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Fourier Analysis --- Hausdorff measures

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Geometric measure theory --- Singular integrals --- Hausdorff measures

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"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"--

Heisenberg uncertainty principle --- Heisenberg, Principe d'incertitude d' --- Hausdorff measures --- Hausdorff, Mesures de --- Nilpotent Lie groups. --- Conformal mapping. --- Markov processes. --- Hausdorff measures. --- Thermodynamics --- Mathematical models.

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Stochastic processes. --- Stochastic differential equations. --- Hausdorff measures --- Probabilities. --- Processus stochastiques --- Equations différentielles stochastiques --- Mesures de Hausdorff --- Probabilités --- Stochastic processes --- Stochastic differential equations --- Probabilities --- Equations différentielles stochastiques --- Probabilités

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The purpose of this book is to present a modern account of mapping theory with emphasis on quasiconformal mapping and its generalizations. The modulus method was initiated by Arne Beurling and Lars Ahlfors to study conformal mappings, and later this method was extended and enhanced by several others. The techniques are geometric and they have turned out to be an indispensable tool in the study of quasiconformal and quasiregular mappings as well as their generalizations. The book is based on recent research papers and extends the modulus method beyond the classical applications of the modulus techniques presented in many monographs.

Boundary value problems. --- Hausdorff measures. --- Moduli theory. --- Quasiconformal mappings. --- Quasiconformal mappings --- Moduli theory --- Calculus --- Applied Mathematics --- Mathematics --- Engineering & Applied Sciences --- Physical Sciences & Mathematics --- Theory of moduli --- Mappings, Quasiconformal --- Mathematics. --- Mathematical analysis. --- Analysis (Mathematics). --- Functional analysis. --- Analysis. --- Functional Analysis. --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- 517.1 Mathematical analysis --- Mathematical analysis --- Math --- Science --- Analytic spaces --- Functions of several complex variables --- Geometry, Algebraic --- Conformal mapping --- Functions of complex variables --- Geometric function theory --- Mappings (Mathematics) --- Global analysis (Mathematics). --- Analysis, Global (Mathematics) --- Differential topology