Iteration of order preserving subhomogeneous maps on a cone. (English) Zbl 1101.37032

Akian, Marianne; Gaubert, Stéphane; Lemmens, Bas; Nussbaum, Roger

Math. Proc. Camb. Philos. Soc. 140, No. 1, 157-176 (2006).

Let \(K\) be a polyhedral cone in a finite-dimensional vector space and \(f:K\to K\) be a continuous order-preserving subhomogeneous map. It is shown that each bounded orbit of \(f\) converges to a periodic orbit and, moreover, an upper bound for the periods of all the periodic orbits is found. This upper bound depends only on \(K\). By constructing examples on the standard positive cone on \(\mathbb R^n\), it is shown that the upper bound is asymptotically sharp.

Reviewer: Adriana Buică (Bellaterra)

MSC:

37E99	Low-dimensional dynamical systems
37C25	Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
47H99	Nonlinear operators and their properties

Keywords:

periodic point; asymptotic behaviour; order-preserving subhomogeneous map; polyhedral cone; finite-dimensional vector space