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Integral points on the Chebyshev dynamical systems. (English) Zbl 1392.37105
Summary: Let \(K\) be a number field and let \(S\) be a finite set of primes of \(K\) containing all the infinite ones. Let \(\alpha_0 \in {\mathbb A}^1 (K) \subset {\mathbb P}^1 (K)\) and let \(\varGamma_0\) be the set of the images of \(\alpha_0\) under especially all Chebyshev morphisms. Then for any \(\alpha \in {\mathbb A}^1 (K)\), we show that there are only a finite number of elements in \(\varGamma_0\) which are \(S\)-integral on \({\mathbb P}^1\) relative to \((\alpha)\). In the light of a theorem of Silverman we also propose a conjecture on the finiteness of integral points on an arbitrary dynamical system on \({\mathbb P}^1\), which generalizes the above finiteness result for Chebyshev morphisms.

MSC:
37P05 Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps
11G50 Heights
14G05 Rational points
14G40 Arithmetic varieties and schemes; Arakelov theory; heights
37P30 Height functions; Green functions; invariant measures in arithmetic and non-Archimedean dynamical systems
37P35 Arithmetic properties of periodic points
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