The Theory of Bernoulli Shifts by Paul C. Shields

By Paul C. Shields

There are numerous degree areas isomorphic to the unit period with Lebesgue
measure, for this reason there are lots of how one can describe measure-preserving transformations
on such areas. for instance, there are translations and automorphisms of
compact metric teams, shifts on series areas (such as these triggered through stationary
processes), and flows bobbing up from mechanical platforms. it's a common question
to ask whilst such changes are isomorphic as measure-preserving
transformations. Such options as ergodicity and combining and the examine of unitary
operators brought on by way of such ameliorations have supplied a few quite coarse
answers to this isomorphism question.

The first significant breakthrough at the isomorphism quesion used to be the introduction
by Kolmogorov in 1958-59 of the concept that of entropy as an invariant for measurepreserving
transformation. In 1970, D. S. Ornstein brought a few new approximation
concepts which enabled him to set up that entropy used to be a whole invariant
for a category of ameliorations often called Bernoulli shifts. next work
by Ornstein and others has proven that a huge category of changes of physical
and mathematical curiosity are isomorphic to Bernoulli shifts.

These lecture notes grew out of my makes an attempt to appreciate and use those new
results approximately Bernoulli shifts. many of the fabric in those notes is anxious with
the facts that Bernoulli shifts with an analogous entropy are isomorphic. This
proof uses a couple of basic rules approximately walls and approximation
by periodic changes. those are conscientiously offered in Chapters 2-6. The
basic effects approximately entropy are sketched in Chapters 7-8. Ornstein's Fundamental
Lemma is proved in bankruptcy nine. this permits one to build walls with
perfect distribution and entropy as regards to these that are virtually excellent, and is the
key to acquiring the isomorphism theorem in bankruptcy 10. Chapters 11-13 contain
extensions of those effects, whereas bankruptcy 1 encompasses a precis of the measure
theory utilized in those notes. For a extra whole account of contemporary extensions of
these principles, the reader is talked about D. S. Ornstein's approaching notes.

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These are the shifts associated with stationary Markov chains, and are defined as follows: Let Π be a k × k-matrix with nonnegative entries, each row of which has sum 1, and let π be a k-tuple of nonnegative numbers which sum to 1. The space X is the set of doubly infinite sequences of the symbols 1, 2, . . 1) µ{x|xi = ti , −m ≤ i ≤ n} = πt−m Πt−m t−m+1 . . Πtn−1 tn . We also assume that πΠ = Π, from which it follows that the shift T defined by (T x)n = xn+1 , n = 0, ±1, ±2, . , is µ-invariant.

It is easy to see that (N ) Πt0 t1 . . ,tN −1 (N ) where (Πij ) is the N th power of Π. ΠtN +m−1 tN +m so that (N ) µ(A ∩ B) − µ(A)µ(B) = µ(A)[Πt0 tN − πtN ]ΠtN tN +1 . . 2) that, uniformly in m, T iP . 2) that P is indeed weak Bernoulli. 1 is established once we have proved the following. 2. A weak Bernoulli partition is finitely determined. This can be proved by generalizing the proof that a partition with independent iterates is finitely determined (Ornstein’s Copying Theorem, Chapter 7). We shall describe instead a weaker condition (called ”very weak Bernoulli”), and give a proof later that very weak Bernoulli implies finitely determined.

We have k n H(T, Q) = lim H(Q| n 1 T i Q) = lim H( n n+k T iP | −k T i P ). −k+1 Note that the definition of conditional entropy gives k n+k T iP | H( −k n+k T i P ) = H( −k+1 which is equal to H(T −k P | n+k T i P ) − H( −k n+k −k+1 T i P ), −k+1 T i P ). Replace P by T k P to obtain n+2k T i P ) = H(T, P ). 1. 2. Given k and > 0, there is a δ > 0 such that if P and Q have k sets and |P − Q| ≤ δ then |H(T, P ) − H(T, Q)| ≤ . Proof. Fix δ < 1/2, and suppose |P − Q| < δ. Let k Pi ∩ Qi , Ri = Pi − R0 , 1 ≤ i ≤ k, R0 = i=1 37 and let R denote the partition {R0 , R1 , R2 , .

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