Download MEASURE-PRESERVING SYSTEMS 1. Introduction Let (Ω,F,µ) be a

MEASURE-PRESERVING SYSTEMS
Abstract. We introduce measure-preserving systems and the isomorphism problem for Bernoulli shifts.
1. Introduction
Let (Ω, F, µ) be a probability space. We say that a map T : Ω → Ω
is measure-preserving if µ ◦ T −1 = µ; that is, for all A ∈ F, we have
µ(A) = µ(T −1 (A)). We call the quadruple (Ω, F, µ, T ) a measurepreserving system. If T has a well-defined inverse, then we say that
the system is invertible.
Remark 1. We will usually assume without mention, that (Ω, F, µ)
is a standard probability space without atoms, so that there exists an
bijection φ : Ω → [0, 1) such that if λ is Lebesgue measure on [0, 1),
then λ is the push-foward of µ under φ, and µ is the push-forward of
λ under φ−1 .
Example 2 (i.i.d. systems). Let [N ] = {0, 1, . . . , N − 1} and p be a
probability measure on [N ]. Set Ω = [N ]Z , F to be the usual product
σ-algebra, and µ = pZ . Thus (Ω, F, µ) is a probability space for a biinfinite sequence of finite valued i.i.d. random variables. Let T be the
left-shift so that (T ω)i = ωi+1 for all i ∈ Z. Then (Ω, F, µ, T ) is a
measure-preserving system.
In Example 2, we may also replace Z with N = {0, 1, 2, . . .}. In case
of Z sometimes Example 2 is called a two-sided Bernoulli shift and
in the case of N sometimes it is called a one-sided Bernoulli shift.
If p is probability measure on a finite set, then we often denote the
corresponding Bernoulli shift by B(p).
Example 3 (Markov chains started at stationarity). Let P be transition matrix for a Markov chain X = (Xi )∞
i=0 taking values on a finite
set S. Since S is finite, by a theorem, there exists a stationary distribution π for P , so that πP = π. Set Ω = S N , F to the be usual product
σ-algebra, µ to be the probability measure given by µ(A) = P(X ∈ A)
for all A ∈ F, and T to be the left-shift. Then (Ω, F, µ, T ) is a measurepreserving system.
Exercise 4. Show that in the set-up of Example 3, the transition matrix P may have more than one stationary distribution. Show that the
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MEASURE-PRESERVING SYSTEMS
convex combination of any two statationary distributions gives another
stationary distribution.
Example
Let S be the unit circle in C, so that
2πiθ 5 (Circle rotation).
S= e
: θ ∈ [0, 1) . Let µ be the normalized Lebesgue measure on
S with the usual Borel σ-algebra B. Let α ∈ R. Define T : S → S via
T (e2πiθ ) = e2πi(θ+α) . Then (S, B, µ, T ) is a measure-preserving system.
2. Recurrence
Theorem 6 (Poincare recurrence theorem). Let (Ω, F, µ, T ) be a measurepreserving system. Let A ∈ F. There exists A0 ∈ F such that A0 ⊂ A,
µ(A\A0 ) = 0, and for all x ∈ A0 there exists n ∈ Z+ such that T n x ∈ A.
Proof. Let F ⊂ A, with x ∈ F if and only if T n x 6∈ A for all n ∈ Z+ .
If z ∈ Ω, then by definition, T z ∈ A if and only if z ∈ T −1 (A). Thus
∞
[
T −n (A).
F =A \
n=1
n
By definition, if x ∈ F , then T x 6∈ F for all n ∈ Z+ , from which
it follows that T −n (F ) and T −m (F ) are disjoint for all nonnegative
integers n 6= m. However, by assumption µ(F ) = µ(T −n (F )), and
since µ(Ω) = 1 < ∞, we are forced to conclude that µ(F ) = 0. Thus
take A0 = A \ F .
A stronger form of recurrence is given by ergodicity. Given a measurepreserving system (Ω, F, µ, T ), we say that it is ergodic or T is ergodic if for all A ∈ F, we have that if µ(A4T −1 (A)) = 0, then
µ(A) ∈ {0, 1}. In Example 2, T is ergodic, but in Example 3, T is
ergodic if and only if P is irreducible.
Exercise 7. Use the Kolmogorov zero-one law to show that T in Example 2 is ergodic.
Exercise 8. Show that in Example 5, the map T is not ergodic if α is
a rational.
Example 9 (Melshalkin matching). Consider the Bernoulli shift on
two symbols {0, 1} with p = ( 21 , 12 ). Holroyd and Peres considered
the following perfect matching of zeros and ones which is based on an
original construction of Meshalkin. The matching is define inductively.
Given a bi-infnite sequence of zeros and ones, we match a zero with one
whenever a zero is immediately to the left of a one. In the next stage,
we remove all the matched pairs, and repeat the same procedure.
