Research article

Monotone Dynamical Systems with Polyhedral Order Cones and Dense Periodic Points

  • Received: 29 November 2016 Accepted: 30 November 2016 Published: 16 December 2016
  • 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.

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

    Related Papers:

  • 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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    [2] M. Hirsch and H. Smith, Monotone Dynamical Systems, Handbook of Differential Equations, volume 2, chapter 4. A. C?nada, P. Drabek & A. Fonda, editors. Elsevier North Holland, 2005.
    [3] S. Kaul, On pointwise periodic transformation groups, Proceedings of the American Mathematical Society 27 (1971), 391-394.
    [4] D. Montgomery, Pointwise periodic homeomorphisms, American Journal of Mathematics 59 (1937), 118-120.
    [5] E. Spanier, Algebraic Topology, McGraw Hill, 1966.
    [6] W. Hurewicz and H. Wallman, Dimension Theory, Princeton University Press, 1941.
    [7] R. Wilder, Topology of Manifolds, American Mathematical Society, 1949.
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  • © 2017 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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