2013/05/13 by Romain Dujardin, Dujardin, Romain, Lyubich Mikhail +1 · 1 citation
Mathematics · #Advanced Differential Equations and Dynamical Systems #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Meromorphic and Entire Functions
paper · pdf · doi:10.48550/arxiv.1305.2898
openalex publication_date 2013/05/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study stability and bifurcations in holomorphic families of polynomial automorphisms of C2. 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).