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Equidistribution towards the bifurcation current I: multipliers and degree d polynomials

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Abstract

In the moduli space \(\mathcal {P}_d\) of degree d polynomials, the set \(\text {Per}_n(w)\) of classes [f] for which f admits a cycle of exact period n and multiplier multiplier w is known to be an algebraic hypersurface. We prove that, given \(w\in {\mathbb C}\), these hypersurfaces equidistribute towards the bifurcation current as n tends to infinity.

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Acknowledgments

We would like to thank Gabriel Vigny for many interesting and helpful discussions. We also would like to thank Vincent Guedj for interesting discussions concerning the comparison principle. The author also thanks the referee whose comments greatly helped to improve the presentation of the paper.

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  1. LAMFA, Université de Picardie Jules Verne, 33 rue Saint-Leu, 80039, Amiens Cedex 1, France

    Thomas Gauthier

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  1. Thomas Gauthier
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Correspondence to Thomas Gauthier.

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Gauthier, T. Equidistribution towards the bifurcation current I: multipliers and degree d polynomials. Math. Ann. 366, 1–30 (2016). https://doi.org/10.1007/s00208-015-1297-6

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  • DOI: https://doi.org/10.1007/s00208-015-1297-6

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