Introduction to Ergodic Theory

Introduction to Ergodic theory

Lecture by Amie Wilkinson Notes by Clark Butler

October 23, 2014

Let Σ be a ﬁnite set. Recall from the previous setup that we have an abelian group G acting on the set ΣG of functions G → Σ. For a function ω ∈ ΣG, this G-action is given by

(g · ω)(h) = ω(g + h), h ∈ G

For today, G = Z, the integers with the group operation of addition. We will think of Z as parametrizing the action of time. We set Ω := ΣZ. We would like to put an interesting metric on Ω, to study the dynamics on Ω of the shift operator σ :Ω → Ω, to study σ-invariant measures, and interesting invariant subdynamics of σ.

We label the elements of Σ with integers in order to distinguish them. Thus we write Σ = {1, . . . , n}. Points of Ω will be represented as inﬁnite strings of elements of Σ, together with a decimal point that marks which element of the string corresponds to 0 ∈ Z.

Ω = {ω = . . . ω−2ω−1.ω0ω1ω2 ··· : ωi ∈ Σ}

Our convention will be that the element to the right of the decimal point corresponds to 0 ∈ Z. We also use ωi to denote the value of the function ω at the integer i. The Z action on Ω has a natural interpretation in this representation. The Z-action is speciﬁed by the action of the generator 1. We will denote the action of 1 by σ :Ω → Ω. σ satisﬁes the formula

[σ(ω)](n) = ω(n + 1)

In the inﬁnite string notation,

σ(. . . ω−2ω−1.ω0ω1ω2 ... ) = . . . ω−2ω−1ω0.ω1ω2 ...

The eﬀect of σ is to shift the decimal point one spot to the right, or equivalently, ω0 is shifted one spot to the left. σk shifts the decimal point to the right k spots (for k > 0) and σ−1 shifts the decimal point to the left one spot.

A string ω is periodic if there is some k 6= 0 such that σkω = ω. All periodic strings of period ` are constructed by taking a finite string j1 . . . j`, ji ∈ Σ, placing the decimal point at the beginning of this string, and then concatenating infinitely many copies of this finite string in both directions. Hence it will appear as

. . . j1 . . . j`.j1 . . . j`j1 . . . j` ...

We want to put a metric on Ω. Σ carries a natural metric, the discrete metric, for which the distance between i, j ∈ Σ is given by δij, the Kronecker delta that is 1 if i = j, and 0 otherwise. Up to scaling, this is the only metric on Σ which is invariant under all permutations of the set {1, . . . , n}. We deﬁne a metric d on Ω by setting, for two strings ω, τ, ∞ X 1 d(ω, τ) = (1 − δ ) 2|i| ωiτi i=−∞

1 Exercise: Prove this is a metric on Ω = ΣZ which induces the product topology.

The metric d compares the values of τ and ω at each integer, giving more weight to values closer to the decimal point. If τi = ωi for |i| ≤ m, then 1 d(τ, ω) ≤ 2m

whereas if τi 6= ωi, then 1 d(τ, ω) ≥ 2|i| So two sequences are close if and only if they agree on all suﬃciently small integers.

Comment: There is no particular reason for choosing the geometric sequence associated to 1/2 in the deﬁnition of d. We could equally well have deﬁned, for any choice of0 < λ < 1,

∞ X |i| dλ(ω, τ) = λ (1 − δωiτi ) i=−∞ What’s important, as we will see later, is that no matter what the choice of λ is, the collection of Hölder continuous functions Ω → R remains the same (though possibly with different Hölderexponents depending on λ).

A basis for the induced topology of this metric is given by cylinder sets. A cylinder set is a clopen subset of Ω determined by j ∈ Z, and a ﬁnite word k0, . . . , k`.

C(j; k0, . . . , k`) = {ω ∈ Ω|ω(j + i) = ki, i = 0, . . . , `}

The integer j centers us at the jth entry of the sequence; C(j; k0, . . . , k`) is the set of all sequences which start with the word k0 . . . k` from the jth spot. Exercise: Let Σ = {1, 2} have two elements. Let C be the standard middle-thirds Cantor set inside of [0, 1]. Construct a biH¨olderhomeomorphism between Ω and C × C, where C × C carries the induced metric from the L1 metric on R2. Remark: Ω is always a Cantor set, in the sense that there is a homeomorphism from Ω to the standard middle-thirds Cantor set. Any metric space X which is compact, totally disconnected, and perfect is home- omorphic to a Cantor set, and Ω is easily seen to satisfy these properties, no matter what the choice of Σ is.

