Distance Expanding Random Mappings, Thermodynamical Formalism, Gibbs Measures and Fractal Geometry PDF Download
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Author: Volker Mayer
Publisher: Springer Science & Business Media
ISBN: 3642236499
Category : Mathematics
Languages : en
Pages : 122
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Book Description
The theory of random dynamical systems originated from stochastic differential equations. It is intended to provide a framework and techniques to describe and analyze the evolution of dynamical systems when the input and output data are known only approximately, according to some probability distribution. The development of this field, in both the theory and applications, has gone in many directions. In this manuscript we introduce measurable expanding random dynamical systems, develop the thermodynamical formalism and establish, in particular, the exponential decay of correlations and analyticity of the expected pressure although the spectral gap property does not hold. This theory is then used to investigate fractal properties of conformal random systems. We prove a Bowen’s formula and develop the multifractal formalism of the Gibbs states. Depending on the behavior of the Birkhoff sums of the pressure function we arrive at a natural classification of the systems into two classes: quasi-deterministic systems, which share many properties of deterministic ones; and essentially random systems, which are rather generic and never bi-Lipschitz equivalent to deterministic systems. We show that in the essentially random case the Hausdorff measure vanishes, which refutes a conjecture by Bogenschutz and Ochs. Lastly, we present applications of our results to various specific conformal random systems and positively answer a question posed by Bruck and Buger concerning the Hausdorff dimension of quadratic random Julia sets.
Author: Volker Mayer
Publisher: Springer Science & Business Media
ISBN: 3642236499
Category : Mathematics
Languages : en
Pages : 122
Get Book Here
Book Description
The theory of random dynamical systems originated from stochastic differential equations. It is intended to provide a framework and techniques to describe and analyze the evolution of dynamical systems when the input and output data are known only approximately, according to some probability distribution. The development of this field, in both the theory and applications, has gone in many directions. In this manuscript we introduce measurable expanding random dynamical systems, develop the thermodynamical formalism and establish, in particular, the exponential decay of correlations and analyticity of the expected pressure although the spectral gap property does not hold. This theory is then used to investigate fractal properties of conformal random systems. We prove a Bowen’s formula and develop the multifractal formalism of the Gibbs states. Depending on the behavior of the Birkhoff sums of the pressure function we arrive at a natural classification of the systems into two classes: quasi-deterministic systems, which share many properties of deterministic ones; and essentially random systems, which are rather generic and never bi-Lipschitz equivalent to deterministic systems. We show that in the essentially random case the Hausdorff measure vanishes, which refutes a conjecture by Bogenschutz and Ochs. Lastly, we present applications of our results to various specific conformal random systems and positively answer a question posed by Bruck and Buger concerning the Hausdorff dimension of quadratic random Julia sets.
Author: Volker Mayer
Publisher: Springer
ISBN: 3642236502
Category : Mathematics
Languages : en
Pages : 122
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Book Description
The theory of random dynamical systems originated from stochastic differential equations. It is intended to provide a framework and techniques to describe and analyze the evolution of dynamical systems when the input and output data are known only approximately, according to some probability distribution. The development of this field, in both the theory and applications, has gone in many directions. In this manuscript we introduce measurable expanding random dynamical systems, develop the thermodynamical formalism and establish, in particular, the exponential decay of correlations and analyticity of the expected pressure although the spectral gap property does not hold. This theory is then used to investigate fractal properties of conformal random systems. We prove a Bowen’s formula and develop the multifractal formalism of the Gibbs states. Depending on the behavior of the Birkhoff sums of the pressure function we arrive at a natural classification of the systems into two classes: quasi-deterministic systems, which share many properties of deterministic ones; and essentially random systems, which are rather generic and never bi-Lipschitz equivalent to deterministic systems. We show that in the essentially random case the Hausdorff measure vanishes, which refutes a conjecture by Bogenschutz and Ochs. Lastly, we present applications of our results to various specific conformal random systems and positively answer a question posed by Bruck and Buger concerning the Hausdorff dimension of quadratic random Julia sets.
Author: Mark Pollicott
Publisher: Springer Nature
ISBN: 3030748634
Category : Mathematics
Languages : en
Pages : 534
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Book Description
This volume arose from a semester at CIRM-Luminy on “Thermodynamic Formalism: Applications to Probability, Geometry and Fractals” which brought together leading experts in the area to discuss topical problems and recent progress. It includes a number of surveys intended to make the field more accessible to younger mathematicians and scientists wishing to learn more about the area. Thermodynamic formalism has been a powerful tool in ergodic theory and dynamical system and its applications to other topics, particularly Riemannian geometry (especially in negative curvature), statistical properties of dynamical systems and fractal geometry. This work will be of value both to graduate students and more senior researchers interested in either learning about the main ideas and themes in thermodynamic formalism, and research themes which are at forefront of research in this area.
