Dense periodicity on graphs (Q2478071)

The paper extends the theorem by \textit{M. Barge} and \textit{J. Martin} [Proc. Am. Math. Soc. 94, 731--735 (1985; Zbl 0567.54024)] for continuous maps of a compact interval to continuous maps of a compact, connected, one-dimensional polyhedron \(G\). An analogous extension from intervals to trees is due to \textit{R. P. Roe} [Topol. Proc. 19, 237--248 (1994; Zbl 0872.54028)]. The main result is that if \(f:G\rightarrow G\) is continuous and has its periodic points dense in \(G\), then there is a positive integer \(N\) and a finite, empty, or countable collection of subgraphs \(G_i \subset G\) with disjoint interiors, each mapped to itself under \(f^N\), the restriction of \(f^N\) to each \(G_i\) is topologically mixing, and \(f^N\) is the identity on the complement of the union of the \(G_i\)'s.