Ergodic Theory and Dynamical Systems

Lifts of Lipschitz maps and horizontal fractals in the Heisenberg group

a1 Department of Mathematics, University of Berne, Sidlerstrasse 5, 3012 Berne, Switzerland (e-mail:,
a2 Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, IL 61801, USA (e-mail:

Article author query
balogh zm   [Google Scholar] 
hoefer-isenegger r   [Google Scholar] 
tyson jt   [Google Scholar] 


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).

(Received January 19 2004)
(Revised February 2 2005)