Quadratic polynomials, multipliers and equidistribution

Xavier Buff (1) , Thomas Gauthier (2)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 LAMFA - Laboratoire Amiénois de Mathématique Fondamentale et Appliquée - UMR CNRS 7352

Given a sequence of complex numbers ρ_n, we study the asymptotic distribution of the sets of parameters c ∈ C such that the quadratic maps z^2 +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.

Mots clés: Mandelbrot set; equidistribution; bifurcations

Domaines: Systèmes dynamiques [math.DS]; Variables complexes [math.CV]

Xavier Buff, Thomas Gauthier. Quadratic polynomials, multipliers and equidistribution. 2013. ⟨hal-00833100⟩