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Let $Rat_d$ be the set of all rational fraction of degree $d$ and $X_d \subset Rat_d$ be the bifurcation locus of rational fractions of degree $d$, i.e. the closure of the set of discontinuity of the map $f \mapsto J(f)$ defined on $Rat_d$, and associating to $f$ its Julia set (with the topology of Hausdorff distance).

Now denote by $M_d$ the quotient of $Rat_d$ under conjugation by homographies, and by $Y_d$ the projection of $X_d$ onto $M_d$. (if you consider polynomials only instead of rational fractions, then $M_2=\mathbb{C}$ and $Y_2$ is the boundary of the Mandelbrot set).

A result of Milnor gives an explicit diffeomorphism from $M_2$ to $\mathbb{C}^2$.

Question : is there somewhere a computer picture of $Y_2$ in the coordinates defined by Milnor ? (like a projection, or a family of cuts). Judging by the aspect of the Mandelbrot's set boundary, one would expect quite a complicated and interesting set.

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What is the explicit biholomorphism? – Per Alexandersson Dec 12 '11 at 20:25
if $\lambda$, $\mu$, $\nu$ are the three fixed points of a quadratic rational map, and $\sigma_i$ are the symetric elementary functions, the map is $\bar{f} \mapsto (\sigma_(\lambda,\mu,\nu), \sigma_2(\lambda,\mu,\nu)$ – glougloubarbaki Dec 12 '11 at 20:50
(that's supposed to be a $\sigma_1$ above) – glougloubarbaki Dec 12 '11 at 20:51
There are many people who have studied the parameter space of quadratic rational maps (Mary Rees is a world expert in this area), and many have made pictures of slices of the parameter space. For example, you can see some on Vladlen Timorin's page: – Lasse Rempe-Gillen Nov 1 '12 at 13:14

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