Duke Mathematical Journal

The dynamical André–Oort conjecture: Unicritical polynomials

D. Ghioca, H. Krieger, K. D. Nguyen, and H. Ye

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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.

Article information

Duke Math. J., Volume 166, Number 1 (2017), 1-25.

Received: 11 May 2015
Revised: 5 February 2016
First available in Project Euclid: 30 September 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37F50: Small divisors, rotation domains and linearization; Fatou and Julia sets
Secondary: 37F05: Relations and correspondences

Mandelbrot set unlikely intersections in dynamics postcritically finite


Ghioca, D.; Krieger, H.; Nguyen, K. D.; Ye, H. The dynamical André–Oort conjecture: Unicritical polynomials. Duke Math. J. 166 (2017), no. 1, 1--25. doi:10.1215/00127094-3673996. https://projecteuclid.org/euclid.dmj/1475266423

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