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June  2008, 21(2): 551-569. doi: 10.3934/dcds.2008.21.551

Generic properties of Lagrangians on surfaces: The Kupka-Smale theorem

Elismar R. Oliveira 1,

1. 

Departamento de Matemática, Universidade Federal do Rio Grande do Sul, Av. Bento Goncalves, 9500, 91509-900, Porto Alegre, RS, Brazil

Received  June 2007 Revised  January 2008 Published  March 2008

We consider generic properties of Lagrangians. Our main result is a Kupka-Smale Theorem for the Lagrangian setting. We show that for convex and superlinear Lagrangians defined on a compact surface and $k\in \mathbb{R}$, then generically, in Mañé's sense, the energy level $k$ is regular and all periodic orbits in this level are nondegenerate at all orders (the linearized Poincaré map, restricted to this energy level, does not have roots of unity as eigenvalues). Moreover, all heteroclinic intersections in this level are transversal. The results that we present are true in dimension $n \geq 2$, with the exception of Theorem 4.5, which we are only able to prove in dimension 2.
Keywords: Lagrangians, Dynamical Systems, Generic Properties, Nondegeneracy of Periodic Orbits., Kupka-Smale Theorem.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
Citation: Elismar R. Oliveira. Generic properties of Lagrangians on surfaces: The Kupka-Smale theorem. Discrete and Continuous Dynamical Systems, 2008, 21 (2) : 551-569. doi: 10.3934/dcds.2008.21.551
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