Lifts of Lipschitz maps and horizontal fractals in the Heisenberg group

Abstract

We consider horizontal iterated function systems in the Heisenberg group $\mathbb{H}^1$, i.e. collections of Lipschitz contractions of $\mathbb{H}^1$ 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 $\mathbb{H}^1$ that are in contrast with the non-existence of horizontal Lipschitz surfaces which was recently proved by Ambrosio and Kirchheim (Rectifiable sets in metric and Banach spaces. Math. Ann.318(3) (2000), 527–555).