Classification problems in ergodic theory / William Parry and Selim Tuncel.

By:

Parry, William, 1934-2006

Contributor(s):

Tuncel, Selim

Material type: Text

TextSeries: London Mathematical Society lecture note series ; 67.Publication details: Cambridge [Cambridgeshire] ; New York : Cambridge University Press, 1982.Description: 1 online resource (101 pages) : illustrationsContent type:

text

Media type:

computer

Carrier type:

online resource

ISBN:

9781107361201
1107361206
9780511892165
0511892160

Report number: 82004350Subject(s): Genre/Form:

Electronic books.

Additional physical formats: Print version:: Classification problems in ergodic theory.DDC classification:

515.4/2 22

LOC classification:

QA313 .P368 1982eb

Other classification:

31.41
SI 320
SK 810

Online resources:

Click here to access online

Contents:

Cover; Title; Copyright; Contents; Preface; Chapter I: Introduction; 1. Motivation; 2. Basic Definitions and Conventions; 3. Processes; 4. Markov Chains; 5. Reduced Processes and Topological Markov Chains; 6. Information and Entropy; 7. Types of Classification; Chapter II: The Information Cocycle; 1. Regular Isomorphisms; 2. Unitary Operators and Cocycles; 3. Information Variance; 4. The Variational Principle for Topological Markov Chains; 5. A Group Invariant; 6. Quasi-regular Isomorphisms and Bounded Codes; 7. Central Limiting Distributions as Invariants; Chapter III: Finitary Isomorphisms

1. The Marker Method2. Finite Expected Code-lengths; Chapter IV: Block-codes; 1. Continuity and Block-codes; 2. Bounded-to-one Codes; 3. Suspensions and Winding Numbers; 4. Computation of the First Cohomology Group; Chapter V: Classifications of Topological Markov Chains; 1. Finite Equivalence; 2. Almost Topological Conjugacy and the Road Problem; 3. Topological Conjugacy of Topological Markov Chains; 4. Invariants and Reversibility; 5. Flow Equivalence; Appendix: Shannon's Work on Maximal Measures; References; Index

Summary: The isomorphism problem of ergodic theory has been extensively studied since Kolmogorov's introduction of entropy into the subject and especially since Ornstein's solution for Bernoulli processes. Much of this research has been in the abstract measure-theoretic setting of pure ergodic theory. However, there has been growing interest in isomorphisms of a more restrictive and perhaps more realistic nature which recognize and respect the state structure of processes in various ways. These notes give an account of some recent developments in this direction. A special feature is the frequent use of the information function as an invariant in a variety of special isomorphism problems. Lecturers and postgraduates in mathematics and research workers in communication engineering will find this book of use and interest.

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Includes bibliographical references (pages 96-99) and index.

Print version record.

The isomorphism problem of ergodic theory has been extensively studied since Kolmogorov's introduction of entropy into the subject and especially since Ornstein's solution for Bernoulli processes. Much of this research has been in the abstract measure-theoretic setting of pure ergodic theory. However, there has been growing interest in isomorphisms of a more restrictive and perhaps more realistic nature which recognize and respect the state structure of processes in various ways. These notes give an account of some recent developments in this direction. A special feature is the frequent use of the information function as an invariant in a variety of special isomorphism problems. Lecturers and postgraduates in mathematics and research workers in communication engineering will find this book of use and interest.

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