In A Primer on Mapping Class Groups by Farb and Margalit theorem 14.19 implies that every pseudo-Anosov homeomorphism $f:S \rightarrow S$ on a compact surface $S$ possesses infinitely many periodic points. I was wondering if this result is also true for compact Surfaces with a finite number of interior points removed (e.g. A sphere with a finite number of punctures).
Does anyone have a reference?
If you put a prong singularity at each puncture, allowing any number of prongs $\ge 1$, then the proof of infinitely many periodic points (in fact, their denseness) goes through with no changes: construct a Markov partition in exactly the same manner, and then apply symbolic dynamics.