Index 1 fixed points of orientation reversing planar homeomorphisms
DOI:
https://doi.org/10.12775/TMNA.2015.044Keywords
Fixed point index, Conley index, orientation reversing homeomorphisms, attractors, stabilityAbstract
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.Downloads
Published
2015-09-01
How to Cite
1.
SALAZAR, José M. and RUIZ DEL PORTAL, Francisco Romero. Index 1 fixed points of orientation reversing planar homeomorphisms. Topological Methods in Nonlinear Analysis. Online. 1 September 2015. Vol. 46, no. 1, pp. 223 - 246. [Accessed 28 March 2024]. DOI 10.12775/TMNA.2015.044.
Issue
Section
Articles
Stats
Number of views and downloads: 0
Number of citations: 0