Conformal graph directed Markov systems on Carnot groups

Vasilis Chousionis, Jeremy Tyson, Mariusz Urbański

Research output: Contribution to journalArticle

Abstract

We develop a comprehensive theory of conformal graph directed Markov systems in the non-Riemannian setting of Carnot groups equipped with asub-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)viii+155
JournalMemoirs of the American Mathematical Society
Volume266
Issue number1291
DOIs
StatePublished - Jul 2020

Keywords

  • Bowen's formula
  • Hausdorff
  • Heisenberg group
  • Iwasawa Carnot group
  • conformal mapping
  • iterated function system
  • open set condition
  • thermodynamic formalism

Fingerprint Dive into the research topics of 'Conformal graph directed Markov systems on Carnot groups'. Together they form a unique fingerprint.

Cite this