Abstract:  We introduce a family of modified baker maps defined in the square, for which the non wandering set is the entire domain. These maps are non uniformly hyperbolic in the interior of the square, and have a dense set of hyperbolic periodic orbits. We use the maps to construct a flow in dimension three with a "fat" non wandering set (non-empty interior) with an attached hyperbolic singularity. The flow is inspired in the construction of the Lorenz geometric attractor.