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  • Cited by 9
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    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Chousionis, Vasilis and Tyson, Jeremy T. 2016. Removable sets for homogeneous linear partial differential equations in Carnot groups. Journal d'Analyse Mathématique, Vol. 128, Issue. 1, p. 215.

    Magnani, Valentino Malý, Jan and Mongodi, Samuele 2015. A Low Rank Property and Nonexistence of Higher-Dimensional Horizontal Sobolev Sets. The Journal of Geometric Analysis, Vol. 25, Issue. 3, p. 1444.

    Miao, Jun Jie and Wu, Xiaonan 2015. Generalizedq-dimension of measures on Heisenberg self-affine sets in the Heisenberg group. Nonlinearity, Vol. 28, Issue. 8, p. 2939.

    Lukyanenko, Anton 2013. Bi-Lipschitz extension from boundaries of certain hyperbolic spaces. Geometriae Dedicata, Vol. 164, Issue. 1, p. 47.

    BALOGH, ZOLTÁN M. BERGER, RETO MONTI, ROBERTO and TYSON, JEREMY T. 2010. Exceptional sets for self-similar fractals in Carnot groups. Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 149, Issue. 01, p. 147.

    Magnani, Valentino 2010. Contact equations, Lipschitz extensions and isoperimetric inequalities. Calculus of Variations and Partial Differential Equations, Vol. 39, Issue. 1-2, p. 233.

    Balogh, Zoltán M. Tyson, Jeremy T. and Warhurst, Ben 2009. Sub-Riemannian vs. Euclidean dimension comparison and fractal geometry on Carnot groups. Advances in Mathematics, Vol. 220, Issue. 2, p. 560.

    Balogh, Zoltán M. Tyson, Jeremy T. and Warhurst, Ben 2008. Gromov's dimension comparison problem on Carnot groups. Comptes Rendus Mathematique, Vol. 346, Issue. 3-4, p. 135.

    Magnani, Valentino 2008. Non-horizontal submanifolds and coarea formula. Journal d'Analyse Mathématique, Vol. 106, Issue. 1, p. 95.


Lifts of Lipschitz maps and horizontal fractals in the Heisenberg group

  • DOI:
  • Published online: 01 April 2006

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).

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Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
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