Iterating Maps on Cellular Complexes
Stephen J. Willson · Transactions of the American Mathematical Society · 1992
Let $K$ be a finite simplicial complex and $f:K \to K$ be a "skeletal" map. A digraph $D$ is defined whose vertices correspond to the simplexes of $K$ and whose arcs give information about the behavior of $f$ on the simplexes. For every walk in $D$ there exists a point of $K$ whose iterates under $f$ mimic the walk. Periodic walks are mimicked by a periodic point. Digraphs with uncountably many infinite walks are characterized; the corresponding maps $f$ exhibit complicated behavior.