Thermodynamic Formalism Beyond Uniformly Hyperbolic Systems

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THERMODYNAMIC FORMALISM BEYOND UNIFORMLY HYPERBOLIC SYSTEMS.

Author : Agnieszka Zelerowicz
Publisher :
ISBN 13 :
Total Pages : pages
Book Rating : 4.:/5 (111 download)

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Book Synopsis THERMODYNAMIC FORMALISM BEYOND UNIFORMLY HYPERBOLIC SYSTEMS. by : Agnieszka Zelerowicz

Download or read book THERMODYNAMIC FORMALISM BEYOND UNIFORMLY HYPERBOLIC SYSTEMS. written by Agnieszka Zelerowicz and published by . This book was released on 2019 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: The aim of this thesis is to study thermodynamics of dierentiable dynamical systems which display various forms of hyperbolicity. In the classical, uniformly hyperbolic setting we propose a new way of constructing equilibrium measures which uses some tools from geometric measure theory. We then use it to construct equilibrium measures for a certain class of partially hyperbolic systems. Almost immediately from the construction one obtains that those measures are unique equilibrium measures with full support which have local product structure. We also consider systems which are nonuniformly hyperbolic We present a systematic study of thermodynamics formalism by methods of inducing. We give a complete overview of known results and prove additional ones. One of the main contributions is the proof of Bernoulli property for equilibrium measures corresponding to a certain class of potential functions. We also prove analyticity of the pressure function for a certain class of families of potential functions. Finally,we study a certain class of nonuniformly hyperbolic attractors. Those are attractors obtained from uniformly hyperbolic systems by the slow down procedure. We study the family of geometric t-potentials on some interval in t. We show existence and uniqueness of equilibrium measures for potentials in this family. We prove that those equilibrium measures satisfy the Central Limit Theorem, have exponential decay of correlations, and the Bernoulli property. Finally, we prove that the pressure function is real analytic.

Thermodynamic Formalism

Author : Mark Pollicott
Publisher : Springer Nature
ISBN 13 : 3030748634
Total Pages : 536 pages
Book Rating : 4.0/5 (37 download)

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Book Synopsis Thermodynamic Formalism by : Mark Pollicott

Download or read book Thermodynamic Formalism written by Mark Pollicott and published by Springer Nature. This book was released on 2021-10-01 with total page 536 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Thermodynamic Formalism and Holomorphic Dynamical Systems

Author : Michel Zinsmeister
Publisher : American Mathematical Soc.
ISBN 13 : 9780821819487
Total Pages : 100 pages
Book Rating : 4.8/5 (194 download)

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Book Synopsis Thermodynamic Formalism and Holomorphic Dynamical Systems by : Michel Zinsmeister

Download or read book Thermodynamic Formalism and Holomorphic Dynamical Systems written by Michel Zinsmeister and published by American Mathematical Soc.. This book was released on 2000 with total page 100 pages. Available in PDF, EPUB and Kindle. Book excerpt: The purpose of thermodynamics and statistical physics is to understand the equilibrium of a gas or the different states of matter. To understand the strange fractal sets appearing when one iterates a quadratic polynomial is one of the goals of the theory of holomorphic dynamical systems. These two theories are strongly linked: The laws of thermodynamics happen to be an extremely powerful tool for understanding the objects of holomorphic dynamical systems. A "thermodynamic formalism" has been developed, bringing together notions that are a priori unrelated. While the deep reasons of this parallelism remain unknown, the goal of this book is to describe this formalism both from the physical and mathematical point of view in order to understand how it works and how useful it can be. This translation is a slightly revised version of the original French edition. The main changes are in Chapters 5 and 6 and consist of clarification of some proofs and a new presentation of the basics in iteration of polynomials.

