Pisot numbers in the neighbourhood of a limit point, I

AbstractLet S denote the set of Pisot numbers. This paper investigates the question of determining effectively all elements of S in a neighbourhood of a limit point of S. To do this, we analyse the structure of an infinite tree T associated with S in which paths to infinity correspond to certain rational functions bounded by one on the unit circle. It is shown that, if f is a path corresponding to a limit point and if Tn(f) denotes the subtree of T which branches off from f at height n then, roughly speaking, Tn(f) converges to a tree T′(f) which is called the derived tree of T at f. When this derived tree is essentially finite for all f associated with a limit point of S, then all elements of S are effectively determined in a small neighbourhood of this limit point. As a corollary, an effective version of a result of Talmoudi is obtained and one can determine all of the points in S ▶ [1, 2 − δ] for anyδ > 0