Stability and bifurcations for dissipative polynomial automorphisms of C^2

By Romain Dujardin, M. Lyubich. 2013

65 citations recorded by the source corpus.

We study stability and bifurcations in holomorphic families of polynomial automorphisms of C^2. We say that such a family is weakly stable over some parameter domain if periodic orbits do not bifurcate there. We first show that this defines a meaningful notion of stability, which parallels in many ways the classical notion of J-stability in one-dimensional dynamics. In the second part of the paper, we prove that under an assumption of moderate dissipativity, the parameters displaying homoclinic tangencies are dense in the bifurcation locus. This confirms one of Palis' Conjectures in the complex setting. The proof relies on the formalism of semi-parabolic bifurcation and the construction of "critical points" in semi-parabolic basins (which makes use of the classical Denjoy-Carleman-Ahlfors and Wiman Theorems).

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/45ccb2d55336021e7a43eb7bc844ff702e821159

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In Applied Mathematics

1. Franccois Berteloot
2. Fabrizio Bianchi
3. Christophe Dupont

In Geometry and Topology

1. Franccois Berteloot
2. Fabrizio Bianchi
3. Christophe Dupont

In Mathematical Physics

1. Christophe Dupont
2. Franccois Berteloot
3. Fabrizio Bianchi

Citations in the corpus

• Dynamical stability and Lyapunov exponents for holomorphic endomorphisms of CP(k) (2014, background): “Let us finally mention that bifurcation phenomena in families of Hénon maps of C(2) have already been studied by Bedford, Lyubich and Smillie [BLS] and by Dinh and Sibony [DS5], the sharpest achievements are due to Dujardin and Lyubich in their recent work on the two dimensional and dissipative case”