Let F be a homeomorphism of 𝕋² = ℝ²/ℤ² isotopic to the identity and f a lift to the universal covering space ℝ². We suppose that κ ∈ H¹(𝕋²,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 𝕋² whose cohomology class is κ.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm189-3-2,
author = {Patrice Le Calvez},
title = {Multivalued Lyapunov functions for homeomorphisms of the 2-torus},
journal = {Fundamenta Mathematicae},
volume = {189},
year = {2006},
pages = {227-253},
zbl = {1134.37340},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm189-3-2}
}
Patrice Le Calvez. Multivalued Lyapunov functions for homeomorphisms of the 2-torus. Fundamenta Mathematicae, Tome 189 (2006) pp. 227-253. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm189-3-2/