Collet, Eckmann and the bifurcation measure
Tóm tắt
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.
Từ khóa
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