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Conformal Graph Directed Markov Systems on Carnot Groups
About this Title
Vasilis Chousionis, Department of Mathematics, University of Connecticut, 196 Auditorium Road U-3009, Storrs, CT 06269-3009, Jeremy Tyson, Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, IL 61801 and Mariusz Urbański, Department of Mathematics, University of North Texas, General Academics Building 435, 1155 Union Circle 311430, Denton, TX 76203-5017
Publication: Memoirs of the American Mathematical Society
Publication Year:
2020; Volume 266, Number 1291
ISBNs: 978-1-4704-4215-6 (print); 978-1-4704-6245-1 (online)
DOI: https://doi.org/10.1090/memo/1291
Published electronically: July 17, 2020
Keywords: Iwasawa Carnot group,
Heisenberg group,
iterated function system,
open set condition,
conformal mapping,
thermodynamic formalism,
Bowen’s formula,
Hausdorff dimension,
Hausdorff measure,
packing measure,
continued fractions
MSC: Primary 30L10, 53C17, 37C40; Secondary 11J70, 28A78, 37B10, 37C30, 37D35, 37F35, 47H10
Table of Contents
Chapters
- Introduction
- 1. Carnot groups
- 2. Carnot groups of Iwasawa type and conformal mappings
- 3. Metric and geometric properties of conformal maps
- 4. Conformal graph directed Markov systems
- 5. Examples of GDMS in Carnot groups
- 6. Countable alphabet symbolic dynamics: foundations of the thermodynamic formalism
- 7. Hausdorff dimension of limit sets
- 8. Conformal measures and regularity of domains
- 9. Examples revisited
- 10. Finer properties of limit sets: Hausdorff, packing and invariant measures
- 11. Equivalent separation conditions for finite GDMS
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.- Daniel Allcock, Identifying models of the octave projective plane, Geom. Dedicata 65 (1997), no. 2, 215–217. MR 1451975, DOI 10.1023/A:1017926505948
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