Distribution of rational maps with a preperiodic critical point

Abstract

Let $\{f_\lambda\}$ be a family of rational maps of a fixed degree, with a marked critical point $c(\lambda)$. Under a natural assumption, we first prove that the hypersurfaces of parameters for which $c(\lambda)$ is periodic converge as a sequence of positive closed $(1,1)$ currents to the bifurcation current attached to $c$ and defined by DeMarco. We then turn our attention to the parameter space of polynomials of a fixed degree $d$. By intersecting the $d-1$ currents attached to each critical point of a polynomial, Bassaneli and Berteloot obtained a positive measure $\mu_{\rm bif}$ of finite mass which is supported on the connectedness locus. They showed that its support is included in the closure of the set of parameters admitting $d-1$ neutral cycles. We show that the support of this measure is precisely the closure of the set of strictly critically finite polynomials (i.e., of Misiurewicz points).