Is it ever possible to predict that two far-apart points will belong to the same orbit? Is it ever possible to predict that an orbit will exhibit 180-degree rotational symmetry? (This is related to the first question, since the orbit of (i,j) has 180-degree rotational symmetry iff (-i,-j) is in the same orbit as (i,j).) What if anything is known about Minsky-ish maps whose orbits approximate hyperbolas rather than circles and ellipses? Jim Propp