Pseudo-Anosov homeomorphisms of punctured non-orientable surfaces with small stretch factor

TL;DR

This paper investigates the minimal stretch factors of pseudo-Anosov homeomorphisms on punctured non-orientable surfaces, showing they decrease proportionally to 1/g as genus increases, extending Thurston's theory to non-orientable cases.

Contribution

It extends the understanding of pseudo-Anosov homeomorphisms to non-orientable surfaces and adapts Thurston's fibered face theory to this setting.

Findings

Minimal stretch factor asymptotically proportional to 1/g

Extension of Thurston's fibered face theory to non-orientable 3-manifolds

Adapts Yazdi's work to non-orientable surfaces

Abstract

We prove that in the non-orientable setting, the minimal stretch factor of a pseudo-Anosov homeomorphism of a surface of genus $g$ with a fixed number of punctures is asymptotically on the order of $\frac{1}{g}$. Our result adapts the work of Yazdi to non-orientable surfaces. We include the details of Thurston's theory of fibered faces for non-orientable 3-manifolds.