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Distribution of postcritically finite polynomials III: Combinatorial continuity

Volume 244 / 2019

Thomas Gauthier, Gabriel Vigny Fundamenta Mathematicae 244 (2019), 17-48 MSC: Primary 37F45; Secondary 37F20, 30D40. DOI: 10.4064/fm220-2-2018 Published online: 23 August 2018

Abstract

In the first part of the present paper, we continue our study of the distribution of postcritically finite parameters in the moduli space of polynomials: we show the equidistribution of PCF Misiurewicz parameters with prescribed combinatorics with respect to the bifurcation measure. Our results essentially rely on a combinatorial description of the escape locus and of the bifurcation measure developed by Kiwi and Dujardin–Favre.

In the second part of the paper, we construct a bifurcation measure for the connectedness locus of the quadratic anti-holomorphic family which is supported by a strict subset of the boundary of the Tricorn. We also establish an approximation property of PCF Misiurewicz parameters in the spirit of the previous one.

Authors

  • Thomas GauthierLAMFA, UFR des Sciences
    Université de Picardie Jules Verne
    33 rue Saint-Leu
    80039 Amiens Cedex 1, France
    e-mail
  • Gabriel VignyLAMFA, UFR des Sciences
    Université de Picardie Jules Verne
    33 rue Saint-Leu
    80039 AMIENS Cedex 1, France
    e-mail

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