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

TL;DR

This paper introduces techniques to demonstrate that certain measure-zero sets of wandering orbits in volume-preserving dynamical systems with non-compact phase space can be characterized by sequences of shrinking hypersurfaces that semi-orbits intersect.

Contribution

It presents a novel method using decreasing hypersurfaces to analyze measure-zero properties of wandering sets in dynamical systems.

Findings

Sets of initial conditions leading to collision are measure zero.

Semi-orbits with non-existent asymptotic velocity are measure zero.

The technique applies to systems with non-compact phase space.

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.