Inventiones mathematicae

, Volume 161, Issue 1, pp 45–89 | Cite as

Kupka-Smale theorem for polynomial automorphisms of ℂ2 and persistence of heteroclinic intersections

  • Gregery T. Buzzard
  • Suzanne Lynch Hruska
  • Yulij Ilyashenko


A map is Kupka-Smale if all periodic points are hyperbolic and the stable and unstable manifolds of any two saddle points are transverse. Here we prove that Kupka-Smale maps form a residual set of full Lebesgue measure in the space \(\mathcal{P}_d\) of polynomial automorphisms of ℂ2 of fixed dynamical degree d≥2. We also prove that a heteroclinic point of two saddle periodic orbits may be continued over (almost) the entire parameter space for this set of maps. This is one of the first persistence theorems proved in holomorphic dynamics in several variables.


Manifold Parameter Space Periodic Orbit Saddle Point Lebesgue Measure 
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© Springer-Verlag 2004

Authors and Affiliations

  1. 1.Department of MathematicsPurdue UniversityWest LafayetteUSA
  2. 2.Department of MathematicsCornell UniversityIthacaUSA
  3. 3.Steklov Math. InstituteMoscow State and Independent UniversitiesMoscowRussia

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