A dichotomy theorem for turbulence

Greg Hjorth

May 27, 2002

1 Introduction

In this note we show:

Theorem 1.1 Let G be a Polish group and X a Polish G-space with the induced orbit equivalence relation E_{G} Borel as a subset of X × X. Then exactly one of the following:

(I) There is a countable language L and a Borel function

θ : X → Mod(L)

such that for all x_{1}, x_{2} ∈ X

x_{1}E_{G}x_{2} ⇔ θ(x_{1}) ≅ θ(x_{2});

or

(II) there is a turbulent Polish G-space Y and a continuous G-embedding

τ : Y ↪ X.

There are various bows and ribbons which can be woven into these statements. We can strengthen (I) by asking that θ also admit a Borel orbit inverse, that is to say some Borel function

ρ : B → X,

for some Borel set B ⊂ Mod(L), such that for all x ∈ X

xE_{G}ρ(θ(x));

and then after having passed to this strengthened version of (I) we still obtain the exact same dichotomy theorem, and hence the conclusion that the two competing versions of (I) are equivalent. Similarly (II) can be relaxed to just asking that τ be a Borel G-embedding, or even simply a Borel reduction of the relevant orbit equivalence relations. It is in fact a consequence of 1.1 that all the plausible weakenings and strengthenings of (I) and (II) are respectively equivalent to one another.

I will not closely examine these possible variations here. The equivalences alluded to above follow from our main theorem and the results of [3]. That monograph had previously shown that (I) and (II) are incompatible, and proved a barbaric forerunner of 1.1, and gone on to conjecture the dichotomy result above.

[3] is the proper place to find further discussion of the notation used in the proofs below. Mod(L) is the space of L-structure on ℕ equipped with the topology generated by quantifier free formulas. E_{G} refers to the orbit equivalence relation arising from the indicated action of G on the indicated space.

2 The Gandy-Harrington topology and a derived Polish G-space

We recall the notions of recursive Polish space and recursive continuous function from [5]. We assume that all our spaces are recursive Polish spaces, our group is a recursive Polish space, and multiplication viewed as a function

G × G → G

is a recursive continuous function, as is the evaluation of the action of G on X viewed as a function

G × X → X.

Indeed in the case of X we want to do more than simply assume that X has a recursive basis; we fix from now on a sequence (U_{ℓ})_{ℓ∈ω} of open subsets of X which forms a basis and such that the assignment

ℓ ↦ U_{ℓ}

is suitably effective; for instance, and this is only one method of many to make this precise, we could assume that

⋃_{ℓ∈ω}{ℓ} × X

is a recursive open subset of ω × X. We also fix a complete compatible metric d_{G}(·, ·) on G and assume it provides a recursive continuous function from G × G to ℝ.

We assume that there is a countable dense subgroup G_{0} = {g_{n} : n ∈ ℕ} of G such that the enumeration n ↦ g_{n} is recursive continuous as a function from ℕ to G and such that g · U_{ℓ} ∈ {U_{k} : k ∈ ℕ} all g ∈ G_{0} and ℓ ∈ ℕ. We then fix a basis for G closed under left and right translation by G_{0} consisting purely of recursive open sets. We assume it includes a neighborhood basis at the identity of symmetric sets. We further assume that E_{G} is Δ_{1}^{1} as a subset of X × X.

Of course in general these assumptions might not be true, and in the arguments below we would need to laboriously replace each instance of “recursive” with “recursive in z” for some parameter z.

Notation Let X_{low} = {x ∈ X : ∀A ∈ Σ_{1}^{1}(x ∈ A ∨ ∃B ∈ Σ_{1}^{1}(x ∈ B ∧ B ∩ A = 0))}. Let τ_{∞} be the topology generated by taking as a basis all Σ_{1}^{1} sets.

In the case that X = 2^{ω} one actually has X_{low} = {x ∈ X : ω_{1}^{ck} = ω_{1}^{ck(x)}}.

Note that X_{low} is a Borel subset of X; on the other hand, while it is lightface Σ_{1}^{1} it is not lightface Δ_{1}^{1}. Recall that Gandy’s low basis theorem states that any non-empty Σ_{1}^{1} contains an element of X_{low}.

The formulation of the next lemma was influenced by some parallel remarks made by Ramez Sami during the course of a seminar. It has been previously proved in [4] using the notion of strong Choquet space; the proof below is longer but more direct.

Lemma 2.1 τ_{∞} is a Polish topology on X_{low}.

Proof We show (X_{low}, τ_{∞}) to be homeomorphic to a G_{δ} subset of some well known Polish space. For ease of exposition we assume X = 2^{ω}; it should be clear how to extend the argument to general recursive Polish spaces.

Let S be the set of Σ_{1}^{1} subsets of X. To each x ∈ X_{low} associate

f_{x} : S → 2

by the formulation

f_{x}(A) = 1

if and only if

x ∈ A.

This map x ↦ f_{x} is clearly one-to-one. The preimage of a subbasic open set in 2^{S} of the form

{x : f_{x}(A) = 1}

is obviously open in (X_{low}, τ_{∞}); the preimage of a subbasic open set of the form

{x : f_{x}(A) = 0}

equals

⋃{D : D ∈ Δ_{1}^{1}, D ∩ A = 0} ∩ X_{low},

and is thus again open with respect to τ_{∞}. Therefore the map

x ↦ f_{x}

is a homeomorphism onto its image, and we need only show that {f_{x} : x ∈ X_{low}} is a G_{δ} subset of 2^{S} in the product topology to complete the proof.

For this purpose let us define at each n ∈ ω the set A_{n}^{∗} to be the set of x ∈ X_{low} for which the n^{th} x-recursive^{1} linear order of ω is not a wellorder of rank less than ω_{1}^{ck}; and then at each m we let C_{n,m}^{∗} be the set of x ∈ X_{low} for which m is in the illfounded part of the n^{th} x-recursive linear order of ω. These sets are of course all Σ_{1}^{1}.

I claim that

f ∈ {f_{x} : x ∈ X_{low}}

if and only if the following four things happen:

1. for all n ∈ ω and basic open W with

f(W) = 1,

there is some U ⊂ X whose diameter is less than \frac{1}{n} with

\overline{U} ⊂ W

and

f(U) = 1;

2. for all k ∈ ω, A_{0}, A_{1}, ..., A_{k}, C_{0}, C_{1}, ..., C_{k} ∈ Σ_{1}^{1}, with

∀i ≤ k(f(A_{i}) = 1, f(C_{i}) = 0),

there is some x ∈ X_{low} with

∀i ≤ k(x ∈ A_{i}, x ∉ C_{i});

(in particular this rules out grievously absurd choices for f : S → {0, 1}, such as the two constant functions);

3. for all n with

f(A_{n}^{∗}) = 1

there exists m with

f(C_{n,m}^{∗}) = 1;

1Here for the only time we use X = 2^{ω}; with suitable care the notion of recursive in x can be translated across to arbitrary recursive spaces.

4. for each Σ_{1}^{1} set A we fix a Lusin scheme

(C_{s}^{A})_{s∈ω^{<ω}}

of Π_{0}^{1} sets such that

A = {x ∈ X|∃h ∈ ω^{ω}∀n ∈ ω(x ∈ C_{h|_{n}}^{A})};

we then demand that

f(A) = 1 ⇒ f(C_{0}^{A}) = 1,

and that for all s ∈ ω^{ω}

f(C_{s}^{A}) = 1 ⇒ ∃k ∈ ω(f(C_{s^{⌢}k}^{A}) = 1).

The first, third, and fourth of these conditions correspond to G_{δ} sets, while the second is closed.

The purpose of the first condition is to ensure that we can indeed find a point x_{f} in some intersection of basic open V with f(V) = 1; and in this light the second condition ensures that this x_{f} meets all the open and closed sets promised by f, and from this we can obtain by the fourth condition and the usual kinds of arguments that x_{f} actually meets every Σ_{1}^{1} set with f(A) = 1. The third condition is the key one. It guarantees that no recursive in x_{f} wellorder has rank exactly ω_{1}^{ck}. □

Notation We let X_{∗low} be the set of all x ∈ X such that there exists some g ∈ G with

g · x ∈ X_{low}.

We endow this with the topology τ_{∞}^{∗} generated by basic open sets of the form

(A ∩ X_{low})^{ΔV}

where A ⊂ X is Σ_{1}^{1} and V ⊂ G is open.

In fact if x ∈ X_{∗low} one does not just obtain one g ∈ G with g · x ∈ X_{low} but a comeager set of g which so move x to a real with ω_{1}^{ck(g·x)} = ω_{1}^{ck}: We first use that

ω_{1}^{ck(x)} = ω_{1}^{ck}

to conclude that there will be a comeager collection of g with ω_{1}^{ck(g,x)} = ω_{1}^{ck}, and any such g will do.

Thus X_{∗low} equals

(X_{low})^{∗G} = (X_{low})^{ΔG},

and is consequently Borel. Note moreover that in this space, for any A ⊂ X_{low} which is Σ_{1}^{1} and for any V ⊂ G open we in fact have a sequence of Δ_{1}^{1} sets

(D_{α})_{α∈ω_{1}^{ck}}

such that

A^{∗V} = X_{∗low} \ ( ⋃_{α∈ω_{1}^{ck}}(D_{α})^{ΔV}),

and thus A^{∗V} is closed.

The next proof crumbles into a sequence of calculations. The first few of these resemble similar points made in the course of the proof of 5.1.5 from [1].

Lemma 2.2 (X_{∗low}, τ_{∞}^{∗}) is a Polish G-space.

Proof The action of G on this space is obviously continuous.

Claim: (X_{∗low}, τ_{∞}^{∗}) is Hausdorff. Proof of claim: τ_{∞}^{∗} includes the subspace topology on X_{∗low} induced by the original topology on X. Since

this original topology is Hausdorff, so is τ_{∞}^{∗}. (□Claim) Claim: (X_{∗low}, τ_{∞}^{∗}) is regular. Proof of claim: Let A^{ΔV} be an open subset of (X_{∗low}, τ_{∞}^{∗}) and x ∈ A^{ΔV}. Then we may find some

W_{0} ⊂ V open and non-empty with x ∈ A^{∗W_{0}}. Appealing to the continuity of the group operations we may find non-empty open sets

W_{2} ⊂ W_{1} ⊂ W_{0}

and an open neighborhood W_{3} of the identity with

W_{1}W_{3}^{−1} ⊂ W_{0}

and

W_{2}W_{3}^{−1} ⊂ W_{1}.

x is then a member of the open set (A^{∗W_{1}})^{ΔW_{3}} since for any g ∈ W_{3} we have

W_{1}g^{−1} ⊂ W_{0}

and

∀^{∗}h ∈ W_{1}g^{−1}(h · (g · x) ∈ A).

Thus the claim will be proved if we show (A^{∗W_{1}})^{ΔW_{3}} is included in the τ_{∞}^{∗}-closed set A^{∗W_{2}}. But here we have that for any h ∈ W_{3} and z with h · z ∈ A^{∗W_{1}}

∀^{∗}g ∈ W_{1}(gh · z ∈ A)
∴ ∀^{∗}g ∈ W_{1}h(g · z ∈ A).

But

W_{2}W_{3}^{−1} ⊂ W_{1}
∴ W_{2} ⊂ W_{1}h
∴ z ∈ A^{∗W_{2}}.
(□Claim)

Since (X_{∗low}, τ_{∞}^{∗}) is clearly separable, we thus have that it is metrizable by Urysohn. To see that the space is Polish we first define a continuous map

π : G × X_{low} → X_{∗low}

by

(g, x) ↦ g · x.

Claim: π is continuous. Proof of claim: We have g · x ∈ A^{ΔV} if and only if x ∈ A^{ΔV g}. (□Claim) Claim: π is an open map. Proof of claim: It suffices to show that whenever A ⊂ X_{low} is Σ_{1}^{1} and V ⊂ G is open,

V · A = ⋃_{W_{0},W_{1} open, W_{0}W_{1}⊂V^{−1}}({y : ∃h ∈ W_{0}(h · y ∈ A)})^{ΔW_{1}}.

There are two parts to demonstrating this. We show these two inclusions separately.

⊂: Let

g · x = z,

g ∈ V, x ∈ A. Then considering the continuity of the action we may choose open V_{0} and V_{1} containing g^{−1} and 1 which satisfy:

V_{0}(V_{1})^{−1}V_{1} ⊂ V^{−1}.

We can then let W_{0} = V_{0}V_{1}^{−1} and W_{1} = V_{1}.

With this arranged we have that for all ϵ ∈ W_{1}

(g^{−1}ϵ^{−1})ϵ · z = (g^{−1}ϵ^{−1})ϵg · x = x ∈ A.

Noting that any such g^{−1}ϵ^{−1} will be in W_{0} we have in particular that

z ∈ ({y : ∃h ∈ W_{0}(h · y ∈ A)})^{∗W_{1}} ⊂ ({y : ∃h ∈ W_{0}(h · y ∈ A)})^{ΔW_{1}}.

⊃: Consider some

z ∈ ({y : ∃h ∈ W_{0}(h · y ∈ A)})^{ΔW_{1}},

where W_{0}W_{1} ⊂ V^{−1}. Then in particular, we can find g_{1} ∈ W_{1} with

g_{1} · z ∈ {y : ∃h ∈ W_{0}(h · y ∈ A)}.

And then we may find some g_{2} ∈ W_{0} with

g_{2}g_{1} · z ∈ A.

But then

g_{2}g_{1} ∈ V^{−1}

by assumption on W_{0} and W_{1}, and hence

x = g_{2}g_{1} · z

and

g = (g_{2}g_{1})^{−1}

witness that z ∈ V · A. (Claim□) In summary, the above claims have shown

π : G × X_{low} → X_{∗low}

to be an open continuous map from a Polish space onto a metric space. Thus by Hausdorff’s theorem, (X_{∗low}, τ_{∞}^{∗}) is Polish. □

3 The split in cases

As above, assume that G is a recursive Polish group and X is a recursive Polish G-space.

