Celestial mechanics is a main parent" of the modern theory of
dynamical systems. Poincare proved non-integrability of the three body
problem when he discovered the homoclinic picture. Alexeev explained the
existence of the oscillatory motions (a planet approaches infinity
always returning to a bounded domain) in Sitnikov model (one of the
restricted versions of the three body problem) using methods of
hyperbolic dynamics.
We show that the structures related to the most recent works in the
smooth dynamical systems (e.g. conservative Henon family, lateral
thickness of a Cantor set, persistent tangencies, splitting of
separatrices) also appear in the three body problem. After we get some
new results in smooth dynamics (parameterized version of conservative
Newhouse phenomena, relation between lateral thicknesses and Hausdorff
dimension of a Cantor set, etc), we prove that in many cases the set of
oscillatory motions has a full Hausdorff dimension.
This is a joint work with V.Kaloshin.