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.
Discrete and Continuous Dynamical Systems
Bernhardt, Chris, "Vertex Maps for Trees: Algebra and Periods of Periodic Orbits" (2006). Mathematics Faculty Publications. 12.
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.