  Given a closed, genus $g$ surface $S$, we consider $Aut(\mathcal{ML})$, the
group of homeomorphism of $\mathcal{ML}$ that preserves the intersection
number. We prove that except in a few special cases, $Aut(\mathcal{ML})$ is
isomorphic to the extended mapping class group. The theorem is a special case
of Ivanov's meta-conjecture which states that any ``sufficiently rich'' object
naturally associated to a surface has automorphism group isomorphic to the
extended mapping class group. Some of the results in this paper can be
generalized to the setting of geodesic currents. To illustrate the challenges
encountered when extending the theorem to the context of currents, we construct
infinite family of pairs of closed curves that have the same marked length
spectra and self intersection number.
