Factors and Joinings
6.4 Kronecker Systems
Proof of Lemma6.8.Suppose that ρ=
Z
ρzdτ(z)
is the ergodic decomposition of ρ from Theorem 6.2, for some probability space (Z,BZ, τ). Recall thatπX:X×Y →Xdenotes the projection ontoX.
Then
μ= (πX)∗ρ=
(πX)∗ρzdτ(z) (6.2) is a decomposition ofμ. The second equality in (6.2) is a consequence of the definitions
μ(B) =ρ(B×Y) =
ρz(B×Y) dτ(z) =
(πX)∗ρz(B) dτ(z).
By ergodicity and Theorem 4.4, it follows that μ = (πX)∗ρz for almost everyz∈Z.By symmetry the same property holds for Y, which proves the
lemma.
In Sect.6.5 we will show that the study of joinings is at least as general as the study of factor maps: Every common factor of two systems Xand Y gives rise to an element ofJ(X,Y) (see also Exercise6.3.3).
Exercises for Sect. 6.3
Exercise 6.3.1.Emulate the proof of Theorem 4.4 to show that J(X,Y) is a convex set, and that the extreme points are ergodic measures for T ×S.
Deduce that there is always an ergodic joining of two ergodic systems. Give an example in which an ergodic joining cannot be a product of invariant measures on the two systems.
Exercise 6.3.2.Suppose X is ergodic andρ ∈J(X,Y). Show thatρ is the trivial joining if and only if ρ is invariant under T ×I : X ×Y → X ×Y where (T×I)(x, y) = (T x, y).
Exercise 6.3.3.Show that ifY is a factor ofX, then there is a joining ofX andYwhich gives measure one to the graph of the factor map.
6.4 Kronecker Systems
Replacing the word “trivially” in Example6.6 with other more interesting dynamical properties, and finding the correspondingσ-algebra, encompasses many of the deepest structural problems in ergodic theory. The first step
in this program describes the largest factor on which a measure-preserving system behaves like a rotation of a compact group.
Lemma 6.9.If(X,B, μ, T)is ergodic then every eigenvalue ofUT is simple, and the set of all eigenvalues ofUT is a subgroup ofS1.
Proof.IfUTf =λf then
f, f=UTf, UTf=λλf, f
soλλ= 1. Iff is an eigenfunction corresponding to the eigenvalueλ, then UT|f|=|UTf|=|λf|=|f|,
so by ergodicity|f|is a non-zero constant almost everywhere.
Thus ifUTf1=λ1f1 andUTf2=λ2f2 thenUT(f1/f2) = (λ1/λ2)(f1/f2), sof1/f2is also an eigenfunction with eigenvalueλ1/λ2. It follows that the set of eigenvalues is a subgroup ofS1. Finally, ifλ1=λ2then, by ergodicity,f1/f2
is a constant so each eigenvalue is simple.
Theorem 6.10 (Kronecker factors). Let(X,B, μ, T)be an invertible er- godic measure-preserving system on a Borel probability space, and let A be the smallest σ-algebra with respect to which all L2μ eigenfunctions of UT are measurable. Then the corresponding factor(Y,BY, ν, S) is the largest factor of (X,B, μ, T) which is isomorphic to a rotation Ra(y) = y+a on some compact abelian groupY.
It will transpire that the compact group Y is monothetic—that is, it is the closure of the subgroup generated by a single element—and indeedY is the closure of the subgroup {an | n ∈ Z} for some a in the multiplicative infinite torus (S1)N. In particular, Y is automatically a metrizable group, so (Y,BY, ν) is a Borel probability space.
Proof of Theorem 6.10. Let {χi | i ∈ N} ⊆ L2μ be an enumeration∗ of the eigenfunctions ofUT normalized so that|χi|= 1 almost everywhere, and letUTχi=λiχi. Define a mapφ:X→SNby
φ(x) = (χ1(x), χ2(x), . . .), and leta= (λ1, λ2, . . .). Then it is clear that
Raφ(x) =φ(T x)
almost everywhere, soν =φ∗μis Ra-invariant and ergodic (because it is a factor of an ergodic transformation; see Exercise 2.3.4). It follows thatν is
∗ By Lemma B.8 the space CC(X) is separable, and hence L2μ is separable since the inclusionCC(X)→L2μis continuous with dense image. It follows that the set of eigenvalues forUT must be countable, since it is an orthonormal set.
6.4 Kronecker Systems 161
the Haar measure for the subgroupY =anSN(see also Theorem 4.14).
