Purely periodic $$\beta $$ β -expansions with Pisot or Salem unit base in $$\mathbb {F}_q((X^{-1}))$$ F q ( ( X - 1 ) )

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

“…Theorem 2.5. [5] Let β be a Pisot or Salem unit series in F q ((X −1 )) and r ∈ F q (X) ∩ D(0, 1). Then d β (r) is purely periodic.…”

“…In [5], the authors proved that the β-expansion of any rational element in the unit disk D(0, 1) is purely periodic when β is a Pisot or Salem unit series in F q ((X −1 )).…”