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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 LattĂ©s 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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