On invariants for measure preserving transformations

G. Hjorth^{∗}

January 31, 2001

Abstract

The classification problem for measure preserving transformations is strictly more complicated than that of graph isomorphism.^{1}

1 Preamble

We consider the group M_{∞} of all invertible measure preserving transformations either on the unit interval or any other reasonable measure space. It seems natural to say that two of these transformations, σ_{1}, σ_{2}, are equivalent or isomorphic if there is a third, π, so that

π ∘ σ_{1} ∘ π^{−1} = σ_{2} a.e.

To what extent can this equivalence relation be considered classifiable? In specific cases – for instance σ_{1}, σ_{2} both Bernoulli or discrete spec-

trum – there are well accepted systems of complete invariants. However in the completely general context of arbitrary measure preserving transformations there is no known satisfactory system of complete invariants nor even a clear statement of what this would entail.

For instance, Halmos in [7] despairs of precisely formulating the problem but at page 1029 suggest that its solution should fulfill the “the vague task of finding a complete set of invariants...” At page 75 of [8] he proposes that the central problem is to “find usable necessary and sufficient conditions for the conjugacy of two measure preserving transformations.” Some time later Weiss at page 670 of [23] raises the problem of finding “a set of invariants large enough so that if all invariants agree for two m.p.t. one can conclude that the m.p.t. are isomorphic.”

G. Hjorth 2

This article considers attempts to make this precise and ask abstractly whether the problem of classifiability could in principle have a positive solution. We present a clearly identifiable lower bound on the classification difficulty of the isomorphism relation for measure preserving transformations.

∗The author gratefully acknowledges support from NSF grants DMS 99-70403 and DMS 96-22977.

1AMS Subject Classification: 04A15. Key words and phrases: Classification; measure preserving transformation; Polish group action.

One precise formulation of the problem would be to understand a classifiable equivalence relation on a Polish space to be one for which we may find a Borel assignment of reals or points in some other standard Borel space as complete invariants. This is the notion of classifiable suggested by the Glimm-Effros dichotomy of [10].

Indeed Feldman in [4] takes exactly that position. Appealing to [18] he observes that the isomorphism relation for Bernoulli shifts allows real numbers to be assigned in a Borel manner as a complete invariant and uses [19] to remark that such an assignment is already impossible for the class of measure preserving having the property of K.

A more generous notion of classification, closer to the kinds we consider below, is already implicit in sources [7], [8], [23]. In each case the results of [9] are accepted as providing a complete classification for the discrete spectrum measure preserving transformations. Here the invariants are not real numbers or single points in a standard Borel space, but rather countable sets of complex numbers. The significance of this is not in the use of complex numbers as against reals – this is immaterial since all uncountable Polish spaces are Borel isomorphic. The significant feature of the invariants from [9] is that they have the form of a countable unordered set of points in a standard Borel space.^{2}

Thus we may in general ask for an equivalence relation E on a standard Borel space X:

Question Q1: Does there exists a countable sequence (f_{i})_{i∈ℕ} of Borel functions, each f_{i} : X → ℂ such that for all x_{1}, x_{2} ∈ X

x_{1}Ex_{2} ⇔ {f_{i}(x_{1}) : i ∈ ℕ} = {f_{i}(x_{2}) : i ∈ ℕ}?^{3}

As suggested by Matt Foreman, we might hope that “usable and sufficient conditions” should at least make the relation of isomorphy Borel in M_{∞} × M_{∞}.

Question Q2: Let Graph(E) ⊆ X × X be the set of (x_{1}, x_{2}) for which x_{1}Ex_{2}. Is Graph(E) Borel in the product Borel structure on X × X? (In future I will refer to this conclusion simply as the statement that E be Borel.)

2Superficially it might be thought that the ability to assign countable sets of points in a standard Borel space as a complete invariant is tantamount to classifiability in the sense of [4]. In fact this is untrue – there is in general no canonical way to encode or parameterize a countable set of complex numbers by a real, or by a point in any other separable completely metrizable (i.e. Polish) space. Indeed this can already been seen in the present context, as it is known that there is no Borel function that assigns to a discrete spectrum measure preserving transformation a complex number as a complete invariant (compare the start of §5 below).

More starkly, and just to allay suspicions that this may be a consequence of restricting to the Borel category, the techniques of [22] are sufficient to demonstrate the consistency of set theory without the axiom of choice along with the non-existence of any function assigning complex numbers as complete invariants for discrete spectrum transformations.

It is intrinsic to the methods of [9] that we obtain countable sets of points as complete invariants, and no amount of modification will squeeze it into the form requested by [4].

3Matt Foreman has shown that the procedure of [9] indeed obtains such a classification, that is to say, in the Borel category, for discrete spectrum mpt’s.

These are in general distinct notions. Given an equivalence relation E on a Polish space X, classifiability in the sense of a single Borel function assigning points as complete invariants implies a positive answer to Q1. A positive answer to Q1 implies one for Q2. Neither of these implications reverse.

Below we consider these questions for the specific case of X = M_{∞} and E the equivalence relation of conjugacy, given by setting σ_{1}Eσ_{2} if there exists π ∈ M_{∞} such that

σ_{1}(x) = π ∘ σ_{2} ∘ π^{−1}(x) a.e.

We place on M_{∞} the customary Borel structure, described by [4] and recalled in section 2 below.

Theorem 1.1 The conjugacy equivalence relation on M_{∞} is non-Borel.

In fact the situation is much worse than this alone would suggest. It turns out that the classification problem for measure preserving transformations encompasses the classification problem for arbitrary countable discrete structures – countable groups, countable linear orderings, graphs, and so on. For L a countable language, Mod(L), the space of all Lstructures whose underlying set is ℕ, is naturally a standard Borel space; the details of this definition are recalled in section 2 below, and discussed at length in many places, such as [15], [13], [12]. A specific example of such a collection is the space of (directed) graphs on ℕ, which by appeal to the corresponding characteristic function of the adjacency relation can be identified with {0, 1}^{ℕ×ℕ} and given a natural topology.

Theorem 1.2 If L is a countable language then there is a Borel function θ: Mod(L) → M_{∞} such that for all M, N ∈ Mod(L)

M ≅ N ⇔ ∃π ∈ M_{∞}(π ∘ θ(M) ∘ π^{−1} = θ(N) a.e.).

It is not known whether 1.2 or 1.1 can be obtained for the ergodic measure preserving transformations – that is to say, whether we may have θ as in 1.2 but with the further requirement that it always assume a value θ(M) ∈ M_{∞} such that every Borel θ(M)-invariant set is either null or co-null.

These further and still open questions are of interest given the ergodic decomposition theorem, stating that every element of M_{∞} may be in some sense written as the integral of its ergodic components (see for instance [8], §2.3 [25], [20]). In this context one could compare the conjugacy equivalence relation on the unitary representations of a discrete group: in the case of irreducible representations this is known to be not only Borel but actually F_{σ} in an appropriate topology. (See [3].)

For the narrow case of ergodic transformations we only know the following:

Theorem 1.3 There is a Polish group G and a “turbulent” Polish Gspace X and a Borel function θ : X → M_{∞} such that

(i) θ(x) is ergodic for all x ∈ X; (ii) x_{0}E_{G}^{X}x_{1} if and only if

∃π ∈ M_{∞}(π ∘ θ(x_{0})π^{−1} = θ(x_{1})

for all x_{0}, x_{1} ∈ X.

In other words, we may embed a turbulent orbit equivalence relation into the isomorphism relation on ergodic measure preserving transformations. In light of the results from [12], this provides a succinct anti-classifiability result. In particular, the reduction in 1.2 does not reverse: The classification problem for measure preserving transformations is strictly more complicated than for discrete countable structures.

Theorem 1.4 For L a countable language, there is no Borel θ_{1} : M_{∞} → Mod(L) such that for all σ_{1}, σ_{2} ∈ M_{∞}

∃π ∈ M_{∞}(π ∘ σ_{1} ∘ π^{−1} = σ_{2} a.e.) ⇔ θ_{1}(σ_{1}) ≅ θ_{1}(σ_{2}).

Indeed there is no such θ_{1} even with domain just the ergodic transformations. Actually we obtain in 1.3(i) that each θ(x) is in the class of rank 2 generalized discrete spectrum (see [6]). This is an important detail: Since the discrete spectrum measure preserving transformations do admit classification by countable sets of complex numbers, and hence by countable models, we might have hoped for instance that the αth level of the generalized discrete spectrum transformations admit complete invariants in something like the αth iteration of the operation of taking all countable subsets applied to ℂ.

It should not be thought that the results above are fragile to the choice of the Borel category. We can define more generous classes of reducibility and show that even with broader but still reasonable classes of functions – of the kind that are encountered in Ulm invariants for abelian p-groups and the Scott analysis for countable structures – there is no reduction of conjugacy on M_{∞} to isomorphism on countable structures, or equality on countable sets of reals, or indeed to any Borel equivalence relation.

Finally I suppose it might be felt that the real problem is not that we are demanding Borel functions but more generally that we are requiring any sort of definability what so ever. In this way we might dream of some manner of classification, only without the invariants being produced in a “effective” manner.

But not even that much can be hoped for. If ≈ is the conjugacy equivalence relation on M_{∞}, ≅ isomorphism on countable structures, then by the techniques of [22] it is consistent with ZF and enough of the axiom of choice to develop most classical mathematics that there be no injection:

M_{∞}/≈ ↪ Mod(L)/≅ .

In particular if P_{ℵ_{0}}(A) denotes the collection of all countable subsets of a set A, then there will be no injection

M_{∞}/≈ ↪ P_{ℵ_{0}}(ℂ),

nor

M_{∞}/≈ ↪ P_{ℵ_{0}}(P_{ℵ_{0}}(ℂ)),

and so on. Similarly it is consistent with ZF and a large fragment (DC) of choice that for any Borel equivalence relation on a Polish space X there is no injection

M_{∞}/≈↪ X/E.

In §2 we give some definitions and present the outline of the proof for 1.2, which is in turn completed in sections 3 and 4. Section 5 embeds a turbulent orbit equivalence relation into the generalized discrete spectrum transformations. §6 gives a proof of a known result to the effect that the natural equivalence relation on cocycles from the measure preserving action of a countable group into a compact group is Borel; this equivalence relation is closely related to the one needed in §5. It is also noted that the collection of measurable transformations conjugating a transformation T to itself is compact if and only if T has discrete spectrum.

In terms of background material needed for reading this paper, formally it does not assume much more than a general knowledge of elementary analysis, of the kind which would be found in an text such as [25]. However as a practical matter it would be more than helpful to have some acquaintance with ergodic theory. A knowledge of classical descriptive set theory in the sense of [15] may also make the paper easier to read. Many of the results appeal to the modern theory of Borel equivalence relations; for this [5] and [1] are good references. The theory of turbulence is developed in [12]; the notation here largely follows the notation there.

We indulge in all the usual sins. A measurable square summable function is identified with its equivalence class in L^{2}. We say “everywhere” when we mean “on all but a null set”.

Acknowledgment: I am grateful to Matthew Foreman for several illuminating discussions in the neighborhood of these topics and for pointing out the relevance of [9].

I am also very much indebted to the referee for an exceptionally thorough and penetrating report, and in particular for finding a serious mathematical error in the first draft. This first draft claimed that one can obtain 1.2 by proving that isomorphism on countable torsion-free abelian groups is Borel complete in the sense of [5]; that proof of the Borel completeness of torsion-free abelian groups was erroneous.

