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Classification of special curves in the space of cubic polynomials. (English) Zbl 1415.14013
Let \(\mathrm{Poly}_{3}\simeq A_{3}\) be the space of cubic polynomials defined by \[ P_{c,a}(z)=\frac{1}{3}z^{3}-\frac{c}{2}z^{2}+a^{3}, \] which is a branched cover of the parameter space of cubic polynomials with marked critical points. The critical points of \(P_{c,a}\) are given by \( c_{0}:=c\) and \(c_{1}:=0\).
The main result of the paper under review is the following:
Theorem A. An irreducible curve \(C\) in the space \(\mathrm{Poly}_{3}\) contains an infinite collection of post-critically finite polynomials if and only if one of the following holds.
1. One of the two critical points is persistently pre-periodic on \(C\), that is, there exist integers \(m>0\) and \(k\geq 0\) such that: \( P_{c,a}^{m+k}(c_{0})=P_{c,a}^{k}(c_{0})\) or \( P_{c,a}^{m+k}(c_{1})=P_{c,a}^{k}(c_{1})\) for all \((c,a)\in C\).
2. There is a persistent collision of the two critical orbits on \(C\), that is, there exist \((m,k)\in\mathbb{N}^{2}\backslash \{(1,1)\}\) such that \(P_{c,a}^{m}(c_{1})=P_{c,a}^{k}(c_{0})\) for all \((c,a)\in C\).
3. The curve \(C\) is given by the equation \(\{(c,a)\), \( 12a^{3}-c^{3}-6c=0\}\), and coincides with the set of cubic polynomials having a non-trivial symmetry, that is, the set of parameters \((c,a)\) for which \(Q_{c}(z):=-z+c\) commutes with \(P_{c,a}\).
Then considering \(\mathrm{Per}_{m}(\lambda )\) – the algebraic curve consisting of those cubic polynomials that admit an orbit of period \(m\) and multiplier \( \lambda \), the authors give a characterization of those \(\mathrm{Per}_{m}(\lambda )\) that contain infinitely many post – critically finite (PCF) polynomials.

14H50 Plane and space curves
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