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.
Similar content being viewed by others
References
B. Alarcón, V. Guíñez and C. Gutierrez. Planar embeddings with a globally attracting fixed point. Nonlinear Anal., 69(1) (2008), 140–150.
B. Alarcón, C. Gutierrez and J. Martínez-Alfaro. Planar maps whose second iterate has a unique fixed point. J. Difference Equ. Appl., 14(4) (2008), 421–428.
A. Bialynicki-Birula and M. Rosenlicht. Injective morphisms of real algebraic varieties. Proc. Amer. Math. Soc., 13 (1962), 200–203.
F. Braun and J.R. dos Santos Filho. The real Jacobian conjecture on ℝ2 is true when one of the components has degree 3. Discrete Contin. Dyn. Syst., 26(1) (2010), 75–87.
H. Cartan. Sur les groupes de transformations analytiques. Actualités Scientifiques et Industrielles, 198 (1935), Hermann, Paris.
M. Chamberland. Characterizing two-dimensional maps whose Jacobians have constant eigenvalues. Canad. Math. Bull., 46(3) (2003), 323–331.
M. Chamberland and G. Meisters. A mountain pass to the Jacobian Conjecture. Canad. Math. Bull., 41(4) (1998), 442–451.
A. Cima, A. Gasull and F. Mañosas. Global linearization of periodic difference equations. Discrete Contin. Dyn. Syst., 32(5) (2012), 1575–1595.
A. Cima, A. Gasull and F. Mañosas. Simple examples of planar involutions with non-global Montgomery-Bochner linearizations. Appl. Math. Lett., (2012), doi: 10.1016/j.aml.2012.05.004
M. Cobo, C. Gutierrez and J. Llibre. On the injectivity of C 1 maps of the real plane. Canad. J. Math., 54(6) (2002), 1187–1201.
A. Constantin and B. Kolev. The theorem of Kerékjártó on periodic homeomorphisms of the disc and the sphere. Enseign. Math., 40(3–4) (1994), 193–204.
A. Fernandes, C. Gutierrez and R. Rabanal. On local diffeomorphisms of ℝn that are injective. Qual. Theory Dyn. Syst., 4(2) (2003), 255–262.
A. Fernandes, C. Gutierrez and R. Rabanal. Global asymptotic stability for differentiable vector fields of ℝn. J. Differential Equations, 206(2) (2004), 470–482.
R.Feßler.A proof of the two-dimensional Markus-Yamabe stability conjecture and a generalization. Ann. Polon. Math., 62(1) (1995), 45–74.
C. Gutierrez. A solution to the bidimensional global asymptotic stability conjecture. Ann. Inst. H. Poincaré Anal. Non Linéaire, 12(6) (1995), 627–671.
C. Gutierrez, X. Jarque, J. Llibre and M.A. Teixeira. Global injectivity of C 1 maps of the real plane, inseparable leaves and the Palais-Smale condition.Canad. Math. Bull., 50(3) (2007), 377–389.
C. Gutierrez and R. Rabanal. Injectivity of differentiable maps ℝ2 → ℝ2 at infinity. Bull. Braz. Math. Soc. 37(2) (2006), 217–239.
C. Gutierrez, B. Pires and R. Rabanal. Asymptotic stability at infinity for differentiable vector fields of the plane. J. Differential Equations, 231(1) (2006), 165–181.
C. Gutierrez and A. Sarmiento. Injectivity of C 1 maps ℝ2 → ℝ2 at infinity and planar vector fields. Astérisque, 287 (2003), 89–102.
C. Gutierrez and M.A. Teixeira. Asymptotic stability at infinity of planar vector fields. Bull. Braz. Mat. Soc., 26(1) (1995), 57–66.
M.W. Hirsch. Fixed-point indices, homoclinic contacts, and dynamics of injective planar maps. Michigan Math. J., 47(1) (2000), 101–108.
R.S. MacKay. Renormalisation in area-preserving maps. Advanced Series in Nonlinear Dynamics, 6, World Scientific Publishing Co., (1993).
S. Mancini, M. Manoel and M.A. Teixeira. Divergent diagrams of folds and simultaneous conjugacy of involutions. Discrete Contin. Dyn. Syst., 12(4) (2005), 657–674.
L. Markus and H. Yamabe. Global stability criteria for differential systems.Osaka Math. J., 12 (1960), 305–317.
K.R. Meyer. Hamiltonian systems with a discrete symmetry. J. Differential Equations, 41(2) (1981), 228–238.
D. Montgomery and L. Zippin. Topological transformation groups. Interscience Publishers (1955).
C. Olech. On the global stability of an autonomous system on the plane. Contributions to Differential Equations, 1 (1993), 389–400.
R. Rabanal. Center type perfomance of differentiable vector fields in the plane. Proc. Amer. Math. Soc., 137(2) (2009), 653–662.
B. Smyth and F. Xavier. Injectivity of local diffeomorphisms from nearly spectral conditions. J. Differential Equations, 130(2) (1996), 406–414.
M.A. Teixeira. Local and simultaneous structural stability of certain diffeomorphisms. Lecture Notes in Math., 898 (1981), Springer.
Author information
Authors and Affiliations
Corresponding author
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).
About this article
Cite this article
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
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00574-012-0030-2