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

We show that given a fully punctured pseudo-Anosov map f:S → 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)|≥ μ⁴ ≈ 6.85408, where μ = {1 + √5}{2}≈ 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.