Quadratic polynomials, multipliers and equidistribution

Buff, Xavier; Gauthier, Thomas

doi:10.1090/S0002-9939-2015-12506-3

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 $2L-2\log 2$.

References

Giovanni Bassanelli and François Berteloot, Distribution of polynomials with cycles of a given multiplier, Nagoya Math. J. 201 (2011), 23–43. MR 2772169, DOI 10.1215/00277630-2010-016
Paul Blanchard, Robert L. Devaney, and Linda Keen, The dynamics of complex polynomials and automorphisms of the shift, Invent. Math. 104 (1991), no. 3, 545–580. MR 1106749, DOI 10.1007/BF01245090
John Milnor, Geometry and dynamics of quadratic rational maps, Experiment. Math. 2 (1993), no. 1, 37–83. With an appendix by the author and Lei Tan. MR 1246482
Thomas Ransford, Potential theory in the complex plane, London Mathematical Society Student Texts, vol. 28, Cambridge University Press, Cambridge, 1995. MR 1334766, DOI 10.1017/CBO9780511623776
Joseph H. Silverman, The arithmetic of dynamical systems, Graduate Texts in Mathematics, vol. 241, Springer, New York, 2007. MR 2316407, DOI 10.1007/978-0-387-69904-2
Saeed Zakeri, On biaccessible points of the Mandelbrot set, Proc. Amer. Math. Soc. 134 (2006), no. 8, 2239–2250. MR 2213696, DOI 10.1090/S0002-9939-06-08559-5