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.