SELF-SIMILAR AND SELF-AFFINE SETS IN THE HEISENBERG GROUP AND OTHER CARNOT GROUPS

Let H denote the three-dimensional Heisenberg group with underlying space R3. The figure shows a set S⊂H which is self-similar in the sub-Riemannian geometry of H, but self-affine in the Euclidean geometry of R3. Its Hausdorff dimension (with respect to either the Carnot-Caratheodory (CC) metric in H or the Euclidean metric in R3) is equal to two. Note that no smooth surface in H can have this equality of dimensions.

Self-similar sets in the Heisenberg group (and more general step two Carnot groups) were first considered by Strichartz [1]. In [2], [3] we studied self-similar and self-affine sets in the Heisenberg group. We show that the above Strichartz set S is the graph of a BV function defined on the unit square [0,1]2. Thus horizontal BV surfaces in H exist.

In [4], [5] we study self-similar sets in arbitrary Carnot groups. We prove a sharp dimension comparison theorem which has applications to the computation of dimension of invariant sets of certain nonlinear, nonconformal Euclidean iterated function systems of polynomial type. The following sequence of images show four three-diemensional projections of a set S⊂G, where G denotes the (topologically four-dimensional) Engel group. S is defined by a set of four contractive mappings (CC similarities); it has dimension two with respect to either the CC or Euclidean metric. The defining equations for the iterated function system involve quadratic polynomials in the underlying Euclidean variables.

The following sequence of images show six of the 20 three-dimensional projections of a set S⊂G, where G denotes a certain six-dimensional Carnot group of step four. S is defined by a set of 16 contractive mappings (CC similarities); it has Euclidean dimension 3 and CC dimension 4. The defining equations for the iterated function system involve cubic polynomials in the underlying Euclidean variables.

[1] R. S. Strichartz, "Self-similarity on nilpotent Lie groups", Geometric Analysis, Contemporary Mathematics 140, AMS (1992), 123-157.

[2] Z. M. Balogh and J. T. Tyson, "Hausdorff dimensions of self-similar and self-affine fractals in the Heisenberg group", Proc. London Math. Soc. 91 (2005), 153-183.

[3] Z. M. Balogh, R. Hofer-Isenegger and J. T. Tyson, "Lifts of Lipschitz maps and horizontal fractals in the Heisenberg group", Ergodic Theory Dynam. Systems 26 (2006), 621-651.

[4] Z. M. Balogh, J. T. Tyson and B. Warhurst, "Gromov's dimension comparison problem on Carnot groups", C. R. Acad. Sci. Paris Ser. I, Math 346 (2008), 135-138.

[5] Z. M. Balogh, J. T. Tyson and B. Warhurst, "Sub-Riemannian vs. Euclidean dimension comparison and fractal geometry on Carnot groups", Advances in Mathematics 220 (2009), 560-619.