Definition A sequence

B⃗ = (B_{α})_{α∈δ}

is suitable if

1. each B_{α} is a subset of the Borel subsets of X; 2. δ < ω_{1}^{ck};

3. the function

α ↦ B_{α}

is Δ_{1}^{1} in the codes for Borel sets;

4. in particular each B_{α} is countable;

5. each B_{α} includes the basic open sets in the original topology on X;

6. each B_{α} is closed under G_{0}-translation.

This definition has glossed over the details of being Δ_{1}^{1} in the codes for Borel sets; a protracted discussion is unlikely to be helpful. The reader will be able to provide their own coding for Borel sets, and any reasonable understanding of “in the codes” should suffice; for instance we can ask that there be some recursive wellorder ≺ of length δ and Δ_{1}^{1} function

F : ω × ω → ω^{ω}

such that for any n ∈ ω and α = ||n||_{≺} we have that

{F (n, m) : m ∈ ω}

gives a set of codes for the Borel sets appearing in the B_{α}.

Given such an admissible sequence, we define a sequence of objects

φ̂B⃗ ; α(x, B^{ΔV_{0}}, V_{1})

around each point x ∈ X and at each α < δ. Definition Given an admissible sequence B⃗ = (B_{β})_{β∈δ} as above, x ∈ X, B ∈ B_{α}, V_{0}, V_{1} ⊂ G basic open, V_{1} containing 1_{G}, V_{1}^{−1} = V_{1}, we define φ̂B⃗ ; β(x, B^{ΔV_{0}}, V_{1}) by induction on β:- BASE CASE: for x ∈ B^{ΔV_{0}}

φ̂B⃗ ; 0(x, B^{ΔV_{0}}, V_{1})

equals the set of ℓ ∈ ω such that there exists k ∈ ω, x_{0}, x_{1}, ..., x_{k} ∈ X such that x_{0} = x;

x′ = x_{k} ∈ U_{ℓ};
each x_{i+1} ∈ V_{1} · x_{i} ∩ B^{ΔV_{0}}.

INDUCTIVE STEP: If β is an ordinal lying strictly between 0 and δ, then

φ̂B⃗ ; β(x, B^{ΔV_{0}}, V_{1})

equals the set of all

⟨φ̂B⃗ ; α(x′, C^{ΔW_{0}}, W_{1}), C, W_{0}, W_{1}⟩,

where

α < β;
C ∈ B_{α};
W_{0}, W_{1} ⊂ G basic open;
W_{1} contains 1_{G}, W_{1}^{−1} = W_{1},
x_{0} = x;
x′ = x_{k} ∈ C^{ΔW_{0}};
each x_{i+1} ∈ V_{1} · x_{i} ∩ B^{ΔV_{0}}.

Notation For ϵ > 0 we let W_{ϵ} be the set

{g ∈ G|d_{G}(g, 1_{G}) < ϵ}.

Definition For A ⊂ X, A ∈ Σ_{1}^{1}, V_{0}, V_{1} ⊂ G basic open, V_{1}^{−1} = V_{1}, 1_{G} ∈ V_{1}, and x ∈ A^{ΔV_{0}}, we let

O(x, A^{ΔV_{0}}, V_{1})

be the set of x′ for which there exists k, x_{0}, x_{1}, ..., x_{k} such that x_{0} = x;

x′ = x_{k};
each x_{i+1} ∈ V_{1} · x_{i} ∩ A^{ΔV_{0}}.

Definition For B⃗ = (B_{α})_{α<δ} suitable, ϵ > 0, A ∈ Σ_{1}^{1}, V_{0}, V_{1} ⊂ G basic open, V_{1}^{−1} = V_{1}, 1_{G} ∈ V_{1}, we say that

(A, V_{0}, V_{1}) is ϵ-stable for B⃗

if for all x, x′, x′′, α < δ, V_{2}, V_{3}, B with 1. x ∈ A^{ΔV_{0}},

2. x′, x′′ ∈ O(x, A^{ΔV_{0}}, V_{1}),

3. α < δ,

4. B ∈ B_{α},

5. V_{2}, V_{3} ⊂ G basic open,

6. V_{3}^{−1} = V_{3},

7. 1_{G} ∈ V_{3}, x′ ∈ B^{ΔV_{2}} implies

∃^{∗}g ∈ W_{ϵ}(g · x′′ ∈ B^{ΔV_{2}} ∧ φ̂B⃗ ; α(x′, B^{ΔV_{2}}, V_{3}) = φ̂B⃗ ; α(g · x′′, B^{ΔV_{2}}, V_{3})).

There are a few remarks about this definition. Firstly that if (A, V_{0}, V_{1}) is \frac{1}{n}-stable then it is ϵ-stable for any larger ϵ. The second observation is that we may replace

∃^{∗}g ∈ W_{ϵ}(g · x′′ ∈ B^{ΔV_{2}} ∧ φ̂B⃗ ; α(x′, B^{ΔV_{2}}, V_{3}) = φ̂B⃗ ; α(g · x′′, B^{ΔV_{2}}, V_{3}))

by

∃g ∈ W_{ϵ}(g · x′′ ∈ B^{ΔV_{2}} ∧ φ̂B⃗ ; α(x′, B^{ΔV_{2}}, V_{3}) = φ̂B⃗ ; α(g · x′′, B^{ΔV_{2}}, V_{3}));

if one g ∈ W_{ϵ} brings about the desired result then it follows from the structure of the definitions that an open set of them will, and hence certainly a non-meager set will as well. Lemma 3.1 If B⃗ is a suitable sequence, n ∈ ω, V_{0}, V_{1} ⊂ G basic open, V_{1}^{−1} = V_{1}, 1_{G} ∈ V_{1}, A ∈ Σ_{1}^{1}, and if

(A, V_{0}, V_{1})

is \frac{1}{n}-stable for B⃗, then there is a Δ_{1}^{1} set D ⊃ A with

(D, V_{0}, V_{1})

\frac{1}{n}-stable for B⃗.

Proof For any Δ_{1}^{1}(x) set C we have that the set of x′′ with

∃g^{∗} ∈ W_{\frac{1}{n}} (g · x′′ ∈ C)

is uniformly Δ_{1}^{1}(x); in particular, the set of x′′ with

∃^{∗}g ∈ W_{ϵ}(g · x′′ ∈ B^{ΔV_{2}} ∧ φ̂B⃗ ; α(x′, B^{ΔV_{2}}, V_{3}) = φ̂B⃗ ; α(g · x′′, B^{ΔV_{2}}, V_{3}))

is uniformly, in the indices for V_{0}, V_{1}, and B, Δ_{1}^{1}(x).

On the other hand, the set

O(x, A^{ΔV_{0}}, V_{1})

is uniformly Σ_{1}^{1}(x).

