Irreducible train tracks for pseudo-Anosov homeomorphisms

TL;DR

This paper presents a method to construct invariant train tracks with irreducible transition matrices for pseudo-Anosov homeomorphisms, filling a gap in the literature by ensuring such train tracks exist.

Contribution

It introduces a novel construction starting from veering triangulations to produce irreducible invariant train tracks for pseudo-Anosov homeomorphisms.

Findings

Constructed invariant train tracks with irreducible transition matrices

Provided a systematic method to bypass obstructions to irreducibility

Filled a gap in the literature regarding the existence of such train tracks

Abstract

We describe a construction of invariant train tracks with irreducible transition matrix for pseudo-Anosov homeomorphisms. This fills what seems to be a gap in the literature concerning the existence of such train tracks. The construction starts with an invariant train track associated to the veering triangulation of the mapping torus of the homeomorphism and then uses the veering property to characterize branches which obstruct irreducibility, finally modifying the track to bypass these obstructions.