An Introduction to Infinite Ergodic TheoryInfinite 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
| 1 | |
Chapter 2 General ergodic and spectral theorems | 53 |
Chapter 3 Transformations with infinite invariant measures | 85 |
Chapter 4 Markov maps | 139 |
Chapter 5 Recurrent events and similarity of Markov shifts | 181 |
Chapter 6 Inner functions | 201 |
Chapter 7 Hyperbolic geodesic flows | 223 |
Chapter 8 Cocycles and skew products | 247 |
| 275 | |
| 281 | |
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Common terms and phrases
analytic B O A Borel sets called compact conservative and ergodic convergence COROLLARY countable defined DEFINITION denote disjoint ergodic measure preserving ergodic theorem f e L'(m factor map finite Fuchsian group geodesic flow Haar measure height function hence Hopf's II(p infinite measure space inner function invariant measure invertible irreducible isomorphic large numbers law of large Lebesgue measure lemma liminf Lipschitz continuous m-absolutely continuous m(An Maharam Markov map Markov tower measurable function measure preserving transformation measure space Möb H Möbius transformations o-finite oo a.e. partition Poincaré series pointwise dual ergodic Polish space probability preserving transformation probability space PROOF proposition regularly varying renewal sequence return sequence S X T satisfying spectral standard measure space standard probability space subsets Suppose T-invariant probability topologically mixing totally dissipative uniformly wandering set weak distortion property weakly whence