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Quadratic polynomials, multipliers and equidistribution

Authors: Xavier Buff and Thomas Gauthier
Journal: Proc. Amer. Math. Soc. 143 (2015), 3011-3017
MSC (2010): Primary 37F10, 37F45
Published electronically: February 25, 2015
MathSciNet review: 3336625
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Abstract: Given a sequence of complex numbers $ \rho _n$, we study the asymptotic distribution of the sets of parameters $ c\in \mathbb{C}$ such that the quadratic map $ z^2+c$ has a cycle of period $ n$ and multiplier $ \rho _n$. Assume $ \frac {1}{n}\log \vert\rho _n\vert\to L$. If $ L\leq \log 2$, they equidistribute on the boundary of the Mandelbrot set. If $ L>\log 2$, they equidistribute on the equipotential outside the Mandelbrot set of level $ 2L-2\log 2$.

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Additional Information

Xavier Buff
Affiliation: Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118, route de Narbonne, 31062 Toulouse Cedex, France

Thomas Gauthier
Affiliation: LAMFA, Université de Picardie Jules Verne, 33 rue Saint-Leu, 80039 Amiens Cedex 1, France

Received by editor(s): August 29, 2013
Received by editor(s) in revised form: March 5, 2014
Published electronically: February 25, 2015
Additional Notes: The research of the first author was supported by the IUF
Communicated by: Nimish Shah
Article copyright: © Copyright 2015 American Mathematical Society

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