How might one construct an element ω of the shift space with dense orbit, i.e., such that {σk(ω): k ∈ Z} = Ω? Two procedures were suggested. First, we could enumerate all ﬁnite strings of numbers from Σ and concatenate them to form a sequence ω. Second, we could take a countable dense collection of points τ j ∈ Ω, and form ω by concatenating successively longer strings of numbers from these points τ j.

We change perspectives now to look at measures on Ω.

Mσ(Ω) = {Borel probability measures µ on Ω such that σ∗µ = µ}

Recall that σ∗µ = µ is equivalent to saying that µ is invariant under σ,

µ(σ−1(A)) = µ(A) = µ(σ(A))

for every Borel set A. The second equality requires that σ is invertible. Invariance in this context means that the measure of a cylinder set C(j; k0, . . . , k`) does not depend on j. The simplest examples of invariant measures for the shift are given by Bernoulli measures. Let

( n ) n−1 n X ∆ = p ∈ R : pi ≥ 0, pi = 1 i=1

2 be the standard n − 1 simplex in Rn. A point p ∈ ∆n−1 will be referred to as a probability vector. Given a probability vector p, we deﬁne µ on cylinder sets by

` Y µ(C(j; k0, . . . , k`)) = pki i=0

and setting µ(Ω) = 1, µ(∅) = 0. We’ve deﬁned µ on cylinder sets now; we need to extend µ to be deﬁned on the full Borel σ-algebra of Ω. The tool we will use for this is the Hahn-Kolmogorov extension theorem.

Some deﬁnitions: For a set X, a σ-algebra is a collection A of subsets of X satisfying

1. ∅ ∈ A 2. A ∈ A → X\A ∈ A S 3. For any countable index set I, if Ai ∈ A for every i ∈ I, then i∈I Ai ∈ A.

S The collection A is called an algebra, if it satisﬁes 1,2, and the weaker condition 3’ that i∈I Ai ∈ A for any ﬁnite index set I.

The Hahn-Kolmogorov extension theorem says the following: Let A0 be an algebra, and let A be the smallest σ-algebra containing A0. Let µ0 : A0 → [0, 1] be a function such that µ0(X) = 1, µ0(∅) = 0, and µ0 is ﬁnitely additive: for any disjoint sets A1,...,An ∈ A,

n ! n [ X µ0 Ai = µ0(Ai) i=1 i=1

Suppose further that µ0 is σ-additive: whenever we have a countable collection {Ai ∈ A}i∈I such that S i∈I Ai ∈ A, we also have ! [ X µ0 Ai = µ0(Ai) i∈I i∈I

Then µ0 extends to a measure on A. Further, this extension is unique if µ0 is σ-finite. We can check that the measure µ constructed on the algebra generated by the cylinder sets satisfies these properties and so extends uniquely to a measure on the Borel σ-algebra of Ω. The uniqueness property of the Hahn-Kolmogorov extension theorem also means that, to check that µ is σ-invariant, it suffices to check that µ is σ-invariant when restricted to unions of cylinder sets, which is clear.

The second example is Markov measures. These are deﬁned by two objects. As before, we have a probability n−1 vector p ∈ ∆ , but we also have an n × n stochastic matrix P . P being stochastic means that Pij ∈ [0, 1] Pn for every i, j and j=1 Pij = 1. We assume also that p · P = p, so that p is a left eigenvector of P with eigenvalue 1. Written out in sums, this says that

n X piPij = pj i=1

for every i, j. P has the property that for every q ∈ ∆n−1, q · P ∈ ∆n−1.

If we take a stochastic matrix P and assume that there is some N > 0 such that all of the entries of P N are strictly positive, then it follows from the Perron-Frobenius theorem that there is a unique left eigenvector p ∈ ∆n−1 whose eigenvalue is the leading eigenvalue of P . We can deduce that the leading eigenvalue of P is 1 by observing that the column vector of all 1’s is a right eigenvector of P , and then noting that eigenvalues are preserved under transposition.

3 Using the pair (P, p) of a stochastic matrix and an associated probability vector, we can deﬁne a measure µ on cylinder sets by `−1 Y µ(C(j; k0, . . . , k`)) = pk0 · Pkiki+1 i=0 Exercise: Prove that µ gives rise to a shift-invariant measure on Ω.