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Integer points in backward orbits. (English) Zbl 1246.37102
Summary: A theorem of J. Silverman states that a forward orbit of a rational map \(\varphi (z)\) on \(\mathbb P^1(K)\) contains finitely many \(S\)-integers in the number field \(K\) when (\(\varphi \circ\varphi )(z)\) is not a polynomial. We state an analogous conjecture for the backward orbits using a general \(S\)-integrality notion based on the Galois conjugates of points. This conjecture is proven for the map \(\varphi (z)=z^d\), and consequently Chebyshev polynomials, by uniformly bounding the number of Galois orbits for \(z^n - \beta \) when \(\beta \neq 0\) is a non-root of unity. In general, our conjecture is true provided that the number of Galois orbits for \(\varphi ^n(z) - \beta \) is bounded independently of \(n\).

MSC:
37P35 Arithmetic properties of periodic points
11S82 Non-Archimedean dynamical systems
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