Homotopy minimal periods for expanding maps on infra-nilmanifolds

Abstract

We prove that the sets of homotopy minimal periods for expanding maps on $n$-dimensional infra-nilmanifolds are uniformly cofinite,i.e., there exists a positive integer $m_0$, which depends only on $n$, such that for any integer $m\ge m_0$, for any $n$-dimensional infra-nilmanifold $M$, and for any expanding map $f$ on $M$, any self-map on $M$ homotopic to $f$ has a periodic point of least period $m$, namely, $[m_0,\mathrm{\infty })\subset \mathrm{H}\mathrm{P}\mathrm{e}\mathrm{r}\left(f\right)$. This extends the main result, Theorem 4.6, of [13] from periods to homotopy periods.