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Portraits of preperiodic points for rational maps. (English) Zbl 1371.11109
Summary: Let \(K\) be a function field over an algebraically closed field \(k\) of characteristic 0, let \(\phi\in K(z)\) be a rational function of degree at least equal to 2 for which there is no point at which is totally ramified and let \(\alpha\in K\). We show that for all but finitely many pairs \((m, n)\in \mathbb{Z}_{\geq0} \times \mathbb{N}\) there exists a place \(\mathfrak{p}\) of \(K\) such that the point \(\alpha\) has preperiod \(m\) and minimum period \(n\) under the action of \(\phi\). This answers a conjecture made by P. Ingram and J. H. Silverman [Math. Proc. Camb. Philos. Soc. 146, No. 2, 289–302 (2009; Zbl 1242.11012)] and X. Faber and A. Granville [J. Reine Angew. Math. 661, 189–214 (2011; Zbl 1290.11019)]. We prove a similar result, under suitable modification, also when has points where it is totally ramified. We give several applications of our result, such as showing that for any tuple \((c_0,\dots,c_{d-2})\in k^{d-1}\) and for almost all pairs \((m_i, n_i)\in \mathbb{Z}_{\geq0}\times\mathbb{N}\) for \(i=0,\dots,d-2\), there exists a polynomial \(f\in k[z]\) of degree \(d\) in normal form such that for each \(i=0,\dots,d-2\), the point \(c_i\) has preperiod \(m_i\) and minimum period \(n_i\) under the action of \(f\).

MSC:
11G50 Heights
37P30 Height functions; Green functions; invariant measures in arithmetic and non-Archimedean dynamical systems
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