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Vertex maps for trees: Algebra and periods of periodic orbits

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  • 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 $k, l \geq 0$.
    Mathematics Subject Classification: 37E05.


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