Conformal graph directed Markov systems on Carnot groups / Vasilionis Chousionis, Jeremy Tyson, Mariusz Urbanski.

Format:

Book

Author/Creator:

Chousionis, Vasilionis, 1980- author.

Tyson, Jeremy T., 1972- author.

Urbański, Mariusz, author.

Series:

Memoirs of the American Mathematical Society ; no. 1291.

Memoirs of the American Mathematical Society, 0065-9266 ; number 1291

Language:

English

Subjects (All):

Nilpotent Lie groups.

Conformal mapping.

Markov processes.

Hausdorff measures.

Thermodynamics--Mathematical models.

Thermodynamics.

Physical Description:

viii, 155 pages ; 26 cm.

Place of Publication:

Providence, RI : American Mathematical Society, [2020]

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"-- Provided by publisher.

Contents:

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.

Notes:

"July 2020, volume 266, number 1291 (first of 6 numbers)."

Includes bibliographical references and index.

ISBN:

9781470442156

1470442159

OCLC:

1160037121