Periodic points of endperiodic maps

TL;DR

This paper proves that spun pseudo-Anosov and Handel--Miller maps minimize the number of high-periodic points in their homotopy class on certain infinite surfaces, but not for low periods.

Contribution

It extends previous results by showing minimization of high-periodic points for these maps, strengthening the understanding of their dynamical properties.

Findings

Spun pseudo-Anosov maps minimize high-periodic points in their homotopy class.

The same minimization property holds for atoroidal Handel--Miller maps for points in stable and unstable laminations.

Examples show minimization does not hold for low-periodic points.

Abstract

Let $g\colon L\rightarrow L$ 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, $g$ 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 a theorem of Landry, Minsky, and Taylor. 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.