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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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References

REFERENCES

[1]Artin, M. & Mazur, B.. On periodic points. Ann. Math. 81 (1965), 82–99.Google Scholar
[2]Bertrand-Mattas, A., Blanchard, F. & Hansel, G.. Sur la fonction ζ de certain systèmes dynamiques symboliques. Preprint (1992).Google Scholar
[3]Blanchard, F.. β-expansions and symbolic dynamics. Theor. Comput. Sci. 65 (1989), 131–141.Google Scholar
[4]Boyd, D.. Salem numbers of degree four have periodic expansions. In: Théorie des Nombres (Quebec 1987), de Gruyter: Berlin, 1989. pp. 57–64.Google Scholar
[5]Boyd, D.W.. Multiplication modulo one by Salem numbers. Preprint (1992).Google Scholar
[6]Brauer, A.. On algebraic equations with all but one root in the interior of the unit circle. Math. Nachr. 4 (1951), 250–257.Google Scholar
[7]Cantor, D. G.. Power series with integral coefficients. Bull. Amer. Math. Soc. 69 (1963), 362–366.CrossRefGoogle Scholar
[8]Carlson, F.. Über Potenzreihen mit ganzzahlingen Koeffizienten. Math. Z. 9 (1921), 1–13.CrossRefGoogle Scholar
[9]Flatto, L.. Z-numbers and β-transformations. In: Symbolic Dynamics and Its Applications (Walters, P., ed), Contemp. Math. 135 (1992).Google Scholar
[10]Flatto, L. & Lagarias, J. C.. Problem 10351. Amer. Math. Monthly 100 (1993), 952.Google Scholar
[11]Flatto, L. & Lagarias, J. C.. On the range of fractional parts {ξ(p/q)n}. Acta Arithmetica. Submitted.Google Scholar
[12]Frougny, C.. Representation of numbers and finite automata. Math. Sys. Th. 25 (1992), 37–60.Google Scholar
[13]Frougny, C. & Solomyak, B.. Finite β-expansions. Ergod. Th. & Dynam. Sys. 12 (1992), 713–723.Google Scholar
[14]Henrici, P.. Applied and Computational Complex Analysis, Vol. I. John Wiley: New York, 1974.Google Scholar
[15]Hofbauer, F.. β-shifts have unique maximal measures. Monat. Math. 85 (1978), 189–198.Google Scholar
[16]Hofbauer, F. & Keller, G.. Zeta-functions and transfer-operators for piecewise linear transformations. J. reine Angew. 352 (1984), 100–113.Google Scholar
[17]Ito, S. & Takahashi, Y.. Markov subshifts and realization of β-expansions. J. Math. Soc. Japan 26 (1974), 33–55.Google Scholar
[18]Milnor, J. & Thurston, W.. On iterated maps of the interval. In: Springer Lecture Notes in Mathematics 1342, Springer: Berlin, 1988. pp. 465–563.Google Scholar
[19]Odlyzko, A. M. & Poonen, B.. Zeros of polynomials with 0, 1 coefficients. Enseign. Math. 39 (1993), 317–348.Google Scholar
[20]Parry, W.. On the β-expansions of real numbers. Acta. Math. Acad. Sci. Hung. 11 (1960), 401–416.CrossRefGoogle Scholar
[21]Parry, W.. Representations for real numbers. Acta Math. Acad. Sci. Hung. 15 (1964), 95–105.CrossRefGoogle Scholar
[22]Polya, G.. Sur les séries entières à coefficients entiers. Proc. London Math. Soc. 21 (1923), 22–38.Google Scholar
[23]Polya, G. & Szego, G.. Problems and Theorems in Analysis, Vol. II. Springer: Berlin, 1976.Google Scholar
[24]Preston, C.. What you need to know to knead. Adv. Math. 78 (1989), 192–252.CrossRefGoogle Scholar
[25]Renyi, A.. Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hung. 8 (1957), 477–493.CrossRefGoogle Scholar
[26]Salem, R.. Power series with integral coefficients. Duke Math. J. 12 (1945), 153–172.Google Scholar
[27]Schmidt, K.. On periodic expansions of Pisot numbers and Salem numbers. Bull. London Math. Soc. 12 (1980), 269–278.Google Scholar
[28]Solomyak, V.. Substitutions, adic transformations and beta-expansions. In: Symbolic Dynamics and Its Applications, (Walters, P., ed), Contemp. Math. 135 (1992a).Google Scholar
[29]Solomyak, V.. On the spectral theory of adic transformations. Adv. Sov. Math. 9 (1992b), 217–230.Google Scholar
[30]Solomyak, V.. Conjugates of beta-numbers and the zero-free domain for a class of analytic functions. Proc. London Math. Soc. (1994), to appear.Google Scholar
[31]Takahashi, Y.. β-transformations and symbolic dynamics. In: Springer Lecture Notes in Mathematics 330, Springer: Berlin, 1973. pp. 455–464.Google Scholar
[32]Takahashi, Y.. Shift with orbit basis and realization of one-dimensional maps. Osaka Math. J. 20 (1983), 269–278.Google Scholar
[33]Titchmarsh, E. C.. The Theory of Functions, 2nd Edn, Oxford University Press: Oxford, 1939.Google Scholar