Index 1 fixed points of orientation reversing planar homeomorphisms - Salazar, José M.; Ruiz del Portal, Francisco Romero (Topological Methods in Nonlinear Analysis, 2015)
Let \(U \subset {\mathbb R}^2\) be an open subset, \(f\colon U → f(U) \subset {\mathbb R}^2\) 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 $φ\colon {\mathbb R}^2 → {\mathbb R}^2$ such that \(f^2=φ\) in a~small neighbourhood of \(0\) and \(\{0\}\) is a~global attractor for \(φ\). 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.