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The dynamical André-Oort conjecture: unicritical polynomials. (English) Zbl 1417.37172
Summary: We establish equidistribution with respect to the bifurcation measure of postcritically finite (PCF) maps in any one-dimensional algebraic family of unicritical polynomials. Using this equidistribution result, together with a combinatorial analysis of certain algebraic correspondences on the complement of the Mandelbrot set $$\mathcal{M}_{2}$$ (or generalized Mandelbrot set $$\mathcal{M}_{d}$$ for degree $$d>2$$), we classify all curves $$C\subset{\mathbb{A}}^{2}$$ defined over $${\mathbb{C}}$$ with Zariski-dense subsets of points $$(a,b)\in C$$, such that both $$z^{d}+a$$ and $$z^{d}+b$$ are simultaneously PCF for a fixed degree $$d\geq2$$. Our result is analogous to the famous result of André regarding plane curves which contain infinitely many points with both coordinates being complex multiplication parameters in the moduli space of elliptic curves and is the first complete case of the dynamical André-Oort phenomenon studied by M. Baker and L. DeMarco [Duke Math. J. 159, No. 1, 1–29 (2011; Zbl 1242.37062); Forum Math. Pi 1, Article ID e3, 35 p. (2013; Zbl 1320.37022)].

##### MSC:
 37F45 Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations (MSC2010) 37F50 Small divisors, rotation domains and linearization in holomorphic dynamics 37F05 Dynamical systems involving relations and correspondences in one complex variable 37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
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