Topics in ergodic theory / William Parry

Contient des exercices

Notes bibliogr. p. [98-107]. Index

This is a collection of the author's favorite topics rather than a comprehensive treatment of ergodic theory. The emphasis is on using algebraic and functional analytic techniques to investigate measure-preserving transformations. Entropy theory is introduced early and developed to the Rokhlin-Sinaĭ characterization of Kolmogorov automorphisms. Measure-theoretic techniques such as Rokhlin towers and Bernoulli theory are not given detailed treatment. However, the author's description of recent work in the subject together with references should easily enable the reader to obtain a balanced view.
After an introduction describing the origins and main ideas of the subject, in Chapter 1 the author uses irrational rotations and their generalizations to examine specific transformations. The chapter concludes with complete proofs of the usual ergodic theorems, including Wiener's Lp-dominated ergodic theorem. The Shannon-McMillan-Breiman theorem on convergence of information is proved in Chapter 2 after dealing with the natural martingale theorem needed. Various notions of mixing are discussed in Chapter 3. Entropy as an invariant and related topics are covered in Chapter 4. The last and most interesting chapter returns to some specific results about perturbing transformations and flows to eliminate eigenfunctions. This chapter also contains some other stimulating examples to complement the previous theory, including Furstenberg's minimal nonergodic homeomorphism of the 2-torus.
The book concludes with a useful appendix on the spectral multiplicity of unitary operators that distills those parts of the theory of interest to ergodic theorists. There is also a good selection of problems, ranging from routine to provokingly cryptic (Exercise 9, p. 17). (MathSciNet)