We prove that the sets of homotopy minimal periods for expanding maps on -dimensional infra-nilmanifolds are uniformly cofinite,i.e., there exists a positive integer , which depends only on , such that for any integer , for any -dimensional infra-nilmanifold , and for any expanding map on , any self-map on homotopic to has a periodic point of least period , namely, . This extends the main result, Theorem 4.6, of  from periods to homotopy periods.
Jong Bum LEE. Xuezhi ZHAO. "Homotopy minimal periods for expanding maps on infra-nilmanifolds." J. Math. Soc. Japan 59 (1) 179 - 184, January, 2007. https://doi.org/10.2969/jmsj/1180135506