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arXiv · 1705.06114

Collet, Eckmann and the bifurcation measure

Matthieu Astorg·Thomas Gauthier·Nicolae Mihalache·Gabriel Vigny
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Abstract

The moduli space $\mathcal{M}_d$ of degree $d\geq2$ rational maps can naturally be endowed with a measure $\mu_\mathrm{bif}$ detecting maximal bifurcations, called the bifurcation measure. We prove that the support of the bifurcation measure $\mu_\mathrm{bif}$ 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 $\mu_\mathrm{bif}$ 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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Matthieu Astorg, Thomas Gauthier, Nicolae Mihalache, Gabriel Vigny. 2017-05-17. Collet, Eckmann and the bifurcation measure. https://arxiv.org/abs/1705.06114

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