Multivalued Lyapunov functions for homeomorphisms of the 2-torus

Patrice Le Calvez¹

¹ Laboratoire Analyse, Géométrie et Applications C.N.R.S.-U.M.R 7539 Institut Galilée Université Paris 13 Avenue J.-B. Clément 93430 Villetaneuse, France

Fundamenta Mathematicae, Tome 189 (2006) no. 3, pp. 227-253.

Résumé

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 $\kappa$.

Détail

DOI : 10.4064/fm189-3-2

Keywords: homeomorphism mathbb mathbb mathbb isotopic identity lift universal covering space mathbb suppose kappa mathbb mathbb cohomology class which positive rotation set prove existence smooth lyapunov function whose derivative lifts non vanishing smooth closed form mathbb whose cohomology class kappa