Lifts of Lipschitz maps and horizontal fractals in the Heisenberg group

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 connectivi...

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Bibliographic Details
Published in:Ergodic theory and dynamical systems 2006-06, Vol.26 (3), p.621-651
Main Authors: BALOGH, ZOLTÁN M., HOEFER-ISENEGGER, REGULA, TYSON, JEREMY T.
Format: Article
Language:English
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Summary: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).
ISSN:0143-3857
1469-4417
DOI:10.1017/S0143385705000593