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15 January 2017 The dynamical André–Oort conjecture: Unicritical polynomials
D. Ghioca, H. Krieger, K. D. Nguyen, H. Ye
Duke Math. J. 166(1): 1-25 (15 January 2017). DOI: 10.1215/00127094-3673996


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 M2 (or generalized Mandelbrot set Md for degree d>2), we classify all curves CA2 defined over C with Zariski-dense subsets of points (a,b)C, such that both zd+a and zd+b are simultaneously PCF for a fixed degree d2. 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 Baker and DeMarco.


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D. Ghioca. H. Krieger. K. D. Nguyen. H. Ye. "The dynamical André–Oort conjecture: Unicritical polynomials." Duke Math. J. 166 (1) 1 - 25, 15 January 2017.


Received: 11 May 2015; Revised: 5 February 2016; Published: 15 January 2017
First available in Project Euclid: 30 September 2016

zbMATH: 06686500
MathSciNet: MR3592687
Digital Object Identifier: 10.1215/00127094-3673996

Primary: 37F50
Secondary: 37F05

Keywords: Mandelbrot set , postcritically finite , unlikely intersections in dynamics

Rights: Copyright © 2017 Duke University Press


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Vol.166 • No. 1 • 15 January 2017
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