On the multiplicity of brake orbits and homoclinics in Riemannian manifolds

Giambò, Roberto ; Giannoni, Fabio ; Piccione, Paolo

Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 16 (2005) no. 2, pp. 73-85.

Abstract (VO)

Let $(M,g)$ be a complete Riemannian manifold, $\Omega \subset M$ an open subset whose closure is diffeomorphic to an annulus. If $\partial \Omega$ is smooth and it satisfies a strong concavity assumption, then it is possible to prove that there are at least two geometrically distinct geodesics in $\overline{\Omega} = \Omega \bigcup \partial \Omega$ starting orthogonally to one connected component of $\partial \Omega$ and arriving orthogonally onto the other one. The results given in [5] allow to obtain a proof of the existence of two distinct homoclinic orbits for an autonomous Lagrangian system emanating from a nondegenerate maximum point of the potential energy, and a proof of the existence of two distinct for a class of Hamiltonian systems. Under a further symmetry assumption, it is possible to show the existence of at least $\text{dim}(M)$ pairs of geometrically distinct geodesics as above, brake orbits and homoclinics.

Riassunto

Sia $(M,g)$ una varietà Riemanniana completa, e $\Omega \subset M$ un aperto la cui chiusura è omeomorfa ad un anello. Se $\partial \Omega$ è liscio e soddisfa un'ipotesi di concavità forte, è possibile dimostrare che esistono almeno due geodetiche geometricamente distinte in $\overline{\Omega} = \Omega \bigcup \partial \Omega$, aventi gli estremi su componenti connesse distinte di $\partial \Omega$ , e velocità iniziale e finale ortogonali a $\partial \Omega$ . I risultati di [5] permettono di ottenere una dimostrazione, nel caso di un sistema Lagrangiano autonomo, dell'esistenza di due distinte curve omocline partenti da un punto di massimo non degenere dell'energia potenziale, e una dimostrazione dell'esistenza di due distinte per una classe di sistemi Hamiltoniani. Sotto ulteriori ipotesi di simmetria, si ottiene l'esistenza di almeno $\text{dim}(M)$ coppie di geodetiche geometricamente distinte, di brake orbits e di curve omocline.

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