Standardly embedded train tracks and pseudo-Anosov maps with minimum expansion factor

TL;DR

The paper establishes a sharp lower bound for the normalized expansion factor $L(S,f)=\lambda(f)^{|\chi(S)|}$ of fully-punctured pseudo-Anosov maps with at least two puncture orbits, proving $L(S,f) \ge \mu^4$ where $\mu$ is the golden ratio. It achieves this by constructing invariant standardly embedded train tracks and analyzing the Thurston symplectic form on their weight spaces, proving the real-edge transition matrix is reciprocal Perron-Frobenius; McMullen’s bound then yields the required inequality. The authors further sharpen the result for even Euler characteristics, relate equality cases to the LT_{1,k} polynomials, and confirm sharpness via explicit train-track foldings and fibered-face realizations; they also discuss braid monodromies, orientable cases, and the necessity of having at least two puncture orbits. The work links dynamical, combinatorial, and 3-manifold techniques to illuminate minimum expansion factors and provides a framework for further refinements across puncture configurations and odd Euler characteristics. Overall, the results deepen the connection between small entropy pseudo-Anosov maps, train-track dynamics, and the geometry of mapping tori, with implications for Teichmüller polynomials and fibered-face theory.

Abstract

We show that given a fully-punctured pseudo-Anosov map $f:S \to S$ whose punctures lie in at least two orbits under the action of $f$, the expansion factor $λ(f)$ satisfies the inequality $λ(f)^{|χ(S)|} \ge μ^4 \approx 6.85408$, where $μ= \frac{1 + \sqrt{5}}{2} \approx 1.61803$ is the golden ratio. The proof involves a study of standardly embedded train tracks, and the Thurston symplectic form defined on their weight space.

Figures (23)

• Figure 1: A train track and its tie neighborhood near a switch.
• Figure 2: A train track map $\tau \to \tau'$ induces an embedding of tie neighborhoods $N \hookrightarrow N'$.
• Figure 3: An example of a standardly embedded train track. The infintesimal polygons are drawn in gray.
• Figure 4: A rectangle in $S$.
• Figure 5: The local picture of $\ell^s$ and $\ell^u$ at a $5$-pronged puncture $x$. We have indicated a side of the unstable star at $x$ in dark blue.

Theorems & Definitions (89)

• Theorem 1.1
• Theorem 1.4: Fried-Thurston Fri82 Fri85 Thu86
• Theorem 1.5: Farb-Leininger-Margalit FLM11
• Theorem 1.6: McMullen McM00
• Theorem 1.7: Kojima-McShane KM18
• Remark 1.8
• Theorem 1.9
• Theorem 1.10
• Theorem 1.11
• Theorem 1.13: McMullen McM15