Periodic points of endperiodic maps

Abstract

Let $g:L\to L$ be an atoroidal, endperiodic map on an infinite-type surface $L$ with no boundary and finitely many ends, each of which is accumulated by genus. By work of Landry, Minsky, and Taylor (2023), $g$ is isotopic to a spun pseudo-Anosov map $f$. We show that spun pseudo-Anosov maps minimize the number of periodic points of period $n$ for sufficiently high $n$ over all maps in their homotopy class, strengthening Theorem 6.1 of Landry, Minsky, and Taylor (2023). We also show that the same theorem holds for atoroidal Handel–Miller maps when one only considers periodic points that lie in the intersection of the stable and unstable laminations. Furthermore, we show via example that spun pseudo-Anosov and Handel–Miller maps do not always minimize the number of periodic points of low period.