Thermodynamic Formalism and Applications to Dimension Theory

Author : Luis Barreira
Publisher : Springer Science & Business Media
ISBN 13 : 3034802064
Total Pages : 300 pages
Book Rating : 4.0/5 (348 download)

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Book Synopsis Thermodynamic Formalism and Applications to Dimension Theory by : Luis Barreira

Download or read book Thermodynamic Formalism and Applications to Dimension Theory written by Luis Barreira and published by Springer Science & Business Media. This book was released on 2011-08-24 with total page 300 pages. Available in PDF, EPUB and Kindle. Book excerpt: This self-contained monograph presents a unified exposition of the thermodynamic formalism and some of its main extensions, with emphasis on the relation to dimension theory and multifractal analysis of dynamical systems. In particular, the book considers three different flavors of the thermodynamic formalism, namely nonadditive, subadditive, and almost additive, and provides a detailed discussion of some of the most significant results in the area, some of them quite recent. It also includes a discussion of the most substantial applications of these flavors of the thermodynamic formalism to dimension theory and multifractal analysis of dynamical systems.

Thermodynamics of Some Nonuniformly Hyperbolic Dynamical Systems

Author : Dominic Veconi
Publisher :
ISBN 13 :
Total Pages : pages
Book Rating : 4.:/5 (125 download)

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Book Synopsis Thermodynamics of Some Nonuniformly Hyperbolic Dynamical Systems by : Dominic Veconi

Download or read book Thermodynamics of Some Nonuniformly Hyperbolic Dynamical Systems written by Dominic Veconi and published by . This book was released on 2021 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: One of the primary tools used to study hyperbolic and expanding dynamical systems are equilibrium measures. These are probability measures that are invariant under the dynamical system, and which maximize the sum of the metric entropy of the dynamics and the integral of a particular function (this maximum is taken over all invariant probability measures). Within the equilibrium measures are the Sinai-Ruelle-Bowen (SRB) measures. These are equilibrium measures [mu] that are "physical" in the sense that for Lebesgue-almost every point x in the phase space X and every continuous real-valued function [phi] on X , the Birkhoff time average is equal to the space of the function [phi]. It is known that if a surface diffeomorphism admits an SRB measure, then this measure is unique. Furthermore, hyperbolic maps admitting singularities have at most countably many ergodic SRB measures. In this dissertation, we consider the SRB measures and thermodynamic properties of three classes of dynamical systems: almost Anosov toral diffeomorphisms; smooth realizations of pseudo-Anosov maps; and singular hyperbolic attractors. Almost Anosov and pseudo-Anosov maps are surface diffeomorphisms admitting a finite number of singular fixed points, where the differential of the map slows to the identity. We develop a thermodynamic formalism for both classes of diffeomorphisms. In particular, we use a Young tower construction to prove the existence and uniqueness of equilibrium measures for a collection of non-Hölder continuous geometric potentials. For both almost and pseudo-Anosov maps, this family of equilibrium measures includes a unique SRB measure and a measure of maximal entropy. This measure of maximal entropy also has exponential decay of correlations and satisfies the Central Limit Theorem. Singular hyperbolic attractors are the invariant supports of physical measures for dissipative hyperbolic systems. Examples of singular hyperbolic attractors include the geometric Lorenz attractor (as well as related Lorenz-type systems), and the Belykh attractor. We establish easily verifiable sufficient conditions guaranteeing that the number of ergodic SRB measures is at most finite, and provide examples and nonexamples showing that the conditions are necessary.

