# zbMATH — the first resource for mathematics

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
Full Text: