• Poincarè-Bendixson theorem

The main topic of this thesis is the study of the geodesics for a meromorphic connection on a Riemann surface, and its application to the study of the dynamics of holomorphic homogeneous vector fields in $\mathbb{C}^n$.

In one complex variable, the iteration theory of a holomorphic function fixing the origin is well established. In particular, the Leau-Fatou flower Theorem gives a precise description for the so-called maps tangent to the identity (i.e., with derivative at the origin equal to 1), asserting the presence of petal-shaped basins of attraction for the origin, and proving that the convergence to zero is possible only tangentially to some precise directions, one for each petal.

A complete generalization of this theorem in $\mathbb{C}^n$ is not known, yet, but there are a lot of partial results, due to \'Ecalle, Hakim, Abate, Bracci, Tovena and others. One thing is sure: the Leau-Fatou Theorem cannot be generalized in a trivial way (i.e., with a multi-dimensional flower figure and the associated directions of convergence) because there are examples of new phenomena arising in several complex variables.

From the topological point of view, a famous theorem by Camacho asserts that every one-dimensional map tangent to the identity is locally topologically conjugated to the time-1 map of a homogeneous vector field, and it is reasonable to suppose that such a statement may be generalized to the several complex variables setting, at least for generic maps. If this were the case, understanding the dynamics of time-1 maps of vector fields would give a description of the topological dynamics of generic maps tangent to the identity in several complex variables. Anyway, time-1 maps of homogeneous vector fields provide an important class of examples to study.

The methods presented in this thesis are mainly due to Abate and Tovena. They are based upon a very precise connection between the real integral curves for homogeneous vector fields in $\mathbb{C}^n$ and the geodesics for a suitable meromorphic connection on the leaves of a foliation in Riemann surfaces of the exceptional divisor obtained blowing up the origin. So, the study splits in two parts: understanding the foliation in surfaces and studying the geodesics for the connection, which are contained in the leaves.

We shall be mainly interested in the second one. In particular, we generalize the theory developed by Abate and Tovena for the Riemann sphere to a generic compact Riemann surface, and we classify all the possible $\omega$-limits of geodesics for meromophic connections in this setting. This problem, apart from its intrinsic interest, is the first step to the understanding of the behavior of the geodesics in a leaf of the foliation. Moreover, we study in detail the holomorphic connections on a complex torus, giving an explicit description of the $\omega$-limits in this case.

Then, we study the behaviour of a geodesic near a singular point for the connection. The theory for the singularities of lower order (0 or 1) is well established, thanks to the existence of holomorphic normal forms for the geodesic field. In the irregular case such a holomorphic classification is not known, yet, and this prevents to get a precise description in this case. Nonetheless, we are able to prove a partial result about geodesics converging to this kind of singular point.

There is a case in which all these methods are particularly efficient (and this is the reason we pay more attention to the problem of understanding the geodesics than to the one of understanding the foliation): in $\mathbb{C}^2$, the exceptional divisor is the Riemann sphere, and so the foliation becomes trivial. So, being able to classify the $\omega$-limits of geodesics of the sphere solves both parts of the problem, allowing to obtain a fairly complete description of the dynamics in this situation.

As an application of this theory, after giving a holomorphic classification of cubic maps tangent to the identity in $\mathbb{C}^2$, we study in detail the dynamics for the holomorphic representatives, in analogy with what was done by Abate and Tovena for the quadratic maps. This allows to obtain a pretty complete description of the possible phenomena arising in this case, and to get concrete examples of maps showing a behaviour very different from their counterparts in one complex variable.