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Bifurcation measures and quadratic rational maps. (English) Zbl 1351.37193

The authors study critical orbits and bifurcations in the moduli space of quadratic rational maps of the projective line over the complex field. The moduli space consists of equivalence classes of maps under conjugacy by Möbius maps (conformal equivalence). The object of study is the family of curves \(\mathrm{Per}_1(\lambda)\) in moduli space, which comprises all the maps that have a fixed point with multiplier \(\lambda\). The authors show that such a curve contains infinitely many postcritically finite maps (that is, maps such that the orbit of each critical point is (pre)-periodic), precisely when the multiplier is zero. This result is analogous to an earlier result by M. Baker and the first author [Forum Math. Pi 1, Article ID e3, 35 p. (2013; Zbl 1320.37022)] for cubic polynomials.
The second object of investigation are the bifurcation points in the curves \(\mathrm{Per}_1(\lambda)\). The authors’ work with an explicit parameterisation of a double cover of this curve, which consist of maps having a fixed point at the origin with multiplier \(\lambda\), and two parameter-independent critical points. Each critical point determines a finite bifurcation measure of the double cover. The main result is that, for all non-zero multipliers, these two measures are distinct.

MSC:

37F45 Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations (MSC2010)
37P30 Height functions; Green functions; invariant measures in arithmetic and non-Archimedean dynamical systems

Citations:

Zbl 1320.37022
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References:

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