We first decribe a compactification of the space of all locally convex surfaces in a hyperbolic compact 3-manifold, which can also be thought of as a quotient of the space of immersions of the disk on the sphere. We then explain that this space can be thought of as a 2-dimensional dynamical system whose properties generalise the chaotic property of the geodesic flow: density of closed leaves, genricity of dense leaves, stability, existence of many transverse invariant measures.