A refinement of the Sharkovsky Theorem for interval maps tells us that there is a forcing relation on cyclic permutations: if a continuous map of an interval has a periodic orbit with a given permutation $A$ (we look at which point is mapped to which) and $A$ forces $B$, then this map has a periodic orbit with permutation $B$. A similar theory exists for orientation preserving disk homeomorphisms, but instead of permutations, one looks at the braid types of the periodic orbits (mapping classes of the disk punctured at the points of the orbit). However, the braid type depends on the behavior of the map off the orbit. We find a class of square homeomorphisms for which all the information about braid types of periodic orbits is encoded in a pair of permutations: vertical order vs. horizontal order, and temporal order vs. horizontal order. We find some simple invariants for braid types of those orbits.
This is a joint research with Ana Rodrigues.