Canonization of equivalence relations classifiable by ...jean.saint-raymond/DST/12/sabok.pdf ·...

Canonical Ramsey theory on Polish spacesPrelude

Applications of canonical Ramsey theoryCanonization of e.r. classifiable by countable structures

Beyond classifiable by countable structures

Canonization of equivalence relationsclassifiable by countable structures

Marcin Sabok (IMPAN)

Paris, December 10, 2012

Marcin Sabok (IMPAN) Canonization of equivalence relations classifiable by countable structures

This is joint work with Vladimir Kanovei and Jindrich Zapletal,part of a book Canonical Ramsey Theory on Polish Spaces.

Canonical Ramsey theory begins with the Erdos–Rado theorem.

Theorem (Erdos–Rado)

For every equivalence relation E on [N]2 there is an infinite setA ⊆ N such that one of the following is true for all pairsp, q ∈ [A]2:

p E q if and only if p = q,

p E q if and only if min(p) = min(q),

p E q if and only if max(p) = max(q),

p E q.

Analogous result is true for equivalence relations on [N]n giving abasis of 2n equivalence relations.

p E q.

Definition

Given a Polish space X, a σ-ideal I of Borel sets on X and twoclasses of equivalence relations E and F we say that E canonizesto F and write

E −→I

if for every Borel set B /∈ I and for every E ∈ E there is C ⊆ B,C /∈ I such that E � C belongs to F.

When F = {F1, . . . , Fn} is finite, then we get a finite basis result.

Definition

When F = {id, ev}, then we talk about total canonization for E.

Definition

E −→I

Definition

E −→I

Definition

Theorem (Mycielski, Silver)

Let I be the family of countable sets on X. Then

analytic −→I

id, ev.

Theorem (Connes–Feldman–Weiss)

If X is a Polish space with a Borel probability measure and I is thefamily of measure zero sets, then

amenable −→I

hyperfinite,

where amenable stands for the class of amenable (countable)non-singular equivalence relations.

Theorem (Mycielski, Silver)

Let I be the family of countable sets on X. Then

analytic −→I

id, ev.

Theorem (Connes–Feldman–Weiss)

If X is a Polish space with a Borel probability measure and I is thefamily of measure zero sets, then

amenable −→I

hyperfinite,

where amenable stands for the class of amenable (countable)non-singular equivalence relations.

Canonization of analytic equivalence relations can be proved byabstract forcing considerations.

Definition

We write PI for the poset of all Borel sets which are not in I,ordered by inclusion. PI leads to a forcing extension of theuniverse by a single real xgen, which lies in the intersection of allBorel sets in the generic filter.

Canonization of analytic equivalence relations can be proved byabstract forcing considerations.

Definition

We write PI for the poset of all Borel sets which are not in I,ordered by inclusion. PI leads to a forcing extension of theuniverse by a single real xgen, which lies in the intersection of allBorel sets in the generic filter.

Definition

A forcing PI is proper if for every countable elementary submodelM ≺∗ V and every B ∈ PI ∩M there is C ⊆ B with C ∈ PI suchthat every x ∈ C is PI -generic over M .

This definition is a restatement of a general definition of properforcing due to Shelah. We consider it as the most generalassumption that encapsulates all the interesting examples that wecan ever be interested in.

Definition

A forcing PI is proper if for every countable elementary submodelM ≺∗ V and every B ∈ PI ∩M there is C ⊆ B with C ∈ PI suchthat every x ∈ C is PI -generic over M .

This definition is a restatement of a general definition of properforcing due to Shelah. We consider it as the most generalassumption that encapsulates all the interesting examples that wecan ever be interested in.

Theorem (Kanovei–S.–Zapletal)

Suppose that PI is proper. Let analytic orbit equivalencerelations stand for the class of analytic equivalence relations givenby Borel actions of Polish groups. Then

analytic orbit equivalence relations→I Borel.

Suppose Gy X is a Polish group action inducing an analyticequivalence relation E. By a result of Becker and Kechris, we canwrite

X =⋃α<ω1

as a union of Borel G-invariant sets such that whenever C ⊆ X iscontained in a union of countably many sets Aα, then E � C isBorel.

Now let B ⊆ X be Borel I-positive. Find a countable elementarymodel M that is big enough to code the group G, the action andthe set B. By properness, find C ⊆ B that consists of genericpoints over M . Let γ = M ∩ ω1. By an easy absoluteness result,

C ⊆⋃α<γ

and hence E � C is Borel.

The above result is not true for arbitrary analytic equivalencerelations in place of the orbit ones.

Example

Let J be an analytic non-Borel ideal on ω. Define the equivalencerelation EJ on (2ω)ω by

x EJ y if {n ∈ ω : x 6= y} ∈ J.

