TY - JOUR

T1 - Iteration of order preserving subhomogeneous maps on a cone

AU - Akian, Marianne

AU - Gaubert, Stéphane

AU - Lemmens, Bas

AU - Nussbaum, Roger

N1 - Copyright:
Copyright 2011 Elsevier B.V., All rights reserved.

PY - 2006/1

Y1 - 2006/1

N2 - We investigate the iterative behaviour of continuous order preserving subhomogeneous maps $f: K → K$, where $K$ is a polyhedral cone in a finite dimensional vector space. We show that each bounded orbit of $f$ converges to a periodic orbit and, moreover, the period of each periodic point of $f$ is bounded by equation given where $N$ is the number of facets of the polyhedral cone. By constructing examples on the standard positive cone in ℝ n, we show that the upper bound is asymptotically sharp. These results are an extension of work by Lemmens and Scheutzow concerning periodic orbits in the interior of the standard positive cone in ℝ n.

AB - We investigate the iterative behaviour of continuous order preserving subhomogeneous maps $f: K → K$, where $K$ is a polyhedral cone in a finite dimensional vector space. We show that each bounded orbit of $f$ converges to a periodic orbit and, moreover, the period of each periodic point of $f$ is bounded by equation given where $N$ is the number of facets of the polyhedral cone. By constructing examples on the standard positive cone in ℝ n, we show that the upper bound is asymptotically sharp. These results are an extension of work by Lemmens and Scheutzow concerning periodic orbits in the interior of the standard positive cone in ℝ n.

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U2 - 10.1017/S0305004105008832

DO - 10.1017/S0305004105008832

M3 - Article

AN - SCOPUS:33244466507

VL - 140

SP - 157

EP - 176

JO - Mathematical Proceedings of the Cambridge Philosophical Society

JF - Mathematical Proceedings of the Cambridge Philosophical Society

SN - 0305-0041

IS - 1

ER -