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A lower bound on end-periodic stretch factors

Marissa Loving, Chenxi Wu

TL;DR

This work derives a Penner-type lower bound for the Handel–Miller stretch factor $\lambda_{HM}$ of end-periodic homeomorphisms on infinite-type surfaces, expressing the bound in terms of the core characteristic $\chi(f)$ as $\log \lambda_{HM} \ge \frac{\log(2)}{3\,\chi(f)}$ when $\lambda_{HM}\neq 1$. The proof combines a direct analysis of the Markov decomposition arising from the intersection $\Lambda^+\cap\Lambda^-$ with a generalized Penner lemma, bounding the complexity by the core data. The authors also connect $\lambda_{HM}$ to the spin-based stretched factor of spun pseudo-Anosov maps and show the bound is sharp by constructing a Penner-like family on the Loch Ness monster surface, with $\lambda_{HM}(f_1)=2$ and $\log \lambda_{HM}(f_d)=\frac{\log 2}{d}$ in the family. This work thus links end-periodic dynamics to spun pseudo-Anosov phenomena and provides concrete, sharp growth rates for end-periodic stretch factors on infinite-type surfaces.

Abstract

Given an end-periodic homeomorphism $f: S \to S$ we give a lower bound on the Handel--Miller stretch factor of $f$ in terms of the core characteristic of $f$, which is a measure of topological complexity for an end-periodic homeomorphism. We also show that the growth rate of this bound is sharp.

A lower bound on end-periodic stretch factors

TL;DR

This work derives a Penner-type lower bound for the Handel–Miller stretch factor of end-periodic homeomorphisms on infinite-type surfaces, expressing the bound in terms of the core characteristic as when . The proof combines a direct analysis of the Markov decomposition arising from the intersection with a generalized Penner lemma, bounding the complexity by the core data. The authors also connect to the spin-based stretched factor of spun pseudo-Anosov maps and show the bound is sharp by constructing a Penner-like family on the Loch Ness monster surface, with and in the family. This work thus links end-periodic dynamics to spun pseudo-Anosov phenomena and provides concrete, sharp growth rates for end-periodic stretch factors on infinite-type surfaces.

Abstract

Given an end-periodic homeomorphism we give a lower bound on the Handel--Miller stretch factor of in terms of the core characteristic of , which is a measure of topological complexity for an end-periodic homeomorphism. We also show that the growth rate of this bound is sharp.

Paper Structure

This paper contains 6 sections, 1 theorem, 9 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. End-periodic homeomorphisms
  4. A key lemma
  5. The proof
  6. A sequence of examples

Key Result

Lemma 2.1

Suppose that $A$ is an $n \times n$ nonnegative integral matrix, where $n> 1$, with leading eigenvalue $> 1$. If $\lambda$ denotes the spectral radius of $A$, then

Figures (4)

  • Figure 1: Note that after finitely many rotations the green and pink curves define a filling pair of multicurves.
  • Figure 2: The Loch Ness monster surface, $S$, is an infinite cyclic cover of $S_d$ which is in turn a $d$-fold cover of $S_1$. Note $S_d$ and $S_1$ are, in fact, homeomorphic. The horizontal and vertical shifts shown on $S$ are used to define the deck transformation $p_d$.
  • Figure 3: The map $f_d^d: S_d \to S_d$, is the composition of the twists about the pink curves composed with a shift by step $1$ along each row in the horizontal direction.
  • Figure 4: Consider the map $f_1 = D \circ h$, where $D$ is the Dehn twist about the curve labeled $D$ and $h$ is the shift map which translates by $1$ to the right. The train track $\tau$ is invariant under $f_1$.

Theorems & Definitions (5)

  • Lemma 2.1: Generalized Penner Lemma
  • proof
  • proof : Proof of Main Theorem
  • Claim 4.1
  • proof : Proof of Claim