We consider horizontal iterated function systems in the Heisenberg group ℍ1, i.e. collections of Lipschitz contractions of ℍ1 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 ℍ1 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).
ASJC Scopus subject areas
- Applied Mathematics