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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Quadratic polynomials, multipliers and equidistribution
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by Xavier Buff and Thomas Gauthier PDF
Proc. Amer. Math. Soc. 143 (2015), 3011-3017 Request permission

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
  • © Copyright 2015 American Mathematical Society
  • 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
  • MathSciNet review: 3336625