Conformal Graph Directed Markov Systems on Carnot Groups

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...

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Hauptverfasser: Chousionis, Vasilis, Tyson, Jeremy, Urbański, Mariusz
Format: E-Book Buch
Sprache:Englisch
Veröffentlicht: Providence, Rhode Island American Mathematical Society 2020
Ausgabe:1
Schriftenreihe:Memoirs of the American Mathematical Society
Schlagworte:
Conformal mapping
Hausdorff measures
Markov processes
Nilpotent Lie groups
Thermodynamics > Mathematical models
Heisenberg group
iterated function system
Bowen’s formula
Hausdorff dimension
Hausdorff measure
continued fractions
Iwasawa Carnot group
conformal mapping
packing measure
open set condition
thermodynamic formalism
ISBN:9781470442156, 1470442159
ISSN:0065-9266, 1947-6221
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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.
AbstractList 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.
The authors 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, they 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. They illustrate their 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.
Author Tyson, Jeremy
Chousionis, Vasilis
Urbański, Mariusz
Author_xml – sequence: 1
  givenname: Vasilis
  surname: Chousionis
  fullname: Chousionis, Vasilis
  email: vasileios.chousionis@uconn.edu
  organization: Department of Mathematics, University of Connecticut, 196 Auditorium Road U-3009, Storrs, CT 06269-3009
– sequence: 2
  givenname: Jeremy
  surname: Tyson
  fullname: Tyson, Jeremy
  email: tyson@illinois.edu
  organization: Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, IL 61801
– sequence: 3
  givenname: Mariusz
  surname: Urbański
  fullname: Urbański, Mariusz
  email: urbanski@unt.edu
  organization: Department of Mathematics, University of North Texas, General Academics Building 435, 1155 Union Circle 311430, Denton, TX 76203-5017
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Keywords Heisenberg group
iterated function system
Bowen’s formula
Hausdorff dimension
Hausdorff measure
continued fractions
Iwasawa Carnot group
conformal mapping
packing measure
open set condition
thermodynamic formalism
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Notes Includes bibliographical references (p. 149-151) and index
July 2020, volume 266, number 1291 (first of 6 numbers).
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Snippet We develop a comprehensive theory of conformal graph directed Markov systems in the non-Riemannian setting of Carnot groups equipped with a sub-Riemannian...
The authors develop a comprehensive theory of conformal graph directed Markov systems in the non-Riemannian setting of Carnot groups equipped with a...
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SubjectTerms Conformal mapping
Hausdorff measures
Markov processes
Nilpotent Lie groups
Thermodynamics -- Mathematical models
TableOfContents Introduction -- Carnot groups -- Carnot groups of Iwasawa type and conformal mappings -- Metric and geometric properties of conformal maps -- Conformal graph directed Markov systems -- Examples of GDMS in Carnot groups -- Countable alphabet symbolic dynamics: foundations of the thermodynamic formalism -- Hausdorff dimension of limit sets -- Conformal measures and regularity of domains -- Examples revisited -- Finer properties of limit sets: Hausdorff, packing and invariant measures -- Equivalent separation conditions for finite GDMS
Chapter 10. Finer properties of limit sets: Hausdorff, packing and invariant measures -- 10.1. Finiteness of Hausdorff measure and positivity of packing measure -- 10.2. Positivity of Hausdorff measure and finiteness of packing measure -- 10.3. Hausdorff and packing measures for continued fraction systems in groups of Iwasawa type -- 10.4. Hausdorff dimensions of invariant measures -- Chapter 11. Equivalent separation conditions for finite GDMS -- Bibliography -- Index -- Back Cover
Cover -- Title page -- Introduction -- Chapter 1. Carnot groups -- 1.1. Carnot groups -- 1.2. Contact mappings -- 1.3. Dimension comparison in Carnot groups -- Chapter 2. Carnot groups of Iwasawa type and conformal mappings -- 2.1. Iwasawa groups of complex, quaternionic or octonionic type -- 2.2. Conformal mappings on Carnot groups -- Chapter 3. Metric and geometric properties of conformal maps -- 3.1. Norm of the horizontal differential and local Lipschitz constant -- 3.2. Koebe distortion theorems for Carnot conformal mappings -- Chapter 4. Conformal graph directed Markov systems -- 4.1. Graph Directed Markov Systems -- 4.2. Carnot conformal graph directed Markov systems -- Chapter 5. Examples of GDMS in Carnot groups -- 5.1. Infinite self-similar iterated function systems -- 5.2. Iwasawa conformal Cantor sets -- 5.3. Continued fractions in Iwasawa groups -- 5.4. Complex hyperbolic Kleinian groups of Schottky type -- Chapter 6. Countable alphabet symbolic dynamics: foundations of the thermodynamic formalism -- 6.1. Subshifts of finite type and topological pressure -- 6.2. Gibbs states, equilibrium states and potentials -- 6.3. Perron-Frobenius Operator -- Chapter 7. Hausdorff dimension of limit sets -- 7.1. Topological pressure, -number, and Bowen's parameter -- 7.2. Hausdorff dimension and Bowen's formula -- 7.3. Characterizations of co-finitely regular and strongly regular Carnot conformal IFS -- 7.4. Dimension spectrum for subsystems of Carnot conformal IFS -- Chapter 8. Conformal measures and regularity of domains -- 8.1. Bounding coding type and null boundary -- 8.2. Regularity properties of domains in Carnot groups -- 8.3. Conformal measure estimates -- Chapter 9. Examples revisited -- 9.1. Infinite self-similar iterated function systems -- 9.2. Continued fractions in groups of Iwasawa type -- 9.3. Iwasawa conformal Cantor sets
Title Conformal Graph Directed Markov Systems on Carnot Groups
URI https://www.ams.org/memo/1291/
https://cir.nii.ac.jp/crid/1130286429696254208
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Volume 266
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