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The dynamics of holomorphic maps near curves of fixed points

Filippo Bracci — 2003

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Let M be a two-dimensional complex manifold and f : M → M a holomorphic map. Let S ⊂ M be a curve made of fixed points of f , i.e.  Fix ( f ) = S . We study the dynamics near  S in case  f acts as the identity on the normal bundle of the regular part of  S . Besides results of local nature, we prove that if  S is a globally and locally irreducible compact curve such that S · S < 0 then there exists a point p ∈ S and a holomorphic f -invariant curve with  p on the boundary which is attracted by  p under the action of  f . These results are achieved...

Local dynamics of holomorphic diffeomorphisms

Filippo Bracci — 2004

Bollettino dell'Unione Matematica Italiana

This is a survey about local holomorphic dynamics, from Poincaré's times to nowadays. Some new ideas on how to relate discrete dynamics to continuous dynamics are also introduced. It is the text of the talk given by the author at the XVII UMI Congress at Milano.

Dynamics of one-resonant biholomorphisms

Filippo Bracci;  Dmitri Zaitsev — 2013

Journal of the European Mathematical Society

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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