Hölder regularity of weak KAM solutions in a priori unstable systems

For {\it a priori} unstable Hamiltonian systems with two and half degrees of freedom, there is a continuous path in $H^1(\mathbb{T}^2,\mathbb{R})$ such that for each cohomology class $c$ in this path, the $c$-minimal measure is supported on a normally hyperbolic cylinder. In this paper, we show that the weak KAM solutions for these classes can be parameterized by the area bounded by the graph of these solutions and obtain the $\frac 14$-Hölder regularity of these solutions in the parameter.

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Published 28 February 2011