Open Access
2015 Index 1 fixed points of orientation reversing planar homeomorphisms
Francisco R. Ruiz del Portal, José M. Salazar
Topol. Methods Nonlinear Anal. 46(1): 223-246 (2015). DOI: 10.12775/TMNA.2015.044


Let \(U \subset {\mathbb R}^2\) be an open subset, \(f\colon U\rightarrow f(U) \subset {\mathbb R}^2\) be an orientationreversing homeomorphism and let \(0 \in U\) be an isolated, as a periodic orbit, fixed point. The main theorem of this paper saysthat if the fixed point indices \(i_{{\mathbbR}^2}(f,0)=i_{{\mathbb R}^2}(f^2,0)=1\) then there exists anorientation preserving dissipative homeomorphism $\varphi\colon{\mathbb R}^2 \rightarrow {\mathbb R}^2$ such that \(f^2=\varphi\)in a small neighbourhood of \(0\) and \(\{0\}\) is a globalattractor for \(\varphi\). As a corollary we have that fororientation reversing planar homeomorphisms a fixed point, whichis an isolated fixed point for \(f^2\), is asymptotically stableif and only if it is stable. We also present an application toperiodic differential equations with symmetries where orientationreversing homeomorphisms appear naturally.


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Francisco R. Ruiz del Portal. José M. Salazar. "Index 1 fixed points of orientation reversing planar homeomorphisms." Topol. Methods Nonlinear Anal. 46 (1) 223 - 246, 2015.


Published: 2015
First available in Project Euclid: 30 March 2016

zbMATH: 1364.37096
MathSciNet: MR3443685
Digital Object Identifier: 10.12775/TMNA.2015.044

Rights: Copyright © 2015 Juliusz P. Schauder Centre for Nonlinear Studies

Vol.46 • No. 1 • 2015
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