Maximal dilatation on nonorientable surfaces

TL;DR

This paper proves that a specific polynomial provides the maximal dilatation for pseudo-Anosov diffeomorphisms with orientable invariant foliations on certain nonorientable surfaces, by establishing its irreducibility.

Contribution

It demonstrates the maximal dilatation is achieved by Liechti-Strenner's polynomial on nonorientable surfaces of even genus, confirming its irreducibility.

Findings

Liechti-Strenner's polynomial is irreducible.

The polynomial yields maximal dilatation on nonorientable surfaces.

Maximal dilatation is characterized for pseudo-Anosov maps with orientable foliations.

Abstract

On each nonorientable surface of even genus $g \geq 4$, we show that the Liechti-Strenner's polynomial in~\cite{LiechtiStrenner18} gives a maximal dilatation among pseudo-Anosov diffeomorphisms with an orientable invariant foliation. This is proved by showing that this polynomial is irreducible.