** 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 **R**^{3}. 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 **R**^{3}. Its Hausdorff dimension (with respect to either
the Carnot-Caratheodory (CC) metric in **H** or the Euclidean
metric in **R**^{3}) 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.