Variational construction for heteroclinic orbits of the N-center problem

It is well-known that the N-center problem is chaotic when N≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\ge 3$$\end{document}. By regularizing collisions, one can associate the dynamics with a symbolic dynamical system which yields infinitely many periodic and chaotic orbits, possibly with collisions. It is a challenging task to construct chaotic orbits without any collision. This paper addresses this problem by considering the planar N-center problem with collinear centers and N≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\ge 3$$\end{document}. We show that, for any fixed nonnegative energy and certain types of periodic free-time minimizers, there are infinitely many collision-free heteroclinic orbits connecting them. Our approach is based on minimization of a normalized action functional over paths within certain topological classes, and the exclusion of collision is based on some recent advances on local deformation methods.