Distribution of postcritically finite polynomials Ⅱ: Speed of convergence

Thomas Gauthier; Gabriel Vigny

doi:10.3934/jmd.2017004

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2017, Volume 11: 57-98. Doi: 10.3934/jmd.2017004

This volume Previous Article Positive metric entropy in nondegenerate nearly integrable systems Next Article On spectra of Koopman, groupoid and quasi-regular representations

Distribution of postcritically finite polynomials Ⅱ: Speed of convergence

Thomas Gauthier and
Gabriel Vigny

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

Received: March 27, 2016

Revised: October 17, 2016

Published: December 2016

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Abstract

In the moduli space of degree $d$ polynomials, we prove the equidistribution of postcritically finite polynomials toward the bifurcation measure. More precisely, using complex analytic arguments and pluripotential theory, we prove the exponential speed of convergence for $\mathscr{C}^2$-observables. This improves results obtained with arithmetic methods by Favre and Rivera-Letellier in the unicritical family and Favre and the first author in the space of degree $d$ polynomials.

We deduce from that the equidistribution of hyperbolic parameters with $(d-1)$ distinct attracting cycles of given multipliers toward the bifurcation measure with exponential speed for $\mathscr{C}^1$-observables. As an application, we prove the equidistribution (up to an explicit extraction) of parameters with $(d-1)$ distinct cycles with prescribed multiplier toward the bifurcation measure for any $(d-1)$ multipliers outside a pluripolar set.

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Mathematics Subject Classification: Primary: 37F45; Secondary: 32U40, 37F10.

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