Conformal graph directed Markov systems on Carnot groups

2016/05/04 by Chousionis, Vasilis, Jeremy T. Tyson, Tyson, Jeremy T. +2 · 1 citation
Computer Science · Mathematics · #11J70 #28A78 #30L10 #37B10 #37C30 #37C40 #37D35 #37F35 #47H10 #53C17 #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Mathematical Dynamics and Fractals #Metric Geometry (math.MG) #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.1605.01127

openalex publication_date 2016/05/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

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.

Citations

• Quasiconformal maps in metric spaces with controlled geometry
• Metriques de Carnot-Caratheodory et Quasiisometries des Espaces Symetriques de rang un

Cited by

• Products of exact dynamical systems and Lorentzian continued fractions

Related

• Non-autonomous conformal graph directed Markov systems
• The Hausdorff dimension spectrum of Conformal Graph Directed Markov Systems and applications to Nearest Integer continued fractions
• Conformal Graph Directed Markov Systems
• Multifractal analysis for conformal graph directed Markov systems
• Asymptotic Counting in Conformal Dynamical Systems