Regularization and topological entropy for the spatial <i>n</i>-center problem

Болотин, С. Н.; Negrini, Piero

doi:10.1017/s0143385701001195

Cited by 21 publications

(25 citation statements)

References 10 publications

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“…[33]). The planar case of N -centre with N ≥ 3 is known to be non integrable on non-negative energy levels and has positive entropy; some partial extensions are available also for the spatial case (see [5,6,7,11,18,19]). Recently, in [24] Soave and Terracini have shown the presence of a chaotic subsystem for the planar N -centre problem also at negative energies.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Scattering Parabolic Solutions for the Spatial N-Centre Problem

Boscaggin

Dambrosio

Terracini

2016

Arch Rational Mech Anal

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Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Scattering Parabolic Solutions for the Spatial N-Centre Problem

Boscaggin

Dambrosio

Terracini

2016

Arch Rational Mech Anal

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“…However, we do assemble some remarkable quantitative and qualitative results related to ours. In [6,8,9] it was shown that for N ≥ 3 the system is non-integrable on non-negative energy levels, and it has positive entropy. The authors introduced a topological (global) regularization by applying the local KS-regularization around each center.…”

Section: Introductionmentioning

confidence: 99%

Topologically Distinct Collision-Free Periodic Solutions for the $${N}$$ N -Center Problem

Castelli¹

2016

Arch Rational Mech Anal

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This work concerns the planar N -center problem with homogeneous potential of degree −α (α ∈ [1, 2)). The existence of infinitely many, topologically distinct, non-collision periodic solutions with a prescribed energy is proved. A notion of admissibility in the space of loops on the punctured plane is introduced so that in any admissible class and for any positive h the existence of a classical periodic solution with energy h for the N -center problem with α ∈ (1, 2) is proven. In case α = 1 a slightly different result is shown: it is the case that there is either a non-collision periodic solution or a collision-reflection solution. The results hold for any position of the centres and it is possible to prescribe in advance the shape of the periodic solutions. The proof combines the topological properties of the space of loops in the punctured plane with variational and geometrical arguments.

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“…Moreover smallness of µ > 0 (or large energy) is not needed, and one can add to the potential a smooth negative function. The proof is given in [4] for d = 2 and in [10] for d = 3. For d = 3 the proof is nontrivial: it is based on the global KS regularization and deep results of Gromov and Paternain.…”

Section: Classical N Center Problemmentioning

confidence: 99%

Degenerate billiards

Болотин

2016

Proc. Steklov Inst. Math.

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In an ordinary billiard system trajectories of a Hamiltonian system are elastically reflected after a collision with a hypersurface (scatterer). If the scatterer is a submanifold of codimension more than one, we say that the billiard is degenerate. Then collisions are rare. We study trajectories of degenerate billiards which have an infinite number of collisions with the scatterer. Degenerate billiards appear as limits of systems with elastic reflections or as limits of systems with singularities in celestial mechanics. We prove the existence of trajectories of such systems shadowing trajectories of the corresponding degenerate billiards. The proofs are based on a version of the method of an antiintegrable limit.

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Regularization and topological entropy for the spatial n-center problem

Abstract: We show that the n-center problem in ℝ3 has positive topological entropy for n ≥ 3. The proof is based on global regularization of singularities and the results of Gromov and Paternain on the topological entropy of geodesic flows. The n-center problem in S3 is also studied

Cited by 21 publications

References 10 publications

Scattering Parabolic Solutions for the Spatial N-Centre Problem

Scattering Parabolic Solutions for the Spatial N-Centre Problem

Topologically Distinct Collision-Free Periodic Solutions for the $${N}$$ N -Center Problem

Degenerate billiards

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