INTEGRAL POINTS ON THE CHEBYSHEV DYNAMICAL SYSTEMS

Title & Authors
INTEGRAL POINTS ON THE CHEBYSHEV DYNAMICAL SYSTEMS
IH, SU-ION;

Abstract
Let K be a number field and let S be a finite set of primes of K containing all the infinite ones. Let $\small{{\alpha}_0{\in}{\mathbb{A}}^1(K){\subset}{\mathbb{P}}^1(K)}$ and let $\small{{\Gamma}_0}$ be the set of the images of $\small{{\alpha}_0}$ under especially all Chebyshev morphisms. Then for any $\small{{\alpha}{\in}{\mathbb{A}}^1(K)}$, we show that there are only a finite number of elements in $\small{{\Gamma}_0}$ which are S-integral on $\small{{\mathbb{P}}^1}$ relative to ($\small{{\alpha}}$). 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 $\small{{\mathbb{P}}^1}$, which generalizes the above finiteness result for Chebyshev morphisms.
Keywords
arithmetical dynamical system;Chebyshev polynomial;exceptional point;integral point;preperiodic point;
Language
English
Cited by
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