No Infinite Spin for Partial Collisions converging to isolated CC on the plane

Abstract

The infinite spin problem is a problem concerning the rotational behavior of total collision orbits in the $n$-body problem. The question makes also sense for partial collision. When a~cluster of bodies tends to a (partial) collision, then its normalized shape curve tends to the set of normalized central configurations, which in the planar case has $SO(2)$ symmetry. This leaves a possibility that the normalized shape curve tends to the circle obtained by rotation of some central configuration instead of a particular point on it. This is the \emph{infinite spin problem}. We show that it is not possible if the limiting circle is isolated from other connected components of set of normalized central configuration. Our approach extends the method from recent work for total collision by Moeckel and Montgomery, which was based on combination of the center manifold theorem with {\L}ojasiewicz inequality. To that we add a shadowing result for pseudo-orbits near normally hyperbolic manifold and careful estimates on the influence of other bodies on the cluster of colliding bodies.