Multivalued Lyapunov functions for homeomorphisms of the 2-torus
Volume 189 / 2006
Fundamenta Mathematicae 189 (2006), 227-253
MSC: 37B25, 37E30, 37E35, 37E45.
DOI: 10.4064/fm189-3-2
Abstract
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 $\kappa$.
