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We develop the Reidemeister theory of iterations of a group endomorphism $\varphi$ and study the asymptotic behavior of the sequence of the Reidemeister numbers of iterations $\{R(\varphi^k)\}$, the essential periodic $[\varphi]$-orbits and the heights of $\varphi$ on poly-Bieberbach groups., Comment: 27 pages v.2 : Theorems 4.2, 4.5, 7.1, 9.2 are corrected; made a few other minor changes in chapter 7; new references are added. arXiv admin note: substantial text overlap with arXiv:1403.7631

In this paper, we show that for every non-nilpotent hyperbolic map $f$ on an infra-nilmanifold, the $\textrm{HPer}(f)$ is cofinite in $\mathbb{N}$. This generalizes a similar result for expanding maps. Moreover, we prove that for every nilpotent map $f$ on an infra-nilmanifold, $\textrm{HPer}(f)=\{1\}$., Comment: 14 pages

We study the homotopical minimal periods for maps on infra-solvmanifolds of type (R) using the density of the homotopical minimal period set in the natural numbers. This extends the result of [10] from flat manifolds to infra-solvmanifolds of type (R). Applying our main result we will list all possible maps on infra-solvmanifolds up to dimension three for which the corresponding density is positive., Comment: 26 pages

We study the asymptotic behavior of the sequence of the Nielsen numbers $\{N(f^k)\}$, the essential periodic orbits of $f$ and the homotopy minimal periods of $f$ by using the Nielsen theory of maps $f$ on infra-solvmanifolds of type $R$. We give a linear lower bound for the number of essential periodic orbits of such a map, which sharpens well-known results of Shub and Sullivan for periodic points and of Babenko and Bogatyi for periodic orbits. We also verify that a constant multiple of infinitely many prime numbers occur as homotopy minimal periods of such a map., Comment: 25 pages v.2 : Theorem 4.1 is corrected; made a few other minor changes in chapter 6