One way to prove that this procedure does define a perfect matching
almost surely is by the following argument. First, note that there
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are only three possibilities for the remaining binary sequence if the
matching procedure fails to produce a perfect matching:
(i)
· · · 111111111111100000000000000000 · · ·
(ii)
· · · 000000000000000 · · ·
(iii)
· · · 1111111111111111 · · ·
Second, note that by symmetry the second and third possibilities must
occur with equal probability, so by the ergodicity, we conclude that
they occur with probability zero. Finally, the Poincare recurrence theorem can also be used to prove that the first possibility occurs with
probability zero.
Exercise 10. Use the Poincare recurrence theorem to eliminate the
first possibility in Example 9.
Proposition 11 (Ergodicity of circle rotation). In Example 5, the map
T is ergodic if α is irrational.
Theorem 12 (Ergodic theorem). Let (Ω, F, µ, T ) be an ergodic measurepreserving system. If f ∈ L1 (Ω), then for µ almost-every ω ∈ Ω, we
have
Z
n−1
1X
k
lim
f (T ω) =
f (ω)dµ(ω).
n→∞ n
Ω
k=0
Remark 13. To compare with Lemma 6, take f = 1A for A ∈ F.
Exercise 14. Use Theorem 12 to deduce the strong law of large numbers.
3. Factors and isomoprhisms
Suppose (Ωi , Fi , µi , Ti ) for i = 1, 2 are measure-preserving systems.
In what follows we ignore sets of measure zero. We say that a measurable map φ : Ω1 → Ω2 is equivariant if φ ◦ T1 = T2 ◦ φ, and is a
factor if µ2 is the pushfoward of µ1 under φ; that is, µ1 ◦ φ−1 = µ2 .
In the case that a factor map exists, we also say that (Ω2 , F2 , µ2 , T2 )
is a factor of (Ω1 , F1 , µ1 , T1 ). If φ is a factor, and φ−1 is also a factor from (Ω2 , F2 , µ2 , T2 ) to (Ω1 , F1 , µ1 , T1 ), then we say that φ is an
isomorphism and the two systems are isomorphic.
Exercise 15. Show that ergodicity is preserved under isomorphisms.
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Exercise 16 (Doubling map). Consider the unit interval Ω = [0, 1)
endowed with the usual Borel σ-algebra and Lebesgue measure µ. Consider the map T : Ω → Ω given by x 7→ 2x mod 1. Sometimes T
is called the doubling map. By taking binary expansions show that
(Ω, B, µ, T ) is isomorphic the one-sided Bernoulli shift B(p) on two
symbols {0, 1} with probability measure p = ( 21 , 12 ).
Presentation topic 17 (Spectral isomorphism). What is a spectral
isomorphism? Show that if two systems are isomorphic, then they are
spectrally isomorphic. When are two circle rotations spectrally isomorphic?
Presentation topic 18. Show that any two Bernoulli shifts are spectrally isomorphic.
4. Entropy
It was a major open problem in the early days of ergodic theory
whether the (two-sided) Bernoulli shifts B( 12 , 21 ) and B( 13 , 31 , 13 ) are isomorphic. Kolmogorov answered this question in the negative via an
isomorphism invariant called entropy, which is assigns a non-negative
real number to an (ergodic) measure-preserving system. In this case of
Bernoulli shifts, entropy has a simple formula:
H(p) = −
N
−1
X
pi log pi .
i=0
Moreover, it turns out that entropy is non-increasing under factors;
that is, if (Ω2 , F2 , µ2 , T2 ) is a factor of (Ω, F1 , µ1 , T1 ), then the entropy
of (Ω2 , F2 , µ2 , T2 ) is no greater than that of the first system. Shortly
after Komogorov introduced entropy into ergodic theory, Sinai proved
that any invertible ergodic measure-preserving system has as factors
all Bernoulli shifts of equal of lesser entropy. We say that two system are weakly isomorphic if they are factors of each other. Thus
Sinai proved that two Bernoulli systems of equal entropy are weakly
isomorphic. Almost a decade later, Ornstein proved that two Bernoulli
systems of equal entopy are isomorphic. We will see later that Ornstein’s theorem is not true for one-sided Bernoulli shifts.
Exercise 19. Let p be a probability measure on [N ]. Show that H(p)
is maximized when p assigns equal weights to each element of [N ].
Presentation topic 20 (Melshalkin’s isomorphism). Let p = ( 21 , 18 , 18 , 18 , 18 )
and q = ( 41 , 14 , 41 , 14 ). Note that H(p) = H(q). Melshalkin gave one of
the first non-trivial isomorphisms between the two Bernoulli shifts B(p)
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and B(q). Present his isomorphism and show that it is indeed an isomophism.
References
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