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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.

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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