Distribution of rational maps with a preperiodic critical point

Details Distribution of rational maps with a preperiodic critical point Distribution of rational maps with a preperiodic critical point

2006-01-25

arxiv.org/abs/math/0601612v3

Mathematics

Dynamical Systems

45 pages. Final version. To appear in American Math. Journal

Scientific paper

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).

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