2 Outline of the proof of 1.2

The concept of “Borel reducibility” is central to the arguments below.

Definition Let E and F be equivalence relations on Polish spaces X and Y. We say that E is Borel reducible to F, written E ≤_{B} F , if there is a Borel function

θ : X → Y

such that for all x_{1}, x_{2} ∈ X we have

x_{1}Ex_{2}

if and only if

θ(x_{1})F θ(x_{2}).

Naturally we write E <_{B} F if E ≤_{B} F holds but F ≤_{B} E fails.

This relation ≤_{B} is clearly transitive and reflexive.

Examples More detail, along with proofs of the various folklore assertions, can be found in [12].

(i) For X a Polish space, id(X) is the identity equivalence relation on X. If E ≤_{B} id(X) for any Polish space X then we say that E is smooth.

(ii) E_{v} is the Vitali equivalence relation on ℝ, given by the cosets of ℚ. Here it is known that E_{v} is not smooth.

(iii) E_{0} the equivalence relation of eventual agreement on infinite binary sequences. It is known that E_{0} ≤_{B} E_{v} ≤_{B} E_{0}.

(iv) Let 2^{ℕ×ℕ} be the space of functions from ℕ × ℕ to {0, 1}, with the topology of point wise convergence.^{4} Following [5] we define F_{2} by

x_{1}F_{2}x_{2}

if and only if

∀n ∈ ℕ ∃m_{1}, m_{2} ∈ ℕ(∀k ∈ ℕ(x_{1}(n, k) = x_{2}(m_{2}, k), x_{1}(m_{1}, k) = x_{2}(n, k))).

Then id(ℝ) <_{B} E_{0} <_{B} F_{2}.

Two important classes of Polish spaces are those consisting of all measure preserving transformations of a Lebesgue space and those consisting of all L-structures on ℕ for some countable language L.

Definition Let M_{∞} be the group of Borel measure preserving bijections from [0, 1] to [0, 1] with identification of maps agreeing on a measure one set. For (O_{n})_{n∈ℕ} a basis of [0, 1] we obtain a separable metric d_{λ} on M_{∞} by setting

d_{λ}(π_{1}, π_{2}) = ∑ 2^{−n}[λ(π_{1}(O_{n})Δπ_{2}(O_{n})) + λ(π_{1}^{−1}(O_{n})Δπ_{2}^{−1}(O_{n}))],

where λ is Lebesgue measure and Δ is used to denote symmetric difference: XΔY = (X \ Y) ∪ (Y \ X).

Let ≈^{∗} denote the conjugacy equivalence relation:

π_{1} ≈^{∗} π_{2} ⇔ ∃σ ∈ M_{∞}(σ ∘ π_{1} ∘ σ^{−1} = π_{2}).

4Here and elsewhere we identify 2 with {0, 1}, and thus 2^{ℕ×ℕ} is the space of all functions from ℕ × ℕ to {0, 1}.

Definition For L a countable language, let Mod(L) be the collection of all different ways we may place an L-structure on the natural numbers ℕ. We then place a topology on this space by taking as subbasic open sets those of the form

{M ∈ Mod(L) | (n_{1}, ..., n_{k}) ∈ R^{M}},
{M ∈ Mod(L) | (n_{1}, ..., n_{k}) ∉ R^{M}},
{M ∈ Mod(L) | f^{M}(n_{1}, ..., n_{k}) = m},
{M ∈ Mod(L) | f^{M}(n_{1}, ..., n_{k}) ≠ m},

where n_{1}, n_{2}, ..., n_{k}, m range over finite sequences from ℕ, R ranges over relation symbols in L, and f ranges over function symbols in L.

We let ≅|_{Mod(L)} denote the isomorphism relation on these L-structures. Thus M_{1} ≅ M_{2} if and only if there is a bijection σ : ℕ → ℕ with

(n_{1}, ..., n_{k}) ∈ R^{M_{1}} ⇔ (σ(n_{1}), ..., σ(n_{k})) ∈ R^{M_{2}}

and

f^{M_{1}}(n_{1}, ..., n_{k}) = m ⇔ f^{M_{2}}(σ(n_{1}), ..., σ(n_{k})) = σ(m)

all relation symbols R, function symbols f, and n_{1}, ..., n_{k}, m ∈ ℕ.

Lemma 2.1 M_{∞} is a Polish group; d_{λ} is complete.

Proof This was shown in [4]; a proof can also be found in chapter 2 of [12]. □

Lemma 2.2 Mod(L) is a Polish space whenever L is a countable language.

Proof This lemma should be obvious, since the space can be naturally identified with a suitable countable product of the Polish spaces ℕ^{{0,1}} and ℕ^{ℕ}. □

We wish to start working towards a proof that for any countable language L we have

≅|_{Mod(L)} ≤_{B}≈^{∗} .

An equivalence relation E on X is said to be Borel if it is Borel as a subset of X × X. It is then easily seen that the Borel equivalence relations are closed downwards under ≤_{B}. And thus since it is well known (see [5] or 6.16 [12]) that for many L one has ≅|_{Mod(L)} non-Borel, a proof of

≈^{∗}≤_{B} ≅|_{Mod(L)}

will in particular give that ≈^{∗} is non-Borel.

It is rather cumbersome to be continually working with the full range of possible ≅|_{Mod(L)} as L ranges over countable languages. Instead it will be convenient to work with a canonical example, which is already known to have maximal complexity in the ≤_{B}-ordering.

For us a graph will be a directed graph where loops are possible but parallel edges are not. Thus we may naturally identify a graph on the underlying set ℕ with a binary relation on ℕ, and this in turn may by consideration of the characteristic function be identified with 2^{ℕ×ℕ}.

Definition Let Mod(Gph) be the Polish space 2^{ℕ×ℕ} equipped with the product topology.

We let the infinite symmetric group, S_{∞}, consisting of all permutations of the natural numbers, act on 2^{ℕ×ℕ} in the following manner: Given σ ∈ S_{∞} and x ∈ 2^{ℕ×ℕ}, we define σ · x by

(σ · x)(n, m) = x(σ^{−1}(n), σ^{−1}(m)).

We then let

E_{S_{∞}}^{Mod(Gph)}

denote the resulting orbit equivalence relation:

x_{1}E_{S_{∞}}^{Mod(Gph)}x_{2}

if and only if

∃σ ∈ S_{∞}(σ · x_{1} = x_{2}).

Thus E_{S_{∞}}^{Mod(Gph)} is the isomorphism relation for the space of all binary relations on ℕ. Of course as a space Mod(Gph) is nothing other than 2^{ℕ×ℕ}; it will be convenient to have this separate notation, to remind ourselves with Mod(Gph) that we are thinking of 2^{ℕ×ℕ} as a Polish S_{∞}-space in a specific way.

Lemma 2.3 If L is a countable language, then

≅|_{Mod(L)} ≤_{B} E_{S_{∞}}^{Mod(Gph)}.

Proof A proof of this well known folklore fact can be found in many places, including [5]. □

Thus the task of showing ≈^{∗} non-Borel has been reduced to showing that for any countable language L we have

≅|_{Mod(L)} ≤_{B}≈^{∗},

which has in turn been reduced to showing

E_{S_{∞}}^{Mod(Gph)} ≤_{B}≈^{∗} .

In order to do this we will introduce one further equivalence relation, E_{G_{∞}}^{Y_{2}} defined shortly, and show in section 3 first that

E_{G_{∞}}^{Y_{2}} ≤_{B}≈^{∗},

and then in section 4 that

E_{S_{∞}}^{Mod(Gph)} ≤_{B} E_{G_{∞}}^{Y_{2}}.

Definition Let M(S_{∞}) = {g ∈ (S_{∞})^{[0,1]} : g Borel} be the group of measurable functions from [0, 1] to S_{∞}, where S_{∞} is the infinite symmetric group on ℕ; we multiply pointwise,

(g_{1}g_{2})(x) = g_{1}(x)g_{2}(x),

and identify functions that agree λ a.e.

Observe then that if g^{−1} is the group theoretic inverse of g ∈ M(S_{∞}) then g^{−1}(x) = (g(x))^{−1} for (λ a.e) x ∈ [0, 1].

Definition Define ψ : M_{∞} → Aut(M(S_{∞})) by the requirement that for π ∈ M_{∞}, x ∈ [0, 1], g ∈ M(S_{∞})

((ψ(π))(g))(x) = g(π^{−1}(x)).

So ψ(π) is the (group theoretic) automorphism of M(S_{∞}) obtained by shift.

In fact these groups all have natural topologies and ψ is a homeomorphism from M_{∞} into the group of continuous automorphisms of M(S_{∞}).

Definition Form the semi-direct product M(S_{∞}) ⋊_{ψ} M_{∞} in the usual way, so that for g_{0}, g_{1} ∈ M(S_{∞}), π_{0}, π_{1} ∈ M_{∞}

(g_{0}, π_{0})(g_{1}, π_{1}) = (g_{0}(ψ(π_{0}))(g_{1}), π_{0}π_{1}).

For short write G_{∞} =_{df} M(S_{∞}) ⋊_{ψ} M_{∞}.

Definition For y ∈ 2^{ℕ×ℕ} and n ∈ ℕ we define y(n, ·) ∈ 2^{ℕ} in the obvious way, by

(y(n, ·))(m) = y(n, m).

Let

B_{2} = {x ∈ 2^{ℕ×ℕ} : n≠ m ⇒ x(n, ·)≠ x(m, ·)};

B_{2} is a G_{δ} subset of the Polish space 2^{ℕ×ℕ} and hence Polish (see [15] 3C). Let

Y_{2} = {y ∈ (B_{2})^{[0,1]} | y Borel}

be the space of measurable functions from [0, 1] to B_{2} where we identify functions agreeing almost everywhere. We give this space the topology of convergence almost every where, so that

f_{n} → f

if for almost every x ∈ [0, 1] we have f_{n}(x) → f(x).

Lemma 2.4 Y_{2} is a Polish space.

Proof Let d′ be a complete compatible metric on B_{2}. We obtain a complete metric d_{2} by setting

d_{2}(y_{0}, y_{1}) = ∫ d′(y_{0}(x), y_{1}(x))λ(x)

for y_{0}, y_{1} ∈ Y_{2}. □

Definition Let G_{∞} act on Y_{2} as follows: Given y ∈ Y_{2},

y : [0, 1] → 2^{ℕ×ℕ},

and

(g, π) ∈ G_{∞},
g : [0, 1] → S_{∞},
π ∈ M_{∞},

we define (g, π) · y : [0, 1] → 2^{ℕ×ℕ} by

(((g, π)(y))(x))(m, n) = (y(π^{−1}(x)))(g^{−1}(x)(m), n).

(Here g^{−1} is intended to be the group theoretic inverse of g ∈ M(S_{∞}); thus g^{−1}(x) = (g(x))^{−1}.) Observing

([(g_{0}, π_{0})((g_{1}, π_{1})(y))](x))(m, n) = [((g_{1}, π_{1})(y))(π_{0}^{−1}(x))]((g_{0}^{−1}(x))(m), n)

= (y(π_{1}^{−1}(π_{0}^{−1}(x))))(g_{1}^{−1}(π_{0}^{−1}(x))(g_{0}^{−1}(x))(m), n)
= (y(π_{1}^{−1}(π_{0}^{−1}(x))))((((ψ(π_{0}))(g_{1}))^{−1}g_{0}^{−1})(x))(m), n).

