Space filling curves and geodesic laminations

In this paper we study a class of connected fractals that admit a space filling curve. We prove that these curves are Hölder 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.