Vertex maps for trees: Algebra and periods of periodic orbits

Chris Bernhardt

doi:10.3934/dcds.2006.14.399

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2006, Volume 14, Issue 3: 399-408. Doi: 10.3934/dcds.2006.14.399

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Vertex maps for trees: Algebra and periods of periodic orbits

Chris Bernhardt^1,

1.
Department of Mathematics and Computer Science, Fairfield University, Fairfield, CT 06824

Received: December 2004

Revised: July 2005

Published: September 2006

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Abstract

Let $T$ be a tree with $n$ vertices. Let $f: T \rightarrow T$ 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 $k, l \geq 0$.

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Mathematics Subject Classification: 37E05.

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