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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.

##### MSC:
 14H50 Plane and space curves
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