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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Bibliographic Details
Main Authors: Chousionis, Vasilis, Tyson, Jeremy, Urbański, Mariusz
Format: eBook Book
Language:English
Published: Providence, Rhode Island American Mathematical Society 2020
Edition:1
Series:Memoirs of the American Mathematical Society
Subjects:
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
Online Access:Get full text
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Summary: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.
Bibliography:Includes bibliographical references (p. 149-151) and index
July 2020, volume 266, number 1291 (first of 6 numbers).
ISBN:9781470442156
1470442159
ISSN:0065-9266
1947-6221
DOI:10.1090/memo/1291