Index 1 fixed points of orientation reversing planar homeomorphisms

Abstract

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.