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Eventually stable rational functions. (English) Zbl 1391.37072

Summary: For a field \(K\), rational function \(\phi \in K(z)\) of degree at least two, and \(\alpha \in \mathbb P^1(K)\), we study the polynomials in \(K [z]\) whose roots are given by the solutions in \(\overline{K}\) to \(\phi^n(z) = \alpha\), where \(\phi^n\) denotes the \(n\)th iterate of \(\phi\). When the number of irreducible factors of these polynomials stabilizes as \(n\) grows, the pair \((\phi, \alpha)\) is called eventually stable over \(K\). We conjecture that \((\phi, \alpha)\) is eventually stable over \(K\) when \(K\) is any global field and \(\alpha\) is any point not periodic under \(\phi\) (an additional non-isotriviality hypothesis is necessary in the function field case). We prove the conjecture when \(K\) has a discrete valuation for which (1) \(\phi\) has good reduction and (2) \(\phi\) acts bijectively on all finite residue extensions. As a corollary, we prove for these maps a conjecture of V. A. Sookdeo [J. Number Theory 131, No. 7, 1229–1239 (2011; Zbl 1246.37102)] on the finiteness of \(S\)-integral points in backward orbits. We also give several characterizations of eventual stability in terms of natural finiteness conditions, and survey previous work on the phenomenon.

MSC:

37P05 Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps
37P15 Dynamical systems over global ground fields
11R32 Galois theory
12E05 Polynomials in general fields (irreducibility, etc.)
37P25 Dynamical systems over finite ground fields
11S82 Non-Archimedean dynamical systems

Citations:

Zbl 1246.37102
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References:

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