Skip to main content

Pólya–Carlson Dichotomy for Dynamical Zeta Functions and a Twisted Burnside–Frobenius Theorem


For the unitary dual mapping of an automorphism of a torsion-free, finite rank nilpotent group, we prove the Pólya–Carlson dichotomy between rationality and the natural boundary for the analytic behavior of its Artin–Mazur dynamical zeta function. We also establish Gauss congruences for the Reidemeister numbers of the iterations of endomorphisms of groups in this class. Our method is the twisted Burnside–Frobenius theorem proven in the paper for automorphisms of this class of groups, and a calculation of the Reidemeister numbers via a product formula and profinite completions.

This is a preview of subscription content, access via your institution.


  1. J. Bell, R. Miles, and Th. Ward, “Towards a Pólya-Carlson Dichotomy for Algebraic Dynamics”, Indag. Math. (N.S.), 25:4 (2014), 652–668.

    MathSciNet  Article  Google Scholar 

  2. Jakub Byszewski and Gunther Cornelissen, “Dynamics on Abelian Varieties in Positive Characteristic”, Algebra Number Theory, 12:9 (2018), 2185–2235.

    MathSciNet  Article  Google Scholar 

  3. A. Fel’shtyn, E. Troitsky, and M. Zietek, “New Zeta Functions of Reidemeister Type and the Twisted Burnside-Frobenius Theory”, Russ. J. Math. Phys., 27:2 (2020), 199–211.

    MathSciNet  Article  Google Scholar 

  4. A. Fel’shtyn, “Dynamical Zeta Functions, Nielsen Theory and Reidemeister Torsion”, Mem. Amer. Math. Soc., 147:699 (2000).

    MathSciNet  Article  Google Scholar 

  5. A. Fel’shtyn and R. Hill, “The Reidemeister Zeta Function with Applications to Nielsen Theory and a Connection with Reidemeister Torsion”, K-Theory, 8:4 (1994), 367–393.

    MathSciNet  Article  Google Scholar 

  6. A. Fel’shtyn and B. Klopsch, “Pólya–Carlson Dichotomy for Coincidence Reidemeister Zeta Functions via Profinite Completions”, arXiv:2102.10900, (2021).

    ADS  Google Scholar 

  7. A. Fel’shtyn and E. Troitsky, “Twisted Burnside-Frobenius Theory for Discrete Groups”, J. Reine Angew. Math., 613 (2007), 193–210.

    MathSciNet  MATH  Google Scholar 

  8. A. Fel’shtyn and E. Troitsky, “Twisted Burnside-Frobenius Theory for Endomorphisms of Polycyclic Groups”, Russian J. Math. Phys., 25:1 (2018), 17–26.

    ADS  MathSciNet  Article  Google Scholar 

  9. A. Fel’shtyn, E. Troitsky, and A. Vershik, “Twisted Burnside Theorem for Type II\({}_1\) Groups: an Example”, Math. Res. Lett., 13:5 (2006), 719–728.

    MathSciNet  Article  Google Scholar 

  10. D. Gonçalves, “The Coincidence Reidemeister Classes on Nilmanifolds and Nilpotent Fibrations”, Topology Appl., 83 (1998), 169–186.

    MathSciNet  Article  Google Scholar 

  11. D. Gonçalves and P. Wong, “Twisted Conjugacy Classes in Nilpotent Groups”, J. Reine Angew. Math., 633 (2009), 11–27.

    MathSciNet  MATH  Google Scholar 

  12. N. Koblitz, p-Adic Numbers, p-Adic Analysis, and Zeta-Functions, vol. 58, Grad. Texts in Math., second edition, Springer-Verlag, New York, 1984.

    Book  Google Scholar 

  13. A. Lubotzky and D. Segal, Subgroup Growth, vol. 212, Birkhäuser Verlag, Basel, 2003.

    Book  Google Scholar 

  14. Derek J. S. Robinson, A Course in the Theory of Groups, vol. 80, Springer-Verlag, New York, 1996.

    Book  Google Scholar 

  15. Sanford L. Segal, Nine Introductions in Complex Analysis, vol. 208, Elsevier Science B.V., Amsterdam, revised edition, 2008.

    MATH  Google Scholar 

  16. A. Weil, Basic Number Theory, Grundlehren Math. Wiss., Band 144, Springer-Verlag New York, Inc., New York, 1967.

    Book  Google Scholar 

Download references


The work of Alexander Fel’shtyn is funded by the Narodowe Centrum Nauki of Poland (NCN) (grant no. 2016/23/G/ST1/04280(Beethoven2)). The work of Evgenij Troitsky was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics “BASIS.”

Author information

Authors and Affiliations


Corresponding authors

Correspondence to A. Fel’shtyn or E. Troitsky.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Fel’shtyn, A., Troitsky, E. Pólya–Carlson Dichotomy for Dynamical Zeta Functions and a Twisted Burnside–Frobenius Theorem. Russ. J. Math. Phys. 28, 455–463 (2021).

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: