On the Purely Periodic β-Expansions for Pisot-Chabauty Series in Function Fields

Abstract

This paper investigates the purely periodic β-expansions of elements in the field of formal Laurent series , where β is a Pisot-Chabauty series. We provide a complete characterization of elements α in  whose β-expansions are purely periodic. Specifically, we prove that the ????-expansion of α is purely periodic if and only if α lies in the intersection of the polynomial ring  and the interval . Our results generalize known results in the real number setting to function fields and highlight the role of Pisot-Chabauty series in the study of numeration systems and dynamical systems.