The Reidemeister and the Nielsen numbers: growth rate, asymptotic behavior, dynamical zeta functions and the Gauss congruences

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 $\{R({\phi }^n,{\psi }^n)\}$ of the Reidemeister coincidence numbers of the tame pair $(\phi ,\psi )$ of endomorphisms of a torsion-free nilpotent group~$G$ of finite Prüfer 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.