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Let K be a number field and let S be a finite set of primes of K containing all the infinite ones. Let α 0 ∈ A 1 (K) ⊂ P 1 (K) and let Γ 0 be the set of the images of α 0 under especially all Chebyshev morphisms. Then for any α ∈ A 1 (K), we show that there are only a finite number of elements in Γ 0 which are S-integral on P 1 relative to (α). 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 P 1 , whichdoi:10.4134/jkms.2015.52.5.955 fatcat:tfbsdrle2vevdpoccerglasl7y