Thermodynamic Formalism

Author : David Ruelle
Publisher : Cambridge University Press
ISBN 13 : 9781139455282
Total Pages : 198 pages
Book Rating : 4.4/5 (552 download)

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Book Synopsis Thermodynamic Formalism by : David Ruelle

Download or read book Thermodynamic Formalism written by David Ruelle and published by Cambridge University Press. This book was released on 2004-11-25 with total page 198 pages. Available in PDF, EPUB and Kindle. Book excerpt: Reissued in the Cambridge Mathematical Library this classic book outlines the theory of thermodynamic formalism which was developed to describe the properties of certain physical systems consisting of a large number of subunits. It is aimed at mathematicians interested in ergodic theory, topological dynamics, constructive quantum field theory, the study of certain differentiable dynamical systems, notably Anosov diffeomorphisms and flows. It is also of interest to theoretical physicists concerned with the conceptual basis of equilibrium statistical mechanics. The level of the presentation is generally advanced, the objective being to provide an efficient research tool and a text for use in graduate teaching. Background material on mathematics has been collected in appendices to help the reader. Extra material is given in the form of updates of problems that were open at the original time of writing and as a new preface specially written for this new edition by the author.

Dynamics Beyond Uniform Hyperbolicity

Author : Christian Bonatti
Publisher : Springer Science & Business Media
ISBN 13 : 3540268448
Total Pages : 390 pages
Book Rating : 4.5/5 (42 download)

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Book Synopsis Dynamics Beyond Uniform Hyperbolicity by : Christian Bonatti

Download or read book Dynamics Beyond Uniform Hyperbolicity written by Christian Bonatti and published by Springer Science & Business Media. This book was released on 2006-03-30 with total page 390 pages. Available in PDF, EPUB and Kindle. Book excerpt: What is Dynamics about? In broad terms, the goal of Dynamics is to describe the long term evolution of systems for which an "infinitesimal" evolution rule is known. Examples and applications arise from all branches of science and technology, like physics, chemistry, economics, ecology, communications, biology, computer science, or meteorology, to mention just a few. These systems have in common the fact that each possible state may be described by a finite (or infinite) number of observable quantities, like position, velocity, temperature, concentration, population density, and the like. Thus, m the space of states (phase space) is a subset M of an Euclidean space M . Usually, there are some constraints between these quantities: for instance, for ideal gases pressure times volume must be proportional to temperature. Then the space M is often a manifold, an n-dimensional surface for some n

Thermodynamic Formalism

Author : David Ruelle
Publisher :
ISBN 13 : 9781107157231
Total Pages : 174 pages
Book Rating : 4.1/5 (572 download)

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Book Synopsis Thermodynamic Formalism by : David Ruelle

Download or read book Thermodynamic Formalism written by David Ruelle and published by . This book was released on 2004 with total page 174 pages. Available in PDF, EPUB and Kindle. Book excerpt: Reissued in the Cambridge Mathematical Library this classic book outlines the theory of thermodynamic formalism which was developed to describe the properties of certain physical systems consisting of a large number of subunits. It is aimed at mathematicians interested in ergodic theory, topological dynamics, constructive quantum field theory, the study of certain differentiable dynamical systems, notably Anosov diffeomorphisms and flows. It is also of interest to theoretical physicists concerned with the conceptual basis of equilibrium statistical mechanics. The level of the presentation is generally advanced, the objective being to provide an efficient research tool and a text for use in graduate teaching. Background material on mathematics has been collected in appendices to help the reader. Extra material is given in the form of updates of problems that were open at the original time of writing and as a new preface specially written for this new edition by the author.

Thermodynamic Formalism for Symbolic Dynamical Systems

Author : Thomas Kempton
Publisher :
ISBN 13 :
Total Pages : 0 pages
Book Rating : 4.:/5 (921 download)

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Book Synopsis Thermodynamic Formalism for Symbolic Dynamical Systems by : Thomas Kempton

Download or read book Thermodynamic Formalism for Symbolic Dynamical Systems written by Thomas Kempton and published by . This book was released on 2011 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Thermodynamic Formalism

Author : David Ruelle
Publisher :
ISBN 13 :
Total Pages : 183 pages
Book Rating : 4.:/5 (11 download)

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Book Synopsis Thermodynamic Formalism by : David Ruelle

Download or read book Thermodynamic Formalism written by David Ruelle and published by . This book was released on 1978 with total page 183 pages. Available in PDF, EPUB and Kindle. Book excerpt:

