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

Proc. Amer. Math. Soc. 143 (2015), 3011-3017

DOI: https://doi.org/10.1090/S0002-9939-2015-12506-3

Published electronically: February 25, 2015

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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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