AIMS Mathematics, 2017, 2(1): 24-27. doi: 10.3934/Math.2017.1.24.

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Monotone Dynamical Systems with Polyhedral Order Cones and Dense Periodic Points

Department of Mathematics, University of Wisconsin, Madison WI 53706, USA

Let $X\subset \mathbb{R}^{n}$ be a set whose interior is connected and dense in $X$, ordered by a closed convex cone $K\subset \mathbb{R}^{n}$ having nonempty interior. Let $T: X\approx X$ be an order-preserving homeomorphism. The following result is proved: Assume the set of periodic points of $T$ is dense in $X$, and  $K$ is a polyhedron.  Then $T$ is periodic. 
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Keywords Dynamical systems; ordered spaces; convex cones; periodic orbits

Citation: MorrisW. Hirsch. Monotone Dynamical Systems with Polyhedral Order Cones and Dense Periodic Points. AIMS Mathematics, 2017, 2(1): 24-27. doi: 10.3934/Math.2017.1.24

References

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  • 2. M. Hirsch and H. Smith, Monotone Dynamical Systems, Handbook of Differential Equations, volume 2, chapter 4. A. Cãnada, P. Drabek & A. Fonda, editors. Elsevier North Holland, 2005.
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  • 7. R. Wilder, Topology of Manifolds, American Mathematical Society, 1949.

 

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