Quadratic polynomials, multipliers and equidistribution

Given a sequence of complex numbers ρn, we study the asymptotic distribution of the sets of parameters c ∈ C such that the quadratic maps z2+c has a cycle of period n and multiplier ρn. Assume 1 n log |ρn| → L. If L ≤ log 2, they equidistribute on the boundary of the Mandelbrot set. If L > log 2 they equidistribute on the equipotential of the Mandelbrot set of level 2L− 2 log 2. Introduction In this article, we study equidistribution questions in the parameters space of the family of quadratic polynomials fc(z) := z 2 + c, c ∈ C. We denote by Kc the filled-in Julia set of fc and by Jc the Julia set: Kc := { z ∈ C | ( f◦n c (z) ) n∈N is bounded } and Jc := ∂Kc. The Mandelbrot set M is the set of parameters c ∈ C such that 0 ∈ Kc. Figure 1. The Mandelbrot set M . The Green functions gc : C→ [0,+∞) and gM : C→ [0,+∞) are defined by gc := lim n→+∞ max ( 1 2n log ∣∣f◦n c (z)∣∣, 0) and gM (c) := gc(c). The research of the first author was supported by the IUF. 1 2 X. BUFF AND T. GAUTHIER The bifurcation measure μbif is defined by μbif := ∆gM . The support of the measure μbif is the boundary of the Mandelbrot set M . Let ρ be a complex number with |ρ| ≤ 1. For n ≥ 1, denote by Xn the set of parameters c ∈ C such that the quadratic polynomial fc has a cycle of period n with multiplier ρ. Let νn be the probability measure