Thus the collection of Σ_{1}^{1} sets A with (A, V_{0}, V_{1}) \frac{1}{n}-stable is uniformly Π_{1}^{1} in the indices.

And thus the lemma follows from Π_{1}^{1} reflection.^{2} □

Definition For B⃗ = (B_{α})_{α∈δ} suitable and n ∈ ω, let A_{B⃗,n} be the set of all x ∈ X for which there is a non-meager collection of g ∈ G with

∀D ∈ Δ_{1}^{1}, V_{0}, V_{1} ⊂ G basic open

(g · x ∈ D^{ΔV_{0}} ⇒ ((D, V_{0}, V_{1}) not \frac{1}{n} − stable for B⃗))).

For a given n and B⃗, the set A_{B⃗,n} is Σ_{1}^{1}.

Notation From now on let us endow X_{∗low} ∩ A_{B⃗,n} with the topology τ_{n,∞}^{∗} generated by all sets of the form

A^{ΔV}

where A ∈ Σ_{1}^{1}, A ⊂ X_{∗low} ∩ A_{B⃗,n}, and V ⊂ G basic open. Let

(Û_{ℓ})_{ℓ∈ω}

enumerate a basis for τ_{n,∞}^{∗}.

Lemma 3.2 (X_{∗low} ∩ A_{b⃗,n}, τ_{n,∞}^{∗}) is a Polish G-space.

Proof Since it is an open subset of (X_{∗low}, τ_{∞}^{∗}). □

At this point the proof splits into two cases. Case(1) below leads to a turbulent Polish G-space Y as in (II) above. Case(2), on the other hand, will finish with a classification of the G-orbits by countable models considered up to isomorphism.

2See [2] for the statement and proof of this theorem.

4 Case(1)

For some suitable B⃗ = (B_{α})_{α∈δ} and n ∈ ω, A_{B⃗,n} is non-empty.

Let us fix such a B⃗ and n. We will press forward, and eventually show that option (II) in 1.1 holds. Following §6.2 of [3] we may define “local invariants” for the Polish G-space (X_{∗low} ∩ A_{B⃗,n}, τ_{n,∞}^{∗}).

Notation For x ∈ X_{∗low} ∩ A_{B⃗,n} and U τ_{n,∞}^{∗}-open, V ⊂ G a symmetric open neighborhood of the identity, we let

O(x, U, V)

be the set of all x′ such that there exists k, x_{0}, x_{1}, ..., x_{k} with

x_{0} = x,
x_{k} = x′,

and at each i < k

x_{i+1} ∈ U ∩ V · x_{i}.

Fix some enumeration (Û_{ℓ})_{ℓ∈ω} of a basis for τ_{∞}^{∗}. We then set off defining

φ_{0}(x, U, V)

be the set of ℓ such that

O(x, U, V) ∩ Û_{ℓ} ≠ 0.

And we continue in this direction by inductively defining

φ_{α}(x, U, V)

to be the set of

⟨φ_{β}(x′, U′, V′), U′, V′⟩,

where β ranges over ordinals less than α, U′ ranges over basic open subsets of U, V′ ranges over symmetric basic open neighborhoods of 1_{G} included in V, and x′ ranges over elements of U′ ∩ O(x, U, V).

In other words, φ_{0}(x, U, V) encodes the closure with respect to τ_{n,∞}^{∗} of the local orbit O(x, U, V); and for α > 0 the local invariant φ_{α}(x, U, V) encodes the set of all possible φ_{β}(x′, U′, V′) we can obtain by reducing α to β, U to U′. V to V′, and passing to some other x′ ∈ O(x, U, V).

Let us fix some particular x ∈ X_{∗low} ∩ A_{B⃗,n}; it in fact does not matter which. Still following the relevant §6.2 of [3] we come, for any given x in the space, to some γ(x) < ω_{1} such that for all U that is τ_{n,∞}^{∗}-open, for all V ⊂ G that is a symmetric basic open neighborhood of the identity, and for all x′.x′′ ∈ [x]_{G}

φ_{ω_{1}}(x′, U, V) = φ_{ω_{1}}(x′′, U, V)

iff

φ_{γ(x)}(x′, U, V) = φ_{γ(x)}(x′′, U, V).

We then obtain a Polish topology on the set

X^{x} =_{df} {y ∈ X_{∗low} ∩ A_{B⃗,n}|φ_{γ(x)+2}(x, X_{∗low} ∩ A_{B⃗,n}, G) = φ_{γ(x)+2}(y, X_{∗low} ∩ A_{B⃗,n}, G)}

by taking as a basis all sets of the form

{y|φ_{γ(x)}(y, U, V) = φ_{γ(x)}(x′, U, V)},

as U ranges over τ_{n,∞}^{∗}-open sets, V ranges over symmetric basic open neighborhoods of the identity, and x′ ranges over [x]_{G}.

From this point on let us understand that X^{x} always refers to X^{x} in the indicated Polish topology.

Lemma 4.1 For any y ∈ X^{x},

\overline{[x]_{G}} = \overline{[y]_{G}}.

Here we understand that the closure operation refers to the indicated topology on the space X^{x}.

Proof An immediate consequence of the definitions. □

Corollary 4.2 For any y ∈ X^{x}, A ⊂ X a Σ_{1}^{1} set,

[x]_{G} ∩ A ≠ 0 iff [y]_{G} ∩ A ≠ 0.

Of course in this corollary it is important that we restrict ourselves to sets A which are lightface Σ_{1}^{1} with respect to the original space X.

Lemma 4.3 For any y ∈ X^{x},

γ(x) = γ(y).

Moreover for any U that is τ_{n,∞}^{∗} basic open and V ⊂ G a symmetric basic open neighborhood of the identity,

O(y, U, V)

is dense (with respect to the topology on X^{x}) in some basic open set of the form

{y|φ_{γ(x)}(y, U, V) = φ_{γ(x)}(x′, U, V)}.

Proof These facts can be verified from the definitions in a reasonably routine manner; or see §6.2 [3] for the details. □

From now until the end of the section, let us just fix some y ∈ X^{x}. We will try to show that [y]_{G} is meager in the space X^{x}.

If we succeed in this task, for some arbitrarily chosen y ∈ X^{x}, then we will have obtained that every orbit in the space must be meager. And then by 4.1 and 4.3 we have that the definition of turbulence is satisfied.

Now turning our attention back towards the fixed, but arbitrary, y ∈ X^{x} we have from 4.2 that there will be some

ŷ ∈ [y]_{G}

such that for all^{3} D ∈ Δ_{1}^{1}, V_{0}, V_{1} ⊂ G basic open, if

ŷ ∈ D^{ΔV_{0}},

then (D, V_{0}, V_{1}) is not \frac{1}{n}-stable for B⃗. Hence by 3.1, for all A ∈ Σ_{1}^{1}, V_{0}, V_{1} ⊂ G basic open, if

ŷ ∈ A^{ΔV_{0}},

then (A, V_{0}, V_{1}) is not \frac{1}{n}-stable for B⃗.

