Foliations and polynomial diffeomorphisms of \({\mathbb{R}^{3}}\)

  • Carlos Gutierrez
  • Carlos MaqueraEmail author


Let \({Y=(f,g,h){:} \mathbb{R}^{3} \to \mathbb{R}^{3}}\) be a C 2 map and let Spec(Y) denote the set of eigenvalues of the derivative DY p , when p varies in \({\mathbb{R}^3}\) . We begin proving that if, for some ϵ > 0, \({Spec(Y)\cap (-\epsilon,\epsilon)=\emptyset,}\) then the foliation \({\mathcal{F}(k),}\) with \({k\in \{f,g,h\},}\) made up by the level surfaces {k = constant}, consists just of planes. As a consequence, we prove a bijectivity result related to the three-dimensional case of Jelonek’s Jacobian Conjecture for polynomial maps of \({\mathbb{R}^n.}\)


Three dimensional vector field Global injectivity Foliation 

Mathematics Subject Classification (2000)

37C85 57R30 


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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Departamento de Matemática, Instituto de Ciências Matemáticas e de ComputaçãoUniversidade de São Paulo, São CarlosSão CarlosBrazil

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