It is clear that the resulting factor is the largest with the required property, since it corresponds to the smallest sub-σ-algebra with the required prop- erty, and any ergodic rotation factor will be generated by eigenfunctions.
Theorem 6.10 suggests that dynamical systems of the sort exhibited—
rotations of compact groups—have many special properties. In this section we show that they have a very prescribed measurable structure.
Definition 6.11.A measure-preserving system (X,BX, μ, T) with the prop- erty that the linear span of the eigenfunctions ofUT in L2μ is dense inL2μ is said to havediscrete spectrum.
We have seen in Lemma 6.9that the eigenvalues of the unitary operator associated to an ergodic measure-preserving transformation form a subgroup ofS1. The next lemma shows the converse.
Lemma 6.12.Given any countable subgroup K S1 there is an ergodic measure-preserving system (X,BX, μ, T) on a Borel probability space with the property that K is the group of eigenvalues ofUT.
Proof. Give K the discrete topology, so that the dual group X = K is a compact metric abelian group; write μ = mX for the normalized Haar measure onX. The mapθ:K→S1defined byθ(κ) =κis a character onK, henceθis an element ofX. DefineT :X→X to be the rotationT(x) =θ·x and letBX be the Borelσ-algebra on X. Then (X,BX, μ, T) is a measure- preserving system, and we claim that it is ergodic (compare this argument with the proof of Proposition 2.16 on p. 26) and that the eigenvalues ofUT
compriseK.
By Pontryagin’s theorem (Theorem C.12) the mapκ→fκdefined by fκ(x) =x(κ)
forκ∈K,x∈X, is an isomorphism fromK toX. For any characterfκ∈X andx∈X,
(UTfκ) (x) =fκ(θ·x) =fκ(θ)fκ(x),
sofκis an eigenfunction ofUT with eigenvaluefκ(θ). Nowfκ(θ) =θ(κ) =κ, soKis a subgroup of the group of eigenvalues ofUT. The setXis a complete orthonormal basis forL2μ (by Theorem C.11), so any eigenfunction f of UT with eigenvalueλcan be writtenf=
κ∈Kcκfκfor Fourier coefficientscκ∈ C. Then (all equalities are inL2μ)
(UTf)(x) =
κ∈K
cκfκ(θ·x)
=
κ∈K
cκfκ(θ)fκ(x)
=
κ∈K
cκκfκ(x)
=λf(x) (since UTf =λf)
=
κ∈K
λcκfκ(x),
so cκκ =λcκ for all κ∈ K. This implies that cκ = 0 unless κ =λ, so we must havef =cλfλ. Thus each eigenfunction ofUT is a scalar multiple of a character ofX. Moreover, each eigenvalue is simple, soT is ergodic.
The following theorem, due to Halmos and von Neumann [139], is the simplest classification theorem for a class of measure-preserving systems; also see Exercise6.4.2. The argument presented here is due to Lema´nczyk [225]
(see also the article by Thouvenot [359]).
Theorem 6.13.Suppose that (X,BX, μ, T) and (Y,BY, ν, S) are two er- godic measure-preserving systems with discrete spectrum. ThenT andS are measurably isomorphic if and only if they have the same group of eigenvalues.
Proof. If T and S are measurably isomorphic then they have the same eigenvalues.
Conversely, letK denote the group of eigenvalues ofUT, and assume this is also the group of eigenvalues of US. We wish to show that any ergodic joining is actually a joining supported on the graph of an isomorphism. By Lemma 6.8we may choose an ergodic joining λ∈J(X,Y). For each κ∈ K there are functionsf ∈L2μ(X) andg∈L2ν(Y) withUTf =κfandUSg=κg.
Writef =f⊗1 for the function onX×Y defined byf(x, y) =f(x) (and similarly defineg= 1⊗g). ThenUT×Sf =κf andUT×Sg=κg, sof and g are eigenfunctions for the ergodic system (X×Y, T ×S, λ) with the same eigenvalue. It follows by Lemma6.9 that there is some c ∈C with f =cg moduloλ. Since the eigenfunctions span a dense set in L2μ and inL2ν, this implies that
L2μ⊗C=
λ C⊗L2ν ⊆L2λ. (6.3) By (6.3) there is a setG⊆X×Y of full λ-measure with the property that
({∅, X} ⊗BY)G= (BX⊗ {∅, Y})G.
We claim thatG is the graph of an isomorphism betweenX and Y. To see this, consider the projection map
πY :X×Y →Y,