Fix this ŷ. We need to “blur” this element of [y]_{G} making it slightly more generic, without altering its important properties; for this purpose let us fix some m > n such that

(W_{\frac{1}{m}})^{±}(W_{\frac{1}{m}})^{±}(W_{\frac{1}{m}})^{±} ⊂ W_{\frac{1}{n}} .

3Here, and everywhere else the issue might emerge, Σ_{1}^{1}, Δ_{1}^{1}, and Π_{1}^{1} refer to Σ_{1}^{1}(X), Δ_{1}^{1}(X), and Π_{1}^{1}(X) respectively.

Claim 4.4 There exists y′ ∈ [y]_{G} such that:

(i) for all A ∈ Σ_{1}^{1}, if y′ ∈ A, then

y′ ∈ A^{∗W}

for some basic open neighborhood W of 1_{G};

(ii) ω_{1}^{ck} = ω_{1}^{ck(y′)}; and hence, for all A ∈ Σ_{1}^{1}, either

y′ ∈ A

or there is a Δ_{1}^{1} set D with

y′ ∈ D

and

D ∩ A = 0;

(iii) for all A ∈ Σ_{1}^{1}, V_{0}, V_{1} ⊂ G basic open, if

y′ ∈ A^{ΔV_{0}},

then (A, V_{0}, V_{1}) is not \frac{1}{m}-stable for B⃗.

Proof Note first of all that since y ∈ X^{x} we may certainly find g ∈ W_{\frac{1}{m}} such that

y′ = g · ŷ

satisfies (i) and (ii); any sufficiently generic g ∈ W_{\frac{1}{m}} will do. For (iii) we need a further subclaim.

Subclaim: If (A, V_{0}, V_{1}) is \frac{1}{m}-stable and g ∈ W_{\frac{1}{m}} with

y′ = g · ŷ ∈ A^{ΔV_{0}},

then there is some \frac{1}{n}-stable (C, V′_{0}, V′_{1}) with

ŷ ∈ C^{ΔV′_{0}}.

Proof of subclaim: Assume not, and then we may assume further that g ∈ G_{0}, since slightly nudging g into G_{0} should not affect O(y′, A^{ΔV_{0}}, V_{1}).

Now consider some arbitrary ẑ ∈ X with

z′ = g · ẑ ∈ A^{ΔV_{0}},

and

ẑ^{∗} ∈ O(ẑ, A^{ΔV_{0}g}, g^{−1}V_{1}g).

Let

z^{∗} = g · ẑ^{∗} ∈ O(z′, A^{ΔV_{0}}, V_{1}).

By assumption on (A, V_{0}, V_{1}), for any α < δ, B ∈ Σ_{1}^{1}, and V_{2}, V_{3} basic open in G, with B^{ΔV_{2}} containing z, (and hence B^{ΔV_{2}g^{−1}} containing z^{∗} = g · ẑ^{∗}) we may find

h ∈ W_{\frac{1}{m}}

with

φ̂B⃗ ; α(hg · ẑ, B^{ΔV_{2}g^{−1}}, gV_{3}g^{−1})

G. Hjorth 13 (which by definition of z′ equals

φ̂B⃗ ; α(h · z′, B^{ΔV_{2}g^{−1}}, gV_{3}g^{−1}))

equal to

φ̂B⃗ ; α(z^{∗}, B^{ΔV_{2}g^{−1}}, gV_{3}g^{−1})

∴ φ̂B⃗ ; α(g^{−1}hg · ẑ, B^{ΔV_{2}}, gV_{3}g^{−1}) = φ̂B⃗ ; α(g^{−1}z^{∗}, B^{ΔV_{2}}, gV_{3}g^{−1}),

which in turn by the definition of z^{∗} equals

φ̂B⃗ ; α(ẑ^{∗}, B^{ΔV_{2}}, gV_{3}g^{−1}).

Thus we have shown (A, V_{0}g, g^{−1}V_{1}g) to be \frac{1}{n}-stable. (Claim□) But this subclaim proves the claim, since ŷ was assumed not to be so captured by any \frac{1}{n}-stable set.

□

Let us then fix y′ as above. Claim 4.5 For all A ∈ Σ_{1}^{1}, V_{0}, V_{1} ⊂ G basic open with 1_{G} ∈ V_{1}, V_{1}^{−1} = V_{1}, if

y′ ∈ A^{ΔV_{0}}

then

∃y_{1}, y_{2} ∈ O(y′, A^{ΔV_{0}}, V_{1})
∃α < δ∃B ∈ B_{α}
∃V_{2}, V_{3} ⊂ G basic open,1_{G} ∈ V_{3}, V_{3}^{−1} = V_{3}

with

y_{1} ∈ B^{ΔV_{2}}

but

∀g ∈ W_{\frac{1}{m}} (g · y_{2} ∈ B^{ΔV_{2}} ⇒ φ̂B⃗ ; α(y_{1}, B^{ΔV_{2}}, V_{3}) ≠ φ̂B⃗ ; α(g · y_{2}, B^{ΔV_{2}}, V_{3})).

Proof Otherwise applying (ii) from our assumptions on y′ we find D ∈ Δ_{1}^{1} containing y′ such that

∀z′ ∈ A^{ΔV_{0}} ∩ D

∀z_{1}, z_{2} ∈ O(z′, A^{ΔV_{0}}, V_{1})
∀α < δ∀B ∈ B_{α}
∀V_{2}, V_{3} ⊂ G basic open, 1_{G} ∈ V_{3}, V_{3}^{−1} = V_{3}

if

z_{1} ∈ B^{ΔV_{2}}

then

∃g ∈ W_{\frac{1}{n}} (g · y_{1} ∈ B^{ΔV_{2}} ∧ φ̂B⃗ ; α(z_{1}, B^{ΔV_{2}}, V_{3}) = φ̂B⃗ ; α(g · z_{2}, B^{ΔV_{2}}, V_{3})),

and hence as prior to 3.1

∃^{∗}g ∈ W_{\frac{1}{n}} (g · y_{1} ∈ B^{ΔV_{2}} ∧ φ̂B⃗ ; α(z_{1}, B^{ΔV_{2}}, V_{3}) = φ̂B⃗ ; α(g · z_{2}, B^{ΔV_{2}}, V_{3})).

By (i) we obtain

y′ ∈ D^{∗W}

G. Hjorth 14 some basic open neighborhood W of 1_{G}. After possibly shrinking W and V_{0} we can assume

W ⊂ V_{1}

and that there is some g_{0} ∈ G_{0} with

g_{0}W = V_{0}.

Here it will be easier if we make a harmless simplifying assumption. I claim that we may assume V_{0} = W and that g_{0} = 1_{G}. The point is that we can always replace A by the set

g_{0}^{−1}A,

and observe that

A^{Δg_{0}W} = (g_{0}^{−1}A)^{ΔW} .

So after making this minor notational reorganization let us assume that 1_{G} = g_{0},

W = V_{0},
1_{G} ∈ V_{0},
V_{0} ⊂ V_{1},
y′ ∈ A^{ΔV_{0}} ∩ D^{∗V_{0}} ⊂ (A ∩ D)^{ΔV_{0}}.

