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Symbolic dynamics for the $N$-centre problem at negative energies

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  • We consider the planar $N$-centre problem, with homogeneous potentials of degree $-\alpha < 0$, $\alpha \in [1,2)$. We prove the existence of infinitely many collisions-free periodic solutions with negative and small energy, for any distribution of the centres inside a compact set. The proof is based upon topological, variational and geometric arguments. The existence result allows to characterize the associated dynamical system with a symbolic dynamics, where the symbols are the partitions of the $N$ centres in two non-empty sets.
    Mathematics Subject Classification: Primary: 70F10, 37N05; Secondary: 70F15, 37J30.


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