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.