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

TL;DR

This paper proves the existence and localization of fixed points for nilpotent groups of plane homeomorphisms under a Lipschitz condition, extending classical results and the Cartwright-Littlewood theorem to broader settings.

Contribution

It introduces a new fixed point theorem for nilpotent group actions on the plane satisfying a Lipschitz condition, generalizing previous results for commuting diffeomorphisms.

Findings

Existence of fixed points for nilpotent group actions on the plane.

Localization of these fixed points within invariant continua.

Extension of the Cartwright-Littlewood theorem to this setting.

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. The 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 continuum preserved by a nilpotent action.