TanDynamical convergence and polynomial vector fields

Let fn → f0 be a convergent sequence of rational maps, preserv-ing critical relations, and f0 be geometrically finite with parabolic points. It is known that for some unlucky choices of sequences fn, the Julia sets J(fn) and their Hausdorff dimensions may fail to converge as n → ∞. Our main result here is to prove the con-vergence of J(fn) and H.dim J(fn) for generic sequences fn. The same conclusion was obtained earlier, with stronger hypotheses on the sequence fn, by Bodart-Zinsmeister and then by McMullen. We characterize those choices of fn by means of flows of appropri-ate polynomial vector fields (following Douady-Estrada-Sentenac). We first prove an independent result about the (s-dimensional) length of separatrices of such flows, and then use it to estimate tails of Poincare ́ series. This, together with existing techniques, provides the desired control of conformal densities and Hausdorff dimensions. Our method may be applied to other problems related to parabolic perturbations. 1