Commuting holomorphic maps in strongly convex domains
The dynamics of holomorphic maps near curves of fixed points
Let be a two-dimensional complex manifold and a holomorphic map. Let be a curve made of fixed points of , i.e. . We study the dynamics near in case acts as the identity on the normal bundle of the regular part of . Besides results of local nature, we prove that if is a globally and locally irreducible compact curve such that then there exists a point and a holomorphic -invariant curve with on the boundary which is attracted by under the action of . These results are achieved...
Local dynamics of holomorphic diffeomorphisms
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.
Holomorphic Evolution: Metamorphosis of the Loewner Equations
Dynamics of one-resonant biholomorphisms
Our first main result is a construction of a simple formal normal form for holomorphic diffeomorphisms in whose differentials have one-dimensional family of resonances in the first eigenvalues, (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.
Punti fissi di mappe olomorfe
Pluripotential theory, semigroups and boundary behavior of infinitesimal generators in strongly convex domains
Aleksandrov-Clark measures and semigroups of analytic functions in the unit disc.
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