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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 (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 + (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 ((0)(p) > − 1 for all p ∈ ℝ2, then φ is globally C 1 conjugate to the linear involution (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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Correspondence to Benito Pires.

Additional information

The first author was supported by FAPESP-BRAZIL (2009/02380-0 and 2008/02841-4).

The second author was supported by FAPESP-BRAZIL (2007/06896-5).

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Pires, B., Teixeira, M.A. On global linearization of planar involutions. Bull Braz Math Soc, New Series 43, 637–653 (2012). https://doi.org/10.1007/s00574-012-0030-2

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Keywords

  • planar involution
  • linearization
  • smooth conjugacy
  • fixed point

Mathematical subject classification

  • Primary: 37C15
  • Secondary: 37C25