The Reidemeister and the Nielsen numbers: growth rate, asymptotic behavior, dynamical zeta functions and the Gauss congruences
DOI:
https://doi.org/10.12775/TMNA.2025.032Keywords
Twisted conjugacy class, Reidemeister coincidence number, coincidence Nielsen number, growth rate, Gauss congruencesAbstract
In the present paper, taking a dynamical point on view, we study the growth rate and asymptotic behavior of the sequences of the Reidemeister numbers and the sequences of the Reidemeister and the Nielsen coincidence numbers. We also prove the Gauss congruences for the sequence $\{R(\varphi^n,\psi^n)\}$ of the Reidemeister coincidence numbers of the tame pair $(\varphi,\psi)$ of endomorphisms of a torsion-free nilpotent group $G$ of finite Pr\"ufer rank. Furthermore, we prove the rationality of the Nielsen coincidence zeta function, the Gauss congruences for the sequence $\{N(f^n, g^n)\}$ of the Nielsen coincidence numbers and show that the growth rate exists for the sequence \{$N(f^n, g^n)\}$ of tame pair of maps $(f,g)$ of a compact nilmanifold to itself.References
I.K. Babenko and S.A. Bogatyi, The behavior of the index of periodic points under iterations of a mapping, Math. USSR Izv. 38 (1992), no. 1, 1–26.
R. Bowen Some systems with unique equilibrium states, Math. Systems Theory 8 (1974), no. 3, 193–202.
J. Buzzi, Specification on the interval, Trans. Amer. Math. Soc. 349 (1997), no. 7, 2737–2754.
K. Chandrasekharan, Introduction to Analytic Number Theory, Springer–Verlag, 1968.
K. Dekimpe, Almost-Bieberbach Groups: Affine and Polynomial Structures, Lecture Notes in Math., vol 1639, 1996.
A. Dold, Fixed point indices of iterated maps, Invent. Math. 74 (1983), 419–435.
A. Fel’shtyn, Dynamical Zeta Functions, Nielsen Theory and Reidemeister Torsion, Mem. Amer. Math. Soc., vol. 147, 2000, no. 699.
A. Fel’shtyn and B. Klopsch, Pólya–Carlson dichotomy for coincidence Reidemeister zeta functions via profinite completions, k Indag. Math. (N.S.) 33 (2022), no. 4, 753–767.
A. Fel’shtyn and J.B. Lee, The Nielsen numbers of iterations of maps on infrasolvmanifolds of type (R) and periodic orbits, J. Fixed Point Theory Appl. 20 (2018), no. 62.
A. Fel’shtyn and E. Troitsky, Twisted Burnside–Frobenius theory for discrete groups, J. Reine Angew. Math. 613 (2007), 193–210.
D.L. Gonçalves, Coincidence Reidemeister classes on nilmanifolds and nilpotent fibrations, Topology Appl. 83 (1998), 169–186.
N.V. Ivanov, Entropy and the Nielsen numbers, Dokl. Akad. Nauk SSSR 265 (1982), no. 2, 284–287 (Russian); English transl.: Soviet Math. Dokl. 26 (1982), 63–66.
J. Jezierski and W. Marzantowicz, Homotopy Methods in Topological Fixed and Periodic Points Theory, Springer, 2006.
I. Kaplansky, Infinite Abelian Groups, University of Michigan Press, 1954.
A. Katok and B. Hasselblatt, k Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, 1995.
W. Marzantowicz and P.M. Przygodzki, Finding periodic points of a map by use of a k-adic expansion, Discrete Contin. Dyn. Syst. 5 (1999), no. 3, 495–514.
R. Miles, Synchronization points and associated dynamical invariants, Trans. Amer. Math. Soc. 365 (2013), 5503–5524.
L. Ribes and P. Zalesskiı̆, Profinite Groups, Springer, 2010.
V. Roman’kov, Twisted conjugace classes in nilpotent groups, J. Pure Appl. Algebra 215 (2010), 664–671.
J.W. Vick, Homology Theory, Springer, 1994.
P. Wong, Reidemeister number, Hirsch rank, coincidences on polycyclic groups and solvmanifolds, J. Reine Angew. Math. 524 (2000), 185–204.
A.V. Zarelua, On congruences for the traces of powers of some matrices, Tr. Mat. Inst. Steklova 263 (2008), 85–105 (Russian); English transl.: Proc. Steklov Inst. Math. 263 (2008), 78–98.
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