The Nielsen numbers of iterations of maps on infra-solvmanifolds of type $R$ and periodic points

TL;DR

This paper investigates the long-term behavior of Nielsen numbers, essential periodic orbits, and minimal periods of maps on infra-solvmanifolds of type R, providing new bounds and demonstrating the occurrence of infinitely many prime-based minimal periods.

Contribution

It introduces a linear lower bound for the number of essential periodic orbits and shows that infinitely many minimal periods are multiples of infinitely many primes, refining previous results.

Findings

Established a linear lower bound for essential periodic orbits.

Proved that infinitely many minimal periods are multiples of infinitely many primes.

Extended Nielsen theory to infra-solvmanifolds of type R.

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 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.