Agol cycles of pseudo-Anosov maps on the 2-punctured torus and 5-punctured sphere

TL;DR

This paper studies Agol cycles of pseudo-Anosov maps on the 2-punctured torus and 5-punctured sphere, providing explicit train tracks, computing cycles, and establishing conjugacy conditions, along with a new dilatation formula.

Contribution

It introduces a method to compute Agol cycles for specific pseudo-Anosov maps and derives a new formula for dilatation, advancing understanding of their structure.

Findings

Computed Agol cycles for the family of maps.

Established criteria for conjugacy of maps.

Derived a new formula for dilatation.

Abstract

Given a periodic splitting sequence of a measured train track, an Agol cycle is the part that constitutes a period up to the action of a pseudo-Anosov map and the rescaling by its dilatation. We consider a family of pseudo-Anosov maps on the 2-punctured torus and on the 5-punctured sphere. We present measured train tracks and compute their Agol cycles. We give a condition under which two maps in the defined family are conjugate or not. In the process, we find a new formula for the dilatation.