A Vision for Dynamics in the 21st Century

Author : Danijela Damjanovic
Publisher : Cambridge University Press
ISBN 13 : 1009278908
Total Pages : 445 pages
Book Rating : 4.0/5 (92 download)

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Book Synopsis A Vision for Dynamics in the 21st Century by : Danijela Damjanovic

Download or read book A Vision for Dynamics in the 21st Century written by Danijela Damjanovic and published by Cambridge University Press. This book was released on 2024-01-31 with total page 445 pages. Available in PDF, EPUB and Kindle. Book excerpt: Leading experts across smooth dynamics and ergodic theory present a broad research perspective and set an agenda for future work.

Entropy in Dynamical Systems

Author : Tomasz Downarowicz
Publisher : Cambridge University Press
ISBN 13 : 1139500872
Total Pages : 405 pages
Book Rating : 4.1/5 (395 download)

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Book Synopsis Entropy in Dynamical Systems by : Tomasz Downarowicz

Download or read book Entropy in Dynamical Systems written by Tomasz Downarowicz and published by Cambridge University Press. This book was released on 2011-05-12 with total page 405 pages. Available in PDF, EPUB and Kindle. Book excerpt: This comprehensive text on entropy covers three major types of dynamics: measure preserving transformations; continuous maps on compact spaces; and operators on function spaces. Part I contains proofs of the Shannon–McMillan–Breiman Theorem, the Ornstein–Weiss Return Time Theorem, the Krieger Generator Theorem and, among the newest developments, the ergodic law of series. In Part II, after an expanded exposition of classical topological entropy, the book addresses symbolic extension entropy. It offers deep insight into the theory of entropy structure and explains the role of zero-dimensional dynamics as a bridge between measurable and topological dynamics. Part III explains how both measure-theoretic and topological entropy can be extended to operators on relevant function spaces. Intuitive explanations, examples, exercises and open problems make this an ideal text for a graduate course on entropy theory. More experienced researchers can also find inspiration for further research.

Mathematics of Complexity and Dynamical Systems

Author : Robert A. Meyers
Publisher : Springer Science & Business Media
ISBN 13 : 1461418054
Total Pages : 1885 pages
Book Rating : 4.4/5 (614 download)

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Book Synopsis Mathematics of Complexity and Dynamical Systems by : Robert A. Meyers

Download or read book Mathematics of Complexity and Dynamical Systems written by Robert A. Meyers and published by Springer Science & Business Media. This book was released on 2011-10-05 with total page 1885 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Multiscale Thermo-Dynamics

Author : Michal Pavelka
Publisher : Walter de Gruyter GmbH & Co KG
ISBN 13 : 3110350955
Total Pages : 292 pages
Book Rating : 4.1/5 (13 download)

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Book Synopsis Multiscale Thermo-Dynamics by : Michal Pavelka

Download or read book Multiscale Thermo-Dynamics written by Michal Pavelka and published by Walter de Gruyter GmbH & Co KG. This book was released on 2018-08-06 with total page 292 pages. Available in PDF, EPUB and Kindle. Book excerpt: One common feature of new emerging technologies is the fusion of the very small (nano) scale and the large scale engineering. The classical environment provided by single scale theories, as for instance by the classical hydrodynamics, is not anymore satisfactory. The main challenge is to keep the important details while still be able to keep the overall picture and simplicity. It is the thermodynamics that addresses this challenge. Our main reason for writing this book is to explain such general viewpoint of thermodynamics and to illustrate it on a very wide range of examples. Contents Levels of description Hamiltonian mechanics Irreversible evolution Reversible and irreversible evolution Multicomponent systems Contact geometry Appendix: Mathematical aspects

Ergodic Theory

Author : Cesar E. Silva
Publisher : Springer Nature
ISBN 13 : 1071623885
Total Pages : 707 pages
Book Rating : 4.0/5 (716 download)

