An Introduction to Infinite Ergodic Theory
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.
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A G B a-finite an(T analytic Borel sets called cocycle compact conservative and ergodic constant convergence Corollary countable defined Definition denote disjoint ergodic measure preserving ergodic theorem factor map fc=i fc=l finite Fuchsian group geodesic flow Haar measure height function hence Hopf's infinite measure space inner function invariant measure invertible irreducible isomorphic Kaluza sequence large numbers law of large Lebesgue measure lemma liminf Lipschitz continuous m-absolutely continuous Maharam Markov map Markov shift Markov tower measurable function measure preserving transformation measure space oo a.e. partition Poincare series pointwise dual ergodic Polish space probability preserving transformation probability space Proof proposition prove regularly varying renewal sequence return sequence satisfying skew products spectral standard measure space standard probability space stochastic matrix subsets Suppose T-invariant probability T-kA T-nB topologically mixing totally dissipative uniformly wandering set weak distortion property weakly whence