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Frontiers in Complex DynamicsIn Celebration of John Milnor's 80th Birthday$
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Araceli Bonifant, Misha Lyubich, and Scott Sutherland

Print publication date: 2014

Print ISBN-13: 9780691159294

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691159294.001.0001

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On Ecalle-Hakim 's theorems in holomorphic dynamics

On Ecalle-Hakim 's theorems in holomorphic dynamics

(p.387) On Ecalle-Hakim 's theorems in holomorphic dynamics
Frontiers in Complex Dynamics

Marco Arizzi

Jasmin Raissy

, Araceli Bonifant, Mikhail Lyubich, Scott Sutherland
Princeton University Press

This chapter provides detailed proofs for the results by M. Hakim regarding the dynamics of germs of biholomorphisms tangent to the identity of order k + 1 ≥ 2 and fixing the origin. One of the main questions in the study of local discrete holomorphic dynamics, i.e., in the study of the iterates of a germ of a holomorphic map of ℂᵖ at a fixed point, which can be assumed to be the origin, is when it is possible to holomorphically conjugate it to a “simple” form, possibly its linear term. It turns out that the answer to this question strongly depends on the arithmetical properties of the eigenvalues of the linear term of the germ.

Keywords:   M. Hakim, biholomorphisms, holomorphic dynamics, resurgence theory, identity germ, eigenvalues

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