A Poincaré–Birkhoff theorem on invariant plane continua

Abstract

We prove that if $F$ is an orientation-preserving homeomorphism of the plane that leaves invariant an irreducible plane separating continuum $\Delta$, then, with the possible exception of three numbers, if $p/q$ is a reduced rational in the interior of the convex hull of the rotation set of $F\vert_{\Delta}$ (with respect to some lift) there are at least two distinct periodic orbits of $F\vert_{\Delta}$ of period $q$ and rotation number $p/q$. This result also applies to certain nonseparating invariant continua.