Fixed points of local actions of nilpotent Lie groups on surfaces. (28th January 2016)

• Record Type: Journal Article
• Authors: HIRSCH, MORRIS W.
• Is Part Of: Ergodic theory and dynamical systems. Volume 37:Number 4(2017)
• Page Start: 1238
• Page End: 1252
• Publication Date: 2016-01-28
• Subjects: Ergodic theory -- Periodicals; Differentiable dynamical systems -- Periodicals; 515.42
• DOI: 10.1017/etds.2015.73
• Languages: English
• ISSNs: 0143-3857

Abstract

Abstract : Let $G$ be a connected nilpotent Lie group with a continuous local action on a real surface $M$, which might be non-compact or have non-empty boundary $\unicode[STIX]{x2202}M$ . The action need not be smooth. Let $\unicode[STIX]{x1D711}$ be the local flow on $M$ induced by the action of some one-parameter subgroup. Assume $K$ is a compact set of fixed points of $\unicode[STIX]{x1D711}$ and $U$ is a neighborhood of $K$ containing no other fixed points. Theorem. If the Dold fixed-point index of $\unicode[STIX]{x1D711}_{t}|U$ is non-zero for sufficiently small $t>0$, then $\mathsf{Fix}(G)\cap K\neq \varnothing$ .