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http://dx.doi.org/10.4134/JKMS.2015.52.5.955

INTEGRAL POINTS ON THE CHEBYSHEV DYNAMICAL SYSTEMS  

IH, SU-ION (Department of Mathematics University of Colorado at Boulder)
Publication Information
Journal of the Korean Mathematical Society / v.52, no.5, 2015 , pp. 955-964 More about this Journal
Abstract
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 ${\Gamma}_0$ 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 ${\Gamma}_0$ 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;
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