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Book Synopsis Ergodic Theory by : Cesar E. Silva

Download or read book Ergodic Theory written by Cesar E. Silva and published by Springer Nature. This book was released on 2023-07-31 with total page 707 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume in the Encyclopedia of Complexity and Systems Science, Second Edition, covers recent developments in classical areas of ergodic theory, including the asymptotic properties of measurable dynamical systems, spectral theory, entropy, ergodic theorems, joinings, isomorphism theory, recurrence, nonsingular systems. It enlightens connections of ergodic theory with symbolic dynamics, topological dynamics, smooth dynamics, combinatorics, number theory, pressure and equilibrium states, fractal geometry, chaos. In addition, the new edition includes dynamical systems of probabilistic origin, ergodic aspects of Sarnak's conjecture, translation flows on translation surfaces, complexity and classification of measurable systems, operator approach to asymptotic properties, interplay with operator algebras

A Dynamical Systems Theory of Thermodynamics

Author : Wassim M. Haddad
Publisher : Princeton University Press
ISBN 13 : 0691190143
Total Pages : 744 pages
Book Rating : 4.6/5 (911 download)

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Book Synopsis A Dynamical Systems Theory of Thermodynamics by : Wassim M. Haddad

Download or read book A Dynamical Systems Theory of Thermodynamics written by Wassim M. Haddad and published by Princeton University Press. This book was released on 2019-06-04 with total page 744 pages. Available in PDF, EPUB and Kindle. Book excerpt: A brand-new conceptual look at dynamical thermodynamics This book merges the two universalisms of thermodynamics and dynamical systems theory in a single compendium, with the latter providing an ideal language for the former, to develop a new and unique framework for dynamical thermodynamics. In particular, the book uses system-theoretic ideas to bring coherence, clarity, and precision to an important and poorly understood classical area of science. The dynamical systems formalism captures all of the key aspects of thermodynamics, including its fundamental laws, while providing a mathematically rigorous formulation for thermodynamical systems out of equilibrium by unifying the theory of mechanics with that of classical thermodynamics. This book includes topics on nonequilibrium irreversible thermodynamics, Boltzmann thermodynamics, mass-action kinetics and chemical reactions, finite-time thermodynamics, thermodynamic critical phenomena with continuous and discontinuous phase transitions, information theory, continuum and stochastic thermodynamics, and relativistic thermodynamics. A Dynamical Systems Theory of Thermodynamics develops a postmodern theory of thermodynamics as part of mathematical dynamical systems theory. The book establishes a clear nexus between thermodynamic irreversibility, the second law of thermodynamics, and the arrow of time to further unify discreteness and continuity, indeterminism and determinism, and quantum mechanics and general relativity in the pursuit of understanding the most fundamental property of the universe—the entropic arrow of time.

Lyapunov Exponents and Smooth Ergodic Theory

Author : Luis Barreira
Publisher : American Mathematical Soc.
ISBN 13 : 0821829211
Total Pages : 166 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Lyapunov Exponents and Smooth Ergodic Theory by : Luis Barreira

Download or read book Lyapunov Exponents and Smooth Ergodic Theory written by Luis Barreira and published by American Mathematical Soc.. This book was released on 2002 with total page 166 pages. Available in PDF, EPUB and Kindle. Book excerpt: A systematic introduction to the core of smooth ergodic theory. An expanded version of an earlier work by the same authors, it describes the general (abstract) theory of Lyapunov exponents and the theory's applications to the stability theory of differential equations, the stable manifold theory, absolute continuity of stable manifolds, and the ergodic theory of dynamical systems with nonzero Lyapunov exponents (including geodesic flows). It could be used as a primary text for a course on nonuniform hyperbolic theory or as supplemental reading for a course on dynamical systems. Assumes a basic knowledge of real analysis, measure theory, differential equations, and topology. c. Book News Inc.