Minimal dynamics for tree maps

We prove that, given a tree pattern P, the set of periods of a minimal representative f: T → T of P is contained in the set of periods of any other representative. This statement is an immediate corollary of the following stronger result: there is a period-preserving injection from the set of periodic points of f into that of any other representative of P. We prove this result by extending the main theorem of [6] to negative cycles.