Read e-book online An outline of ergodic theory PDF

By Steven Kalikow

ISBN-10: 0521194407

ISBN-13: 9780521194402

This casual creation specializes in the department of ergodic thought referred to as isomorphism concept. workouts, open difficulties, and beneficial tricks actively have interaction the reader and inspire them to take part in constructing proofs independently. excellent for graduate classes, this e-book can be a beneficial reference for the pro mathematician.

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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. 161. Definition. Let ( , A, μ) be a probability space. Write A ∼ B if μ(A B) = 0. 162. Exercise. Show that ∼ is an equivalence relation. 163. Definition. For A ∈ A, write A for the equivalence class of A under ∼ and write A for the family of equivalence classes.

Suppose that this system is not ergodic. Then there exists A ∈ A with 0 < μ(A) < 1 such that μ(A T −1 A) = 0. By Corollary 71, the algebra of cylinder sets generates A mod 0. 150. Exercise. For any > 0 there exist two cylinder sets C1 and C2 such that each Ci approximates A up to and yet C1 and C2 are independent. Use this fact to complete the proof. Formerly we saw how a process gives rise to a measure-preserving transformation. Now we shall see how to run this correspondence in reverse. 151. Definition.

183. Definition. Let ( , A, μ) be a probability space. An event S and a measurable partition P = { p1 , p2 , . } are independent if S and pi are independent for each i. If Q = {q1 , q2 , . } is another partition, we say that Q and P are independent if pi and q j are independent for all i, j. 184. Theorem. ) Let ( , A, μ, T ) be a non-periodic measure-preserving system, let N ∈ N and let > 0. For any finite measurable partition P, there exists some S ∈ A that 32 That is, E = {x ∈ \ ∞ i j i=−∞ T C : |x − T x| > δ, 1 ≤ j ≤ M} and one has μ(E) > 0.

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An outline of ergodic theory by Steven Kalikow

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