The zeta function of the beta transformation

Abstract

AbstractThe β-transformation ƒβ(x) = βx(mod 1), for β > 1, has a symbolic dynamics generalizing radix expansions to an integer base. Two important invariants of ƒβ are the (Artin-Mazur) zeta functionwhere Pk counts the number of fixed points of , and the lap-counting function where Lk counts the number of monotonic pieces of the kth iterate . For β-transformations these functions are related by ζβ(z) = (1 − z)Lβ(z). The function ζβ(z) is meromorphic in the unit disk, is holomorphic in {z: |z| < 1/β}, has a simple pole at z = 1/β, and has no other singularities with |z| = 1/β. Let M(β) denote the minimum modulus of any pole of ζβ(z) in |z| < 1 other than z = 1/β, and set M(β) = 1 if no other pole exists with |z| < 1. Then Pk = βk + O((M(β)−1+ε)k) for any ε > 0. This paper shows that M(β) is a continuous function, that ( for all β, and that An asymptotic formula is derived for M(β) as β → 1+, which implies that M(β) < 1 for all β in an interval (1, 1 + c0). The set is shown to have properties analogous to the set of Pisot numbers. It is closed, totally disconnected, has smallest element ≥ 1 + C0 and contains infinitely many β falling in each interval [n, n + 1) for n ∈ ℤ+. All known members of are algebraic integers which are either Pisot or Salem numbers.