Note that EJ is an analytic non-Borel equivalence relation.

One can show that if I be the σ-ideal associated with thecountable-support iteration of the Sacks forcing, then EJ is nonBorel after restriction to any Borel positive set.

Example

x EJ y if {n ∈ ω : x 6= y} ∈ J.

Example

x EJ y if {n ∈ ω : x 6= y} ∈ J.

In the above example, the classes of EJ are not Borel. On theother hand, in case of orbit equivalence relations, all classes arealways Borel

If PI is proper, then

countable analytic→I Borel.

In the above example, the classes of EJ are not Borel. On theother hand, in case of orbit equivalence relations, all classes arealways Borel

If PI is proper, then

countable analytic→I Borel.

Question

Let E be an analytic equivalence relation with Borel classes and PIbe proper. Is it true that

E →I Borel?

The answer to the above question is “yes” if suitable largecardinals exist.

A positive answer to a weaker question also follows from thefollowing

Subquestion

Let A ⊆ X ×X be analytic. Is it true that

{x ∈ X : Ax is Π0α} is Π1

Question

E →I Borel?

Subquestion

Question

E →I Borel?

Subquestion

Question

E →I Borel?

Subquestion

Definition

Given two equivalence relation E and F on X, Y respectively anda Borel probability measure µ on X we say that E is F ,µ-ergodicif every homomorphism f : X → Y from E to F maps a set ofmeasure 1 to a single F -class

For example, E0 is id-ergodic with respect to the usual Lebesguemeasure

This is a measure analogue of the notion of generically ergodicequivalence relations, by Hjorth and Kechris.

Definition

E2 is the equivalence relation on 2ω with

x E2 y if∑{ 1

n+ 1: x(n) 6= y(n)} <∞

E2 is Borel bi-reducible with the orbit relation of `1 y Rω and ingeneral, for p ≤ ∞, we write `p for the orbit equivalence relation of`p y Rω.

Definition

F2 is the equivalence relation =+, defined on (2ω)ω by

(xn : n < ω) F2 (yn : n < ω) if {xn : n < ω} = {yn : n < ω}.

Definition

x E2 y if∑{ 1

n+ 1: x(n) 6= y(n)} <∞

Definition

x E2 y if∑{ 1

n+ 1: x(n) 6= y(n)} <∞

Definition

Application 1 (Kanovei–S.–Zapletal)

If E is classifiable by countable structures, then

E2 is E, λ-ergodic

where λ is the usual Lebesgue measure on 2ω.

If E is an analytic equivalence relation, then exactly one of thefollowing holds:

F2 ≤B E,

F2 is E,µ-ergodic and E-generically ergodic,

where µ is the usual product measure on (2ω)ω

If E is classifiable by countable structures, then

E2 is E, λ-ergodic

where λ is the usual Lebesgue measure on 2ω.

If E is an analytic equivalence relation, then exactly one of thefollowing holds:

F2 ≤B E,

F2 is E,µ-ergodic and E-generically ergodic,

where µ is the usual product measure on (2ω)ω

Total canonizationCountable equivalence relationsClassifiable by countable structuresFat treesE2 canonization

I has total canonization for smooth equivalence relations if

smooth −→I

id, ev

Definition

A forcing notion P adds a minimal real degree if every real in thegeneric extension is either a ground model real, or is interdefinablewith the generic filter.

This means that there are no intermediate forcing extensionsbetween V and V [G], which are generated by a single real.

A forcing notion P adds a minimal extension if there are nointermediate models of ZFC between V and V [G].

smooth −→I

id, ev

Definition

smooth −→I

id, ev

Definition

smooth −→I

id, ev

Definition

Suppose PI is proper. The following are equivalent:

I has total canonization for smooth equivalence relations

PI adds a minimal real degree,

A question that motivated some part of this research is to whatextent total canonization for analytic equivalence relations isconnected with PI adding a minimal forcing extension.

Suppose PI is proper. The following are equivalent:

I has total canonization for smooth equivalence relations

PI adds a minimal real degree,

A question that motivated some part of this research is to whatextent total canonization for analytic equivalence relations isconnected with PI adding a minimal forcing extension.

Definition

A forcing notion is nowhere ccc if it is not ccc under any condition.

A (proper) forcing is ℵ0-distributive if it does not add reals.

Definition

A forcing notion is nowhere ccc if it is not ccc under any condition.

A (proper) forcing is ℵ0-distributive if it does not add reals.

Suppose PI is proper, adds minimal forcing extension and isnowhere ccc. Then

essentially countable→I {id, ev}.

