Pseudo-Anosov Maps and Pairs of Filling Simple Closed Geodesics on Riemann Surfaces, II

Abstract

Let $S$ be a Riemann surface containing at least two punctures $z$ and $z_0$. Let $\mathscr{F}(S)$ be the set of pseudo-Anosov maps of $S$ that are isotopic to the identity on $S\cup \{z\}$. We show that for any $f\in \mathscr{F}(S)$ and any twice punctured disk $\Delta$ enclosing $z$ and $z_0$, the pair $(\partial \Delta, f(\partial \Delta))$ fills $S$, where $\partial \Delta$ denotes the boundary of $\Delta$. Fix such a $\Delta$, and denote by $\mathscr{T}(\Delta)$ the set of twice punctured disks $\Delta'$ on $S$ enclosing $z$ and $z_0$ with the property that $(\partial \Delta, \partial \Delta')$ fills $S$. Let $\Delta_0\in \mathscr{T}(\Delta)$. We describe all possible pseudo-Anosov maps $f$ in $\mathscr{F}(S)$ sending $\Delta$ to $\Delta_0$, and classify elements of $\mathscr{F}(S)$ in terms of $\mathscr{T}(\Delta)$. We also show that there are infinitely many elements $f_k\in \mathscr{F}(S)$ with $f_k(\Delta)=\Delta_0$ such that their dilatations $\lambda(f_k)\rightarrow +\infty$ as $k\rightarrow +\infty$.