Multivalued Lyapunov functions for homeomorphisms of the 2-torus

Volume 189 / 2006

Patrice Le Calvez Fundamenta Mathematicae 189 (2006), 227-253 MSC: 37B25, 37E30, 37E35, 37E45. DOI: 10.4064/fm189-3-2


Let $F$ be a homeomorphism of $\mathbb T^2=\mathbb R^2/\mathbb Z^2$ isotopic to the identity and $f$ a lift to the universal covering space $\mathbb R^2$. We suppose that $\kappa\in H^1(\mathbb T^2,\mathbb R)$ is a cohomology class which is positive on the rotation set of $f$. We prove the existence of a smooth Lyapunov function of $f$ whose derivative lifts a non-vanishing smooth closed form on $\mathbb T^2$ whose cohomology class is $\kappa$.


  • Patrice Le CalvezLaboratoire Analyse,
    Géométrie et Applications
    C.N.R.S.-U.M.R 7539
    Institut Galilée
    Université Paris 13
    Avenue J.-B. Clément
    93430 Villetaneuse, France

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