On global linearization of planar involutions

Let $\varphi :{\mathbb{R}}^2\to {\mathbb{R}}^2$ be an orientation–preserving $C^1$ involution such that $\varphi (0)=0$ and let ${\rm Spc},(\phi)={{\rm Eigenvalues,,of},, D\phi(p)\mid p\in\R^2}$\mathrm{.}Weprovethatif${\rm Spc},{(\phi)}\subset\R$or${\rm Spc},(\phi)\cap [1,1+\epsilon)=\emptyset$forsome$\epsilon>0$then$\phi$isglobally$C^1$conjugatetothelinearinvolution$D\phi(0)$ via the conjugacy $h=(I+D\varphi (0)\varphi )\mathrm{/}2$, 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 $\mathrm{T}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(D\varphi (0)D\varphi (p))>-1$ 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.