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Collet, Eckmann and the bifurcation measure. (English) Zbl 07089444
Summary: 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.

##### MSC:
 37F45 Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations (MSC2010) 37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets 32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) 37P45 Families and moduli spaces in arithmetic and non-Archimedean dynamical systems
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