Index 1 fixed points of orientation reversing planar homeomorphisms

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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.