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Let T be a tree with n vertices. Let f : T --> T be continuous and suppose that the n vertices form a periodic orbit under f. The combinatorial information that comes from possible permutations of the vertices gives rise to an irreducible representation of S-n. Using the algebraic information it is shown that f must have periodic orbits of certain periods. Finally, a family of maps is defined which shows that the result about periods is best possible if n = 2(k) + 2(l) for k, l >= 0.


A copy of this article is posted here with the permission of the copyright holder: (2006) American Institute of Mathematical Sciences, publisher of Discrete and Continuous Dynamical Systems.

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Discrete and Continuous Dynamical Systems

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Bernhardt, C. Vertex Maps for Trees: Algebra and Periods of Periodic Orbits, Discrete & Continuous Dynamical Systems, Vol. 14, No.3, (March 2006) p. 399-408.

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