Author: Mariusz Urbański
Publisher: Walter de Gruyter GmbH & Co KG
ISBN: 3110769875
Category : Mathematics
Languages : en
Pages : 440
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Book Description
Author: Feliks Przytycki
Publisher: Cambridge University Press
ISBN: 0521438004
Category : Mathematics
Languages : en
Pages : 365
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Book Description
A one-stop introduction to the methods of ergodic theory applied to holomorphic iteration that is ideal for graduate courses.
Author: Utkir A Rozikov
Publisher: World Scientific
ISBN: 9811251258
Category : Mathematics
Languages : en
Pages : 367
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Book Description
This book presents recently obtained mathematical results on Gibbs measures of the q-state Potts model on the integer lattice and on Cayley trees. It also illustrates many applications of the Potts model to real-world situations in biology, physics, financial engineering, medicine, and sociology, as well as in some examples of alloy behavior, cell sorting, flocking birds, flowing foams, and image segmentation.Gibbs measure is one of the important measures in various problems of probability theory and statistical mechanics. It is a measure associated with the Hamiltonian of a biological or physical system. Each Gibbs measure gives a state of the system.The main problem for a given Hamiltonian on a countable lattice is to describe all of its possible Gibbs measures. The existence of some values of parameters at which the uniqueness of Gibbs measure switches to non-uniqueness is interpreted as a phase transition.This book informs the reader about what has been (mathematically) done in the theory of Gibbs measures of the Potts model and the numerous applications of the Potts model. The main aim is to facilitate the readers (in mathematical biology, statistical physics, applied mathematics, probability and measure theory) to progress into an in-depth understanding by giving a systematic review of the theory of Gibbs measures of the Potts model and its applications.
Author: R. Daniel Mauldin
Publisher: Cambridge University Press
ISBN: 9780521825382
Category : Mathematics
Languages : en
Pages : 302
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Book Description
The main focus of this book is the exploration of the geometric and dynamic properties of a far reaching generalization of a conformal iterated function system - a Graph Directed Markov System. These systems are very robust in that they apply to many settings that do not fit into the scheme of conformal iterated systems. The basic theory is laid out here and the authors have touched on many natural questions arising in its context. However, they also emphasise the many issues and current research topics which can be found in original papers. For example the detailed analysis of the structure of harmonic measures of limit sets, the examination of the doubling property of conformal measures, the extensive study of generalized polynomial like mapping or multifractal analysis of geometrically finite Kleinian groups. This book leads readers onto frontier research in the field, making it ideal for both established researchers and graduate students.
Author: Mark Pollicott
Publisher: Springer
ISBN: 9783030748623
Category : Mathematics
Languages : en
Pages : 536
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Book Description
This volume arose from a semester at CIRM-Luminy on “Thermodynamic Formalism: Applications to Probability, Geometry and Fractals” which brought together leading experts in the area to discuss topical problems and recent progress. It includes a number of surveys intended to make the field more accessible to younger mathematicians and scientists wishing to learn more about the area. Thermodynamic formalism has been a powerful tool in ergodic theory and dynamical system and its applications to other topics, particularly Riemannian geometry (especially in negative curvature), statistical properties of dynamical systems and fractal geometry. This work will be of value both to graduate students and more senior researchers interested in either learning about the main ideas and themes in thermodynamic formalism, and research themes which are at forefront of research in this area.
Author: Robert A. Meyers
Publisher: Springer Science & Business Media
ISBN: 1461418054
Category : Mathematics
Languages : en
Pages : 1885
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Book Description
Mathematics of Complexity and Dynamical Systems is an authoritative reference to the basic tools and concepts of complexity, systems theory, and dynamical systems from the perspective of pure and applied mathematics. Complex systems are systems that comprise many interacting parts with the ability to generate a new quality of collective behavior through self-organization, e.g. the spontaneous formation of temporal, spatial or functional structures. These systems are often characterized by extreme sensitivity to initial conditions as well as emergent behavior that are not readily predictable or even completely deterministic. The more than 100 entries in this wide-ranging, single source work provide a comprehensive explication of the theory and applications of mathematical complexity, covering ergodic theory, fractals and multifractals, dynamical systems, perturbation theory, solitons, systems and control theory, and related topics. Mathematics of Complexity and Dynamical Systems is an essential reference for all those interested in mathematical complexity, from undergraduate and graduate students up through professional researchers.
Author: Alain Connes
Publisher: Springer Science & Business Media
ISBN: 9783540203575
Category : Mathematics
Languages : en
Pages : 372
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Book Description
Noncommutative Geometry is one of the most deep and vital research subjects of present-day Mathematics. Its development, mainly due to Alain Connes, is providing an increasing number of applications and deeper insights for instance in Foliations, K-Theory, Index Theory, Number Theory but also in Quantum Physics of elementary particles. The purpose of the Summer School in Martina Franca was to offer a fresh invitation to the subject and closely related topics; the contributions in this volume include the four main lectures, cover advanced developments and are delivered by prominent specialists.
