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

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 $(i(\gamma )+1{)}^{1\mathrm{/}5}$ 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 $\log (k)$.