## Tesi etd-02112010-094712 |

Thesis type

Tesi di dottorato di ricerca

Author

RAISSY, JASMIN

URN

etd-02112010-094712

Title

Geometrical methods in the normalization of germs of biholomorphisms

Settore scientifico disciplinare

MAT/03

Corso di studi

MATEMATICA

Supervisors

**tutor**Prof. Abate, Marco

Parole chiave

- Small divisors
- Resonances
- Normalization problem
- Linearization problem
- Discrete local holomorphic dynamical systems
- Commuting holomorphic maps
- Brjuno condition
- Torus actions

Data inizio appello

26/02/2010;

Consultabilità

Completa

Riassunto analitico

In this thesis we use geometrical methods to study the linearization and the normalization problems for germs of biholomorphisms in several complex variables, discussing both the formal level and convergence issues.

We first present a survey on local holomorphic discrete dynamics, focusing our attention on linearization and normalization problems. After fixing the setting and the notation, we deal with the one-dimensional case, and then with the multi-dimensional case. Among other things, we present a new proof of a linearization result in presence of resonances, originally proved by R\"ussmann under a slightly different arithmetic hypothesis.

We then deal with the linearization problem in presence of resonances. In particular we find, under certain arithmetic conditions on the eigenvalues and some restrictions on the resonances, that a necessary and sufficient condition for holomorphic linearization in presence of resonances is the existence of a particular invariant complex manifold. Moreover such a result has as corollaries most of the known linearization results.

Next, we explore in our setting the consequences of the general heuristic principle saying that if a map f commutes with a map g, then some properties of g might be inherited by f, and we show how commuting with a linearizable germ gives us information on the germs that can be conjugated to a given one.

For instance, one possible generalization of the linearization problem is to ask when a given set of m > 1 germs of biholomorphisms at the same fixed point, which we may place at the origin, are simultaneously holomorphically linearizable, i.e., there exists a local holomorphic change of coordinates conjugating each of them to its linear part. We find that if the germs of biholomorphisms have diagonalizable linear part and are such that the first one commutes with all the others, under certain arithmetic conditions on the eigenvalues and some restrictions on their resonances, the germs are simultaneously holomorphically linearizable if and only if there exists a particular complex manifold invariant under them.

Finally we study commutations with a particular kind of linearizable object: torus actions. We find out in a complete and computable manner what kind of structure a torus action must have in order to give a Poincar\'e-Dulac holomorphic normalization, studying the possible torsion phenomena. In particular, we link the eigenvalues of the linear part of our germ of biholomorphism to the weight matrix of the action. The link and the structure we find are more complicated than what one would expect; a detailed study is needed to completely understand the relations between torus actions, holomorphic Poincar\'e-Dulac normalizations, and torsion phenomena. We end our work giving an example of techniques that can be used to construct torus actions.

We first present a survey on local holomorphic discrete dynamics, focusing our attention on linearization and normalization problems. After fixing the setting and the notation, we deal with the one-dimensional case, and then with the multi-dimensional case. Among other things, we present a new proof of a linearization result in presence of resonances, originally proved by R\"ussmann under a slightly different arithmetic hypothesis.

We then deal with the linearization problem in presence of resonances. In particular we find, under certain arithmetic conditions on the eigenvalues and some restrictions on the resonances, that a necessary and sufficient condition for holomorphic linearization in presence of resonances is the existence of a particular invariant complex manifold. Moreover such a result has as corollaries most of the known linearization results.

Next, we explore in our setting the consequences of the general heuristic principle saying that if a map f commutes with a map g, then some properties of g might be inherited by f, and we show how commuting with a linearizable germ gives us information on the germs that can be conjugated to a given one.

For instance, one possible generalization of the linearization problem is to ask when a given set of m > 1 germs of biholomorphisms at the same fixed point, which we may place at the origin, are simultaneously holomorphically linearizable, i.e., there exists a local holomorphic change of coordinates conjugating each of them to its linear part. We find that if the germs of biholomorphisms have diagonalizable linear part and are such that the first one commutes with all the others, under certain arithmetic conditions on the eigenvalues and some restrictions on their resonances, the germs are simultaneously holomorphically linearizable if and only if there exists a particular complex manifold invariant under them.

Finally we study commutations with a particular kind of linearizable object: torus actions. We find out in a complete and computable manner what kind of structure a torus action must have in order to give a Poincar\'e-Dulac holomorphic normalization, studying the possible torsion phenomena. In particular, we link the eigenvalues of the linear part of our germ of biholomorphism to the weight matrix of the action. The link and the structure we find are more complicated than what one would expect; a detailed study is needed to completely understand the relations between torus actions, holomorphic Poincar\'e-Dulac normalizations, and torsion phenomena. We end our work giving an example of techniques that can be used to construct torus actions.

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