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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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
DOI: https://doi.org/10.1090/S0002-9939-2015-12506-3
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 $\frac {1}{n}\log |\rho _n|\to L$. 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
Email: xavier.buff@math.univ-toulouse.fr

Thomas Gauthier
Affiliation: LAMFA, Université de Picardie Jules Verne, 33 rue Saint-Leu, 80039 Amiens Cedex 1, France
MR Author ID: 1019319
Email: thomas.gauthier@u-picardie.fr

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