Stability and bifurcations for dissipative polynomial automorphisms of C2

• 成果类型: Article
• 署名作者: Dujardin, Romain; Lyubich, Mikhail
• 署名单位: Universite Gustave-Eiffel; Universite Paris-Est-Creteil-Val-de-Marne (UPEC); State University of New York (SUNY) System; Stony Brook University; State University of New York (SUNY) System; Stony Brook University
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-014-0535-y
• 发表日期: 2015
• 页码: 439-511
• 关键词: analytic transformations; holomorphic motions; local-structure; diffeomorphisms; HYPERBOLICITY; DYNAMICS; approximation; connectivity; families; entropy

摘要

We study stability and bifurcations in holomorphic families of polynomial automorphisms of . 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 -stability in one-dimensional dynamics. Define the bifurcation locus to be the complement of the weak stability locus. 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).