Purely periodic β-expansions in cubic Salem base in Fq((x−1))

Abstract

Let Fq be the finite field with q elements and ? Salem series in Fq((X?1)). It is proved in [15] that, in this case, all elements in Fq(X,?) have purely periodic ?-expansion. We characterize the formal power series f in Fq(X,?) with purely periodic ?-expansions by the conjugate vector ~f when ? is a cubic unit. No similar results exist in the real case.