  We study the problem of classifying stationary measures and orbit closures
for non-abelian action on a surface with a given smooth invariant measure.
Using a result of Brown and Rodriguez Hertz, we show that under a certain
finite verifiable average growth condition, the only nonatomic stationary
measure is the given smooth invariant measure, and every orbit closure is
either finite or dense. Moreover, every point with infinite orbit
equidistributes on the surface with respect to the smooth invariant measure.
This is analogous to the results of Benoist-Quint and Eskin-Lindenstrauss in
the homogeneous setting, and the result of Eskin-Mirzakhani in the setting of
moduli spaces of translation surfaces. We then apply this result to two
concrete settings, namely discrete perturbation of the standard map and
Out($F_2$)-action on a certain character variety. We verify the growth
condition analytically in the former setting, and verify numerically in the
latter setting.
