A pseudo-Anosov foliation of a compact orientable surface $F$ is a one whose class in the space $\mathcal{PMF}(F)$ of projective measured foliations is preserved by some pseudo-Anosov homeomorphism of $F$. I saw it casually mentioned that pseudo-Anosov foliations are dense in $\mathcal{PMF}(F)$. What is a proper reference for that result?


2 Answers 2


The pseudo-Anosov foliations form a subset of $\mathcal{PMF}(F)$ which is invariant under the action of the mapping class group $MCG(F)$, because if $\Lambda_+(\phi) \in \mathcal{PMF}(F)$ is the stable lamination of a pseudo-Anosov $\phi \in MCG(F)$ then $\psi(\Lambda_+(\phi)) = \Lambda_+(\psi\phi\psi^{-1})$ is the stable lamination of the pseudo-Anosov $\psi\phi\psi^{-1}$.

So the fact that the pseudo-Anosov foliations are dense is an immediate corollary of Theorem 6.1 from "Thurston's Work on Surfaces" (originally published in 1979 as "Travaux de Thurston sur les Surfaces"):

The action of $MCG(F)$ on $\mathcal{PMF}(F)$ is minimal, meaning that every orbit is dense.


I am sure this was known earlier (check Ivanov's book, most likely, it is there), but you can refer to the following theorem of Lindenstraus and Mirzakhani (see their paper Ergodic theory of the space of measured laminations):

Let $\mu$ be a nonzero measured lamination on a hyperbolic surface $S$ whose support contains no closed geodesics. Then the $Mod(S)$-orbit of $\mu$ is dense in $ML(S)$.

Now, apply this to $\mu$ which is the stable lamination of a pseudo-Anosov homeomorphism.

One can also show that in $PML(S)\times PLM(S)$ the subset of pairs $(\mu_+(h),\mu_-(h))$ is dense, where $\mu_\pm$ are stable/unstable laminations of $h\in Mod(S)$, which is pseudo-Anosov.


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