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$$S$$-integral preperiodic points for dynamical systems over number fields. (English) Zbl 1243.11073
From the introduction: Let $$K$$ be a number field with algebraic closure $$\overline K$$, let $$S$$ be a finite set of places of $$K$$ including all of the archimedean places, and let $$\mathcal O_S$$ denote the ring of $$S$$-integers in $$K$$. Given two points $$P,Q \in \mathbb P^1(\overline K)$$, we say that $$Q$$ is $$S$$-integral with respect to $$P$$ if the Zariski closures of $$P$$ and $$Q$$ do not meet in the model $$\mathbb P^1_{\mathcal O_S}$$ for $$\mathbb P^1$$. Intuitively, this means that $$P$$ and $$Q$$ have distinct reduction with respect to each place of $$\overline K$$ lying over the places of $$K$$ outside $$S$$.
Let $$\phi: \mathbb P^1\to\mathbb P^1$$ be a rational map of degree at least two defined over $$K$$. A point $$P\in \mathbb P^1(\overline K)$$ is called periodic (with respect to $$\phi$$) if $$\phi^n(P) = P$$ for some $$n\geq 1$$, where $$\phi^n$$ denotes the map $$\phi$$ composed with itself $$n$$ times. More generally, $$P$$ is called preperiodic if some iterate $$\phi^n(P)$$ of it is periodic. By analogy with several well-known results in Diophantine geometry, S.-i. Ih has conjectured the following finiteness property for $$S$$-integral preperiodic points.
Conjecture 1 (Ih). Let $$K$$ be a number field, let $$\phi: \mathbb P^1\to\mathbb P^1$$ be a rational map of degree at least two defined over $$K$$, and let $$S$$ be a finite set of places of $$K$$ including all of the archimedean places. Let $$P\in \mathbb P^1(\overline K)$$ be a nonpreperiodic point. Then the set of preperiodic points in $$\mathbb P^1(\overline K)$$ which are $$S$$-integral with respect to $$P$$ is finite.
M. Baker, S.-i. Ih and R. Rumely [Algebra Number Theory 2, No. 2, 217–248 (2008; Zbl 1182.11030)] have proved this conjecture in the following cases: (a) $$\phi(z) = z^n$$ for some $$n\geq 2$$, where $$z$$ denotes the standard affine coordinate on $$\mathbb P^1$$; (b) $$\phi$$ is a Lattés map associated to an elliptic curve $$E/K$$; and (c) $$\phi$$ is a Chebyshev map (although in this latter case they do not work out the details in their paper). In all three of these cases the map $$\phi$$ is defined via the multiplication-by-$$n$$ map on a commutative algebraic group $$G$$ (either $$\mathbb G_m$$ or $$E$$), in the latter two cases by replacing $$G$$ with a quotient of itself by some nontrivial automorphism. The $$\phi$$-preperiodic points on $$\mathbb P^1$$ are thus closely related to the torsion points on the group $$G$$.
In this paper we prove a special case of Conjecture 1 which holds for an arbitrary map $$\phi$$, but which requires certain local conditions on the non-preperiodic point $$P\in \mathbb P^1(\overline K)$$. For each place $$v$$ of $$K$$, the map $$\phi: \mathbb P^1(\mathbb C_v)\to \mathbb P^1(\mathbb C_v)$$ determines a decomposition $$\mathbb P^1(\mathbb C_v) = \mathcal F_v(\phi) \cup\mathcal J_v(\phi)$$ of the projective line into two disjoint subsets, the $$v$$-adic Fatou set $$\mathcal F_v(\phi)$$ and Julia set $$\mathcal J_v(\phi)$$. …Given a point $$P\in \mathbb P^1(\overline K)$$ and a place $$v$$ of $$K$$, we say that $$P$$ is a totally Fatou point at $$v$$ (with respect to $$\phi$$) if each of the $$[K(P) : K]$$ embeddings of $$P$$ into $$\mathbb P^1(\mathbb C_v)$$ is contained in the Fatou set $$\mathcal F_v(\phi)$$.
Theorem 1. Let $$K, S$$, $$\phi: \mathbb P^1\to \mathbb P^1$$, and $$P\in \mathbb P^1(\overline K)$$ satisfy the same hypotheses as in Conjecture 1. Assume in addition that $$P$$ is a totally Fatou point at all places $$v$$ of $$K$$. Then the set of preperiodic points $$\mathbb P^1(\overline K)$$ which are $$S$$-integral with respect to $$P$$ is finite.
In the proof the authors make use of the equidistribution theorem for small points with respect to the canonical height associated to $$\phi$$.

##### MSC:
 11G50 Heights 37P30 Height functions; Green functions; invariant measures in arithmetic and non-Archimedean dynamical systems 37P40 Non-Archimedean Fatou and Julia sets 11R04 Algebraic numbers; rings of algebraic integers
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