Topological Methods in Nonlinear Analysis

Index 1 fixed points of orientation reversing planar homeomorphisms

Francisco R. Ruiz del Portal and José M. Salazar

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Abstract

Let \(U \subset {\mathbb R}^2\) be an open subset, \(f\colon U \rightarrow f(U) \subset {\mathbb R}^2\) be an orientation reversing homeomorphism and let \(0 \in U\) be an isolated, as a periodic orbit, fixed point. The main theorem of this paper says that if the fixed point indices \(i_{{\mathbb R}^2}(f,0)=i_{{\mathbb R}^2}(f^2,0)=1\) then there exists an orientation 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 global attractor for \(\varphi\). As a corollary we have that for orientation reversing planar homeomorphisms a fixed point, which is an isolated fixed point for \(f^2\), is asymptotically stable if and only if it is stable. We also present an application to periodic differential equations with symmetries where orientation reversing homeomorphisms appear naturally.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 46, Number 1 (2015), 223-246.

Dates
First available in Project Euclid: 30 March 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1459343892

Digital Object Identifier
doi:10.12775/TMNA.2015.044

Mathematical Reviews number (MathSciNet)
MR3443685

Zentralblatt MATH identifier
1364.37096

Citation

Ruiz del Portal, Francisco R.; Salazar, José M. Index 1 fixed points of orientation reversing planar homeomorphisms. Topol. Methods Nonlinear Anal. 46 (2015), no. 1, 223--246. doi:10.12775/TMNA.2015.044. https://projecteuclid.org/euclid.tmna/1459343892


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