Periodic points of nonexpansive maps and nonlinear generalizations of the Perron-Frobenius theory |
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Authors: | R.D. Nussbaum M. Scheutzow S.M. Verduyn Lunel |
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Affiliation: | Department of Mathematics, Rutgers University, Piscataway, NJ 08854-8019, USA, e-mail: nussbaum@math.rutgers.edu, US Fachbereich Mathematik, Technische Universit?t Berlin, Berlin, Germany, e-mail: ms@math.tu-berlin.de, DE Faculteit Wiskunde en Informatica, Vrije Universiteit, NL-1081 HV Amsterdam, The Netherlands, e-mail: verduyn@cs.vu.nl, NL
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Abstract: | Let and suppose that f : K n →K n is nonexpansive with respect to the l 1-norm, , and satisfies f (0) = 0. Let P 3(n) denote the (finite) set of positive integers p such that there exists f as above and a periodic point 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 P 3(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 : D g→D g, where is a lattice (or lower semilattice) and g is a lattice (or lower semilattice) homomorphism. |
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Keywords: | . Nonexpansive maps periods of periodic points of maps Perron-Frobenius theory. |
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