Discrete Mathematics

Download e-book for kindle: An outline of ergodic theory by Steven Kalikow

By Steven Kalikow

ISBN-10: 0511676972

ISBN-13: 9780511676970

ISBN-10: 0511679483

ISBN-13: 9780511679483

ISBN-10: 0511681461

ISBN-13: 9780511681462

ISBN-10: 0511801602

ISBN-13: 9780511801600

ISBN-10: 0521194407

ISBN-13: 9780521194402

An advent to ergodic conception for graduate scholars, and an invaluable reference for the pro mathematician.

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Whereas probability theory is the study of probability spaces, ergodic theory is the study of measure-preserving systems. In other words, the most basic object of study for a probabilist is ( , A, μ), while the most basic object of study for an ergodic theorist is ( , A, μ, T ). 129. Convention. In this book we will primarily deal with invertible measurepreserving systems. Accordingly, we may not always say “invertible” though we generally mean it unless we specify otherwise. 130. Definition. e.

The finite sequence ( pi )i=0 157. Comment. In practice, one usually chooses an alphabet ={λ1 , λ2 , . } in one-to-one correspondence with a partition P = { p1 , p2 , . } and writes the ∞ . P-name of ω as (λi )i=−∞ 158. Theorem. If P generates T then the (P, T ) process is isomorphic27 to ( , A, μ, T ). Sketch of proof. 28 159. Exercise. Show that π is an isomorphism. 160. Corollary. If two partitions P and Q each generate T then the (P, T ) process is isomorphic to the (Q, T ) process. 4. Rohlin tower theorem In this section we give three versions of the Rohlin tower theorem, which is one of the fundamental tools of constructive ergodic theory.

6. Birkhoff ergodic theorem and the strong law 39 Ni digits in their binary expansions. The superimposition of the Pi supplies the desired partition; error sets can be obnoxious but Borel–Cantelli can dispose of all but finitely many of them. Sketch of proof. Let ( , A, μ, T ) be the system in question. 200. Exercise. Show that we may, without loss of generality, assume that ( , A, μ) is [0, 1] with Lebesgue measure. e. (see Definition 154) such that {T i p : i ∈ Z, p ∈ P} separates points mod 0.

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