Remarks on the Liechti-Strenner's examples having small dilatations

TL;DR

This paper investigates specific examples of pseudo-Anosov homeomorphisms on surfaces, demonstrating their minimal dilatations under certain algebraic and topological constraints, thereby advancing understanding of surface dynamics.

Contribution

It proves that Liechti-Strenner's examples minimize dilatation among classes with particular polynomial coefficient conditions, clarifying their optimality in surface homeomorphism dynamics.

Findings

Liechti-Strenner's example for nonorientable surfaces minimizes dilatation.

Their orientable surface example also minimizes dilatation under specified polynomial conditions.

The results specify algebraic constraints under which these examples are optimal.

Abstract

We show that the Liechti-Strenner's example for the closed nonorientable surface in \cite{LiechtiStrenner18} minimizes the dilatation within the class of pseudo-Anosov homeomorphisms with an orientable invariant foliation and all but the first coefficient of the characteristic polynomial of the action induced on the first cohomology nonpositive. We also show that the Liechti-Strenner's example of orientation-reversing homeomorphism for the closed orientable surface in \cite{LiechtiStrenner18} minimizes the dilatation within the class of pseudo-Anosov homeomorphisms with an orientable invariant foliation and all but the first coefficient of the characteristic polynomial $p(x)$ of the action induced on the first cohomology nonpositive or all but the first coefficient of $p(x) (x \pm 1)^2$, $p(x) (x^2 \pm 1)$, or $p(x) (x^2 \pm x + 1)$ nonpositive.