The zeta function of the beta transformation

Abstract

Abstract The β-transformation ƒ β ( x ) = β x (mod 1), for β > 1, has a symbolic dynamics generalizing radix expansions to an integer base. Two important invariants of ƒ β are the (Artin-Mazur) zeta function where P k counts the number of fixed points of , and the lap-counting function where L k counts the number of monotonic pieces of the k th iterate . For β-transformations these functions are related by ζβ( z ) = (1 − z ) L β ( 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 P k = β k + 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 + c 0 ). The set is shown to have properties analogous to the set of Pisot numbers. It is closed, totally disconnected, has smallest element ≥ 1 + C 0 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.