Then for any

z ∈ (A ∩ D)^{ΔV_{0}},
z_{1}, z_{2} ∈ O(z, (A ∩ D)^{ΔV_{0}}, V_{1}),

and for any

α < δ
B ∈ B_{α}
V_{2}, V_{3} ⊂ G basic open

with

1_{G} ∈ V_{3}, V_{3}^{−1} = V_{3}
z_{1} ∈ B^{ΔV_{2}}

we may find h_{0} ∈ V_{0} ⊂ V_{1} with

z′ =_{df} h_{0} · z ∈ D ∩ (A ∩ D)^{ΔV_{0}},
∴ O(z′, (A ∩ D)^{ΔV_{0}}, V_{1}) = O(z, (A ∩ D)^{ΔV_{0}}, V_{1}),
∴ z_{1}, z_{2} ∈ O(z′, (A ∩ D)^{ΔV_{0}}, V_{1}) ⊂ O(z′, A^{ΔV_{0}}, V_{1}),

and hence by the assumptions on D and the fact that z′ ∈ D we have

∃^{∗}g ∈ W_{\frac{1}{m}} (g · y_{1} ∈ B^{ΔV_{2}} ∧ φ̂B⃗ ; α(z_{1}, B^{ΔV_{2}}, V_{3}) = φ̂B⃗ ; α(g · z_{2}, B^{ΔV_{2}}, V_{3})).

And thus we have seen that (A ∩ D, V_{0}, V_{1}) is \frac{1}{m}-stable. Which contradicts assumption (iii) on y′. □

But the statement of this lemma is catastrophic with respect to y’s hope of having a G_{δ} orbit. It implies in particular that for all A ∈ Σ_{1}^{1}, V_{0}, V_{1} ⊂ G basic open with 1_{G} ∈ V_{1}, V_{1}^{−1} = V_{1}, if

y′ ∈ A^{ΔV_{0}}

then

∃y_{1}, y_{2} ∈ A^{ΔV_{0}} ∩ [y]_{G}

with

O(y_{1}, A^{ΔV_{0}}, V_{1}) = O(y_{2}, A^{ΔV_{0}}, V_{1}),
∴ φ_{γ(x)}(y_{1}, A^{ΔV_{0}}, V_{1}) = φ_{γ(x)}(y_{2}, A^{ΔV_{0}}, V_{1}) = φ_{γ(x)}(y′, A^{ΔV_{0}}, V_{1})

but

∀g ∈ G(d_{G}(g, 1) < \frac{1}{m} ⇒ g · y_{1} ≠ y_{2}).

But now since the sets of the form

{z : φ_{γ(x)}(z, A^{ΔV_{0}}, V_{1}) = φ_{γ(x)}(x′, A^{ΔV_{0}}, V_{1})}

(as x′ ranges over [x]_{G}, A ranges over Σ_{1}^{1}, V_{0}, V_{1} range over basic open sets with V_{1} a symmetric neighborhood of the identity) form a basis of the topology on X_{x}, we have that the induced map

G → [y]_{G}
g ↦ g · y

is not open. Thus by the Effros lemma, [y]_{G} is not G_{δ}.

Thus we have shown that no orbit in X_{x} is G_{δ}. And hence all the orbits are meager.

And hence as in §6.2 of [3], all the orbits in X_{x} are dense, the local orbits of the form

O(y, A^{ΔV_{0}}, V_{1})

are all dense inside the corresponding open set

{z : φ_{γ(x)}(z, A^{ΔV_{0}}, V_{1}) = φ_{γ(x)}(x′, A^{ΔV_{0}}, V_{1})}

for some x′ ∈ [x]_{G} with φ_{γ(x)}(y, A^{ΔV_{0}}, V_{1}) = φ_{γ(x)}(x′, A^{ΔV_{0}}, V_{1}), and by the calculation above every orbit is meager. Thus the action is turbulent, and we are done with this case.

5 Case (2)

Not case (1).

Claim 5.1 There is a sequence

B⃗ = (B_{α})_{α∈ω_{1}^{ck}}

such that (a) α ↦ B_{α} is Δ_{1}^{1} in the codes;

(b) for each δ < ω_{1}^{ck} the sequence B⃗|_{δ} = (B_{α})_{α∈δ} is suitable;
(c) for each δ < ω_{1}^{ck}, ϵ > 0, x ∈ X there is some D ∈ B_{δ} along with basic open
V_{0}, V_{1} ⊂ G,
V_{1} a symmetric open neighborhood of the identity, (D, V_{0}, V_{1}) ϵ-stable for B⃗|_{δ} and
x ∈ D^{ΔV_{0}}.

Proof Recalling from Spector-Gandy that a subset of L_{ω_{1}^{ck}} is Π_{1}^{1} in the codes if and only if it is Σ_{1}^{L_{ω{ck \atop 1}}} , it

suffices to show that there is a total Σ_{1}^{L_{ω{ck \atop 1}}} function

α ↦ B_{α},

with each B_{α} included in the indices for Δ_{1}^{1} sets, such that if we let B_{α} be the corresponding collection of actual Δ_{1}^{1} sets encoded by the B_{α}, then (B_{α})_{α∈δ} satisfies (c) above and 1, 2, 4, 5, 6 from the definition

of suitability all δ < ω_{1}^{ck}; note that these requirements are all Π_{1}^{1} and hence Σ_{1}^{L_{ω{ck \atop 1}}} . Since the total Σ_{1}^{L_{ω{ck \atop 1}}}

functions are closed under transfinite Σ_{1}^{L_{ω{ck \atop 1}}} recursion, it suffices to show that given any (B_{α})_{α∈δ} ∈ L_{ω_{1}^{ck}} which gives rise to (B_{α})_{α∈δ} with the desired properties we can find some B_{δ} ∈ L_{ω_{1}^{ck}} such that the slightly extended corresponding (B_{α})_{α∈δ+1} still has (c) above and 1, 2, 4, 5, 6.

We know from the case assumption that given any x ∈ X, ϵ > 0 there will be some D ∈ Δ_{1}^{1} and basic open V_{0}, V_{1} with x ∈ D^{ΔV_{0}} and (D, V_{0}, V_{1}) having the Π_{1}^{1}(ϵ) property of ϵ-stability. The problem is to find a single set of such indices in L_{ω_{1}^{ck}} which works for all such x and ϵ.

Towards this purpose let P ⊂ ω × X be Π_{1}^{1} and universal for Π_{1}^{1} subsets of X. We let

f : ω × X → LO

be recursive continuous such that (n, x) ∈ P if and only if f(n, x) ∈ WO. At each β < ω_{1}^{ck} and n ∈ ω we let P_{β,n} = {x ∈ X : f(x) ∈ WO, ||f(x)|| < β}^{4}. Note that the Δ_{1}^{1} subsets of X are exactly the {P_{α,n} : α < ω_{1}^{ck}, n ∈ ℕ}. By the case assumption and the number uniformization theorem^{5}, we may find a Δ_{1}^{1} function

(ϵ, x) ↦ (α_{ϵ,x}, n_{ϵ,x})

such that each (P_{α_{ϵ,x},n_{ϵ,x}}, V_{0}, V_{1}) is ϵ-stable (some choice of basic open V_{0}, V_{1}) for (B_{α})_{α∈δ}, and

x ∈ P_{α_{ϵ,x},n_{ϵ,x}}^{ΔV_{0}}.

