Topological approach to the generalized $n$-centre problem

This paper considers a natural Hamiltonian system with two degrees of freedom and Hamiltonian $H=\|p\|^2/2+V(q)$. The configuration space $M$ is a closed surface (for non-compact $M$ certain conditions at infinity are required). It is well known that if the potential energy $V$ has $n>2\chi(M)$ Newtonian singularities, then the system is not integrable and has positive topological entropy on the energy level $H=h>\sup V$. This result is generalized here to the case when the potential energy has several singular points $a_j$ of type $V(q)\sim {-}\operatorname{dist}(q,a_j)^{-\alpha_j}$. Let $A_k=2-2k^{-1}$, $k\in\mathbb{N}$, and let $n_k$ be the number of singular points with $A_k\leqslant \alpha_j<A_{k+1}$. It is proved that if
$$ \sum_{2\leqslant k\leqslant\infty}n_kA_k>2\chi(M), $$
then the system has a compact chaotic invariant set of collision-free trajectories on any energy level $H=h>\sup V$. This result is purely topological: no analytical properties of the potential energy are used except the presence of singularities. The proofs are based on the generalized Levi-Civita regularization and elementary topology of coverings. As an example, the plane $n$-centre problem is considered.
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