A Poincaré-Bendixson theorem for homogeneous vector fields and meromorphic connections

Jeudi 1er janvier 2009 14:00-15:00 - Abate Marco - Université de Pise

Résumé : In one complex dimension, a holomorphic germ tangent to the identity is locally topologically conjugated to the time-1 map of a homogeneous vector field. In particular, the study of the real flow of (complex) homogeneous vector fields in (complex) dimension one provides a large amount of informations on the local dynamics of functions tangent to the identity. This suggests that the study of the real flow of complex homogeneous vector fields might also help to understand the local dynamics of holomorphic map tangent to the identity in complex dimension two, at least in generic cases. In this talk I shall describe how it is possible to reduce the study of the real 1-dimensional flow of a complex 2-dimensional homogenous vector field to the study of the geodesic flow of a meromorphic connection on the complex projective line, and how to use this reduction and a new Poincaré-Bendixson theorem for meromorphic connections to describe the recurrent behavior of such flows. Together with a study of the local dynamics of the geodesic flow nearby the poles of the connection this yields a complete description of the topological dynamics of a large class of time-1 maps of homogeneous vector fields in dimension two. (Joint work with F. Tovena)

Lieu : bât. 425 - 113-115

A Poincaré-Bendixson theorem for homogeneous vector fields and meromorphic connections  Version PDF