The Nielsen and Reidemeister theories of iterations on infra-solvmanifolds of type (𝑅) and poly-Bieberbach groups

Abstract

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 develop the Reidemeister theory for the iterations of any endomorphism ϕ on an arbitrary group and study the asymptotic behavior of the sequence of the Reidemeister numbers {R(ϕ k )}, the essential periodic [ϕ]-orbits and the heights of ϕ on poly-Bieberbach groups.

Minimal number of periodic points of smooth boundary-preserving self-maps of simply-connected manifolds

“…Finding minimal number of r -periodic points in the homotopy class (for a fixed r ) is an important challenge in modern homotopy periodic point theory, with an increasing number of valuable results obtained in the last decade in many particular cases [1][2][3][4][5][6][7].…”