In the study of complex dynamics, the so called equidistribution phenomena (of pullback of points, preperiodic points, and so on) towards the canonical measure supported on the chaotic part of dynamics have played a fundamental role in the recent development. Such equidistribution phenomena also occur in the parameter space of holomorphic family of rational functions, and have been studied by many researchers including Levin, Favre--Rivera-Letelier, Dujardin--Favre, Buff--Gauthier, and Gauthier--Vigny. In a spirit of the Nevanlinna theory, it seems interesting to study how those equidistribution phenomena would be quantified. In this talk, we will survey a history of quantitative equidistribution in complex dynamics including a recent remarkable Gauthier--Vigny's in the parameter space of polynomial families, and will give a precision of their result in the specific (monic and centered) unicritical polynomials family.
