无穷遍历理论是研究无穷测度空间中的保测变换的理论。本书着重介绍了无穷保测变换的特殊性质。本书适合对遍历理论、动力系统和概率论感兴趣的研究生以及数学研究人员阅读参考。 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 behavior, existence of invariant measures, ergodic theorems, and spectral theory. A wide range of possible “ergodic behavior” 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.

前辅文
Preface
Chapter 1. Non-singular transformations
  §1.0 Standard measure spaces
  §1.1 Recurrence and conservativity
  §1.2 Ergodicity
  §1.3 The dual operator
  §1.4 Invariant measures
  §1.5 Induced transformations and applications
  §1.6 Group actions and flows
Chapter 2. General ergodic and spectral theorems
  §2.1 von Neumann's mean ergodic theorem
  §2.2 Pointwise ergodic theorems
  §2.3 Converses to Birkhoff's theorem
  §2.4 Transformations with infinite invariant measures
  §2.5 Spectral properties
  §2.6 Eigenvalues
  §2.7 Ergodicity of Cartesian products
Chapter 3. Transformations with infinite invariant measures
  §3.1 Isomorphism, factors, and similarity
  §3.2 Intrinsic normalising constants and laws of large numbers
  §3.3 Rational ergodicity
  §3.4 Maharam transformations
  §3.5 Category theorems
  §3.6 Asymptotic distributional behaviour
  §3.7 Pointwise dual ergodicity
  §3.8 Wandering rates
Chapter 4. Markov maps
  §4.1 Markov partitions
  §4.2 Graph shifts
  §4.3 Distortion properties
  §4.4 Ergodic properties of Markov maps with distortion properties
  §4.5 Markov shifts
  §4.6 Schweiger's jump transformation
  §4.7 Smooth Probenius-Perron operators and the Gibbs property
  §4.8 Non-expanding interval maps
  §4.9 Additional reading
Chapter 5. Recurrent events and similarity of Markov shifts
  §5.1 Renewal sequences
  §5.2 Markov towers and recurrent events
  §5.3 Kaluza sequences
  §5.4 Similarity of Markov towers
  §5.5 Random walks
Chapter 6. Inner functions
  §6.1 Inner functions on the unit disc
  §6.2 Inner functions on the upper half plane
  §6.3 The dichotomy
  §6.4 Examples
Chapter 7. Hyperbolic geodesic flows
  §7.1 Hyperbolic space models
  §7.2 The geodesic flow of H
  §7.3 Asymptotic geodesies
  §7.4 Surfaces
  §7.5 The Poincare series
  §7.6 Further results
Chapter 8. Cocycles and skew products
  §8.1 Skew Products
  §8.2 Persistencies and essential values
  §8.3 Coboundaries
  §8.4 Skew products over Kronecker transformations
  §8.5 Joinings of skew products
  §8.6 Squashable skew products over odometers
Bibliography
Index