In this talk we shall associate space filling curves to connected fractals, obtained as the fixed point of an iterated function systems (IFS) satisfying the common point property and other conditions. These curves are H\"older continuous and measure preserving. To these space filling curves we associate geodesic laminations satisfying among other properties that points joined by geodesics have the same image in the fractal under the space filling curve. The laminations help us to understand the geometry of the curves. We define an expanding dynamical system on the laminations.