Lifts of Lipschitz maps and horizontal fractals in the Heisenberg group

We consider horizontal iterated function systems in the Heisenberg group H-1, i.e. collections of Lipschitz contractions of H-1 with respect to the Heisenberg metric. The invariant sets for such systems are so-called horizontal fractals. We study questions related to connectivity of horizontal fractals and regularity of functions whose graph lies within a horizontal fractal. Our construction yields examples of horizontal BV (bounded variation) surfaces in H-1 that are in contrast with the non-existence of horizontal Lipschitz surfaces which was recently proved by Ambrosio and Kirchheim.