Boundedness gives

sup{α_{ϵ,x} : ϵ > 0, x ∈ X} < ω_{1}^{ck},

and so some such recursive ordinal γ > sup{α_{ϵ,x} : ϵ > 0, x ∈ X} will give {P_{β,n} : n ∈ ℕ, β ≤ γ} as required. Under any reasonable system of indexing Δ_{1}^{1} sets we can find B_{δ} ∈ L_{ω_{1}^{ck}} such that B_{δ} exactly gives the indices of the Δ_{1}^{1} sets appearing in {P_{β,n} : n ∈ ℕ, β ≤ γ}. □

Claim 5.2 For all x, y ∈ X

φ̂B⃗ ; ω_{1}^{ck}(x, X, G) = φ̂B⃗ ; ω_{1}^{ck}(y, X, G) ⇒ xE_{G}y.

Proof Suppose not. Then we may find x, y ∈ X with

φ̂B⃗ ; ω_{1}^{ck}(x, X, G) = φ̂B⃗ ; ω_{1}^{ck}(y, X, G)

but

x¬E_{G}y.

4Here ||w|| denotes the ordinal isomorphic to w for w ∈ WO.

5I.e. that if A ⊂ ω^{ω} × ω is Π_{1}^{1} with all vertical sections non-empty, then there is Δ_{1}^{1} π : ω^{ω} → ω such that for all z we have (z, π(n)) ∈ A.

But then using for the first time that E_{G} is Δ_{1}^{1} we may apply the low basis theorem^{6} for Σ_{1}^{1} and find x, y ∈ X with

ω_{1}^{ck(x,y)} = ω_{1}^{ck},

φ̂B⃗ ; ω_{1}^{ck}(x, X, G) = φ̂B⃗ ; ω_{1}^{ck}(y, X, G),

x¬E_{G}y.

Now choose a recursive ϵ_{0} > 0 such that

ϵ_{0} < \frac{1}{2}

and

∀g ∈ W_{ϵ_{0}}(d_{X}(y, g · y) < \frac{1}{2}).

Subclaim: For all β < ω_{1}^{ck} there exists

x_{0} ∈ [x]_{G}

such that for all α < β, B ∈ B_{α}, V_{2}, V_{3} ⊂ G basic open with V_{3} a symmetric neighborhood of 1_{G},

∃^{∗}h ∈ W_{ϵ_{0}}(φ̂B⃗ ; α(x_{0}, B^{ΔV_{2}}, V_{3}) = φ̂B⃗ ; α(h · y, B^{ΔV_{2}}, V_{3})).

Proof of subclaim: Choose A ∈ B_{β}, V_{0}, V_{1} ⊂ G basic open sets with V_{1} a symmetric neighborhood of the identity, (A, V_{0}, V_{1}) ϵ_{0}-stable for B⃗|_{β}, and

y ∈ A^{ΔV_{0}}.

Then choose x_{0}E_{G}x with

φ̂B⃗ ; β(x_{0}, A^{ΔV_{0}}, V_{1}) = φ̂B⃗ ; β(y, A^{ΔV_{0}}, V_{1}).

It follows from the iterative structure of the definition of the φ̂B⃗ ; β(·, ·, ·) invariants that for each B, V_{2}, V_{3}, α as in statement of subclaim that

∃ŷ ∈ O(y, A^{ΔV_{0}}, V_{1})

with

φ̂B⃗ ; α(x_{0}, B^{ΔV_{2}}, V_{3}) = φ̂B⃗ ; α(ŷ, B^{ΔV_{2}}, V_{3}).

Then by definition of stability we have that

∃^{∗}h ∈ W_{ϵ_{0}}(φ̂B⃗ ; α(h · y, B^{ΔV_{2}}, V_{3}) = φ̂B⃗ ; α(ŷ, B^{ΔV_{2}}, V_{3}) = φ̂B⃗ ; α^{ck}(x_{0}, B^{ΔV_{2}}, V_{3})).

(Subclaim□)

Now we may apply overspill^{7} to find some

x_{0} ∈ [x]_{G}

such that for all α < ω_{1}^{ck}, B ∈ B_{α}, V_{2}, V_{3} ⊂ G basic open with V_{3} a symmetric neighborhood of 1_{G},

∃^{∗}h ∈ W_{ϵ_{0}}(φ̂B⃗ ; α(x_{0}, B^{ΔV_{2}}, V_{3}) = φ̂B⃗ ; α(h · y, B^{ΔV_{2}}, V_{3})).

By the low basis theorem we may assume

ω_{1}^{ck(x_{0},x,y)} = ω_{1}^{ck(x,y)} = ω_{1}^{ck}.

6I.e. any non-empty Σ_{1}^{1} contains an x with ω_{1}^{ck(x)} = ω_{1}^{ck}.

7I.e. the converse of the usual boundedness theorem for Σ_{1}^{1} subset of the recursive ordinals.

At this stage we can choose a recursive real δ_{0} such that

δ_{0} < \frac{1}{2}

∀g ∈ W_{δ_{0}}(d_{X}(x_{0}, g · x_{0}) < \frac{1}{2}).

Subclaim: For all β < ω_{1}^{ck} there exists

y_{0} ∈ W_{ϵ_{0}} · y

such that for all α < β, B ∈ B_{α}, V_{2}, V_{3} ⊂ G basic open with V_{3} a symmetric neighborhood of 1_{G},

∃^{∗}h ∈ W_{δ_{0}}(φ̂B⃗ ; α(y_{0}, B^{ΔV_{2}}, V_{3}) = φ̂B⃗ ; α(h · x_{0}, B^{ΔV_{2}}, V_{3})).

Proof of subclaim: Again we may choose some (A, V_{0}, V_{1}) which is δ_{0}-stable for B⃗|_{β} with x_{0} ∈ A^{ΔV_{0}}. And then we may choose

y_{0} ∈ W_{ϵ_{0}} · y

with

φ̂B⃗ ; β(x_{0}, A^{ΔV_{0}}, V_{1}) = φ̂B⃗ ; β(y, A^{ΔV_{0}}, V_{1}),

and we may complete the argument as in the proof of the last subclaim. (Subclaim□)

Once more overspill allows us to find y_{0} ∈ W_{ϵ_{0}} such that for all α < ω_{1}^{ck}, B ∈ B_{α}, V_{2}, V_{3} ⊂ G basic open with V_{3} a symmetric neighborhood of 1_{G},

∃^{∗}h ∈ W_{δ_{0}}(φ̂B⃗ ; α(y_{0}, B^{ΔV_{2}}, V_{3}) = φ̂B⃗ ; α(h · x_{0}, B^{ΔV_{2}}, V_{3})).

