# Interior periodic points of area preserving homeomorphisms of a pair of pants

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.

## 2 Answers

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).

Thurston's decomposition theory for surface mapping classes gives a nice condition which does not require $\phi$ to be area preserving, nor does it require $P$ to be a pair of pants, pretty much any compact, oriented surface-with-boundary will do.

Namely, suppose that $X \subset \text{int}(P)$ is a finite $\phi$-invariant subset; the only restriction is that the interior of $P-X$ is not a sphere with $\le 2$ punctures.

Consider the restriction $\phi \,\bigm|\, P-X$, and consider the element of the mapping class group $\Phi \in \text{MCG}(P-X)$ which is represented by that restriction. Suppose that $\Phi$ is of infinite order, and suppose that $\Phi$ is irreducible in the sense that for each nontrivial, nonperipheral simple closed curve $C \subset P-X$, and for each $k \ge 1$, the curve $\phi^k(C)$ is not isotopic to $C$.

Then $\phi$ has infinitely many interior periodic points on $P$.

This kind of thing is exploited in joint papers of Franks and Handel.