Quadratic polynomials, multipliers and equidistribution

2013/06/12 by Xavier Buff, Buff, Xavier, Thomas Gauthier +1
Mathematics · #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #math.CV #math.DS

paper · pdf · doi:10.48550/arxiv.1306.2736

arxiv created 2013/06/12 · arxiv updated 2013/06/13

Abstract

Given a sequence of complex numbers ρ_n, we study the asymptotic distribution of the sets of parameters c ε C such that the quadratic maps z² +c has a cycle of period n and multiplier ρ_n. Assume 1/n.log|ρ_n| tends to 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.

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