TWO REMARKS ON SPECIAL ORBITS OF AREA PRESERVING MAPS NEAR ELLIPTIC PERIODIC POINTS

ANATOLE KATOK

Our first remark is an elaboration and strengthening of an observation made by G.D Birkhoff in his classical treatise ``Dynamical systems'' about existence of quasi-periodic orbits that appear as limits of elliptic periodic points of growing periods. In the general style of the time and his work Birkhoff was quite vague. What he had in mind was neither invariant curves, nor disconnected Aubrey-Mather sets but what in the modern language  is called odometers. In a joint work we Kurt Vinhage we prove that for any odometer OD the set of C^r, r>3 area preserving diffeomorphisms with an elliptic fixed point that have an invariant set  with dynamics isomorphic to OD that appears  naturally as  a limit of elliptic period points is residual in C^2 topology. We will also have partial results on simultaneous existence on many odometers.  

Our second remark  is the following statement:
Among  C-infinity area preserving diffeomorphisms of a surface that have an elliptic periodic point those that have infinitely many ergodic components of positive Lebesgue measure are dense in C^r topology for any r. The proof is based on an application of the approximation-by-conjugation method.