Quasisymmetric robustness of the Collet-Eckmann condition in the quadratic family

We consider quasisymmetric reparametrizations of the parameter space of the quadratic family. We prove that the set of quadratic maps which are either regular or Collet-Eckmann with polynomial recurrence of the critical orbit has full Lebesgue measure. (The intent of this work is just to be a rigorous reference for the companion preprint Statistical properties of unimodal maps: smooth families with negative Schwarzian derivative. It is based on the LaTeX file of the paper Statistical properties of unimodal maps: the quadratic family. The proofs are very similar and in many places differ just by change of constants.)