A two-dimensional polynomial mapping with a wandering Fatou component. (English) Zbl 1368.37055
Sullivan’s “non-wandering domain theorem” asserts that every Fatou component of each rational map is eventually periodic. This result is fundamental for the study of dynamics of rational maps since it opens the way to a complete description of the dynamics in the Fatou set of rational maps and also introduces quasi-conformal mappings as a new tool in this research area. This claim has been generalized to the case of entire mappings with finitely many singular values. On the other hand, various examples of entire mappings with wandering Fatou components have been also constructed. In this paper, the authors apply the parabolic implosion technique and an idea of M. Lyubich to show that there exists an endomorphism \(P:\;\mathbb{P}^2(\mathbb{C})\rightarrow\mathbb{P}^2(\mathbb{C})\), induced by a polynomial skew-product mapping \(P:\;\mathbb{C}^2\rightarrow\mathbb{C}^2\), possessing a wandering Fatou component. They also give real examples with wandering domains in \(\mathbb{R}^2\).
Reviewer: Guoping Zhan (Hangzhou)
MSC:
| 37F50 | Small divisors, rotation domains and linearization in holomorphic dynamics |
| 37F10 | Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets |
| 37C70 | Attractors and repellers of smooth dynamical systems and their topological structure |
| 30C20 | Conformal mappings of special domains |
| 30C65 | Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations |
Keywords:
Fatou set; holomorphic dynamics; parabolic implosion; polynomial mapping; skew-product; wandering Fatou componentReferences:
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