The last equality uses that the group operations for M(S_{∞}) are calculated pointwise, and hence

g_{1}^{−1}(π_{0}^{−1}(x)) = (g_{1}(π_{0}^{−1}(x)))^{−1}

and

[(((ψ(π_{0}))(g_{1}))^{−1}g_{0}^{−1})](x) = [((ψ(π_{0}))(g_{1}^{−1}))(x)][g_{0}^{−1}(x)].

Thus,

[(((g_{0}, π_{0})((g_{1}, π_{1})(y)))(x))](m, n) = [y(π_{1}^{−1}(π_{0}^{−1}(x)))](((((ψ(π_{0}))(g_{1}))^{−1}g_{0}^{−1})(x))(m), n)

= [((g_{0}(ψ(π_{0})(g_{1})), π_{0}π_{1})(y))(x)](m, n),

which establishes this to be an action.

We then let E_{G_{∞}}^{Y_{2}} be the orbit equivalence relation on Y_{2} resulting from this action.

Lemma 2.5 For y_{0}, y_{1} ∈ Y_{2} we have

y_{0}E_{G_{∞}}^{Y_{2}}y_{1}

if and only if there is some π ∈ M_{∞} such that

λ({x : {(y_{0}(x))(n, ·) : n ∈ ℕ} = {(y_{1}(π^{−1}(x)))(n, ·) : n ∈ ℕ}}) = 1.

Proof The “only if” part of the lemma is trivial. The “if” direction uses the well known fact, a proof of which can be found in 18A [15], that any Borel set in the plane may be uniformized by a Lebesgue measurable function. □

Notation Following the Kuratowski-Mycielski theorem of 19A [15], choose C ⊂ [0, 1] to be a perfect set such that for any k ∈ ℕ and

x_{1}, x_{2}, ..., x_{k} ∈ C

we have that if x_{i} ≠ x_{j} all i < j ≤ k then x_{1}, x_{2}, ..., x_{k} are rationally independent. Fix a continuous injection

φ_{C} : 2^{ℕ} ↪ C.

To each element y of Y_{2} we wish to associate a measure preserving transformation

T_{y} : [0, 1] × (ℝ/ℤ)^{ℕ} → [0, 1] × ℝ/ℤ^{ℕ}

whose ergodic components have the form

{x} × (ℝ/ℤ)^{ℕ}

for x ∈ [0, 1]; on each such ergodic component we will have a discrete spectrum measure preserving transformation with eigenvalues {e^{2πiφ_{C} ((y(x))(n,·))} : n ∈ ℕ}. The proof that y_{1}E_{G_{∞}}^{Y_{2}}y_{2} if and only if T_{y_{1}} and T_{y_{2}} are conjugate is then a consequence of the well known fact that two measure preserving transformations are conjugate if and only if there is a measure one set on which their ergodic components are individually conjugate component by component. Partly for the convenience of the reader, and partly because there seems no easy source listing exactly the facts we need in exactly the form we need them, we write out the proof in §3 without assuming any familiarity with the ergodic decomposition of a measure preserving transformation.

3 E_{G_{∞}}^{Y_{2}} ≤_{B}≈^{∗}

Definition Let T : (X, B, μ) → (X, B, μ) be Borel a measure preserving map, X a Polish space, μ a Borel probability measure on X, B the collection of Borel subsets of X. A non-zero f ∈ L^{2}(X, μ)(=_{df} the Hilbert space of all square integrable complex valued functions on (X, μ), subject to the usual identification of functions that agree almost everywhere) is said to be an eigenfunction for T if for some λ ∈ ℂ we have f ∘ T = λf a.e; we then also say that λ is an eigenvalue. T is said to be ergodic if all T -invariant Borel sets are either null or conull with respect to μ.

Lemma 3.1 T : (X, B, μ) → (X, B, μ) is ergodic if and only if the space of eigenfunctions with eigenvalue 1 is one dimensional.

Proof If f : X → ℂ is a non-constant eigenfunction for the eigenvalue 1, then for some U ⊂ ℂ the pullbacks f^{−1}[U] and f^{−1}[ℂ \ U] are disjoint and non-null. □

Hence if T is ergodic then no eigenvalue can have corresponding eigenspace with dimension greater than 1 – for if f_{1}, f_{2} : X → ℂ are linearly independent non-zero functions with f_{1} ∘ T = cf_{1} and f_{2} ∘ T = cf_{2} then

f_{2}/f_{1} : X → ℂ

would be a non-constant function with eigenvalue 1.

Definition Let X = [0, 1] × ℝ/ℤ^{ℕ} with the product of Lebesgue measure on [0, 1] and Haar measure ν on each copy of ℝ/ℤ (so that ν({xℤ : a ≤ x ≤ b}) = b − a for any 0 < a < b ≤ 1). For y ∈ Y_{2} we define

T_{y} : X → X

by

T_{y}(x, z_{0}, z_{1}, ...) = (x, φ_{C}((y(x))(0, ·)) ⊕ z_{0}, φ_{C}((y(x))(1, ·)) ⊕ z_{1}, ...),

where ⊕ is addition modulo 1. For x ∈ [0, 1] we let T_{y}^{x} : (ℝ/ℤ)^{ℕ} → (ℝ/ℤ)^{ℕ} be the map resulting from restriction to the fiber above x:

T_{y}^{x}(z_{0}, z_{1}, ...) = (φ_{C}((y(x))(0, ·)) ⊕ z_{0}, φ_{C}((y(x))(1, ·)) ⊕ z_{1}, ...)

Lemma 3.2 For all y ∈ Y_{2} and x ∈ [0, 1]:

(i) the set of eigenvalues of T_{y}^{x} is the subgroup of the complex unit circle generated by {e^{2πiφ_{C} ((y(x))(n,·)} : n ∈ ℕ}; (ii) T_{y}^{x} : (ℝ/ℤ)^{ℕ} → (ℝ/ℤ)^{ℕ} is ergodic.

Proof For x such that {(y(x))(n, ·) : n ∈ ℕ} is a rationally independent set we can use the Stone-Weierstrass theorem to see that the sums of the finite multiples of the projection functions and their inverses

P r_{k} : (ℝ/ℤ)^{ℕ} → ℂ,
z⃗ ↦ e^{2πiz_{k}}

are dense in L^{2}(ℝ/ℤ^{ℕ}, μ^{ℕ}). Hence the functions of the form

(P r_{k_{1}})^{n_{1}} · (P r_{k_{2}})^{n_{2}}...(P r_{k_{p}})^{n_{p}},

as ⟨n_{1}, n_{2}, ..., n_{p}⟩ and ⟨k_{1}, ..., k_{p}⟩ range over finite sequences from ℤ and ℕ, form a Hilbert basis for L^{2}(ℝ/ℤ^{ℕ}, μ^{ℕ}). The rational independence property assumed for C implies

(P r_{k_{1}})^{n_{1}} · (P r_{k_{2}})^{n_{2}}...(P r_{k_{p}})^{n_{p}}

and

(P r_{l_{1}})^{m_{1}} · (P r_{l_{2}})^{m_{2}}...(P r_{l_{q}})^{m_{q}}

have distinct eigenvalues whenever these functions are distinct, any eigenfunctions must be a finite multiple of these coordinate functions

P r_{k} : (ℝ/ℤ)^{ℕ} → ℂ,
z⃗ ↦ e^{2πiz_{k}}.

Thus up to scalar multiplication the only eigenfunctions are 1 and the finite multiples of the coordinate projections, {P r_{k} : k ∈ ℕ}. □

Lemma 3.3 If y ∈ Y_{2} and A ⊂ X is Borel and T_{y}-invariant then there is a Borel B ⊂ X so that A = B × (ℝ/ℤ)^{ℕ} modulo some null set.

Proof For each x ∈ [0, 1] we have that

A^{x} = {z⃗ : (x, z⃗) ∈ A}

is T_{y}^{x}-invariant. Thus by 3.2

ν^{ℕ}(A^{x}) ∈ {0, 1}

(where ν^{ℕ} is the ℕ-fold product of Haar measure ν on ℝ/ℤ). Thus for B = {x ∈ [0, 1] : ν^{ℕ}(A^{x}) = 1} we have A = B × (ℝ/ℤ)^{ℕ} off of a null set by Fubini. □

Lemma 3.4 If y_{1}, y_{2} ∈ Y_{2}, x_{1}, x_{2} ∈ [0, 1], then the transformations T_{y_{1}}^{x_{1}} and T_{y_{2}}^{x_{2}} are conjugate if and only if

{(y_{1}(x_{1}))(n, ·) : n ∈ ℕ} = {(y_{2}(x_{2}))(n, ·) : n ∈ ℕ}.

Proof By 3.2, the transformations T_{y_{1}}^{x_{1}} and T_{y_{2}}^{x_{2}} are conjugate only if the set of eigenvalues are equal, which is to say that the multiplicative subgroup of the complex unit circle generated by {e^{2πiφ_{C} ((y_{1}(x_{1})(n,·))} : n ∈ ℕ} and {e^{2πiφ_{C} ((y_{2}(x_{2})(n,·))} : n ∈ ℕ} are equal; and this in turn, by the assumptions on C, holds only if

{e^{2πiφ_{C} ((y_{1}(x_{1})(n,·))} : n ∈ ℕ} = {e^{2πiφ_{C} ((y_{2}(x_{2})(n,·))} : n ∈ ℕ}.

The converse direction is trivial. □

Lemma 3.5 If y_{1}, y_{2} ∈ Y_{2} with T_{y_{1}}, T_{y_{2}} conjugate then we may find g ∈ G_{∞} with g · y_{1} = y_{2}.

Proof If π : X → X is measure preserving and conjugates T_{y_{1}} and T_{y_{2}} then we use 3.3 to find some measure one set M such that for all basic open O ⊂ [0, 1] there is B_{O}, D_{O} ⊂ [0, 1] so that

π[(O × (ℝ/ℤ)^{ℕ}) ∩ M] = (B_{O} × (ℝ/ℤ)^{ℕ}) ∩ M,
π^{−1}[(O × (ℝ/ℤ)^{ℕ}) ∩ M] = (D_{O} × (ℝ/ℤ)^{ℕ}) ∩ M.

We may also assume that M is invariant under π, T_{y_{1}}, T_{y_{2}}. Thus we have that on the measure one set M for all (x, z⃗) ∈ M there is some π̂(x) ∈ [0, 1] so that π|_{({x}×ℝ/ℤ^{ℕ})} conjugates

T_{y_{1}}|_{({x}×ℝ/ℤ^{ℕ}∩M)}

and

T_{y_{2}}|_{({π̂(x)}×ℝ/ℤ^{ℕ}∩M)}.

This π̂ : [0, 1] → [0, 1] is measure preserving since for a.e. (x, z⃗) ∈ [0, 1] × 𝕋^{ℕ} we have

π(x, z⃗) = (π̂(x), z′),

some z′ ∈ 𝕋^{ℕ}. Thus we are finished by 2.5. □

Lemma 3.6 If g ∈ G_{∞} with g · y_{1} = y_{2} then T_{y_{1}} and T_{y_{2}} are conjugate.

Proof This is simply unpacking the definitions.

First consider the case that g = (1, π̂), some π̂ ∈ M_{∞}. Then

(y_{2}(x))(n, ·) = ((g · y_{1})(x))(n, ·) =_{df} (y_{1}(π̂^{−1}(x)))(n, ·)

for a.e. x ∈ [0, 1] and all n ∈ ℕ. Thus we can define π : X → X by π(x, z⃗) = (π̂(x), z⃗) to obtain

T_{y_{2}} = π ∘ T_{y_{1}} ∘ π^{−1}.

