Dilatation versus self-intersection number for point-pushing pseudo-Anosov homeomorphisms

TL;DR

This paper explores the relationship between the self-intersection number of a filling curve on a surface and the dilatation of the associated point-pushing pseudo-Anosov homeomorphism, establishing bounds and growth rates.

Contribution

It provides new bounds on dilatation in terms of self-intersection number and analyzes minimal dilatation and entropy growth for point-pushing pseudo-Anosov maps.

Findings

Dilatation is bounded between (i(γ)+1)^{1/5} and 9^{i(γ)}.

Least dilatation tends to infinity with genus.

Minimal entropy grows like log(k) for self-intersection number k.

Abstract

A filling curve $\gamma$ on a based surface $S$ determines a pseudo-Anosov homeomorphism $P(\gamma)$ of $S$ via the process of "point-pushing along $\gamma$." We consider the relationship between the self-intersection number $i(\gamma)$ of $\gamma$ and the dilatation of $P(\gamma)$; our main result is that the dilatation is bounded between $(i(\gamma)+1)^{1/5}$ and $9^{i(\gamma)}$. We also bound the least dilatation of any pseudo-Anosov in the point-pushing subgroup of a closed surface and prove that this number tends to infinity with genus. Lastly, we investigate the minimal entropy of any pseudo-Anosov homeomorphism obtained by pushing along a curve with self-intersection number $k$ and show that, for a closed surface, this number grows like $\log(k)$.