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Rational preimages in families of dynamical systems. (English) Zbl 1302.37060
Summary: Let \({\phi}\) be a rational function of degree at least two defined over a number field \(k\). Let \({a \in \mathbb{P}^1(k)}\) and let \(K\) be a number field containing \(k\). We study the cardinality of the set of rational iterated preimages Preim\({(\phi, a, K) = \{x_{0} \in \mathbb{P}^1(K) | \phi^{N} (x_0) = a \text{ for some } N \geq 1\}}\). We prove two new results (Theorems 2 and 4) bounding \({|\mathrm {Preim}(\phi, a, K)|}\) as \({\phi}\) varies in certain families of rational functions. Our proofs are based on unit equations and a method of Runge for effectively determining integral points on certain affine curves. We also formulate and state a uniform boundedness conjecture for Preim\({(\phi, a, K)}\) and prove that a version of this conjecture is implied by other well-known conjectures in arithmetic dynamics.

MSC:
37P15 Dynamical systems over global ground fields
14G25 Global ground fields in algebraic geometry
37P05 Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps
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