2011/11/09 by Jean-François Mattéi, Mattei, Jean-François, Julio C. Rebelo +3
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Geometry and complex manifolds #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.1111.2364
openalex publication_date 2011/11/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We show that generically a pseudogroup generated by holomorphic diffeomorphisms defined about 0 ∈ ℂ is free in the sense of pseudogroups even if the class of conjugacy of the generators is fixed. This result has a number of consequences on the topology of leaves for a (singular) holomorphic foliation defined on a neighborhood of an invariant curve. In particular in the classical and simplest case arising from local foliations possessing a unique separatrix that is given by a cusp of the form \y2-x2k+1=0\, our results allow us to settle the problem of showing that a generic foliation possesses only countably many non-simply connected leaves and that this countable set is, indeed, infinite.