A Case of the Dynamical André-oort Conjecture
نویسنده
چکیده
We prove a special case of the Dynamical André-Oort Conjecture formulated by Baker and DeMarco [3]. For any integer d ≥ 2, we show that for a rational plane curve C parametrized by (t, h(t)) for some non-constant polynomial h ∈ C[z], if there exist infinitely many points (a, b) ∈ C(C) such that both zd+a and zd+b are postcritically finite maps, then h(z) = ξz for a (d − 1)-st root of unity ξ. As a key step in our proof, we show that the Mandelbrot set is not the filled Julia set of any polynomial g ∈ C[z].
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