Collet, Eckmann and the bifurcation measure

  • Matthieu Astorg
  • Thomas GauthierEmail author
  • Nicolae Mihalache
  • Gabriel Vigny


The moduli space \(\mathcal {M}_d\) of degree \(d\ge 2\) rational maps can naturally be endowed with a measure \(\mu _{\text{ bif }}\) detecting maximal bifurcations, called the bifurcation measure. We prove that the support of the bifurcation measure \(\mu _{\text{ bif }}\) has positive Lebesgue measure. To do so, we establish a general sufficient condition for the conjugacy class of a rational map to belong to the support of \(\mu _{\text{ bif }}\) and we exhibit a large set of Collet–Eckmann rational maps which satisfy this condition. As a consequence, we get a set of Collet–Eckmann rational maps of positive Lebesgue measure which are approximated by hyperbolic rational maps.



We thank the anonymous referee for his careful reading and his suggestions, in particular his formulation of the large scale condition.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Matthieu Astorg
    • 1
  • Thomas Gauthier
    • 2
    • 3
    Email author
  • Nicolae Mihalache
    • 4
  • Gabriel Vigny
    • 2
  1. 1.MAPMOUniversité d’OrléansOrléans Cedex 2France
  2. 2.LAMFAUPJVAmiens Cedex 1France
  3. 3.CMLS, École polytechnique, CNRSUniversité Paris-SaclayPalaiseau CedexFrance
  4. 4.LAMAUPECCréteil CedexFrance

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