Improbability of Wandering Orbits Passing Through a Sequence of Poincar\'e Surfaces of Decreasing Size

Abstract

Given a volume preserving dynamical system with non-compact phase space, one is sometimes interested in special subsets of its wandering set. One example from celestial mechanics is the set of initial values leading to collision. Another one is the set of initial values of semi-orbits, whose asymptotic velocity does not exist as a limit. We introduce techniques that can be helpful in showing that these sets are of measure zero: by defining a sequence of hypersurfaces, that are eventually hit by each of those semi-orbits and whose total surface area decreases to zero.