Vertex Maps for Trees: Algebra and Periods of Periodic Orbits

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 Sn. 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 = 2k + 2l for k, l ≥ 0.