Conformal Graph Directed Markov Systems on Carnot Groups

The authors develop a theory of conformal graph directed Markov systems in non-Riemannian Carnot groups, showing that the limit set has Hausdorff dimension given by Bowen's parameter. They illustrate their results for various examples, including Heisenberg continued fractions and Kleinian and Schottky groups.

Format: Paperback / softback
Length: 153 pages
Publication date: 30 October 2020
Publisher: American Mathematical Society

The authors delve into the realm of conformal graph directed Markov systems, specifically in the non-Riemannian setting of Carnot groups adorned with a sub-Riemannian metric. Their comprehensive theory encompasses various aspects, including the development of a thermodynamic formalism. Through meticulous analysis, they demonstrate that under natural assumptions, the limit set of an Carnot conformal GDMS possesses a Hausdorff dimension determined by Bowen's parameter. This groundbreaking revelation sheds light on the structure and behavior of these systems. To illustrate their findings, the authors present a diverse range of examples, encompassing both linear and nonlinear iterated function systems and graph directed Markov systems. These examples encompass a wide spectrum, including the Heisenberg continued fractions introduced by Lukyanenko and Vandehey, as well as Kleinian and Schottky groups associated with the non-real classical rank one hyperbolic spaces. Through their meticulous exploration, the authors unveil the intricate connections and properties of these systems, providing valuable insights into the complex dynamics and behavior of non-Riemannian Carnot groups.

Weight: 316g
Dimension: 177 x 253 x 15 (mm)
ISBN-13: 9781470442156