On global linearization of planar involutions

Abstract

Let φ: ℝ2 → ℝ2 be an orientation-preserving C 1 involution such that φ(0) = 0. Let Spc(φ) = {Eigenvalues of Dφ(p) | p ∈ ℝ2}. We prove that if Spc(φ) ⊂ ℝ or Spc(φ) ∩ [1, 1 + ε) = ∅ for some ε > 0, then φ is globally C 1 conjugate to the linear involution D φ(0) via the conjugacy h = (I + Dφ(0)φ)/2,where I: ℝ2 → ℝ2 is the identity map. Similarly, we prove that if φ is an orientation-reversing C 1 involution such that φ(0) = 0 and Trace (Dφ(0)Dφ(p) > − 1 for all p ∈ ℝ2, 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