On the density of mechanical Lagrangians in T^{2} without continuous invariant graphs in all supercritical energy levels

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September 2008, 10(2&3, September): 661-679. doi: 10.3934/dcdsb.2008.10.661

On the density of mechanical Lagrangians in $T^{2}$ without continuous invariant graphs in all supercritical energy levels

1.	Departamento de Matemática, Pontificia Universidade Católica do Rio de Janeiro, Rua Marqués de São Vicente 225, Gávea, Rio de Janeiro, Brazil

Received September 2006 Revised June 2007 Published June 2008

Abstract
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We show that the set of $C^{\infty}$ mechanical Lagrangians $L(p,v)$ in $T^{2}$ without continuous invariant graphs in all supercritical energy levels is dense in the $C^{1}$ topology. A mechanical Lagrangian $L: T$$T^{2} \rightarrow \mathbb R$ is a function in the tangent space of the torus $T$$T^{2}$ given by $L(p,v)=\frac{1}{2}g(v,v)-U(p)$, where $g$ is a Riemannian metric and $U$ is a smooth potential.

Keywords: Mather set., critical level, mechanical Lagrangian, Invariant graph.

Mathematics Subject Classification: 37J40, 37J50, 53D2.

Citation: Rafael O. Ruggiero. On the density of mechanical Lagrangians in $T^{2}$ without continuous invariant graphs in all supercritical energy levels. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 661-679. doi: 10.3934/dcdsb.2008.10.661

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