TITLE: Interior periodic points of area preserving homeomorphisms of a pair of pants
QUESTION [8 upvotes]: A celebrated result of Franks shows that any area preserving homeomorphism of the closed annulus $A$ with at least one periodic point (possibly along the boundary) has infinitely many interior periodic points. Let $\phi$ be an area preserving homeomorphism of the pair of pants P (viewed as a sphere minus three open discs). 
Question: What are some natural conditions which guarantee that $\phi$ has infinitely many periodic points?  
For example is there some behavior near the boundary which guarantees this? An obvious example would be that it agrees with an irrational rotation near one of the boundary components so that we could extend it to A by a rotation. Perhaps there are more subtle statements.

REPLY [6 votes]: It always has infinitely many periodic points, since you can collapse one of the boundary components to get a homeomorphism of the closed annulus with a fixed point, and then you can use Franks' theorem (the area-preserving condition can be replaced by the existence of a non-atomic invariant measure of full support; Franks proves the results using these assumptions, and in any case this is equivalent to being area-preserving after a topological conjugation due to the Oxtoby-Ulam theorem).