On global linearization of planar involutions

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

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