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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.

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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