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Distribution of rational maps with a preperiodic critical point. (English) Zbl 1246.37071

Summary: 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 \(\mu_{\mathrm{bif}}\) 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).

MSC:

37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables
32U40 Currents
37F45 Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations (MSC2010)
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