On global linearization of planar involutions

الأصل · EN

Let ϕ:²→² be an orientation--preserving C¹ involution such that ϕ(0)=0 and let Spc(ϕ)={ Eigenvaluesof Dϕ(p) p∈²}. We prove that if Spc(ϕ)⊂ or Spc(ϕ)∩ [1,1+ε)= for some ε>0 then ϕ is globally C¹ conjugate to the linear involution Dϕ(0) via the conjugacy h=(I+Dϕ(0)ϕ)/2, where I:²→² is the identity map. Similarly, if ϕ is an orientation-reversing C¹ involution such that ϕ(0)=0 and Trace(Dϕ(0)Dϕ(p))>-1 for all p∈² then ϕ is globally C¹ 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.