Asymptotic Counting in Conformal Dynamical Systems

• Autore: Pollicott Mark
• Pubblicazione: Providence : , : American Mathematical Society, , 2021; ©2021
• Edizione: 1st ed.
• Descrizione fisica: 1 online resource (152 pages)
• Disciplina: 515/.39
• Soggetto topico: Conformal geometry; Dynamical systems and ergodic theory -- Complex dynamical systems -- Conformal densities and Hausdorff dimension; Dynamical systems and ergodic theory -- Dynamical systems with hyperbolic behavior -- Thermodynamic formalism, variational principles, equilibrium states; Convex and discrete geometry -- Discrete geometry -- Circle packings and discrete conformal geometry; Probability theory and stochastic processes -- Limit theorems -- Central limit and other weak theorems
• Classificazione: 37F3537D3552C2660F05
• Altri autori: UrbańskiMariusz
• Nota di bibliografia: Includes bibliographical references.
• Nota di contenuto: Cover -- Title page -- Chapter 1. Introduction -- 1.1. Short General Introduction -- 1.2. Asymptotic Counting Results -- 1.3. Examples -- 1.4. Statistical Results -- 1.5. Historical Overview of Applications and Examples -- Chapter 2. Attracting Conformal Graph Directed Markov Systems -- 2.1. Thermodynamic Formalism of Subshifts of Finite Type with Countable Alphabet -- Preliminaries -- 2.2. Attracting Conformal Countable Alphabet Graph Directed Markov Systems (GDMSs) and Countable Alphabet Attracting Iterated Function Systems (IFSs) -- Preliminaries -- 2.3. Complex Ruelle-Perron-Frobenius Operators -- Spectrum and D-Genericity -- 2.4. Asymptotic Results for Multipliers -- Statements and First Preparations -- 2.5. Complex Localized Poincaré Series ᵨ -- 2.6. Asymptotic Results for Multipliers -- Concluding of Proofs -- 2.7. Asymptotic Results for Diameters -- Chapter 3. Parabolic Conformal Graph Directed Markov Systems -- 3.1. Parabolic GDMS -- Preliminaries -- 3.2. Poincaré's Series for \cS*, the Associated Countable Alphabet Attracting GDMS -- 3.3. Asymptotic Results for Multipliers -- 3.4. Asymptotic Results for Diameters -- Chapter 4. Central Limit Theorems -- 4.1. Central Limit Theorems for Multipliers and Diameters: Attracting GDMSs with Invariant Measure _{ _{\cS}} -- 4.2. Central Limit Theorems for Multipliers and Diameters: Parabolic GDMSs with Finite Invariant Measure _{ _{\cS}} -- 4.3. Central Limit Theorems: Asymptotic Counting Functions for Attracting GDMSs -- 4.4. Central Limit Theorems: Asymptotic Counting Functions for Parabolic GDMSs -- Chapter 5. Examples and Applications, I -- 5.1. Attracting/Expanding Conformal Dynamical Systems -- 5.2. Conformal Parabolic Dynamical Systems -- Chapter 6. Examples and Applications, II: Kleinian Groups -- 6.1. Finitely Generated Classical Schottky Groups with no Tangencies. ; 6.2. Generalized (allowing tangencies) Classical Schottky Groups -- 6.3. Fuchsian Groups -- Bibliography -- Back Cover.
• Sommario/riassunto: "In this monograph we consider the general setting of conformal graph directed Markov systems modeled by countable state symbolic subshifts of finite type. We deal with two classes of such systems: attracting and parabolic. The latter being treated by means of the former. We prove fairly complete asymptotic counting results for multipliers and diameters associated with preimages or periodic orbits ordered by a natural geometric weighting. We also prove the corresponding Central Limit Theorems describing the further features of the distribution of their weights. These results have direct applications to a wide variety of examples, including the case of Apollonian Circle Packings, Apollonian Triangle, expanding and parabolic rational functions, Farey maps, continued fractions, Mannenville-Pomeau maps, Schottky groups, Fuchsian groups, and many more. This gives a unified approach which both recovers known results and proves new results. Our new approach is founded on spectral properties of complexified Ruelle- Perron-Frobenius operators and Tauberian theorems as used in classical problems of prime number theory"--
• Titolo autorizzato: Asymptotic Counting in Conformal Dynamical Systems
• ISBN: 9781470466329; 1470466325
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9910953752003321
• Lo trovi qui: Univ. Federico II
• Serie: Memoirs of the American Mathematical Society