An introduction to infinite ergodic theory / Jon Aaronson.

Format:

Book

Author/Creator:

Aaronson, Jon, 1949- author.

Series:

Mathematical surveys and monographs ; no. 50.

Mathematical surveys and monographs, 0076-5376 ; volume 50

Language:

English

Subjects (All):

Ergodic theory.

Physical Description:

1 online resource (298 p.)

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [1997]

Language Note:

English

Summary:

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.

Contents:

""Contents""; ""Preface""; ""Chapter 1. Non-singular transformations""; ""Â1.0 Standard measure spaces""; ""Â1.1 Recurrence and conservativity""; ""Â1.2 Ergodicity""; ""Â1.3 The dual operator""; ""Â1.4 Invariant measures""; ""Â1.5 Induced transformations and applications""; ""Â1.6 Group actions and flows""; ""Chapter 2. General ergodic and spectral theorems""; ""Â2.1 von Neumann's mean ergodic theorem""; ""Â2.2 Pointwise ergodic theorems""; ""Â2.3 Converses to Birkhoff's theorem""; ""Â2.4 Transformations with infinite invariant measures""; ""Â2.5 Spectral properties""

""Â2.6 Eigenvalues""""Â2.7 Ergodicity of Cartesian products""; ""Chapter 3. Transformations with infinite invariant measures""; ""Â3.1 Isomorphism, factors, and similarity""; ""Â3.2 Intrinsic normalising constants and laws of large numbers""; ""Â3.3 Rational ergodicity""; ""Â3.4 Maharam transformations""; ""Â3.5 Category theorems""; ""Â3.6 Asymptotic distributional behaviour""; ""Â3.7 Pointwise dual ergodicity""; ""Â3.8 Wandering rates""; ""Chapter 4. Markov maps""; ""Â4.1 Markov partitions""; ""Â4.2 Graph shifts""; ""Â4.3 Distortion properties""

""Â4.4 Ergodic properties of Markov maps with distortion properties""""Â4.5 Markov shifts""; ""Â4.6 Schweiger's jump transformation""; ""Â4.7 Smooth Probenius-Perron operators and the Gibbs property""; ""Â4.8 Non-expanding interval maps""; ""Â4.9 Additional reading""; ""Chapter 5. Recurrent events and similarity of Markov shifts""; ""Â5.1 Renewal sequences""; ""Â5.2 Markov towers and recurrent events""; ""Â5.3 Kaluza sequences""; ""Â5.4 Similarity of Markov towers""; ""Â5.5 Random walks""; ""Chapter 6. Inner functions""; ""Â6.1 Inner functions on the unit disc""

""Â6.2 Inner functions on the upper half plane""""Â6.3 The dichotomy""; ""Â6.4 Examples""; ""Chapter 7. Hyperbolic geodesic flows""; ""Â7.1 Hyperbolic space models""; ""Â7.2 The geodesic flow of H""; ""Â7.3 Asymptotic geodesies""; ""Â7.4 Surfaces""; ""Â7.5 The Poincare series""; ""Â7.6 Further results""; ""Chapter 8. Cocycles and skew products""; ""Â8.1 Skew Products""; ""Â8.2 Persistencies and essential values""; ""Â8.3 Coboundaries""; ""Â8.4 Skew products over Kronecker transformations""; ""Â8.5 Joinings of skew products""; ""Â8.6 Squashable skew products over odometers""

""Bibliography""""Index""; ""A""; ""B""; ""C""; ""D""; ""E""; ""F""; ""G""; ""H""; ""I""; ""J""; ""K""; ""L""; ""M""; ""N""; ""O""; ""P""; ""R""; ""S""; ""T""; ""U""; ""W""

Notes:

Description based upon print version of record.

Includes bibliographical references (pages 275-280) and index.

Description based on print version record.

ISBN:

1-4704-1281-0