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Vertex maps for trees: algebra and periods of periodic orbits. (English) Zbl 1110.37033

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 \(k,l\geq 0\).

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
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