A local-global principle in the dynamics of quadratic polynomials (Q2833095)

Let \(K\) be a number field, \(f\in K[x]\) a quadratic polynomials, and \(n\in \{1, 2, 3\}\). The author shows that if \(f\) has a point of period \(n\) in every non-Archimedean completion of \(K\), then \(f\) has a point of period \(n\) in \(K\). For \(n\in \{4, 5\}\), the author showed that there exist at most finitely many linear conjugacy classes of quadratic polynomials over \(K\) for which this local-global principle fails. By considering a stronger form of this principle, the author strengthened global results obtained by \textit{P. Morton} [Acta Arith. 87, No. 2, 89--102 (1998; Zbl 1029.12002)] and \textit{E. V. Flynn} et al. [Duke Math. J. 90, No. 3, 435--463 (1997; Zbl 0958.11024)] in the case \(K=\mathbb Q\). More precisely, the author shows that for every quadratic polynomials \(f\in \mathbb Q[x]\) there exist infinitely many primes \(p\) such that \(f\) does not have a point of period four in the \(p\)-adic field \(\mathbb Q_p\). Conditional on knowing all rational points on a particular curve of genus \(11\), the same result is proved for points of period five.