Calculus of Variations and Geometric Measure Theory
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Z. Balogh - R. Hoefer-Isenegger - J. Tyson

Lifts of Lipschitz maps and horizontal fractals in the Heisenberg group

created on 06 Apr 2004
modified by zoltan on 13 Aug 2007


Published Paper

Inserted: 6 apr 2004
Last Updated: 13 aug 2007

Journal: Ergodic Theory and Dyn. Syst.
Volume: 26
Number: 3
Pages: 621-651
Year: 2006


We consider horizontal iterated function systems in the Heisenberg group $\Qua^1$, i.e., collections of Lipschitz contractions of $\Qua^1$ with respect to the Heisenberg metric. The invariant sets for such systems are so-called {\it 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 surfaces in $\Qua^1$ that is in contrast with the nonexistence of horizontal Lipschitz surfaces which was recently proved by Ambrosio and Kirchheim.

Keywords: Heisenberg group, lipschitz maps, Invariant sets, Fractals


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