Bernhardt, Chris Vertex maps for trees: algebra and periods of periodic orbits. (English) Zbl 1110.37033 Discrete Contin. Dyn. Syst. 14, No. 3, 399-408 (2006). Summary: Let \(T\) be a tree with \(n\) vertices. Let \(f:T\to T\) 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 \(k,l\geq 0\). Cited in 1 ReviewCited in 11 Documents MSC: 37E25 Dynamical systems involving maps of trees and graphs 37E15 Combinatorial dynamics (types of periodic orbits) 37B20 Notions of recurrence and recurrent behavior in topological dynamical systems 20B30 Symmetric groups Keywords:tree maps; periods of orbits; Sharkovsky’s theorem PDFBibTeX XMLCite \textit{C. Bernhardt}, Discrete Contin. Dyn. Syst. 14, No. 3, 399--408 (2006; Zbl 1110.37033) Full Text: DOI