Rational preimages in families of dynamical systems

Abstract

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 $${(\phi, a, K) = \{x_{0} \in \mathbb{P}^1(K) | \phi^{N} (x_0) = a {\rm for some} N \geq 1\}}$$ . We prove two new results (Theorems 2 and 4) bounding $${|{\rm Preim}(\phi, a, K)|}$$ 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.