)
• Resumen

  • Let Kn={x∈Rn∣xi≥0, 1≤i≤n} and suppose that f : K n →K n is nonexpansive with respect to the l1-norm, ∥x∥1=∑ni=1|xi| , and satisfies f (0) = 0. Let P3(n) denote the (finite) set of positive integers p such that there exists f as above and a periodic point ξ∈Kn of f of minimal period p. For each n≥ 1 we use the concept of 'admissible arrays on n symbols' to define a set of positive integers Q(n) which is determined solely by number theoretical and combinatorial constraints and whose computation reduces to a finite problem. In a separate paper the sets Q(n) have been explicitly determined for 1 ≤n≤ 50, and we provide this information in an appendix. In our main theorem (Theorem 3.1) we prove that P3(n) = Q(n) for all n≥ 1. We also prove that the set Q(n) and the concept of admissible arrays are intimately connected to the set of periodic points of other classes of nonlinear maps, in particular to periodic points of maps g : Dg→Dg, where Dg⊂Rn is a lattice (or lower semilattice) and g is a lattice (or lower semilattice) homomorphism.