Continuing in a like minded fashion we may produce sequences

g_{0}, g_{1}, ⋯,
x_{0}, x_{1}, ⋯,
g′_{0}, g′_{1}, ⋯,
y_{0}, y_{1}, ⋯,
δ_{0}, δ_{1}, ⋯,
ϵ_{0}, ϵ_{1}, ⋯,

such that:

(0) for all j ∈ ω,

ω_{1}^{ck(x_{j},y_{j})} = ω_{1}^{ck(x_{j+1},y_{j})} = ω_{1}^{ck};

(i) for all j ∈ ω, α < ω_{1}^{ck}, B ∈ B_{α}, V_{2}, V_{3} ⊂ G basic open with V_{3} a symmetric neighborhood of 1_{G},

∃^{∗}h ∈ W_{δ_{j}}(φ̂B⃗ ; α(y_{j}, B^{ΔV_{2}}, V_{3}) = φ̂B⃗ ; α(h · x_{j}, B^{ΔV_{2}}, V_{3}));

(ii) for all j ∈ ω, α < ω_{1}^{ck}, B ∈ B_{α}, V_{2}, V_{3} ⊂ G basic open with V_{3} a symmetric neighborhood of 1_{G},

∃^{∗}h ∈ W_{δ_{j+1}}(φ̂B⃗ ; α(x_{j+1}, B^{ΔV_{2}}, V_{3}) = φ̂B⃗ ; α(h · y_{j}, B^{ΔV_{2}}, V_{3}));

(iii) each x_{j} equals g_{j−1}g_{j−2} ⋯ g_{0} · x_{0};
(iv) each y_{j} equals g′_{j−1}g′_{j−2} ⋯ g′_{0} · y_{0};

(v) each g_{j} ∈ W_{δ_{j}}; (vi) each g′_{j} ∈ W_{ϵ_{j+1}}; (vii) at each j we have ∀g ∈ W_{δ_{j}}

d_{G}(gg_{j−1}g_{j−2} ⋯ g_{0}, g_{j−1} ⋯ g_{0}) < 2^{−j−1};

(viii) at each j we have ∀g ∈ W_{ϵ_{j}}

d_{G}(gg′_{j−1}g′_{j−2} ⋯ g′_{0}, g′_{j−1} ⋯ g′_{0}) < 2^{−j−1};

(ix) at each j

∀g ∈ W_{δ_{j}}(d_{X}(x_{j}, g · x_{j}) < 2^{−j−1});

(x) at each j

∀g ∈ W_{ϵ_{j+1}}(d_{X}(y_{j}, g · y_{j}) < 2^{−j−1}).

All the relevant sequences are Cauchy, so we can find g_{∞}, g′_{∞} with

g_{j} → g_{∞},
g′_{j} → g′_{∞},

and hence by the continuity of the action

x_{j} → g_{∞}x_{0},
y_{j} → g′_{∞}y_{0}.

We will be finished with the proof of the claim if we show that

g_{∞}x_{0} = g′_{∞}y_{0}.

But note that at any j we can apply (i) to find h ∈ W_{δ_{j}} with

d_{X}(h · x_{j}, y_{j}) < 2^{−j−1};

which by (v) yields

d_{X}(x_{j}, y_{j}) ≤ d_{X}(x_{i}, h · x_{i}) + d_{X}(h · x_{i}, y_{i}) < 2^{−j−1} + 2^{−j−1} = 2^{−j}.

And so

x_{0}E_{G}y_{0}
∴ xE_{G}y,

after all. □

Notation Let HC denote the collection of hereditarily countable sets. TC(a) denotes the transitive closure of a set a.

Definition Let us say that

θ : X → HC

is Δ_{1}^{1} in the codes if there is an actual Δ_{1}^{1} function

Θ : X → 2^{ω×ω}

such that for all x ∈ X we have that Θ(x) is a well founded and extensional relation on ω whose transitive collapse is isomorphic to

TC(θ(x)) ∪ {θ(x)};

that is to say, Θ(x) encodes the smallest transitive set containing θ(x) as an element.

Claim 5.3 For all α < ω_{1}^{ck}, B ∈ B_{α}, V_{0}, V_{1} ⊂ G basic open with V_{1} a symmetric neighborhood of the identity, the function

x ↦ φ̂B⃗ ; α(x, B^{ΔV_{0}}, V_{1})

is Δ_{1}^{1} in the codes.

Proof This follows routinely by induction on α. □

Thus we may obtain a Borel function

θ : X → 2^{ω×ω}

such that for all x ∈ X the relation θ(x) on ω is wellfounded, extensional, and has transitive collapse equal to

{φ̂B⃗ ; α(x, X, G) : α < ω_{1}^{ck}} ∪ {TC(φ̂B⃗ ; α(x, X, G)) : α < ω_{1}^{ck}}.

Hence

xE_{G}y

if and only if θ(x) and θ(y) are isomorphic as relations on ω. Thus E_{G} is reducible to

≅|_{Mod(L)},

for L a language generated by a single binary relation.

Finally some words on how we can extend the conclusion of this case to obtain a reduction that allows an orbit inverse.

To start we may apply theorem 5.1.5 of [1] to produce a Polish topology τ̂ on X which is compatible with the original Borel structure, includes all sets of the form

B^{ΔV}

where B ∈ B_{α} some α < ω_{1}^{ck}, and is such that the action of G is still continuous with respect to this space.

From here we can consider the Polish G-space (X, τ̂). For this space we may directly evoke the generalized Scott invariants

x ↦ φ_{x}

from §6.2 of [3]. Traipsing through the definitions there one sees that

φ_{x} = φ_{y} ⇒ ∀α < ω_{1}(φ̂B⃗ ; α(x, X, G) = φ̂B⃗ ; α(y, X, G)),

and so by 5.2

x ↦ φ_{x}

gives a complete invariant. It was shown in §6.2[3] that whenever we have these φ_{x}’s serving as complete invariants then we can indeed obtain a Borel reduction to isomorphism on a class of countable structures which admits an orbit inverse. And thus case(2) leads to a strengthened version of (I) as its outcome.

References

[1] H. Becker, A.S. Kechris, The descriptive set theory of Polish group actions, London Mathematical Society.

[2] L.A. Harrington, D. Marker, S. Shelah, Borel orders, Transactions of the American Mathematical Society, vol. 310(1988), pp. 293-302.

[3] G. Hjorth, Classification and orbit equivalence relations, American Mathematical Society, Rhode Island, 2000.

[4] A.S. Kechris, A. Louveau, The classification of hypersmooth Borel equivalence relations, Journal of the American Mathematical Society, vol. 10(1997), pp. 215–242.

[5] Y.N. Moschovakis, Descriptive set theory, North-Holland, Amsterdam, 1980.

greg@math.ucla.edu

MSB 6363 UCLA CA 90095-1555