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Periodic points of endperiodic maps

Ellis Buckminster

TL;DR

This work extends the classical pseudo-Anosov periodic-point minimization to endperiodic maps on infinite-type surfaces. It proves that spun pseudo-Anosov representatives minimize the number of periodic points of period $n$ for all large $n$ within their homotopy class, and it extends a parallel result to atoroidal Handel–Miller maps for the core dynamics on the HM laminations. The arguments hinge on lifts to $\mathbb{H}^2$, Nielsen equivalence of periodic points, and Lefschetz–Hopf theory, together with HM laminations and principal region structure to handle the infinite-type setting. The paper also provides sharpness results via explicit constructions showing the $N$-threshold cannot be removed and demonstrates how fixed-point behavior can persist in spA and HM representatives even when a nearby map can be fixed-point-free.

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.

Periodic points of endperiodic maps

TL;DR

This work extends the classical pseudo-Anosov periodic-point minimization to endperiodic maps on infinite-type surfaces. It proves that spun pseudo-Anosov representatives minimize the number of periodic points of period for all large within their homotopy class, and it extends a parallel result to atoroidal Handel–Miller maps for the core dynamics on the HM laminations. The arguments hinge on lifts to , Nielsen equivalence of periodic points, and Lefschetz–Hopf theory, together with HM laminations and principal region structure to handle the infinite-type setting. The paper also provides sharpness results via explicit constructions showing the -threshold cannot be removed and demonstrates how fixed-point behavior can persist in spA and HM representatives even when a nearby map can be fixed-point-free.

Abstract

Let be an atoroidal, endperiodic map on an infinite type surface with no boundary and finitely many ends, each of which is accumulated by genus. By work of Landry, Minsky, and Taylor, is isotopic to a spun pseudo-Anosov map . We show that spun pseudo-Anosov maps minimize the number of periodic points of period for sufficiently high 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.
Paper Structure (16 sections, 11 theorems, 15 equations, 5 figures)

This paper contains 16 sections, 11 theorems, 15 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Background
  4. Endperiodic homeomorphisms
  5. Spun pseudo-Anosov maps
  6. Periodic points and leaves
  7. The Lefschetz-Hopf theorem
  8. The proof of Theorem A
  9. Recurrent fixed points
  10. The Handel--Miller case
  11. Handel--Miller theory
  12. The proof of Theorem C
  13. Sharpness of results
  14. Constructing the endperiodic map $g$
  15. The spA and Handel--Miller representatives of $g$ have fixed points.
  16. ...and 1 more sections

Key Result

Theorem A

Let $f\colon L\rightarrow L$ be a spun pseudo-Anosov map. For some $N\in \mathop{\mathrm{\mathbb{N}}}\nolimits$, every homeomorphism $g$ homotopic to $f$ has no fewer points of period $n$ than $f$ for all $n> N$.

Figures (5)

  • Figure 1: The options for $C$ in Lemma \ref{['lem:rec']}. Type (1) is on the left, and type (2) is on the right. In blue are flow lines of $\varphi$, and in red is $\ell$.
  • Figure 2: The setup in Lemma \ref{['lem:index']}.
  • Figure 3: The red region is the closure of a lift of a positive principal region, and the blue region is the closure of a lift of a negative principal region. The point $q$ must lie on one of the vertices, shown in black, of the polygonal intersection of these two regions.
  • Figure 4: On the left is the surface $S$. On the right is a system of $g$-junctures on $L$. The arcs are drawn on $S$ and should be doubled to get closed curves on $L$. Positive $g$-junctures are drawn in red, and negative $g$-junctures are drawn in blue.
  • Figure 5: The region $R$ in $C$ is the region foliated in orange. The singular foliation $\mathcal{F}$ of $C$ by hyperbolas of the form $xy=a$ for $a\in[-1,1]$ is shown in blue.

Theorems & Definitions (19)

  • Theorem A
  • Corollary B
  • Theorem C
  • Lemma 2.1
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 9 more