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

TL;DR

This paper establishes a lower bound on the expansion factor of certain fully-punctured pseudo-Anosov maps using standardly embedded train tracks and Thurston's symplectic form, revealing new geometric constraints.

Contribution

It introduces a novel approach combining train track embeddings and symplectic geometry to derive bounds on pseudo-Anosov map expansion factors.

Findings

Proves a lower bound of approximately 6.85408 for the expansion factor.

Uses standardly embedded train tracks and Thurston's symplectic form in the analysis.

Provides insights into the geometric structure of pseudo-Anosov maps with punctures in multiple orbits.

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 $\lambda(f)$ satisfies the inequality $\lambda(f)^{|\chi(S)|} \ge \mu^4 \approx 6.85408$, where $\mu = \frac{1 + \sqrt{5}}{2} \approx 1.61803$ is the golden ratio. The proof involves a study of standardly embedded train tracks, and the Thurston symplectic form defined on their weight space.