  We develop a comprehensive theory of conformal graph directed Markov systems
in the non-Riemannian setting of Carnot groups equipped with a sub-Riemannian
metric. In particular, we develop the thermodynamic formalism and show that,
under natural hypotheses, the limit set of an Carnot conformal GDMS has
Hausdorff dimension given by Bowen's parameter. We illustrate our results for a
variety of examples of both linear and nonlinear iterated function systems and
graph directed Markov systems in such sub-Riemannian spaces. These include the
Heisenberg continued fractions introduced by Lukyanenko and Vandehey as well as
Kleinian and Schottky groups associated to the non-real classical rank one
hyperbolic spaces.
