Discrete orbits, recurrence and solvable subgroups of Diff(C^2,0)

We discuss the local dynamics of a subgroup of Diff(C^2,0) possessing locally discrete orbits as well as the structure of the recurrent set for more general groups. It is proved, in particular, that a subgroup of Diff(C^2,0) possessing locally discrete orbits must be virtually solvable. These results are of considerable interest in problems concerning integrable systems.Comment: The first version of this paper and "A note on integrability and finite orbits for subgroups of Diff(C^n,0)" are an expanded version of our paper "Discrete orbits and special subgroups of Diff(C^n,0)". An intermediate version re-submitted to the journal on March 2015 is available at http://www.fep.up.pt/docentes/hreis/publications.htm where there is also a comparison between these 3 version

Foliaciones de codimensión uno Newton no degeneradas

The main topic of this research is the study of “Newton non-degenerate codimension one foliations”. The non-degenerate singularities for hypersurfaces have been described classically by A. Kouchnirenko in [35]; let us give a quick description of them.We consider a germ of hypersurface in (Cn, 0), defined locally by a reduced equation f = 0 in local coordinates z = (z1, z2, . . . , zn). We take the Taylor’s expansion of f, we consider the convex hull of the 2 Rn 0 such that 6= 0 and we add to it the first orthant Rn 0. In this way it is obtained the Newton polyhedron 1 of f. We consider its compact boundary and we say that a singularity is “non-degenerate” if the coefficients are “generic” in a sense that we will define later. This class of singularities is open and dense in the space of coefficients when is fixed. Also M. Oka does a study in [36] of the non-degenerate singularities for the case of complete intersections. Taking a logarithmic point of view, we can define a Newton polyhedron associated to a germ of differential form or vector field, once we fix a system of coordinates. From a more geometrical approach, the fact of considering a normal crossings divisor in the ambient space determines the coordinates we are going to consider. On this way, we can define not just a single polyhedron, but a whole polyhedra system, each one associated to one of the strata naturally given by the divisor as we will see in Chapter 2. A foliated space consists of a codimension one foliation F in a complex analytic space M, together with a normal crossings divisor E M. Most of the definitions, properties and results we present in this work concerns the foliated space (M,E,F) and not just to the foliation F. In the general theory established in Chapter 4, we introduce the concept of “Newton non-degenerate foliated space” which, of course, coincides with the classical one for germs of hypersurfaces, when we consider germs of foliations having a holomorphic first integral. On the other hand, once we have a normal crossings divisor in the ambient space, we can talk about “combinatorial blowing-ups”. They are blowing-ups centered at the closure of one of the strata determined by the divisor. We extend the definition introduced by M.I.T. Camacho and F. Cano in [9] and we say that a codimension one foliation is of “toric type” if we obtain only “simple points” after a combinatorial sequence of blowing-ups, that is, if it has a “combinatorial desingularization”.Departamento de Algebra, Geometría y TopologíaDoctorado en Matemática