Discrete Mathematics

An outline of ergodic theory by Steven Kalikow

By Steven Kalikow

This casual advent specializes in the department of ergodic thought often called isomorphism conception. workouts, open difficulties, and important tricks actively have interaction the reader and inspire them to take part in constructing proofs independently. perfect for graduate classes, this publication is usually a worthwhile reference for the pro mathematician.

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Let C = C ∪ (I ∩ E). 177. Exercise. Show that C ∈ S, C C and C ∼ C, a contradiction. 178. Definition. Let ( , A, μ, T ) be an invertible measure-preserving system and suppose S ∈ A with S, T S, T 2 S, . . , T N −1 S pairwise disjoint. We call {S, T S, T 2 S, . . , T N −1 S} a Rohlin tower of height N . The sets T i S are called N −1 i T S is called the rungs of the tower, and S is called the base. The set \ i=0 the error set. 179. Comment. In all arguments, we shall assume that the error set is sufficiently small in measure.

Discussion. Put = Z , where an immortal monkey type any infinite sequence λ1 , λ2 , . . 38 We can use this sequence to construct a shift-invariant measure on as follows. 41 37 Translation: let (X )∞ i i=−∞ be a stationary process on a countable alphabet . Form the associated measure-preserving system ( , A, μ, T ), where = Z , etc. ( , A, μ, T ) is ergodic if and only if for every k ∈ N and every k-tuple w = (λ0 , λ1 , . . e. ,xn+k−1 =λk−1 }| lim N →∞ = f w . ,X n+k−1 =λk−1 }| to say that with probability 1, lim N →∞ = fw .

We also say that the system ( , A , μ , T ) is a factor of the system ( , A, μ, T ), and that the system ( , A, μ, T ) is an extension of the system ( , A , μ , T ). 131. Definition. Let ( , A, μ, T ) and ( , A , μ , T ) be measure-preserving is a homomorphism. If there systems and assume that π : → such that the restriction of exist full measure sets X ⊂ and X ⊂ π to X is a bimeasurable bijection between X and X , we say that π is an isomorphism and that the systems ( , A, μ, T ) and ( , A , μ , T ) are isomorphic.

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