# American Institute of Mathematical Sciences

July  2006, 14(3): 399-408. doi: 10.3934/dcds.2006.14.399

## Vertex maps for trees: Algebra and periods of periodic orbits

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

Received  December 2004 Revised  July 2005 Published  December 2005

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 $k, l \geq 0$.
Citation: Chris Bernhardt. Vertex maps for trees: Algebra and periods of periodic orbits. Discrete and Continuous Dynamical Systems, 2006, 14 (3) : 399-408. doi: 10.3934/dcds.2006.14.399
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