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Lifts of Lipschitz maps and horizontal fractals in the Heisenberg group

Published online by Cambridge University Press:  18 April 2006

ZOLTÁN M. BALOGH
Affiliation:
Department of Mathematics, University of Berne, Sidlerstrasse 5, 3012 Berne, Switzerland (e-mail: zoltan.balogh@math.unibe.ch, regula.hoefer-isenegger@math.unibe.ch)
REGULA HOEFER-ISENEGGER
Affiliation:
Department of Mathematics, University of Berne, Sidlerstrasse 5, 3012 Berne, Switzerland (e-mail: zoltan.balogh@math.unibe.ch, regula.hoefer-isenegger@math.unibe.ch)
JEREMY T. TYSON
Affiliation:
Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, IL 61801, USA (e-mail: tyson@math.uiuc.edu)

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).

Type
Research Article
Copyright
2006 Cambridge University Press

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