Comments on the proof

The assumption gives total canonization for smooth equivalencerelations.

We need to prove that essentially countable canonize to smooth.We pick an ess. countable equivalence relation E.

On general grounds, we are able to decompose the PI -extensioninto a two step iteration P ∗Q (both correspond to E)

If E does not canonize to smooth, then P is ℵ0-distributive and Qis ccc. This contradicts the assumptions.

Definition

A forcing notion adds finitely many real degrees if in every forcingextension there are finitely many reals r1, . . . , rn such that everyother real is inter-definable with one of them.

In particular, if PI has total canonization for smooth equivalencerelations, then it adds finitely many real degrees (i.e. exactly one).

Typically, if I has total canonization for smooth equivalencerelations and In is a finite product of n-many copies of I, then PIn

adds finitely many (exactly, 2n) real degrees.

Definition

Suppose PI is proper and adds finitely many real degrees. Then

classifiable by countable structures→I essentially countable.

Corollary 1

If PI is proper and has total canonization for essentially countableequivalence relations, then it has total canonization for relationsclassifiable by countable structures.

Suppose PI is proper and adds finitely many real degrees. Then

classifiable by countable structures→I essentially countable.

Corollary 1

If PI is proper and has total canonization for essentially countableequivalence relations, then it has total canonization for relationsclassifiable by countable structures.

Corollary 2

If PI is proper, nowhere ccc and adds a minimal forcing extension.Then

classifiable by countable structures→I {id, ev}.

Definition

E is nontrivially I-ergodic if for every Borel A,B /∈ Ithere are x ∈ A, y ∈ B for which x E y holds,

there are x ∈ A, y ∈ B for which x E y fails,

A typical example of ergodicity is the case of E = E0 and I thefamily of meager (or null) sets. In these cases, total canonizationfails badly.

Definition

E is nontrivially I-ergodic if for every Borel A,B /∈ Ithere are x ∈ A, y ∈ B for which x E y holds,

there are x ∈ A, y ∈ B for which x E y fails,

A typical example of ergodicity is the case of E = E0 and I thefamily of meager (or null) sets. In these cases, total canonizationfails badly.

Definition

We say that an equivalence relation E belongs to the spectrum ofI if for every I-positive Borel set B the restriction E � B isbireducible with E.

The spectrum of I consists of equivalence relations which are “notinvertible” and has to appear on every canonical list.

Definition

We say that an equivalence relation E belongs to the spectrum ofI if for every I-positive Borel set B the restriction E � B isbireducible with E.

The spectrum of I consists of equivalence relations which are “notinvertible” and has to appear on every canonical list.

There is a σ-ideal I such that

PI is proper, nowhere ccc, adds a minimal forcing extension,

E2 belongs to the spectrum of I,

I is nontrivially I-ergodic.

Corollary

The σ-ideal I above has total canonization for e.r. classifiable bycountable structures.

There is a σ-ideal I such that

PI is proper, nowhere ccc, adds a minimal forcing extension,

E2 belongs to the spectrum of I,

I is nontrivially I-ergodic.

Corollary

The σ-ideal I above has total canonization for e.r. classifiable bycountable structures.

Comment of the proof

The construction of I above uses the so-called fat trees and aconcentration of measure trick

Corollary

Application 1 follows abstractly from the above theorem, bycanonizing an equivalence relation which contains E2.

Comment of the proof

The construction of I above uses the so-called fat trees and aconcentration of measure trick

Corollary

Application 1 follows abstractly from the above theorem, bycanonizing an equivalence relation which contains E2.

Definition

Let I be the family of Borel sets B ⊆ 2ω such that E2 � B isessentially countable.

It is not difficult to show that I is closed under taking countableunions

Definition

Let I be the family of Borel sets B ⊆ 2ω such that E2 � B isessentially countable.

It is not difficult to show that I is closed under taking countableunions

Let `p equivalences denote the class of Borel equivalence relationsreducible to some `p with p <∞. Then

`p equivalences −→I

id, ev, E2.

Corollary

classifiable by countable structures −→I

id, ev

Let `p equivalences denote the class of Borel equivalence relationsreducible to some `p with p <∞. Then

`p equivalences −→I

id, ev, E2.

Corollary

classifiable by countable structures −→I

id, ev

Let E be an analytic equivalence relation with classes in I.Suppose PI is proper. If PI adds a minimal forcing extension, then

either

E canonizes to id,

or E � C is ergodic for some Borel C /∈ I.

Let E be an analytic equivalence relation with classes in I.Suppose PI is proper. If PI adds a minimal forcing extension, then

either

E canonizes to id,

or E � C is ergodic for some Borel C /∈ I.

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