Multisummability of unfoldings of tangent to the identity diffeomorphisms

We prove the multisummability of the infinitesimal generator of unfoldings of finite codimension tangent to the identity 1-dimensional local complex analytic diffeomorphisms. We also prove the multisummability of Fatou coordinates and extensions of the Ecalle-Voronin invariants associated to these unfoldings. The quasi-analytic nature is related to the parameter variable. As an application we prove an isolated zeros theorem for the analytic conjugacy problem. The proof is based on good asymptotics of Fatou coordinates and the introduction of a new auxiliary tool, the so called multi-transversal flows. They provide the estimates and the combinatorics of sectors typically associated to summability. The methods are based on the study of the infinitesimal stability properties of the unfoldings.