Similarly if σ ∈ M(S_{∞}) with (σ, 1) · y_{1} = y_{2} then we can define π : X → X by

(x, z_{0}, z_{1}, z_{2}, ...) ↦ (x, z_{(σ(x))(0)}, z_{(σ(x))(1)}, ...).

The above terminates the proof, since any g ∈ G_{∞} can be written in the form g = (σ, 1)(1, π̂). □

Definition Following earlier notation, let M_{∞}(X) be the group of Borel measure preserving bijections from X to X, again identifying two maps that agree a.e with respect to the measure λ × ν^{ℕ}. For (O_{n})_{n∈ℕ} a basis of X we obtain a complete separable metric d_{X} on M_{∞}(X) by setting

d_{X}(π_{1}, π_{2}) = ∑ 2^{−n}[λ×ν^{ℕ}(π_{1}(O_{n})Δπ_{2}(O_{n}))+λ×ν^{ℕ}(π_{1}^{−1}(O_{n})Δπ_{2}^{−1}(O_{n}))].

Note then that M_{∞}(X) is a Polish group under composition. It is in fact isomorphic to M_{∞}, since we can find a measure isomorphism Φ : (X, λ × ν^{ℕ}) → ([0, 1], λ) and then take the induced isomorphism

Φ̂ : M_{∞}(X) → M_{∞},
π ↦ Φ^{−1} ∘ π ∘ Φ.

Let ≈^{∗∗} denote the conjugacy equivalence relation on M_{∞}(X).

Lemma 3.7 The map

Y_{2} → M_{∞}(X),
y ↦ T_{y},

is continuous.

Proof Let M(λ × ν^{ℕ}) be the algebra of measurable subsets of (X, λ × ν^{ℕ}) subject to the usual identification of sets which agree off of a null set; this a Polish space in the metric d(A, B) = λ × ν^{ℕ}(AΔB) (see for instance 2.2 [12]). First note almost immediately from the definitions that if A ⊂ [0, 1]×(ℝ/ℤ)^{ℕ} is a basic open set of the form {x⃗ : x(i) ∈ J} for some open interval J included in either [0, 1] or ℝ/ℤ, then the resulting map into the measure algebra

Y_{2} → M(λ × ν^{ℕ}),
y ↦ T_{y}(A)

is continuous. Since the subalgebra of the Boolean algebra of measurable sets in (X, λ × ν^{ℕ}) generated by the cylinders is dense, we obtain that y ↦ T_{y} is continuous. □

Proposition 3.8 E_{G_{∞}}^{Y_{2}} ≤_{B}≈^{∗} .

Proof X and [0, 1] are two non-atomic, standard Borel probability spaces, and hence^{5} they are isomorphic as measure spaces. Thus it suffices to show

E_{G_{∞}}^{Y_{2}} ≤_{B}≈^{∗∗},

which is exactly the content of the last three lemmas. □

There are some details here which were not needed in developing a proof of 3.8 but which might have independent interest. Namely, the group G_{∞} is a Polish group, and its action on Y_{2} is not only continuous but also turbulent in the sense of [12].

5See for instance 17.41 of [15]

4 E_{S_{∞}}^{Mod(Gph)} ≤_{B} E_{G_{∞}}^{Y_{2}}

Notation From now until the end of the section fix continuous one-to-one

f_{0} : 2^{ℕ} ↪ 2^{ℕ}
f_{1} : 2^{ℕ} × 2^{ℕ} ↪ 2^{ℕ}

with

f_{0}[2^{ℕ}] ∩ f_{1}[2^{ℕ} × 2^{ℕ}] = ∅.

It is easily seen that such a pair f_{0}, f_{1} exists. For instance

(f_{0}(y))(n + 1) = y(n),
(f_{0}(y))(0) = 0,
(f_{1}(y_{1}, y_{2}))(2n + 1) = y_{1}(n),
(f_{1}(y_{1}, y_{2}))(2n + 2) = y_{2}(n),
(f_{1}(y_{1}, y_{2}))(0) = 1.

Notation For z⃗ = (z_{i})_{i∈ℕ} ∈ (2^{ℕ})^{ℕ} and x ∈ Mod(Gph) we let

P(2^{ℕ}, x, z⃗) = {y ∈ 2^{ℕ} | ∃n(y = f_{0}(z_{n}))}∪{y ∈ 2^{ℕ} | ∃n, m(y = f_{1}(z_{n}, z_{m}), x(n, m) = 1)}.

We have previously defined B_{2} to be the set of w ∈ 2^{ℕ×ℕ} such that for all n_{1} ≠ n_{2} we have w(n_{1}, ·)≠ w(n_{2}, ·).

Lemma 4.1 There is a Borel function

φ_{eva} : Mod(Gph) × (2^{ℕ})^{ℕ} → B_{2}

such that for all x ∈ Mod(Gph) and z⃗ ∈ (2^{ℕ})^{ℕ} with z_{i} ≠ z_{j} all i ≠ j we have

{(φ_{eva}(x, z⃗))(n, ·) | n ∈ ℕ} = P(2^{ℕ}, x, z⃗).

Proof We may partition Mod(Gph) into Borel sets

A_{0}, A_{1}, A_{2}, ..., A_{ℵ_{0}}

such that for each κ ∈ {0, 1, 2, ..., ℵ_{0}} and each x ∈ A_{κ} there are exactly κ many pairs (n, m) with x(n, m) = 1. It suffices then to show φ_{eva}|_{A_{κ}×(2^{ℕ})^{ℕ}} is Borel for each κ.

Fixing κ, we divide ℕ into sets {a_{i} : i ∈ ℕ}, {b_{j} : j < κ}. Then for a given x ∈ A_{κ} we can let (m_{j}, n_{j})_{j<κ} enumerate, in the ordering obtained by comparing maximums and then adjudicating ties lexicographically, the pairs (m, n) with x(m, n) = 1. We can then let

φ_{eva}(x, z⃗) ∈ 2^{ℕ×ℕ}

be defined by

(φ_{eva}(x, z⃗))(a_{i}, m) = (f_{0}(z_{i}))(m),
(φ_{eva}(x, z⃗))(b_{j}, m) = (f_{1}(z_{m_{j}} , z_{n_{j}}))(m).

The verification that the resulting function is Borel is routine. □

Notation Let μ be the usual product measure on 2^{ℕ}, so that for each n we have

μ(\{x ∈ 2^{ℕ} | x(n) = 1\}) = \frac{1}{2} .

Let μ^{ℕ} be its corresponding product measure on (2^{ℕ})^{ℕ}. Let N_{2} = {z⃗ ∈ (2^{ℕ})^{ℕ} | ∀i ≠ j(z_{i} ≠ z_{j})}, and note that μ^{ℕ}(N_{2}) = 1.

Since ((2^{ℕ})^{ℕ}, μ^{ℕ}) and ([0, 1], Lebesgue measure) are both non-atomic and both have standard Borel structures, we can find a Borel measure preserving bijection

φ_{iso} : N_{2} → [0, 1].

Notation For x ∈ Mod(Gph) define

ψ_{x} : [0, 1] → 2^{ℕ×ℕ}

by

ψ_{x}(α) = φ_{eva}(x, φ_{iso}^{−1}(α)).

Lemma 4.2 The function

x ↦ ψ_{x}

is a Borel function from Mod(Gph) to Y_{2}.

Proof The key point is that we can apply 4.1 to see that if

f(·, ·) = φ_{eva}(·, φ_{iso}^{−1}(·))

then for each x the set

{ψ_{x}} = {ψ ∈ Y_{2} | ψ(β) = f(x, β) a.e.β}

is a uniformly in x Borel singleton; thus the assignment of ψ_{x} to x is Borel by the uniformization theorem for Borel subsets of the plane with countable sections. □

Lemma 4.3 If x_{1}, x_{2} ∈ Mod(Gph) with

x_{1}E_{S_{∞}}^{Mod(Gph)}x_{2},

then

ψ_{x_{1}}E_{G_{∞}}^{Y_{2}}ψ_{x_{2}}.

Proof Fix σ ∈ S_{∞} with

σ · x_{1} = x_{2}.

Thus for all (n, m) ∈ ℕ × ℕ we have

x_{1}(n, m) = x_{2}(σ(n), σ(m)).

And so if we define

σ̂ : (2^{ℕ})^{ℕ} → (2^{ℕ})^{ℕ}

by

(σ̂(z⃗))_{n} = z⃗_{σ(n)}

then σ̂ is an invertible measure preserving transformation such that at each z⃗

P(2^{ℕ}, x_{1}, z⃗) = P(2^{ℕ}, x_{2}, σ̂(z⃗)).

From this we obtain ψ_{x_{1}}E_{S_{∞}}^{Y_{2}}ψ_{x_{2}} by 2.5. □

Lemma 4.4 If x_{1}, x_{2} ∈ Mod(Gph) with

ψ_{x_{1}}E_{G_{∞}}^{Y_{2}}ψ_{x_{2}},

then

x_{1}E_{G_{∞}}^{Mod(Gph)}x_{2}.

Proof The assumption that ψ_{x_{1}}E_{G_{∞}}^{Y_{2}}ψ_{x_{2}} in particular implies the existence of some z^{1} and z^{2} with

P(2^{ℕ}, x_{1}, z^{1}) = P(2^{ℕ}, x_{2}, z^{2})

and for all n ≠ m

(z^{1})_{n} ≠ (z^{1})_{m},
(z^{2})_{n} ≠ (z^{2})_{m}.

This implies

{f_{0}((z^{1})_{i}) : i ∈ ℕ} = {f_{0}((z^{2})_{i}) : i ∈ ℕ},

and so we can find some σ ∈ S_{∞} with

(z^{1})_{n} = (z^{2})_{σ(n)}

all n ∈ ℕ. At this point the assumptions on f_{0} and f_{1} give that for all n, m

x_{1}(n, m) = x_{2}(σ(n), σ(m)),

and so

σ · x_{1} = x_{2}.
□

Proposition 4.5 E_{S_{∞}}^{Mod(Gph)} ≤_{B} E_{G_{∞}}^{Y_{2}}

Proof This is exactly what the last three lemmas show. □

5 Turbulence for the generalized discrete spectrum transformations

Below we consider the isomorphism relation for generalized discrete spectrum transformations, or what [2] calls the “measure-distal” actions. The results below show non-classifiability by countable structures, but perhaps raise more questions than they answer. For instance it is not known if there is a way to embed isomorphism of countable models into the generalized discrete spectrum transformations, or more modestly just embed the equivalence relation

F_{α},

arising from the αth iteration of the operation “countable subset of” applied to some standard Borel space^{6} into some appropriate level of the generalized discrete spectrum hierarchy.

Just by way of comparison, I should mention that for the very simple subclass consisting of the transformations having completely discrete spectrum the situation is totally understood. Here [9] shows that we may assign countable subsets of ℂ as complete invariants. Indeed Foreman and Louveau have observed that even in the Borel context this precisely encapsulates the classification difficulty of the discrete spectrum maps.

