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Preimages of quadratic dynamical systems. (English) Zbl 1258.37076
Let $$K$$ be a number field, and consider a quadratic polynomial $$f_c(x)=x^2+c$$, with $$c\in K$$, and a point $$a\in K$$. Let $$N(c, a)$$ denote the number of points $$x\in K$$ such that $a\in \left\{f_c(x), f_c(f_c(x)), f_c(f_c(f_c(x))),\dots \right\},$ a natural condition when considering the dynamics of $$f_c(x)$$. It is relatively clear that $$N(c, a)$$ is finite, for any $$c, a\in K$$, but it turns out that $$N(c, a)$$ is uniformly bounded as $$c\in K$$ varies, for a fixed $$a\in K$$ [X. Faber et al., Math. Res. Lett. 16, No. 1, 87–101 (2009; Zbl 1222.11086)]. This article examines this bound more closely, or, more specifically, it examines the largest value $$N(c, a)$$ attained by infinitely many $$c\in K$$, denoted by $$\tilde{\kappa}(a, K)$$. The main result is that $$\tilde{\kappa}(a, K)$$ is 10 if $$a=-1/4$$; it equals 6 or 8 if $$256a^3+368a^2+104a+23=0$$; it is 4 if $$a$$ comes from a certain finite (but not explicitly known) set $$S$$; otherwise it equals 6.

##### MSC:
 37P05 Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps 14G05 Rational points 37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
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