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Finer Thermodynamic Formalism - Distance Expanding Maps and Countable State Subshifts of Finite Type, Conformal GDMSs,... Finer Thermodynamic Formalism - Distance Expanding Maps and Countable State Subshifts of Finite Type, Conformal GDMSs, Lasota-Yorke Maps and Fractal Geometry (Hardcover)
Mariusz Urbanski, Mario Roy, Sara Munday
R4,412 Discovery Miles 44 120 Ships in 12 - 19 working days

This book consists of three volumes. The first volume contains introductory accounts of topological dynamical systems, fi nite-state symbolic dynamics, distance expanding maps, and ergodic theory of metric dynamical systems acting on probability measure spaces, including metric entropy theory of Kolmogorov and Sinai. More advanced topics comprise infi nite ergodic theory, general thermodynamic formalism, topological entropy and pressure. Thermodynamic formalism of distance expanding maps and countable-alphabet subshifts of fi nite type, graph directed Markov systems, conformal expanding repellers, and Lasota-Yorke maps are treated in the second volume, which also contains a chapter on fractal geometry and its applications to conformal systems. Multifractal analysis and real analyticity of pressure are also covered. The third volume is devoted to the study of dynamics, ergodic theory, thermodynamic formalism and fractal geometry of rational functions of the Riemann sphere.

Ergodic Theory - Finite and Infinite, Thermodynamic Formalism, Symbolic Dynamics and Distance Expanding Maps (Hardcover):... Ergodic Theory - Finite and Infinite, Thermodynamic Formalism, Symbolic Dynamics and Distance Expanding Maps (Hardcover)
Mariusz Urbanski, Mario Roy, Sara Munday
R4,765 Discovery Miles 47 650 Ships in 12 - 19 working days

The book contains a detailed treatment of thermodynamic formalism on general compact metrizable spaces. Topological pressure, topological entropy, variational principle, and equilibrium states are presented in detail. Abstract ergodic theory is also given a significant attention. Ergodic theorems, ergodicity, and Kolmogorov-Sinai metric entropy are fully explored. Furthermore, the book gives the reader an opportunity to find rigorous presentation of thermodynamic formalism for distance expanding maps and, in particular, subshifts of finite type over a finite alphabet. It also provides a fairly complete treatment of subshifts of finite type over a countable alphabet. Transfer operators, Gibbs states and equilibrium states are, in this context, introduced and dealt with. Their relations are explored. All of this is applied to fractal geometry centered around various versions of Bowen's formula in the context of expanding conformal repellors, limit sets of conformal iterated function systems and conformal graph directed Markov systems. A unique introduction to iteration of rational functions is given with emphasize on various phenomena caused by rationally indifferent periodic points. Also, a fairly full account of the classicaltheory of Shub's expanding endomorphisms is given; it does not have a book presentation in English language mathematical literature.

Meromorphic Dynamics: Volume 1 - Abstract Ergodic Theory, Geometry, Graph Directed Markov Systems, and Conformal Measures... Meromorphic Dynamics: Volume 1 - Abstract Ergodic Theory, Geometry, Graph Directed Markov Systems, and Conformal Measures (Hardcover)
Janina Kotus, Mariusz Urbanski
R3,575 Discovery Miles 35 750 Ships in 12 - 19 working days

This text, the first of two volumes, provides a comprehensive and self-contained introduction to a wide range of fundamental results from ergodic theory and geometric measure theory. Topics covered include: finite and infinite abstract ergodic theory, Young's towers, measure-theoretic Kolmogorov-Sinai entropy, thermodynamics formalism, geometric function theory, various kinds of conformal measures, conformal graph directed Markov systems and iterated functions systems, semi-local dynamics of analytic functions, and nice sets. Many examples are included, along with detailed explanations of essential concepts and full proofs, in what is sure to be an indispensable reference for both researchers and graduate students.

Meromorphic Dynamics: Volume 2 - Elliptic Functions with an Introduction to the Dynamics of Meromorphic Functions (Hardcover):... Meromorphic Dynamics: Volume 2 - Elliptic Functions with an Introduction to the Dynamics of Meromorphic Functions (Hardcover)
Janina Kotus, Mariusz Urbanski
R3,867 Discovery Miles 38 670 Ships in 12 - 19 working days

This text, the second of two volumes, builds on the foundational material on ergodic theory and geometric measure theory provided in Volume I, and applies all the techniques discussed to describe the beautiful and rich dynamics of elliptic functions. The text begins with an introduction to topological dynamics of transcendental meromorphic functions, before progressing to elliptic functions, discussing at length their classical properties, measurable dynamics and fractal geometry. The authors then look in depth at compactly non-recurrent elliptic functions. Much of this material is appearing for the first time in book or paper form. Both senior and junior researchers working in ergodic theory and dynamical systems will appreciate what is sure to be an indispensable reference.

Conformal Graph Directed Markov Systems on Carnot Groups (Paperback): Vasileios Chousionis, Jeremy T. Tyson, Mariusz Urbanski Conformal Graph Directed Markov Systems on Carnot Groups (Paperback)
Vasileios Chousionis, Jeremy T. Tyson, Mariusz Urbanski
R2,307 Discovery Miles 23 070 Ships in 12 - 19 working days

The authors 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, they 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. They illustrate their 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.

Distance Expanding Random Mappings, Thermodynamical Formalism, Gibbs Measures and Fractal Geometry (Paperback, 2011 ed.):... Distance Expanding Random Mappings, Thermodynamical Formalism, Gibbs Measures and Fractal Geometry (Paperback, 2011 ed.)
Volker Mayer, Bartlomiej Skorulski, Mariusz Urbanski
R1,348 Discovery Miles 13 480 Ships in 10 - 15 working days

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."

Open Conformal Systems and Perturbations of Transfer Operators (Paperback, 1st ed. 2017): Mark Pollicott, Mariusz Urbanski Open Conformal Systems and Perturbations of Transfer Operators (Paperback, 1st ed. 2017)
Mark Pollicott, Mariusz Urbanski
R1,462 Discovery Miles 14 620 Ships in 10 - 15 working days

The focus of this book is on open conformal dynamical systems corresponding to the escape of a point through an open Euclidean ball. The ultimate goal is to understand the asymptotic behavior of the escape rate as the radius of the ball tends to zero. In the case of hyperbolic conformal systems this has been addressed by various authors. The conformal maps considered in this book are far more general, and the analysis correspondingly more involved. The asymptotic existence of escape rates is proved and they are calculated in the context of (finite or infinite) countable alphabets, uniformly contracting conformal graph-directed Markov systems, and in particular, conformal countable alphabet iterated function systems. These results have direct applications to interval maps, rational functions and meromorphic maps. Towards this goal the authors develop, on a purely symbolic level, a theory of singular perturbations of Perron--Frobenius (transfer) operators associated with countable alphabet subshifts of finite type and Hoelder continuous summable potentials. This leads to a fairly full account of the structure of the corresponding open dynamical systems and their associated surviving sets.

Asymptotic Counting in Conformal Dynamical Systems (Paperback): Mark Pollicott, Mariusz Urbanski Asymptotic Counting in Conformal Dynamical Systems (Paperback)
Mark Pollicott, Mariusz Urbanski
R2,221 Discovery Miles 22 210 Ships in 12 - 19 working days
Conformal Fractals - Ergodic Theory Methods (Paperback): Feliks Przytycki, Mariusz Urbanski Conformal Fractals - Ergodic Theory Methods (Paperback)
Feliks Przytycki, Mariusz Urbanski
R2,276 Discovery Miles 22 760 Ships in 12 - 19 working days

This is a one-stop introduction to the methods of ergodic theory applied to holomorphic iteration. The authors begin with introductory chapters presenting the necessary tools from ergodic theory thermodynamical formalism, and then focus on recent developments in the field of 1-dimensional holomorphic iterations and underlying fractal sets, from the point of view of geometric measure theory and rigidity. Detailed proofs are included. Developed from university courses taught by the authors, this book is ideal for graduate students. Researchers will also find it a valuable source of reference to a large and rapidly expanding field. It eases the reader into the subject and provides a vital springboard for those beginning their own research. Many helpful exercises are also included to aid understanding of the material presented and the authors provide links to further reading and related areas of research.

Graph Directed Markov Systems - Geometry and Dynamics of Limit Sets (Hardcover, New): R. Daniel Mauldin, Mariusz Urbanski Graph Directed Markov Systems - Geometry and Dynamics of Limit Sets (Hardcover, New)
R. Daniel Mauldin, Mariusz Urbanski
R3,427 Discovery Miles 34 270 Ships in 12 - 19 working days

The main focus of this book is on the development of the theory of Graph Directed Markov Systems. This far-reaching generalization of the theory of conformal iterated systems can be applied in many situations, including the theory of dynamical systems. Dan Mauldin and Mariusz Urbanski include much of the necessary background material to increase the appeal of this book to graduate students as well as researchers. They also include an extensive list of references for further reading.

Meromorphic Dynamics 2 Volume Hardback Set (Hardcover): Janina Kotus, Mariusz Urbanski Meromorphic Dynamics 2 Volume Hardback Set (Hardcover)
Janina Kotus, Mariusz Urbanski
R5,872 Discovery Miles 58 720 Ships in 12 - 19 working days

This two-volume set provides a comprehensive and self-contained approach to the dynamics, ergodic theory, and geometry of elliptic functions mapping the complex plane onto the Riemann sphere. Volume I discusses many fundamental results from ergodic theory and geometric measure theory in detail, including finite and infinite abstract ergodic theory, Young's towers, measure-theoretic Kolmogorov-Sinai entropy, thermodynamics formalism, geometric function theory, various conformal measures, conformal graph directed Markov systems and iterated functions systems, classical theory of elliptic functions. In Volume II, all these techniques, along with an introduction to topological dynamics of transcendental meromorphic functions, are applied to describe the beautiful and rich dynamics and fractal geometry of elliptic functions. Much of this material is appearing for the first time in book or even paper form. Both researchers and graduate students will appreciate the detailed explanations of essential concepts and full proofs provided in what is sure to be an indispensable reference.

Reasoning - Games, Cognition, Logic (Paperback): Mariusz Urbanski, Tomasz Skura, Pawel Lupowski Reasoning - Games, Cognition, Logic (Paperback)
Mariusz Urbanski, Tomasz Skura, Pawel Lupowski
R569 Discovery Miles 5 690 Ships in 10 - 15 working days
Geometry and Dynamics in Gromov Hyperbolic Metric Spaces - With an Emphasis on Non-Proper Settings (Hardcover): Tushar Das,... Geometry and Dynamics in Gromov Hyperbolic Metric Spaces - With an Emphasis on Non-Proper Settings (Hardcover)
Tushar Das, David Simmons, Mariusz Urbanski
R3,442 Discovery Miles 34 420 Ships in 12 - 19 working days

This book presents the foundations of the theory of groups and semigroups acting isometrically on Gromov hyperbolic metric spaces. Particular emphasis is paid to the geometry of their limit sets and on behavior not found in the proper setting. The authors provide a number of examples of groups which exhibit a wide range of phenomena not to be found in the finite-dimensional theory. The book contains both introductory material to help beginners as well as new research results, and closes with a list of attractive unsolved problems.

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