We study the dynamics of the group of holomorphic automorphisms of the affine cubic surfaces

S_{A,B,C,D} = {(x,y,z)  \in C^3   :    x^2 + y^2 + z^2 +xyz = Ax + By+Cz+D},

where A,B,C, and D are complex parameters.  It arises naturally in the dynamics on character varieties and it also describes the monodromy of the famous Painlevé 6 differential equation.  We explore the Fatou/Julia dichotomy for this group action (defined analogously to the usual definition for iteration of a single rational map) and also the Locally Discrete / Non-Discrete dichotomy (a non-linear version from the classical discrete/non-discrete dichotomy for Lie groups). The interplay between these two dichotomies allow us to prove several results about the topological dynamics of this group.  Moreover, we show the coexistence of non-empty Fatou sets and Julia sets with non-trivial interior for a large set of parameters.