Journal of the
Korean Mathematical Society JKMS

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 ${\it \Gamma}_0$ 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 ${\it \Gamma}_0$ 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.

Keywords: arithmetical dynamical system, Chebyshev polynomial, exceptional point, integral point, preperiodic point

MSC numbers: 11G50, 14G05, 14G40, 37P05, 37P30, 37P35