Degenerate characteristic directions for maps tangent to the identity. (English) Zbl 1280.37041
Summary: Let \(F:(\mathbb{C}^2,O)\to(\mathbb{C}^2,O)\) be a holomorphic germ tangent to the identity. Assume \(F\) has a characteristic direction \([v]\). In [Duke Math. J. 92, No. 2, 403–428 (1998; Zbl 0952.32012)], M. Hakim gives conditions to guarantee the existence of an attracting basin to the origin along \([v]\), in the case of \([v]\) a non-degenerate characteristic direction. In this paper, we give conditions to guarantee the existence of basins along \([v]\) in the case of \([v]\) a degenerate characteristic direction.
MSC:
| 37F10 | Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets |
| 32H50 | Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables |
| 32H02 | Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables |
| 37C70 | Attractors and repellers of smooth dynamical systems and their topological structure |
| 37C25 | Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics |
