Ergodic Theory Of Numbers

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Ergodic Theory of Numbers

Karma Dajani,Cor Kraaikamp
Ergodic Theory of Numbers Book PDF

Author : Karma Dajani,Cor Kraaikamp
Publisher : American Mathematical Soc.
Page : 190 pages
File Size : 41,7 Mb
Release : 2002-12-31
Category : Mathematics
ISBN : 9780883850343

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Ergodic Theory of Numbers by Karma Dajani,Cor Kraaikamp Pdf

Ergodic Theory of Numbers looks at the interaction between two fields of mathematics: number theory and ergodic theory (as part of dynamical systems). It is an introduction to the ergodic theory behind common number expansions, like decimal expansions, continued fractions, and many others. However, its aim does not stop there. For undergraduate students with sufficient background knowledge in real analysis and graduate students interested in the area, it is also an introduction to a "dynamical way of thinking". The questions studied here are dynamical as well as number theoretical in nature, and the answers are obtained with the help of ergodic theory. Attention is focused on concepts like measure-preserving, ergodicity, natural extension, induced transformations, and entropy. These concepts are then applied to familiar expansions to obtain old and new results in an elegant and straightforward manner. What it means to be ergodic and the basic ideas behind ergodic theory will be explained along the way. The subjects covered vary from classical to recent, which makes this book appealing to researchers as well as students.

Ergodic Theory

Manfred Einsiedler,Thomas Ward
Ergodic Theory Book PDF

Author : Manfred Einsiedler,Thomas Ward
Publisher : Springer Science & Business Media
Page : 481 pages
File Size : 46,6 Mb
Release : 2010-09-11
Category : Mathematics
ISBN : 9780857290212

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Ergodic Theory by Manfred Einsiedler,Thomas Ward Pdf

This text is a rigorous introduction to ergodic theory, developing the machinery of conditional measures and expectations, mixing, and recurrence. Beginning by developing the basics of ergodic theory and progressing to describe some recent applications to number theory, this book goes beyond the standard texts in this topic. Applications include Weyl's polynomial equidistribution theorem, the ergodic proof of Szemeredi's theorem, the connection between the continued fraction map and the modular surface, and a proof of the equidistribution of horocycle orbits. Ergodic Theory with a view towards Number Theory will appeal to mathematicians with some standard background in measure theory and functional analysis. No background in ergodic theory or Lie theory is assumed, and a number of exercises and hints to problems are included, making this the perfect companion for graduate students and researchers in ergodic theory, homogenous dynamics or number theory.

Infinite Ergodic Theory of Numbers

Marc Kesseböhmer,Sara Munday,Bernd Otto Stratmann
Infinite Ergodic Theory of Numbers Book PDF

Author : Marc Kesseböhmer,Sara Munday,Bernd Otto Stratmann
Publisher : Walter de Gruyter GmbH & Co KG
Page : 204 pages
File Size : 55,5 Mb
Release : 2016-10-10
Category : Mathematics
ISBN : 9783110430851

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Infinite Ergodic Theory of Numbers by Marc Kesseböhmer,Sara Munday,Bernd Otto Stratmann Pdf

By connecting dynamical systems and number theory, this graduate textbook on ergodic theory acts as an introduction to a highly active area of mathematics, where a variety of strands of research open up. The text explores various concepts in infinite ergodic theory, always using continued fractions and other number-theoretic dynamical systems as illustrative examples. Contents: Preface Mathematical symbols Number-theoretical dynamical systems Basic ergodic theory Renewal theory and α-sum-level sets Infinite ergodic theory Applications of infinite ergodic theory Bibliography Index

Ergodic Theory and Dynamical Systems in their Interactions with Arithmetics and Combinatorics

Sébastien Ferenczi,Joanna Kułaga-Przymus,Mariusz Lemańczyk
Ergodic Theory and Dynamical Systems in their Interactions with Arithmetics and Combinatorics Book PDF

Author : Sébastien Ferenczi,Joanna Kułaga-Przymus,Mariusz Lemańczyk
Publisher : Springer
Page : 434 pages
File Size : 52,9 Mb
Release : 2018-06-15
Category : Mathematics
ISBN : 9783319749082

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Ergodic Theory and Dynamical Systems in their Interactions with Arithmetics and Combinatorics by Sébastien Ferenczi,Joanna Kułaga-Przymus,Mariusz Lemańczyk Pdf

This book concentrates on the modern theory of dynamical systems and its interactions with number theory and combinatorics. The greater part begins with a course in analytic number theory and focuses on its links with ergodic theory, presenting an exhaustive account of recent research on Sarnak's conjecture on Möbius disjointness. Selected topics involving more traditional connections between number theory and dynamics are also presented, including equidistribution, homogenous dynamics, and Lagrange and Markov spectra. In addition, some dynamical and number theoretical aspects of aperiodic order, some algebraic systems, and a recent development concerning tame systems are described.

Ergodic Theory

Karl E. Petersen,Karl Petersen
Ergodic Theory Book PDF

Author : Karl E. Petersen,Karl Petersen
Publisher : Cambridge University Press
Page : 348 pages
File Size : 50,5 Mb
Release : 1989-11-23
Category : Mathematics
ISBN : 0521389976

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Ergodic Theory by Karl E. Petersen,Karl Petersen Pdf

The study of dynamical systems forms a vast and rapidly developing field even when one considers only activity whose methods derive mainly from measure theory and functional analysis. Karl Petersen has written a book which presents the fundamentals of the ergodic theory of point transformations and then several advanced topics which are currently undergoing intense research. By selecting one or more of these topics to focus on, the reader can quickly approach the specialized literature and indeed the frontier of the area of interest. Each of the four basic aspects of ergodic theory - examples, convergence theorems, recurrence properties, and entropy - receives first a basic and then a more advanced, particularized treatment. At the introductory level, the book provides clear and complete discussions of the standard examples, the mean and pointwise ergodic theorems, recurrence, ergodicity, weak mixing, strong mixing, and the fundamentals of entropy. Among the advanced topics are a thorough treatment of maximal functions and their usefulness in ergodic theory, analysis, and probability, an introduction to almost-periodic functions and topological dynamics, a proof of the Jewett-Krieger Theorem, an introduction to multiple recurrence and the Szemeredi-Furstenberg Theorem, and the Keane-Smorodinsky proof of Ornstein's Isomorphism Theorem for Bernoulli shifts. The author's easily-readable style combined with the profusion of exercises and references, summaries, historical remarks, and heuristic discussions make this book useful either as a text for graduate students or self-study, or as a reference work for the initiated.

Recurrence in Ergodic Theory and Combinatorial Number Theory

Harry Furstenberg
Recurrence in Ergodic Theory and Combinatorial Number Theory Book PDF

Author : Harry Furstenberg
Publisher : Princeton University Press
Page : 216 pages
File Size : 54,7 Mb
Release : 2014-07-14
Category : Mathematics
ISBN : 9781400855162

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Recurrence in Ergodic Theory and Combinatorial Number Theory by Harry Furstenberg Pdf

Topological dynamics and ergodic theory usually have been treated independently. H. Furstenberg, instead, develops the common ground between them by applying the modern theory of dynamical systems to combinatories and number theory. Originally published in 1981. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

A First Course in Ergodic Theory

Karma Dajani,Charlene Kalle
A First Course in Ergodic Theory Book PDF

Author : Karma Dajani,Charlene Kalle
Publisher : CRC Press
Page : 268 pages
File Size : 52,7 Mb
Release : 2021-07-04
Category : Mathematics
ISBN : 9781000402773

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A First Course in Ergodic Theory by Karma Dajani,Charlene Kalle Pdf

A First Course in Ergodic Theory provides readers with an introductory course in Ergodic Theory. This textbook has been developed from the authors’ own notes on the subject, which they have been teaching since the 1990s. Over the years they have added topics, theorems, examples and explanations from various sources. The result is a book that is easy to teach from and easy to learn from — designed to require only minimal prerequisites. Features Suitable for readers with only a basic knowledge of measure theory, some topology and a very basic knowledge of functional analysis Perfect as the primary textbook for a course in Ergodic Theory Examples are described and are studied in detail when new properties are presented.

Infinite Ergodic Theory of Numbers

Marc Kesseböhmer,Sara Munday,Bernd Otto Stratmann
Infinite Ergodic Theory of Numbers Book PDF

Author : Marc Kesseböhmer,Sara Munday,Bernd Otto Stratmann
Publisher : Walter de Gruyter GmbH & Co KG
Page : 204 pages
File Size : 54,6 Mb
Release : 2016-10-10
Category : Mathematics
ISBN : 9783110439427

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Infinite Ergodic Theory of Numbers by Marc Kesseböhmer,Sara Munday,Bernd Otto Stratmann Pdf

By connecting dynamical systems and number theory, this graduate textbook on ergodic theory acts as an introduction to a highly active area of mathematics, where a variety of strands of research open up. The text explores various concepts in infinite ergodic theory, always using continued fractions and other number-theoretic dynamical systems as illustrative examples. Contents: Preface Mathematical symbols Number-theoretical dynamical systems Basic ergodic theory Renewal theory and α-sum-level sets Infinite ergodic theory Applications of infinite ergodic theory Bibliography Index

An Introduction to Infinite Ergodic Theory

Jon Aaronson
An Introduction to Infinite Ergodic Theory Book PDF

Author : Jon Aaronson
Publisher : American Mathematical Soc.
Page : 298 pages
File Size : 52,5 Mb
Release : 1997
Category : Ergodic theory
ISBN : 9780821804940

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An Introduction to Infinite Ergodic Theory by Jon Aaronson Pdf

Infinite ergodic theory is the study of measure preserving transformations of infinite measure spaces. The book focuses on properties specific to infinite measure preserving transformations. The work begins with an introduction to basic nonsingular ergodic theory, including recurrence behaviour, existence of invariant measures, ergodic theorems, and spectral theory. A wide range of possible "ergodic behaviour" is catalogued in the third chapter mainly according to the yardsticks of intrinsic normalizing constants, laws of large numbers, and return sequences. The rest of the book consists of illustrations of these phenomena, including Markov maps, inner functions, and cocycles and skew products. One chapter presents a start on the classification theory.

Operator Theoretic Aspects of Ergodic Theory

Tanja Eisner,Bálint Farkas,Markus Haase,Rainer Nagel
Operator Theoretic Aspects of Ergodic Theory Book PDF

Author : Tanja Eisner,Bálint Farkas,Markus Haase,Rainer Nagel
Publisher : Springer
Page : 628 pages
File Size : 45,6 Mb
Release : 2015-11-18
Category : Mathematics
ISBN : 9783319168982

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Operator Theoretic Aspects of Ergodic Theory by Tanja Eisner,Bálint Farkas,Markus Haase,Rainer Nagel Pdf

Stunning recent results by Host–Kra, Green–Tao, and others, highlight the timeliness of this systematic introduction to classical ergodic theory using the tools of operator theory. Assuming no prior exposure to ergodic theory, this book provides a modern foundation for introductory courses on ergodic theory, especially for students or researchers with an interest in functional analysis. While basic analytic notions and results are reviewed in several appendices, more advanced operator theoretic topics are developed in detail, even beyond their immediate connection with ergodic theory. As a consequence, the book is also suitable for advanced or special-topic courses on functional analysis with applications to ergodic theory. Topics include: • an intuitive introduction to ergodic theory • an introduction to the basic notions, constructions, and standard examples of topological dynamical systems • Koopman operators, Banach lattices, lattice and algebra homomorphisms, and the Gelfand–Naimark theorem • measure-preserving dynamical systems • von Neumann’s Mean Ergodic Theorem and Birkhoff’s Pointwise Ergodic Theorem • strongly and weakly mixing systems • an examination of notions of isomorphism for measure-preserving systems • Markov operators, and the related concept of a factor of a measure preserving system • compact groups and semigroups, and a powerful tool in their study, the Jacobs–de Leeuw–Glicksberg decomposition • an introduction to the spectral theory of dynamical systems, the theorems of Furstenberg and Weiss on multiple recurrence, and applications of dynamical systems to combinatorics (theorems of van der Waerden, Gallai,and Hindman, Furstenberg’s Correspondence Principle, theorems of Roth and Furstenberg–Sárközy) Beyond its use in the classroom, Operator Theoretic Aspects of Ergodic Theory can serve as a valuable foundation for doing research at the intersection of ergodic theory and operator theory

Dynamical Systems and Ergodic Theory

Mark Pollicott,Michiko Yuri
Dynamical Systems and Ergodic Theory Book PDF

Author : Mark Pollicott,Michiko Yuri
Publisher : Unknown
Page : 128 pages
File Size : 52,9 Mb
Release : 2013-07-13
Category : Electronic
ISBN : 1299733905

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Dynamical Systems and Ergodic Theory by Mark Pollicott,Michiko Yuri Pdf

Essentially a self-contained text giving an introduction to topological dynamics and ergodic theory.

Equilibrium States in Ergodic Theory

Gerhard Keller
Equilibrium States in Ergodic Theory Book PDF

Author : Gerhard Keller
Publisher : Cambridge University Press
Page : 196 pages
File Size : 49,6 Mb
Release : 1998-01-22
Category : Mathematics
ISBN : 0521595347

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Equilibrium States in Ergodic Theory by Gerhard Keller Pdf

Based on a one semester course, this book provides a self contained introduction to the ergodic theory of equilibrium states.

Generalized Notions of Continued Fractions

Juan Fernández Sánchez,Jerónimo López-Salazar Codes,Juan B. Seoane Sepúlveda,Wolfgang Trutschnig
Generalized Notions of Continued Fractions Book PDF

Author : Juan Fernández Sánchez,Jerónimo López-Salazar Codes,Juan B. Seoane Sepúlveda,Wolfgang Trutschnig
Publisher : CRC Press
Page : 154 pages
File Size : 51,5 Mb
Release : 2023-07-20
Category : Mathematics
ISBN : 9781000907582

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Generalized Notions of Continued Fractions by Juan Fernández Sánchez,Jerónimo López-Salazar Codes,Juan B. Seoane Sepúlveda,Wolfgang Trutschnig Pdf

Ancient times witnessed the origins of the theory of continued fractions. Throughout time, mathematical geniuses such as Euclid, Aryabhata, Fibonacci, Bombelli, Wallis, Huygens, or Euler have made significant contributions to the development of this famous theory, and it continues to evolve today, especially as a means of linking different areas of mathematics. This book, whose primary audience is graduate students and senior researchers, is motivated by the fascinating interrelations between ergodic theory and number theory (as established since the 1950s). It examines several generalizations and extensions of classical continued fractions, including generalized Lehner, simple, and Hirzebruch-Jung continued fractions. After deriving invariant ergodic measures for each of the underlying transformations on [0,1] it is shown that any of the famous formulas, going back to Khintchine and Levy, carry over to more general settings. Complementing these results, the entropy of the transformations is calculated and the natural extensions of the dynamical systems to [0,1]2 are analyzed. Features Suitable for graduate students and senior researchers Written by international senior experts in number theory Contains the basic background, including some elementary results, that the reader may need to know before hand, making it a self-contained volume

Conformal Fractals

Feliks Przytycki,Mariusz Urbański
Conformal Fractals Book PDF

Author : Feliks Przytycki,Mariusz Urbański
Publisher : Cambridge University Press
Page : 365 pages
File Size : 49,8 Mb
Release : 2010-05-06
Category : Mathematics
ISBN : 9780521438001

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Conformal Fractals by Feliks Przytycki,Mariusz Urbański Pdf

A one-stop introduction to the methods of ergodic theory applied to holomorphic iteration that is ideal for graduate courses.

An Outline of Ergodic Theory

Steven Kalikow,Randall McCutcheon
An Outline of Ergodic Theory Book PDF

Author : Steven Kalikow,Randall McCutcheon
Publisher : Cambridge University Press
Page : 183 pages
File Size : 45,8 Mb
Release : 2010-03-25
Category : Mathematics
ISBN : 9781139484251

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An Outline of Ergodic Theory by Steven Kalikow,Randall McCutcheon Pdf

This informal introduction provides a fresh perspective on isomorphism theory, which is the branch of ergodic theory that explores the conditions under which two measure preserving systems are essentially equivalent. It contains a primer in basic measure theory, proofs of fundamental ergodic theorems, and material on entropy, martingales, Bernoulli processes, and various varieties of mixing. Original proofs of classic theorems - including the Shannon–McMillan–Breiman theorem, the Krieger finite generator theorem, and the Ornstein isomorphism theorem - are presented by degrees, together with helpful hints that encourage the reader to develop the proofs on their own. Hundreds of exercises and open problems are also included, making this an ideal text for graduate courses. Professionals needing a quick review, or seeking a different perspective on the subject, will also value this book.