On global linearization of planar involutions

Abstract

Let $\varphi :{\mathbb{R}}^2\to {\mathbb{R}}^2$ be an orientation--preserving $C^1$ involution such that $\varphi (0)=0$ and let $\mathrm{Spc}(\varphi )=\{\mathrm{Eigenvalues}\mathrm{of}D\varphi (p)\mid p\in {\mathbb{R}}^2\}$. We prove that if $\mathrm{Spc}(\varphi )\subset \mathbb{R}$ or $\mathrm{Spc}(\varphi )\cap [1,1+\epsilon )=\varnothing$ for some $\epsilon >0$ then $\varphi$ is globally $C^1$ conjugate to the linear involution $D\varphi (0)$ via the conjugacy $h=(I+D\varphi (0)\varphi )/2$, where $I:{\mathbb{R}}^2\to {\mathbb{R}}^2$ is the identity map. Similarly, if $\varphi$ is an orientation-reversing $C^1$ involution such that $\varphi (0)=0$ and $\mathrm{Trace}(D\varphi (0)D\varphi (p))>-1$ for all $p\in {\mathbb{R}}^2$ then $\varphi$ is globally $C^1$ conjugate to the linear involution $D\varphi (0)$ via the conjugacy $h$. Finally, we show that $h$ may fail to be a global linearization of $\varphi$ if the above conditions are not fulfilled.