Collet, Eckmann and the bifurcation measure

Abstract : The moduli space M_d of degree d ≥ 2 rational maps can naturally be endowed with a measure µ_bif detecting maximal bifurcations, called the bifurcation measure. We prove that the support of the bifurcation measure µ_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 µ_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.
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Submitted on : Thursday, October 4, 2018 - 9:09:05 PM
Last modification on : Wednesday, March 27, 2019 - 4:10:22 PM
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  • HAL Id : hal-01888305, version 1
  • ARXIV : 1705.06114


Matthieu Astorg, Thomas Gauthier, Nicolae Mihalache, Gabriel Vigny. Collet, Eckmann and the bifurcation measure. 2018. ⟨hal-01888305⟩



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