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

##### 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
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