## A natural boundary for the dynamical zeta function for commuting group automorphisms

HTML articles powered by AMS MathViewer

- by Richard Miles PDF
- Proc. Amer. Math. Soc.
**143**(2015), 2927-2933 Request permission

## Abstract:

For an action $\alpha$ of $\mathbb {Z}^d$ by homeomorphisms of a compact metric space, D. Lind introduced a dynamical zeta function and conjectured that this function has a natural boundary when $d\geqslant 2$. In this note, under the assumption that $\alpha$ is a mixing action by continuous automorphisms of a compact connected abelian group of finite topological dimension, it is shown that the upper growth rate of periodic points is zero and that the unit circle is a natural boundary for the dynamical zeta function.## References

- M. Artin and B. Mazur,
*On periodic points*, Ann. of Math. (2)**81**(1965), 82–99. MR**176482**, DOI 10.2307/1970384 - J. Bell, R. Miles, and T. Ward. Towards a Polyá–Carlson dichotomy for algebraic dynamical zeta functions.
*Indagationes Mathematicae*, to appear. - Fritz Carlson,
*Über ganzwertige Funktionen*, Math. Z.**11**(1921), no. 1-2, 1–23 (German). MR**1544479**, DOI 10.1007/BF01203188 - Pietro Corvaja, Zéev Rudnick, and Umberto Zannier,
*A lower bound for periods of matrices*, Comm. Math. Phys.**252**(2004), no. 1-3, 535–541. MR**2104888**, DOI 10.1007/s00220-004-1184-6 - Pietro Corvaja and Umberto Zannier,
*A lower bound for the height of a rational function at $S$-unit points*, Monatsh. Math.**144**(2005), no. 3, 203–224. MR**2130274**, DOI 10.1007/s00605-004-0273-0 - Manfred Einsiedler and Douglas Lind,
*Algebraic $\Bbb Z^d$-actions of entropy rank one*, Trans. Amer. Math. Soc.**356**(2004), no. 5, 1799–1831. MR**2031042**, DOI 10.1090/S0002-9947-04-03554-8 - G. Everest, V. Stangoe, and T. Ward,
*Orbit counting with an isometric direction*, Algebraic and topological dynamics, Contemp. Math., vol. 385, Amer. Math. Soc., Providence, RI, 2005, pp. 293–302. MR**2180241**, DOI 10.1090/conm/385/07202 - Douglas Lind, Klaus Schmidt, and Tom Ward,
*Mahler measure and entropy for commuting automorphisms of compact groups*, Invent. Math.**101**(1990), no. 3, 593–629. MR**1062797**, DOI 10.1007/BF01231517 - D. A. Lind,
*A zeta function for $\textbf {Z}^d$-actions*, Ergodic theory of $\textbf {Z}^d$ actions (Warwick, 1993–1994) London Math. Soc. Lecture Note Ser., vol. 228, Cambridge Univ. Press, Cambridge, 1996, pp. 433–450. MR**1411232**, DOI 10.1017/CBO9780511662812.019 - Anthony Manning,
*Axiom $\textrm {A}$ diffeomorphisms have rational zeta functions*, Bull. London Math. Soc.**3**(1971), 215–220. MR**288786**, DOI 10.1112/blms/3.2.215 - Richard Miles,
*Zeta functions for elements of entropy rank-one actions*, Ergodic Theory Dynam. Systems**27**(2007), no. 2, 567–582. MR**2308145**, DOI 10.1017/S0143385706000794 - Richard Miles,
*Synchronization points and associated dynamical invariants*, Trans. Amer. Math. Soc.**365**(2013), no. 10, 5503–5524. MR**3074380**, DOI 10.1090/S0002-9947-2013-05829-1 - Richard Miles and Thomas Ward,
*Periodic point data detects subdynamics in entropy rank one*, Ergodic Theory Dynam. Systems**26**(2006), no. 6, 1913–1930. MR**2279271**, DOI 10.1017/S014338570600054X - G. Pólya,
*Über gewisse notwendige Determinantenkriterien für die Fortsetzbarkeit einer Potenzreihe*, Math. Ann.**99**(1928), no. 1, 687–706 (German). MR**1512473**, DOI 10.1007/BF01459120 - Klaus Schmidt,
*Dynamical systems of algebraic origin*, Modern Birkhäuser Classics, Birkhäuser/Springer Basel AG, Basel, 1995. [2011 reprint of the 1995 original] [MR1345152]. MR**3024809** - Sanford L. Segal,
*Nine introductions in complex analysis*, Revised edition, North-Holland Mathematics Studies, vol. 208, Elsevier Science B.V., Amsterdam, 2008. MR**2376066** - T. B. Ward,
*Periodic points for expansive actions of $\textbf {Z}^d$ on compact abelian groups*, Bull. London Math. Soc.**24**(1992), no. 4, 317–324. MR**1165372**, DOI 10.1112/blms/24.4.317

## Additional Information

**Richard Miles**- Affiliation: School of Mathematics, University of East Anglia, Norwich, NR4 7TJ, United Kingdom
- Email: r.miles@uea.ac.uk
- Received by editor(s): September 19, 2013
- Received by editor(s) in revised form: January 9, 2014
- Published electronically: February 25, 2015
- Communicated by: Nimish Shah
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**143**(2015), 2927-2933 - MSC (2010): Primary 37A45, 37B05, 37C25, 37C30, 37C85, 22D40
- DOI: https://doi.org/10.1090/S0002-9939-2015-12515-4
- MathSciNet review: 3336617