Conformal equivalence of analytic flows

Classification of complex polynomial vector fields in one complex variable

“…The second school and the one this paper belongs to is studying holomorphic vector fields in their own right (classification) in, for example, Brickman and Thomas [4] and Douady et al [8], and in the study of quadratic differentials in Jenkins [11] and Strebel [18]. Quadratic differentials and holomorphic vector fields can both be viewed as foliations with singularities.…”

“…In [4], the authors use 'two numerical invariants' (one being the multiplicity of the zero, and the other coinciding with the dynamical residue defined in Section 4 of this paper) associated with the analytic function defining the vector field, which 'classify the induced flow up to conformal equivalence'. However, this is a local, not global, result.…”

“…However, this is a local, not global, result. In this vein, [4,10] also give the local normal forms for such flows.…”

“…It seems natural to consider the dynamical residues (defined below) of the equilibrium points as analytic invariants, as these determine the local normal form of the vector field in a neighbourhood of each equilibrium point (see for instance [4]), but we will see that there is in fact a better choice (see Definition 4.2).…”

“…The vector fields j M and F * ðj M Þ ¼ Pðd=dwÞ must have the same number of zeros with the same multiplicities (and at the images Fð1 ½l Þ and Fð1 {½l } Þ) and the same dynamical residues (for example, Theorem 1 in [4]), hence the same analytic invariants.…”