Minimal dynamics for tree maps

Abstract

We prove that, given a tree pattern $\mathcal{P}$, the set ofperiods of a minimal representative $f: T\rightarrow T$ of $\mathcal{P}$ is containedin the set of periods of any other representative. This statement is animmediate corollary of the following stronger result: there is aperiod-preserving injection from the set of periodic points of $f$ into thatof any other representative of $\mathcal{P}$. We prove this result byextending the main theorem of [6] to negative cycles.