Distribution of rational maps with a preperiodic critical point
Abstract
Let {f_t} be any algebraic family of rational maps of a fixed degree, with a marked critical point c(t). We first prove that the hypersurfaces of parameters for which c(t) 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 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).
Keywords
Cite
@article{arxiv.math/0601612,
title = {Distribution of rational maps with a preperiodic critical point},
author = {Romain Dujardin and Charles Favre},
journal= {arXiv preprint arXiv:math/0601612},
year = {2007}
}
Comments
45 pages. Final version. To appear in American Math. Journal