Dynamics of one-resonant biholomorphisms
Filippo Bracci, Dmitri Zaitsev
TL;DR
This paper develops a formal normal form for certain holomorphic diffeomorphisms with specific resonance structures, identifies invariants, and explores conditions for basins of attraction, with applications illustrating the results.
Contribution
It introduces a simple formal normal form for one-resonant biholomorphisms and provides criteria for basins of attraction, expanding understanding of their dynamics.
Findings
Constructed a formal normal form for one-resonant biholomorphisms.
Identified invariants and conditions for basins of attraction.
Provided examples demonstrating the sharpness of the conditions.
Abstract
Our first main result is a construction of a simple formal normal form for holomorphic diffeomorphisms in C^n whose differentials have one-dimensional family of resonances in the first m eigenvalues, m <= n (but more resonances are allowed for other eigenvalues). Next, we provide invariants and give conditions for the existence of basins of attraction. Finally, we give applications and examples demonstrating the sharpness of our conditions.
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