A note on homotopy minimal periods for hyperbolic maps on infra-nilmanifolds

TL;DR

This paper investigates the set of homotopy minimal periods for hyperbolic and nilpotent maps on infra-nilmanifolds, establishing that non-nilpotent hyperbolic maps have cofinite homotopy minimal periods, while nilpotent maps have a singleton set.

Contribution

It generalizes previous results on expanding maps to hyperbolic maps on infra-nilmanifolds and characterizes the homotopy minimal periods for nilpotent maps.

Findings

Non-nilpotent hyperbolic maps have cofinite homotopy minimal periods.

Nilpotent maps have a singleton homotopy minimal period set {1}.

Results extend understanding of periodic points in infra-nilmanifold dynamics.

Abstract

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, $\textrm{HPer}(f)=\{1\}$.