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Fixed points of local actions of nilpotent Lie groups on surfaces

Published online by Cambridge University Press:  28 January 2016

MORRIS W. HIRSCH
MORRIS W. HIRSCH*
Affiliation:
Mathematics Department, University of Wisconsin, Madison, WI, USA University of California, Berkeley, CA, USA email [email protected]
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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$\mathsf{Fix}(G)\cap K\neq \varnothing$.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 37 , Issue 4 , June 2017 , pp. 1238 - 1252
Copyright
© Cambridge University Press, 2016 

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