Discrete Mathematics

Download e-book for iPad: 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 idea for graduate scholars, and an invaluable reference for the pro mathematician.

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T M ω} that are outside S is usually a small fraction of (b − a)M, because the terms are bounded above by B and the number of bad points is usually a small fraction of (b−a) B . | f2| summed over these same points is usually a small fraction of M, since | f 2 | is a small fraction of M. This proves that the average of all the points in the interval is usually much closer to b than to a. A similar argument shows that the average is usually much closer to a than to b, leading to a contradiction.

Is a sequence of functions whose sum con124. Exercise. e. verges in a dominated way. Show E( i=1 125. Exercise. e. 126. Exercise. e. e. e. e. Hint: for (b), let f = P(A|B1 ) and B = B2 . Apply part (a) and the previous exercise. 20 That is, E( f |X, Y, Z ) = E f |B(X, Y, Z ) . e. 1. Systems and homomorphisms In this subchapter, we give basic definitions concerning measure-preserving systems and homomorphisms between them. 127. Definition. Let ( , A, μ) be a probability space and assume that T : → is a measure-preserving transformation.

Process composed of real-valued X i . e. Sketch of proof. Denote by (Z , B, ν) the probability space on which the process is defined. Let n > 0 be large and let = { ni : i ∈ Z}, which we view 36 as a countable alphabet. Next, define f : R → by f (x) = nx n . Then let Yi = f ◦ X i . ∞ is an independent stationary process on a countable 211. Exercise. (Yi )i=−∞ alphabet. • Let ( , A, μ, T ) be the measure-preserving system associated with (Yi ). By Theorem 149, ( , A, μ, T ) is ergodic. Define a measurable function ∞ = f (x0 ).

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