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The zeta function of the beta transformation

Published online by Cambridge University Press:  19 September 2008

Leopold Flatto ,
Jeffrey C. Lagarias  and
Bjorn Poonen
Leopold Flatto
Affiliation:
AT & T Bell Laboratories, Murray Hill, NJ 07974, USA
Jeffrey C. Lagarias
Affiliation:
AT & T Bell Laboratories, Murray Hill, NJ 07974, USA
Bjorn Poonen
Affiliation:
University of California at Berkeley, Berkeley, CA 94720, USA
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Abstract

The 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 function

where 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.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 14 , Issue 2 , June 1994 , pp. 237 - 266
Copyright
Copyright © Cambridge University Press 1994

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