Constructing Birkhoff sections for pseudo-Anosov flows with controlled complexity

TL;DR

This paper presents a novel method linking veering triangulations to construct Birkhoff sections for pseudo-Anosov flows, enabling explicit control over their topological complexity and boundary components.

Contribution

It introduces a new construction technique for Birkhoff sections using veering triangulations, providing explicit control over their complexity and boundary structure.

Findings

Any transitive pseudo-Anosov flow admits a Birkhoff section with two boundary components.

The method allows explicit construction and control over the Euler characteristic of Birkhoff sections.

The approach connects pseudo-Anosov flows with veering triangulations for topological analysis.

Abstract

We introduce a new method of constructing Birkhoff sections for pseudo-Anosov flows, which uses the connection between pseudo-Anosov flows and veering triangulations. This method allows for explicit constructions, as well as control over the Birkhoff section in terms of its Euler characteristic and the complexity of the boundary orbits. In particular, we show that any transitive pseudo-Anosov flow has a Birkhoff section with two boundary components.