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.
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