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Journal of Modern Dynamics (JMD)
 

Distribution of postcritically finite polynomials II: Speed of convergence

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

 
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Thomas Gauthier - LAMFA, Université de Picardie Jules Verne, 33 rue Saint-Leu, 80039 Amiens Cedex 1, France (email)
Gabriel Vigny - LAMFA, Université de Picardie Jules Verne, 33 rue Saint-Leu, 80039 Amiens Cedex 1, France (email)

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 $\mathfrak{C}^2$-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 $\mathfrak{C}^1$-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.

Keywords:  Bifurcation measure, polynomial dynamics, equidistribution, hyperbolic polynomials, speed estimate, postcritically finite polynomials.
Mathematics Subject Classification:  Primary: 37F45; Secondary: 32U40, 37F10.

Received: March 2016;      Revised: October 2016;      Available Online: January 2017.

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