Space-filling curves and geodesic laminations. II: Symmetries

In 2008, the author introduced a class of space-filling curves associated to fractals that satisfy the a special property. These structures admit geodesic laminations on the disc, which help to understand the geometrical and the dynamical properties of the space-filling curves. In the present article we study the relation between the symmetries of the laminations and the fractals. In particular we prove that the group of symmetries of the lamination is isomorphic to a subgroup of the full group of symmetries of the fractal. We extend the results to a larger class of fractals using the concept of sub-IFS.