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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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