Dynamics of Multi-Resonant Biholomorphisms

Filippo Bracci (1) , Jasmin Raissy (2) , Dmitri Zaitsev (3)

1 DIPMAT - Dipartimento di Matematica [Roma II]
2 Dipartimento di Matematica e Applicazioni [Milano]
3 Trinity College Dublin

The goal of this paper is to study the dynamics of holomorphic diffeomorphisms in $\mathbb{C}^n$ such that the resonances among the first $1 \le r \le n$ eigenvalues of the differential are generated over $\mathbb{N}$ by a finite number of $\mathbb{Q}$-linearly independent multi-indices (and more resonances are allowed for other eigenvalues). We give sharp conditions for the existence of basins of attraction where a Fatou coordinate can be defined. Furthermore, we obtain a generalization of the Leau-Fatou flower theorem, providing a complete description of the dynamics in a full neighborhood of the origin for 1-resonant parabolically attracting holomorphic germs in Poincaré-Dulac normal form.

Domaines: Systèmes dynamiques [math.DS]; Variables complexes [math.CV]

Filippo Bracci, Jasmin Raissy, Dmitri Zaitsev. Dynamics of Multi-Resonant Biholomorphisms. International Mathematics Research Notices, 2013, 2013 (20), pp.4772-4797. ⟨hal-00968269⟩