Theorem 5.1 (Foreman, Louveau) Let D ⊂ M_{∞} be the class of discrete spectrum measure preserving transformations. Then:

(i) The set D is a Borel subset of M_{∞}. (ii) There is a sequence of Borel functions

f_{n} : D → ℝ

such that for all π_{1}, π_{2} ∈ D

π_{1} ≈^{∗} π_{2}

if and only if

{f_{n}(π_{1}) : n ∈ ℕ} = {f_{n}(π_{2}) : n ∈ ℕ};

thus there is a Borel

θ_{1} : D → 2^{ℕ×ℕ}

such that for all π_{1}, π_{2} ∈ D

π_{1} ≈^{∗} π_{2} ↔ {(θ_{1}(π_{1}))(n, ·) : n ∈ ℕ} = {(θ_{1}(π_{2}))(n, ·) : n ∈ ℕ},

so in other words we have

≈^{∗} |_{D} ≤_{B} F_{2}.

(iii) And conversely,

F_{2} ≤_{B}≈^{∗} |_{D}

6See [5].

in the sense that there is a Borel function

θ_{2} : 2^{ℕ×ℕ} → D

such that for all x_{1}, x_{2} ∈ 2^{ℕ×ℕ}

{x_{1}(n, ·) : n ∈ ℕ} = {x_{2}(n, ·) : n ∈ ℕ} ⇔ θ_{2}(x_{1}) ≈^{∗} θ_{2}(x_{2}).

In particular the isomorphism relation on the discrete spectrum transformations is non-smooth.

Notation Let 𝕋 = {e^{2πix} : x ∈ [0, 1]}, the complex unit circle, be viewed as a group under multiplication. For the remainder of this section let λ be the usual Lebesgue measure on 𝕋 normalized so that λ(𝕋) = 1.

Let H_{0} = {f : 𝕋 → 𝕋 | f is Lebesgue measurable}, where we identify f_{0}, f_{1} ∈ H_{0} if they agree λ a.e. For f_{0}, f_{1} ∈ H_{0} the product f_{0}f_{1} ∈ H_{0} is defined by pointwise multiplication:

(f_{0}f_{1})(ζ) = (f_{0}(ζ))(f_{1}(ζ)).

We give this group the topology of a.e. pointwise convergence; which is to say the topology induced by the metric

d_{0}(f_{0}, f_{1}) = 1/2 ∫ |f_{0}(ξ) − f_{1}(ξ)|dλ,

where |·| is the usual Euclidean distance in ℂ. The technical advantage of just this choice for d_{0} is that whenever f_{0} and f_{1} differ on a set of measure less than ϵ we must have

d_{G}((1, ¯0, f_{0}), (1, ¯0, f_{1})) < ϵ.

In future I will use 1 to denote the function in H_{0} which constantly takes the value 1 for all ζ ∈ 𝕋. This is the group identity, which of course creates some notational conflict with H_{0} being commutative.

Lemma 5.2 H_{0} is an abelian Polish group.

Proof The metric d_{0} is easily checked to be complete, continuous with respect to the group action, and separable since L^{1}(𝕋, λ) is separable. □

Notation Let H_{1} = 𝕋 × ℤ_{2} be the direct product of the groups 𝕋 and ℤ_{2}. We define

φ : H_{1} → Aut(H_{0})

by the requirement that

((φ(ζ, ¯1))(f))(ξ) = (f(ξζ))^{−1}
((φ(ζ, ¯0))(f))(ξ) = (f(ξζ)).

Note that H_{1} is a compact Polish group and φ is a group homomorphism.

I will write φ_{(ζ,k̄)} for the homomorphism φ(ζ, k̄) : H_{0} → H_{0}. The map φ is continuous in the following sense:

Lemma 5.3 The function

H_{1} × H_{0} → H_{0}
((ζ, ī), f) ↦ φ_{(ζ,ī)}(f)

is continuous as a map from H_{1} × H_{0} to H_{0}.

Proof Recall that the step functions consisting of finite linear combinations of the characteristic functions of intervals are dense in L^{1}(𝕋, λ). This rapidly implies that for f_{0} ∈ H_{0} and ϵ > 0 there is g ∈ H_{0} of the form

g = ∑_{j=1}^{j=k} c_{j}χ_{A_{j}}

with

d_{0}(g, f_{0}) < \frac{ϵ}{3} ,

some k ∈ ℕ, c_{1}, c_{2}, ..., c_{k} ∈ ℂ, of absolute value 1, and measurable subsets A_{1}, ..., A_{k} of 𝕋, each given by

A_{n} = {e^{2πix} | a_{n} ≤ x < a_{n+1}}.

For continuity it is enough to check that for

(ζ, j̄) = (e^{2πiy}, j̄)

sufficiently close to the identity and f ∈ H_{0} sufficiently close to f_{0} we have

d_{0}(φ_{(e^{2πiy},j̄)}(f), f_{0}) < ϵ.

But if j̄ = ¯0 and y is close enough to 0 that

|c_{n}y| = |y| < \frac{ϵ}{3k} ,

n = 1, 2, ..., k, then

d_{0}(φ_{(e^{2πiy},¯0)}(g), g) < \frac{ϵ}{3} ,

since at each n ≤ k

λ(A_{n}Δ{e^{2πy}ζ|ζ ∈ A_{n}}) ≤ |y|.

φ_{(e^{2πiy},¯0)} is an isometry, so we have for f close to f_{0} that

d_{0}(φ_{(e^{2πiy},¯0)}(g), φ_{(e^{2πiy},¯0)}(f)) < \frac{ϵ}{3} ,

and thus by the triangle inequality

d_{0}(φ_{(e^{2πiy},j̄)}(f), f_{0}) ≤ d_{0}(φ_{(e^{2πiy},j̄)}(f), φ_{(e^{2πiy},¯0)}(g))+d_{0}(φ_{(e^{2πiy},¯0)}(g), g)+d_{0}(g, f_{0})

< \frac{ϵ}{3} + \frac{ϵ}{3} + \frac{ϵ}{3} = ϵ.

□

Notation Let G = H_{1} ⋊_{φ} H_{0} be the semi-direct product of H_{1} and H_{0} along φ. Thus for (ζ_{0}, ī_{0}, f_{0}), (ζ_{1}, ī_{1}, f_{1}) ∈ G

(ζ_{0}, ī_{0}, f_{0})(ζ_{1}, ī_{1}, f_{1}) = (ζ_{0}ζ_{1}, ī_{0} + ī_{1}, (φ(ζ_{1}, ī_{1}))(f_{0})f_{1}).

This order of taking the semi-direct product does give us a group since H_{1} is abelian.

Here and elsewhere I simply write (ζ, ī, f) for an arbitrary element of G, instead of the more cumbersome but perhaps more formally correct ((ζ, ī), f).

Fix complete metrics d_{0} and d_{1} on H_{0} and H_{1}.

Lemma 5.4 G is a Polish group.

Proof We obtain a complete and compatible metric d_{G} on G by

d_{G}((ζ_{0}, ī_{0}, f_{0}), (ζ_{1}, ī_{1}, f_{1})) = d_{1}((ζ_{0}, ī_{0}), (ζ_{1}, ī_{1})) + d_{0}(f_{0}, f_{1}).

It follows from 5.3 and H_{0} and H_{1} being topological groups that the group operation of multiplication is continuous on G. Since G is Polish as a space it follows from say [21] that

g ↦ g^{−1}

is continuous as well. □

Notation Let X = {h : 𝕋 → 𝕋 | h is Lebesgue measurable}, where we identity functions agreeing a.e. For future reference we let d_{X} be a metric given on X by d_{X}(h_{0}, h_{1}) = 1/2(∫ |h_{0}(ξ) − h_{1}(ξ)|dλ).

We let G act on X as follows:

((ζ, ī, f) · h)(ξ) = [f(ξ/ζ)h(ξ/ζ)(f(e^{2πi√2}ξ/ζ))^{−1}]^{(−1)^{i}}.

(Of course literally as a space X is the same as H_{0}. But here we are thinking of X as coming equipped with a G-action, while we think of H_{0} as presented with a group structure.)

Lemma 5.5 This is an action.

Proof It is trivial to confirm that

(1, ¯0, 1) · h = h

all h ∈ X; the main task is to show that the associativity properties of the action.

Let Ĥ_{0} be subgroup of G consisting of elements of the form

(1, ¯0, f)

and let Ĥ_{1} consist of those of the form

(ζ, ī, 1).

Every element g of G can be written in the form

g = h_{0}h_{1} = h_{1}^{∗}h_{0}^{∗}

for suitable h_{0}, h_{0}^{∗} ∈ Ĥ_{0} and h_{1}, h_{1}^{∗} ∈ Ĥ_{1}. Hence it suffices to show purely for k_{1}, k_{2} ∈ Ĥ_{0} ∪ Ĥ_{1} that we have for all x ∈ X that

k_{1} · (k_{2} · x) = (k_{1}k_{2}) · x;

the point is that given arbitrary g_{1}, g_{2} ∈ G we can write

g_{1} = h_{0,1}h_{1,1}
g_{2} = h_{1,2}h_{0,2}

(for suitable h_{0,i} ∈ Ĥ_{0}, h_{1,i} ∈ Ĥ_{1}) and then steadily multiply through to get

g_{1} · (g_{2} · x) = h_{0,1} · h_{1,1} · h_{1,2} · h_{0,2} · x
= h_{0,1} · (h_{1,1}h_{1,2}) · h_{0,2} · x
= (h_{0,1}h_{1,1}h_{1,2}) · h_{0,2} · x = h_{1}^{∗}h_{0}^{∗} · h_{0,2} · x

(suitable h_{0}^{∗} ∈ Ĥ_{0}, h_{1}^{∗} ∈ Ĥ_{1})

h_{1}^{∗} · (h_{0}^{∗}h_{0,2}) · x = (h_{1}^{∗}h_{0}^{∗}h_{0,2}) · x.

So we are left only with checking the associativity of the action for k_{1}, k_{2} ∈ Ĥ_{0} ∪ Ĥ_{1}. There are four possibilities, but only the case k_{2} ∈ Ĥ_{1}, k_{1} ∈ Ĥ_{0} requires close inspection.

Here however we see that for any h ∈ X, (ζ, ī) ∈ H_{1}, f ∈ H_{0},

(((1, ¯0, f)(ζ, ī, 1)) · h)(ξ) = ((ζ, ī, φ_{(ζ,ī)}(f)) · h)(ξ)
= [(φ_{(ζ,ī)}(f))(ξ/ζ))h(ξ/ζ)((φ_{(ζ,ī)}(f))(ξe^{2πi√2}/ζ))^{−1}]^{(−1)^{i}}
= [(f(ζξ/ζ))^{(−1)^{i}}h(ξ/ζ)(f(ζe^{2πi√2}ξ/ζ))^{(−1)^{i+1}}]^{(−1)^{i}}
= (f(ξ))^{(−1)^{2i}}(h(ξ/ζ))^{(−1)^{i}}(f(ξe^{2πi√2}))^{(−1)^{2i+1}}
= f(ξ)(h(ξ/ζ))^{(−1)^{i}}(f(ξe^{2πi√2}))^{−1}

= f(ξ)(((ζ, ī, 0) · h)(ξ))(f(e^{2πi√2}ξ))^{−1} = ((1, ¯0, f) · (ζ, ī, 1) · h)(ξ)

as required. □

The action is clearly continuous, and so X is a Polish G-space.

Notation Let E_{G}^{X} denote the orbit equivalence relation induced by this action.

Notation From now until the end of the section M_{∞} is used to denote M_{∞}(𝕋^{2}, λ^{2}), the group of invertible λ^{2} measure preserving functions π : 𝕋^{2} → 𝕋^{2}, subject to the usual identification in the event of agreement almost everywhere.

Thus we are using M_{∞} to denote a different Polish group to the one from section 2, but since these two are naturally isomorphic the identification would seem harmless.

