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November  2002, 8(4): 873-892. doi: 10.3934/dcds.2002.8.873

Global regularization for the $n$-center problem on a manifold

Sergey V. Bolotin 1, and  Piero Negrini 2,

1. 

Department of Mathematics, University of Wisconsin, Madison, United States
2. 

Department of Mathematics, La Sapienza, University of Rome, Italy

Received  August 2001 Revised  February 2002 Published  July 2002

We describe a global version of the KS regularization of the $n$-center problem on a closed 3-dimensional manifold. The regularized configuration manifold turns out to be 4 or 5 dimensional closed manifold depending on whether $n$ is even or odd. As an application, we show that the $n$ center problem in $S^3$ has positive topological entropy for $n\ge 5$ and energy greater than the maximum of the potential energy. The proof is based on the results of Gromov and Paternain on the topological entropy of geodesic flows. This paper is a continuation of [6], where global regularization of the $n$-center problem in $\mathbf R^3$ was studied.
Keywords: topological entropy., Newtonian singularity, Lagrangian system, regularization.
Mathematics Subject Classification: 37B40, 37J15, 27J30, 70F1.
Citation: Sergey V. Bolotin, Piero Negrini. Global regularization for the $n$-center problem on a manifold. Discrete and Continuous Dynamical Systems, 2002, 8 (4) : 873-892. doi: 10.3934/dcds.2002.8.873
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