Fixed points for nilpotent actions on the plane and the Cartwright–Littlewood theorem

Abstract

The goal of this paper is proving the existence and then localizing global fixed points for nilpotent groups generated by homeomorphisms of the plane satisfying a certain Lipschitz condition and having a bounded orbit. The Lipschitz condition is inspired in a classical result of Bonatti for commuting diffeomorphisms of the \(2\)-sphere and in particular it is satisfied by diffeomorphisms, not necessarily of class \(C^{1}\), whose linear part at every point is uniformly close to the identity. In this same setting we prove a version of the Cartwright–Littlewood theorem, obtaining fixed points in any full continuum preserved by a nilpotent action.