Notation For h ∈ X, let T_{h} : 𝕋^{2} → 𝕋^{2} be given by

(ζ, ξ) ↦ (ζe^{2πi√2}, ξh(ζ)).

Lemma 5.6 The function h ↦ T_{h} is a continuous function from X to M_{∞}.

Proof In general “skew products” of this form give rise to measure preserving transformations (compare §2 [2] or chapter 1 of [20]). The further facts that T_{h} is invertible and that the assignment h ↦ T_{h} is continuous follow almost immediately from the definitions. □

Lemma 5.7 Every G-orbit in X is dense; in fact, for any h ∈ X

{(1, ¯0, f) · h|f ∈ H_{0}}

is dense in X.

Proof Fix h_{0}, h_{1} ∈ X and ϵ > 0. Following the Kakutani-Rokhlin lemma (see page 48 [20]) we may find A ⊆ 𝕋 so that for some n

(i) A, e^{2πi√2}A, e^{4πi√2}A,... e^{2nπi√2}A, are pairwise disjoint^{7};

(ii) λ(⋃_{l<n} e^{2lπi√2}A) > 1 − ϵ.

To obtain the existence of this set A we apply Kakutani-Rokhlin to any n > \frac{2}{ϵ} to obtain A so that

λ( ⋃_{l≤n} e^{2lπi√2}A) > 1 − \frac{ϵ}{2}

and the sets

A, e^{2πi√2}A, e^{4πi√2}A, ..., e^{2nπi√2}A

are disjoint as in (i). Then we must have λ(A) < \frac{ϵ}{2} and (ii) follows as well.

Now we may simply step around this sequence A, e^{2πi√2}A, e^{4πi√2}A,... e^{2nπi√2}A, defining f|_{e^{2lπi^{√}2}A} by induction on l so that at each ξ ∈ e^{2lπi√2}A we have f(ξ)h_{0}(ξ)(f(ξe^{2πi√2}))^{−1} = h_{1}(ξ).

More formally, we let f|_{A} just be constantly 1. Assuming inductively that l < n and f|_{e^{2lπi^{√}2}A} has been defined, we let

f(ξe^{2πi√2}) = f(ξ)h_{0}(ξ)(h_{1}(ξ))^{−1}

for any ξ ∈ e^{2lπi√2}A.

Finally we have that for all ξ ∈ ⋃_{l<n} e^{2lπi√2}A

f(ξ)h_{0}(ξ)(f(ξe^{2πi√2}))^{−1} = h_{1}(ξ),

and thus

d_{X}((1, ¯0, f) · h_{0}, h_{1}) < ϵ.
□

7Here e^{2πi√2}A = {e^{2πi√2}ζ : ζ ∈ A}.

Notation Let ≈ denote the conjugacy equivalence relation on M_{∞} – so that for σ_{1}, σ_{2} ∈ M_{∞} we write σ_{1} ≈ σ_{2} if there is a π ∈ M_{∞} such that for a.e. (ζ, ξ) ∈ 𝕋^{2}

π ∘ σ_{1} ∘ π^{−1} = σ_{2}.

Lemma 5.8 For all h_{0}, h_{1} ∈ X

h_{0}E_{G}^{X}h_{1} ⇒ T_{h_{0}} ≈ T_{h_{1}}.

Proof We may break this down into three cases.

(i) (ζ_{0}, ¯0, 1) · h_{0} = h_{1}, so that h_{1}(ξ) = h_{0}(ξ/ζ_{0}) for λ a.e. ξ. Then we may define π : 𝕋^{2} → 𝕋^{2} by

π : (ζ, ξ) ↦ (ζζ_{0}, ξ).

We then have for a.e. (ζ, ξ)

(π ∘ T_{h_{0}} ∘ π^{−1})(ζ, ξ) = (π ∘ T_{h_{0}})(ζ/ζ_{0}, ξ) = (π(e^{2πi√2}ζ/ζ_{0}), h_{0}(ζ/ζ_{0})ξ)

= (e^{2πi√2}ζ, ξh_{1}(ζ)) = T_{h_{1}}(ζ, ξ).

(ii) (1, ¯1, 1) · h_{0} = h_{1}, so that h_{1}(ξ) = (h_{0}(ξ))^{−1} a.e. Then define π : 𝕋^{2} → 𝕋^{2} by

(ζ, ξ) ↦ (ζ, ξ^{−1})

and note that

πT_{h_{0}}π^{−1}(ζ, ξ) = π(e^{2πi√2}ζ, h_{0}(ζ)/ξ) = (e^{2πi√2}ζ, (h_{0}(ζ))^{−1}ξ) = (e^{2πi√2}ζ, h_{1}(ζ)ξ).

(iii) (1, ¯0, f) · h_{0} = h_{1}, so that f(ζ)h_{0}(ζ)(f(ζe^{2πi√2}))^{−1} = h_{1}(ζ) a.e. Define π : 𝕋^{2} → 𝕋^{2} by

(ζ, ξ) ↦ (ζ, f(ζ)ξ).

Note then that

π^{−1}T_{h_{0}}π(ζ, ξ) = π^{−1}(e^{2πi√2}ζ, h_{0}(ζ)f(ζ)ξ)
= (e^{2πi√2}ζ, (f(ζe^{2πi√2}))^{−1}h_{0}(ζ)f(ζ)ξ) = T_{h_{1}}(ζ, ξ).
□

Lemma 5.9 The set of h ∈ X for which T_{h} is ergodic is a dense G_{δ}.

Proof Recall (compare §2 [2]) that ergodicity is a G_{δ} condition on an element of M_{∞}: π ∈ M_{∞} is ergodic if and only if for all open A, B ⊆ 𝕋^{2} arising as finite unions of basic open sets there is m with

λ(π^{m}(A) ∩ B) > 1/4λ(A)λ(B).

Since h ↦ T_{h} is continuous, the set of h for which T_{h} is ergodic is again G_{δ}.

However there is some h for which T_{h} is ergodic – for instance h : 𝕋 → 𝕋, ζ ↦ ζ (see here [2] or [6]). Thus by 5.7, the set of h for which T_{h} ergodic is a dense G_{δ}. □

Notation Let X_{0} = {h ∈ X : T_{h} is ergodic }.

By 5.8 we have that X_{0} is G-invariant; since it is G_{δ} we have that it is a Polish G-space in its own right.

For the convenience of the reader, we give the next definition only in the narrow context that is directly relevant. The more general definitions can be found in [24] or [20] §2.4.

Definition Let

ρ : 𝕋^{2} → 𝕋^{2}

be an invertible measure preserving transformation of the form

(ζ, ξ) ↦ (ζe^{2πi√2}, ρ̂(ζ, ξ))

where ρ̂ : 𝕋^{2} → 𝕋. Then a non-zero function f ∈ L^{2}(𝕋^{2}) is said to be a generalized eigenfunction for ρ if there is some g ∈ L^{2}(𝕋), called a generalized eigenvalue, with the property that for all (ζ, ξ) ∈ 𝕋^{2} we have

f ∘ ρ(ζ, ξ) = g(ζ)(f(ζ, ξ));

in other words, f ∘ ρ = ĝf for ĝ defined by

ĝ(ζ, ξ) = g(ζ).

The next couple of lemmas are standard; more general results, along with related facts, can be found in [24].

Lemma 5.10 Let h ∈ X_{0} and let f_{1}, f_{2} ∈ L^{2}(𝕋^{2}) be generalized eigenfunctions for T_{h} with a common generalized eigenvalue. Then f_{1} is a linear multiple of f_{2}.

Proof f_{1}^{−1}f_{2} is invariant under T_{h}, and hence must be a constant function by ergodicity. □

Lemma 5.11 Let h ∈ X_{0}. Then the only generalized eigenfunctions are

(ζ, ξ) ↦ ξ^{n}k(ζ)

for some measurable function

k : 𝕋 → ℂ.

Proof Note that by Stone-Weierstrass, every function f ∈ L^{2}(𝕋^{2}) can be written as

f : (ζ, ξ) ↦ ∑_{n∈ℤ}ξ^{n}k_{n}(ζ)

for some k_{n} ∈ L^{2}(𝕋). Moreover decomposition is unique, since

(ζ, ξ) ↦ ξ^{n}k_{n}(ζ)

and

(ζ, ξ) ↦ ξ^{m}k_{m}(ζ)

are orthogonal for n ≠ m.

Now let us suppose that

f : (ζ, ξ) ↦ ∑_{n∈ℤ}ξ^{n}k_{n}(ζ)

has g as its generalized eigenfunction. Then

f ∘ T_{h} : (ζ, ξ) ↦ ∑_{n∈ℤ}ξ^{n}h(ζ)^{n}k_{n}(ζe^{2πi√2}).

Then the uniqueness of the decomposition of f ∘ T_{h} gives for a.e. ζ ∈ 𝕋 that

ξ^{n}h(ζ)^{n}k_{n}(ζe^{2πi√2}) = g(ζ)ξ^{n}k_{n}(ζ).

This means that any k_{n} not identically zero gives rise to

(ζ, ξ) ↦ ξ^{n}k_{n}(ζ)

as a function with generalized eigenfunction g; thus by 5.10 we have k_{n} ≡ 0 for all but a single n. □

Lemma 5.12 For all h_{0}, h_{1} ∈ X_{0}

T_{h_{0}} ≈ T_{h_{1}} ⇒ h_{0}E_{G}^{X}h_{1}.

Proof Fix h_{0}, h_{1} with T_{h_{0}} ≈ T_{h_{1}}, and let π ∈ M_{∞} witness this – in that πT_{h_{0}}π^{−1} = T_{h_{1}} λ^{2} a.e.

By ergodicity

(ζ, ξ) ↦ ζ

is up to scalar multiples the only eigenfunction with eigenvalue e^{2πi√2} for both T_{h_{0}} and T_{h_{1}}. Thus

π(ζ, ξ) = (ζ_{0}ζ, ρ̂(ζ, ξ))

for some fixed ζ_{0} ∈ 𝕋 and suitable ρ̂. By replacing h_{0} by ((ζ_{0})^{−1}, ¯0, 1) · h_{0} we may assume that ζ_{0} = 1 and thus

π(ζ, ξ) = (ζ, ρ̂(ζ, ξ)).

By 5.11 we have that the only generalized eigenfunctions for T_{h_{i}} (i equal to either 0 or 1) are of the form

(ζ, ξ) ↦ ξ^{n}k(ζ)

for some n ∈ ℤ and measurable k : 𝕋 → ℂ. Thus we see that the generalized eigenfunctions of the form

(ζ, ξ) ↦ ξk(ζ)
(ζ, ξ) ↦ ξ^{−1}k(ζ)

have a privileged status, as the only generalized eigenfunctions which are able to generated the space L^{2}(𝕋^{2}, λ^{2}) by the operations of multiplication, addition, and multiplication by linear combinations of the eigenfunctions (ζ, ξ) ↦ ζ^{m} some m ∈ ℤ (here by generate, I mean that they are dense in the sense of the Hilbert space norm on L^{2}). Note then that (ζ, ξ) ↦ ξ must be sent to a generalized eigenfunction for T_{h_{1}} of the form

(ζ, ξ) ↦ ξ^{j}k(ζ)

where j is either 1 or −1.

Thus we may assume

π(ζ, ξ) = (ζ, ξ^{(−1)^{i}}k(ζ))

for some measurable k : 𝕋 → 𝕋, and so

π^{−1}(ζ, ξ) = (ζ, ξ^{(−1)^{i}}(k(ζ))^{(−1)^{i+1}}).

