Distribution of rational maps with a preperiodic critical point

2006/01/25 by Dujardin, Romain, Favre, Charles · 3 citations
#32H50 #37F45 #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.math/0601612

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).

Cited by

• On the multiplier spectrum of polynomials
• Rigidity in Complex Dynamics: Multiplier Spectrum and Dynamical André-Oort Conjecture
• Analyticity of the Hausdorff dimension and metric structures on Misiurewicz families of polynomials

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