Conformal Graph Directed Markov Systems on Carnot Groups

Vasilis Chousionis, Jeremy Tyson, Mariusz Urbański

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)1-169
Number of pages169
JournalMemoirs of the American Mathematical Society
Issue number1291
StatePublished - Jul 2020


  • Bowen's formula
  • Conformal mapping
  • Continued fractions
  • Hausdorff dimension
  • Hausdorff measure
  • Heisenberg group
  • Iterated function system
  • Iwasawa carnot group
  • Open set condition
  • Packing measure
  • Thermodynamic formalism

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics


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