Thus

(e^{2πi√2}ζ, h_{0}(ζ)ξ) = π^{−1}T_{h_{1}}π(ζ, ξ) = π^{−1}T_{h_{1}}(ζ, k(ζ)(ξ)^{(−1)^{i}})

= π^{−1}(ζe^{2πi√2}, k(ζ)h_{1}(ζ)(ξ)^{(−1)^{i}}) = (ζe^{2πi√2}, (k(ζe^{2πi√2}))^{(−1)^{i+1}}(h_{1}(ζ))^{(−1)^{i}}k(ζ)^{(−1)^{i}}ξ).

Thus for a.e. (ζ, ξ) we have

[(k(ζe^{2πi√2})^{−1}h_{1}(ζ)k(ζ)]^{(−1)^{i}} = h_{0}(ζ),

and so h_{0}E_{G}^{X}h_{1} as required. □

Lemma 5.13 Every orbit in X is meager.

Proof Let

A_{0} = 𝕋 × \{e^{2πix} | \frac{1}{4} ≤ x ≤ \frac{3}{4} \}.

Claim: For any O_{1}, O_{2} ⊂ X open, non-empty and A measurable, there exist h_{1} ∈ O_{1}, h_{2} ∈ O_{2}, k ∈ ℕ, k > 0. with

λ^{2}(AΔT_{h_{1}}^{k}(A)) < \frac{1}{13} ,

λ^{2}(A_{0}ΔT_{h_{2}}^{k}(A_{0})) > \frac{1}{4} .

Proof of Claim: Let ĥ_{1} be the function on 𝕋

ĥ_{1} : ζ ↦ 1,

and let ĥ_{√3} be the function

ĥ_{√3} : ζ ↦ e^{2πi√3}.

Then by 5.7 we may find (1, ¯0, f_{1}), (1, ¯0, f_{2}) ∈ G with

h_{1} =_{df} (1, ¯0, f_{1}) · ĥ_{1} ∈ O_{1},
h_{2} =_{df} (1, ¯0, f_{2}) · ĥ_{√3} ∈ O_{2}.

Since the continuous functions are dense in L^{1} we may actually assume that f_{1} and f_{2} are continuous. Then

h_{1}(e^{2πix}) = f_{1}(e^{2πix})ĥ_{1}(e^{2πix})(f_{1}(e^{2πi(x+√2)}))^{−1}
= f_{1}(e^{2πix})(f_{1}(e^{2πi(x+√2)}))^{−1}

and thus

T_{h_{1}} : (e^{2πix}, ξ) ↦ (e^{2πi(x+√2)}, f_{1}(e^{2πix})(f_{1}(e^{2πi(x+√2)}))^{−1}ξ)

and

(T_{h_{1}})^{k} : (e^{2πix}, ξ) ↦ (e^{2πi(x+k√2)}, f_{1}(e^{2πix})(f_{1}(e^{2πi(x+k√2)}))^{−1}ξ).

An exactly similar calculation gives

(T_{h_{2}})^{k} : (e^{2πix}, ξ) ↦ (e^{2πi(x+k√2)}, f_{2}(e^{2πix})e^{2kπi√3}(f_{2}(e^{2πi(x+k√2)}))^{−1}ξ).

The continuity of f_{1}, f_{2} guarantees for each δ some corresponding δ̂ > 0 such that whenever

|e^{2kπi√2} − 1| < δ̂

then for all x ∈ [0, 1]

|f_{1}(e^{2πix}) − f_{1}(e^{2πi(x+k√2)})|, |f_{2}(e^{2πix}) − f_{2}(e^{2πi(x+k√2)})| < δ.

Since √2 and √3 are rationally independent, we can therefore apply Kronecker’s lemma (theorem 28 of [16]) and find k with

|f_{1}(e^{2πix}) − f_{1}(e^{2πi(x+k√2)})|,
|f_{2}(e^{2πix}) − f_{2}(e^{2πi(x+k√2)})|

both arbitrarily small for all x ∈ [0, 1] and e^{2kπi√3} arbitrarily close to e^{πi}. Such k clearly suffices. (Claim□)

Now let us choose a sequence of measurable sets (B_{i})_{i} such that for all π ∈ M_{∞} there is some i ∈ ℕ with

λ^{2}(π(A_{0})ΔB_{i}) < \frac{1}{13} .

Let V_{i} be

\{π ∈ M_{∞} | λ^{2}(π(A_{0})ΔB_{i}) < \frac{1}{13} \}.

By 5.8 suffices to show that for each i the set

{(h_{1}, h_{2}) ∈ X × X | ∃π ∈ V_{i}(π^{−1} ∘ T_{h_{1}} ∘ π = T_{h_{2}})}

is nowhere dense.

But given any non-empty open O_{1}, O_{2} ⊂ X × X we can by the above claim find some non-empty open U_{1} ⊂ O_{1}, U_{2} ⊂ O_{2} and k ∈ ℕ such that for all h_{1} ∈ U_{1}, h_{2} ∈ U_{2}

λ^{2}(B_{i}ΔT_{h_{1}}^{k}(B_{i})) < \frac{1}{13} ,

λ^{2}(A_{0}ΔT_{h_{2}}^{k}(A_{0})) > \frac{1}{4} .

Fixing such h_{1}, h_{2} and π ∈ V_{i} we need to show

π^{−1} ∘ T_{h_{1}} ∘ π ≠ T_{h_{2}}.

But since π is measure preserving we have

λ^{2}(A_{0}Δ(π^{−1}T_{h_{1}}π)^{k}(A_{0})) = λ^{2}(A_{0}Δπ^{−1}(T_{h_{1}}^{k}(π(A_{0}))))
= λ^{2}(π(A_{0})ΔT_{h_{1}}^{k}(π(A_{0}))),

which by the triangle inequality is bounded by

λ^{2}(π(A_{0}), B_{i}) + λ^{2}(B_{i}, T_{h_{1}}^{k}(B_{i})) + λ^{2}(T_{h_{1}}^{k}(B_{i}), T_{h_{1}}^{k}(π(A_{0})))
= λ^{2}(π(A_{0}), B_{i}) + λ^{2}(B_{i}, T_{h_{1}}^{k}(B_{i})) + λ^{2}(B_{i}, π(A_{0}))

since T_{h_{1}}^{k} is measure preserving, which in turn is bounded by

\frac{1}{13} + \frac{1}{13} + \frac{1}{13} < \frac{1}{4}

by assumption of h_{1} ∈ U_{1} and π ∈ V_{i}.

This is as required to show

λ^{2}(A_{0}Δπ^{−1}(T_{h_{1}}^{k}(π(A_{0})))) ≠ λ^{2}(A_{0}ΔT_{h_{2}}^{k}(A_{0})).
□

Definition Let H be a Polish group and Y a Polish H-space. The action of H on Y is said to be turbulent if

(i) every orbit is dense; (ii) every orbit is meager; (iii) for all x, y ∈ Y, U ⊆ Y, V ⊆ H open with x ∈ U, 1 ∈ V, there exists y_{0} ∈ [y]_{H} (the orbit of y) and such that for all open U_{0} containing y_{0} there is k ∈ ℕ, (h_{i})_{i<k} ⊆ V, (x_{i})_{i≤k} ⊆ U with

x_{0} = x,
x_{i+1} = h_{i} · x_{i},

and

x_{k} ∈ U_{0}.

The usefulness of this concept is that it gives a sufficient condition for a degree of non-classifiability: As in [12] no turbulent action allows a Borel – or even Baire measurable – function reducing its orbit equivalence relation to isomorphism on countable structures. More generally, any equivalence relation into which we can embed a turbulent orbit equivalence relation will similarly be unclassifiable by countable structures considered up to isomorphism.

Lemma 5.14 The action of G on X_{0} is turbulent.

Proof We already established that every orbit is dense and meager, so we are only left to show the “local density” condition at (iii) from the definition of turbulence.

For this purpose, fix h_{0}, h_{1} ∈ X and ϵ > 0. It suffices to show there is some n ∈ ℕ and there are some g_{0}, g_{1}, ..., g_{n−1} ∈ G such that

h_{0,0} = h_{0},
h_{0,l+1} = g_{l} · h_{0,l},

we have

(i) d_{G}(g_{l}, 1_{G}) < ϵ each l < n;
(ii) d_{X}(h_{0,l}, h_{0}) < d_{X}(h_{0}, h_{1}) + ϵ each l ≤ n;
(iii) d_{X}(h_{0,n}, h_{1}) < ϵ.

We will do this in a manner resembling the proof of 5.7.

Choose n ∈ ℕ such that n > 3/ϵ. Appealing to Kakutani-Rokhlin we find A ⊆ 𝕋 with

(iv) A, e^{2πi√2}A, e^{4πi√2}A,... e^{2nπi√2}A all disjoint;

(v) ∴ λ(A) < ϵ/3;
(vi) λ(⋃_{l≤n} e^{2lπi√2}A) > 1 − ϵ/3.

We now define f_{l} : 𝕋 → 𝕋 by induction on l < n. f_{0} is constantly 1. Given the definition of f_{l} we let

f_{l+1}(e^{2πi√2}ξ) = f_{l}(ξ)h_{0}(ξ)(h_{1}(ξ))^{−1}

for ξ ∈ e^{2lπi√2}A, and

f_{l+1}(e^{2πi√2}ξ) = 1

otherwise. We then let g_{i} = (1, ¯0, f_{i}). We then let h_{0,0} = h_{0} and h_{0,l+1} = g_{l}h_{0,l} as indicated above.

At once we have (i), since each f_{i} is not equal to 1 only on a set of measure < ϵ. For (ii), note that if k < n then ∀ξ ∈ ⋃_{l<k} e^{2lπi√2}A

h_{0,k}(ξ) = h_{1}(ξ),

∀ξ ∈ ⋃_{l>k,l<n} e^{2lπi√2}A

h_{0,k}(ξ) = h_{0}(ξ).

Thus

d_{X}(h_{0,k}, h_{0}) = ∫ 1/2|h_{0,k}(ξ) − h_{0}(ξ)|dλ
< λ(e^{2kπi√2}A) + λ(e^{2nπi√2}A) + ϵ/3 + ∫ 1/2|h_{1}(ξ) − h_{0}(ξ)|dλ
< ϵ + d_{X}(h_{0}, h_{1}).

Finally for (iii) note that h_{0,n} and h_{1} agree except on the set

𝕋 \ ⋃_{l<n} e^{2lπi√2}A,

which has measure less than ϵ. □

Summarizing what has been proved:

Theorem 5.15 There is a Polish group G and a Polish G-space X_{0} and a Borel function θ : X_{0} → M_{∞} such that:

(i) the action of G on X_{0} is turbulent (5.14);

(ii) for each h ∈ X_{0} the transformation θ(h) ∈ M_{∞} is ergodic (5.9, 5.12); (iii) in fact each θ(h) is “measure-distal” (in the sense of [2]), and in fact it is rank 2 generalized discrete spectrum (granting (ii), this is an immediate consequence of the definition of the assignment h ↦ T_{h}); (iv) for all h_{0}, h_{1} ∈ X_{0}

h_{0}E_{G}^{X}h_{1}

if and only if

θ(h_{0}) ≈ θ(h_{1}),

where ≈ is the equivalence relation of conjugacy (5.8, 5.12).

Corollary 5.16 (a) There is no countable language L and Borel

ρ : M_{∞} → Mod(L)

such that for all σ_{1}, σ_{2} ∈ M_{∞}

σ_{1} ≈ σ_{2} ⇔ ρ(σ_{1}) ≅ ρ(σ_{2}).

(b) In fact, if B ⊆ M_{∞} is the subclass consisting of rank 2 generalized discrete spectrum ergodic transformations then there is no Borel

ρ : B → Mod(L)

such that for all σ_{1}, σ_{2} ∈ M_{∞}

σ_{1} ≈ σ_{2} ⇔ ρ(σ_{1}) ≅ ρ(σ_{2}).

Proof At once by the results of §3.2, [12]. □

In actual fact there is no compulsion to restrict ourselves so carefully to the Borel category. The methods of [12] are sufficient to obtain nonreducibility to isomorphism on countable structures even using very broad classes of functions, such as C-measurable, absolutely Δ ∼_{2}^{1}, and universally Baire measurable. As mentioned in the introduction, we may even obtain the consistency of ZF+DC along with the non-existence of any injection

i : P/≈ ↪ Mod(L).

6 Remarks on the equivalence of cocycles

There is obviously a close relation between the arguments of §5 and the isomorphism relation on cocycles. It might be worth pausing before the finish of this paper to consider what can be drawn out in this fashion.

Definition Let (Ω, B, μ) be a probability space and H a countable group acting by measure preserving transformations on Ω. For later purposes assume that Ω is a Lebesgue space (that is to say measurably isomorphic to ([0, 1], Borel, λ)). Let K be a compact metric group. A map measurable

α : H × Ω → K

is a cocycle if for all h, h′ ∈ H and s ∈ Ω

α(hh′, s) = α(h, h′s)α(h′, s).

(Here the measurability requirement is that for all h ∈ H we have s ↦ α(s, h) measurable.)

In the case that H = ℤ the cocycle condition becomes especially transparent, since we can exactly specify a cocyle by its value on a generator of ℤ. Hence we can naturally identify a cocycle for ℤ with a measurable function from the space Ω to K.

The perspective of [24] is to only consider the case that K is a compact metric group; the remarks below persist in some form even in the more general context that K is a locally compact Polish group with an invariant two sided metric. If d_{K} is a compatible complete metric on a compact metric group K then we obtain an invariant metric with d_{K,inv}(g_{0}, g_{1}) set equal to

∫_{K×K} d_{K}(h_{0}g_{0}h_{1}, h_{0}g_{1}h_{1})μ_{K} × μ_{K}.

Definition Let (Ω, B, μ), K, H be as above. Two cocycles

α, β : H × Ω → K

are said to be equivalent if there is a measurable function

φ : Ω → K

such that for every h ∈ H and a.e. every s ∈ Ω

φ(h · s)^{−1}α(h, s)φ(s) = β(h, s).

Note – as exploited in §5 – this equivalence relation is induced by a Polish group action.

Notation Let (Ω, B, μ), K, H, be as above. Let d_{K} be an invariant metric on K; by possibly replacing it with d_{K}/(1 + d_{K}) we may assume it is bounded by 1. Let X(Ω, K, H, μ) be the space of all cocycles from H × Ω to K. Let G(Ω, K, H, μ) be the group of all measurable

φ : Ω → K

under the operation of pointwise multiplication. We let G(Ω, K, H, μ) act on X(Ω, K, H, μ) by the specification that

(φ · α)(h, s) = φ(h · s)^{−1}α(h, s)φ(s).

For α ∈ X(Ω, K, H, μ) use G(Ω, K, H, μ)_{α} to denote the stabilizer of α – that is to say the set of φ for which

φ · α = α.

The action of G(Ω, K, H, μ) on X(Ω, K, H, μ) is exactly chosen so that the resulting orbit equivalence relation is the cocycle equivalence relation. G(Ω, K, H, μ) and X(Ω, K, H, μ) are Polish spaces, and X(Ω, K, H, μ) is a Polish G(Ω, K, H, μ)-space. For the group we obtain a complete and in fact invariant metric by

d_{G(Ω,K,H,μ)}(φ_{0}, φ_{1}) = ∫_{Ω} d_{K}(φ_{0}(s), φ_{1}(s))dμ.

For the cocycles we can choose an enumeration (h_{i})_{i∈ℕ} of H and let

d_{X(Ω,K,H,μ)}(α, β) = ∑_{i∈ℕ}2^{−i} ∫_{Ω} d_{K}(α(h_{i}, s), β(h_{i}, s))dμ.

A special case of the above is when H = ℤ, Ω = 𝕋, K = 𝕋, and the action of ℤ on 𝕋 is given by

l · ζ = e^{2lπi√2}ζ.

Clearly the arguments of §5 are sufficient to obtain that in this case – and in many others – the action of G(Ω, K, H, μ) on X(Ω, K, H, μ) is turbulent. In showing conditions (i) and (iii) from the definitions of turbulence for G we only used the G(Ω, K, H, μ) part in 5.7 and 5.14. Clearly the property of every orbit being meager goes down to this sub-action by G(Ω, K, H, μ). Thus in general the cocycle equivalence relation refuses classification by countable models.

In the presence of ergodicity, the stabilizers are all compact.

Lemma 6.1 Let (Ω, B, μ), K, H be as above. Suppose that the action of H on Ω is ergodic.

Then for every α ∈ X(Ω, K, H, μ) we have that G(Ω, K, H, μ)_{α} is compact.

Proof G(Ω, K, H, μ)_{α} is a complete metric space, so we just need to show that it is ϵ-bounded for each ϵ.

Let (φ_{i})_{i∈ℕ} be a countable dense subset of G(Ω, K, H, μ)_{α}. Using the fact that K is a compact metric space we may find a finite sequence of balls B_{0}, B_{1}, ...B_{n} of radius < ϵ/3 covering K. For each i_{0}, i_{1} ∈ ℕ, j ≤ n let A_{i_{0},i_{1},j} = {s ∈ Ω : φ_{i_{0}}(s), φ_{i_{1}}(s) ∈ B_{j}}. Let A be the countable collection {A_{i_{0},i_{1},j} : i_{0}, i_{1} ∈ ℕ, j ≤ n}. We may clearly find s_{0} ∈ Ω such that for all A ∈ A, if s_{0} ∈ A then A is not null.

Now we may choose a finite collection φ_{l_{0}}, φ_{l_{1}}, ..., φ_{l_{k}} from our sequence (φ_{i})_{i∈ℕ} such that for any j ≤ n, if there exists i ∈ ℕ with φ_{i}(s_{0}) ∈ B_{j} then there is some i′ ≤ k with φ_{l_{i′}} (s_{0}) ∈ B_{j}.

Claim. If φ_{i}(s_{0}) ∈ B_{j} and φ_{l_{i′}} (s_{0}) ∈ B_{j} then for a.e. s ∈ Ω,

d_{K}(φ_{i}(s), φ_{l_{i′}} (s)) ≤ 2ϵ/3.

Proof of claim: Let A = {s ∈ Ω : d_{K}(φ_{i}(s), φ_{l_{i′}} (s)) ≤ 2ϵ/3}. By assumption on s_{0} this set A has non-zero measure. By ergodicity it suffices to show A is H-invariant.

But for any h ∈ H and s ∈ A

α(h, s) = (φ_{i} · α)(h, s) = φ_{i}(h · s)^{−1}α(h, s)φ_{i}(s)
∴ φ_{i}(h · s) = α(h, s)φ_{i}(s)α(h, s)^{−1},

and similarly

φ_{l_{i′}} (h · s) = α(h, s)φ_{l_{i′}} (s)α(h, s)^{−1}.

Thus by the invariance of the metric d_{K}(φ_{i}(h · s), φ_{l_{i′}} (h · s)) ≤ 2ϵ/3. (Claim□)

But then it is immediate from the definition of the complete metric on G(Ω, K, H, μ) and the density of the set (φ_{i})_{i∈ℕ} that every element in G(Ω, K, H, μ)_{α} is within ϵ of some φ_{l_{i′}}. □

Thus for any two cocycles α and β the set of φ ∈ G(Ω, K, H, μ) with

φ · α = β

is either empty or compact. Therefore the set

{(α, β) ∈ X(Ω, K, H, μ)^{2} : ∃φ ∈ G(Ω, K, H, μ)(φ · α = β)}

is the projection of a Borel set all of whose sections are compact, and is, by the Arsenin-Kunugui theorem (see §18 [15] and §4F [17]), itself Borel.

We consequently have a short proof of the equivalence relation being Borel for ergodic actions. I believe a much deeper proof of this fact has been previously extracted by Foreman and Weiss from the results of [24].

Lemma 6.1 should not be thought of as implying that the measure preserving transformations considered in §5 themselves have compact stabilizers in the natural action of M_{∞} (= group of invertible measure preserving transformations on the unit interval) on M_{∞}. In fact the ergodic transformations having compact stabilizer in M_{∞} are exactly the discrete spectrum transformations.

Lemma 6.2 Let T ∈ M_{∞} be ergodic. Then the set

{π ∈ M_{∞} : π ∘ T ∘ π^{−1} = T }

is compact if and only if T is a discrete spectrum transformation.

Proof First let us take the case that T is discrete spectrum. Then by [9] we can assume that there is a compact abelian metric group G with a Haar measure μ and corresponding g ∈ G with

{n · g : n ∈ ℤ}

dense in G and T as a measure preserving transformation on [0, 1] isomorphic to translation of (G, μ) by g. For each k ∈ G let

T_{k} : G → G
h ↦ kh

be the transformation given by k-translation. Let M_{∞}(G) be the group of invertible measure preserving transformation on (G, μ). Since the assignment

G → M_{∞}(G)
k ↦ T_{k}

is continuous we need only show

{π ∈ M_{∞}(G) : π ∘ T_{g} ∘ π^{−1} = T_{g}}

equals {T_{k} : k ∈ G}.

It is clear that the compact group of measure preserving transformations

{T_{k} : k ∈ G}

is included in {π ∈ M_{∞}(G) : π ∘ T_{g} ∘ π^{−1} = T_{g}}, so let us fix π with π ∘ T_{g} ∘ π^{−1} = T_{g} a.e. To each x ∈ G let h_{x} ∈ G be such that

h_{x}x = π(x).

We then have

h_{x}(T_{g}x) = h_{x}(gx)

by definition of T_{g}, which in turn equals

gh_{x}x

by G abelian, which now equals

gπ(x) = T_{g}π(x) = (T_{g} ∘ π)(x),

which by assumption on π equals

(π ∘ T_{g})(x) = π(T_{g}(x))

on a measure one set; and thus the function

x ↦ h_{x}

is a T_{g}-invariant function on a measure one set, and hence by ergodicity constant almost everywhere. Hence π = T_{h_{x}} for some x ∈ G on a measure one set, and we are done.

Conversely, if

{π ∈ M_{∞} : π ∘ T ∘ π^{−1} = T }

is compact, then we have that T is an element of a compact subgroup of M_{∞}. Thus by the Peter-Weyl theorem, as for instance found in [25], we may find a sequence of finite dimensional subspaces

H_{0}, H_{1}, H_{2}, ⋯

of L^{2}([0, 1], λ) which are invariant under the unitary operator f ↦ f ∘ T and which jointly sum up to give

L^{2}([0, 1], λ = ⨁_{n} H_{n}.

Then diagonalizing this unitary on each of these finite dimensional subspaces we finish. □

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6363 MSB Mathematics UCLA CA90095-1555 greg@math.ucla.edu www.math.ucla.edu/˜ greg