the complexity of classification problems in ergodic theory€¦ · t has discrete spectrum , i.e.,...
TRANSCRIPT
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The complexity of classification problems inergodic theory
Alexander S. Kechris
June 27, 2011
Dedicated to the memory of Greg Hjorth (1963-2011)
The complexity of classification problems in ergodic theory
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I. INTRODUCTION
The complexity of classification problems in ergodic theory
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The last two decades have seen the emergence of a theory of settheoretic complexity of classification problems in mathematics. Inthis talk I will survey recent developments concerning theapplication of this theory to classification problems in ergodictheory.
The complexity of classification problems in ergodic theory
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Measure preserving transformations
Definition
A standard measure space is a measure space (X,µ), where X is aPolish space and µ a non-atomic Borel probability measure on X.
All such spaces are isomorphic to the unit interval with Lebesguemeasure.
Definition
A measure preserving transformation on (X,µ) is a measurablebijection T such that µ(T (A)) = µ(A), for any Borel set A.
The complexity of classification problems in ergodic theory
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Measure preserving transformations
Definition
A standard measure space is a measure space (X,µ), where X is aPolish space and µ a non-atomic Borel probability measure on X.
All such spaces are isomorphic to the unit interval with Lebesguemeasure.
Definition
A measure preserving transformation on (X,µ) is a measurablebijection T such that µ(T (A)) = µ(A), for any Borel set A.
The complexity of classification problems in ergodic theory
![Page 6: The complexity of classification problems in ergodic theory€¦ · T has discrete spectrum , i.e., there is a basis ... (countable) subgroup of T. It turns out that up to isomorphism](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0438a67e708231d40ce90f/html5/thumbnails/6.jpg)
Measure preserving transformations
Definition
A standard measure space is a measure space (X,µ), where X is aPolish space and µ a non-atomic Borel probability measure on X.
All such spaces are isomorphic to the unit interval with Lebesguemeasure.
Definition
A measure preserving transformation on (X,µ) is a measurablebijection T such that µ(T (A)) = µ(A), for any Borel set A.
The complexity of classification problems in ergodic theory
![Page 7: The complexity of classification problems in ergodic theory€¦ · T has discrete spectrum , i.e., there is a basis ... (countable) subgroup of T. It turns out that up to isomorphism](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0438a67e708231d40ce90f/html5/thumbnails/7.jpg)
Measure preserving transformations
Examples
X = T with the usual measure; T (z) = az, where a ∈ T, i.e.,T is a rotation.
X = 2Z, T (x)(n) = x(n− 1), i.e., the shift transformation.
Definition
A mpt T is ergodic if every T -invariant set has measure 0 or 1.
Any irrational, modulo π, rotation and the shift are ergodic.
The ergodic decomposition theorem shows that every mpt can becanonically decomposed into a (generally continuous) direct sum ofergodic mpt’s.
The complexity of classification problems in ergodic theory
![Page 8: The complexity of classification problems in ergodic theory€¦ · T has discrete spectrum , i.e., there is a basis ... (countable) subgroup of T. It turns out that up to isomorphism](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0438a67e708231d40ce90f/html5/thumbnails/8.jpg)
Measure preserving transformations
Examples
X = T with the usual measure; T (z) = az, where a ∈ T, i.e.,T is a rotation.
X = 2Z, T (x)(n) = x(n− 1), i.e., the shift transformation.
Definition
A mpt T is ergodic if every T -invariant set has measure 0 or 1.
Any irrational, modulo π, rotation and the shift are ergodic.
The ergodic decomposition theorem shows that every mpt can becanonically decomposed into a (generally continuous) direct sum ofergodic mpt’s.
The complexity of classification problems in ergodic theory
![Page 9: The complexity of classification problems in ergodic theory€¦ · T has discrete spectrum , i.e., there is a basis ... (countable) subgroup of T. It turns out that up to isomorphism](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0438a67e708231d40ce90f/html5/thumbnails/9.jpg)
Measure preserving transformations
Examples
X = T with the usual measure; T (z) = az, where a ∈ T, i.e.,T is a rotation.
X = 2Z, T (x)(n) = x(n− 1), i.e., the shift transformation.
Definition
A mpt T is ergodic if every T -invariant set has measure 0 or 1.
Any irrational, modulo π, rotation and the shift are ergodic.
The ergodic decomposition theorem shows that every mpt can becanonically decomposed into a (generally continuous) direct sum ofergodic mpt’s.
The complexity of classification problems in ergodic theory
![Page 10: The complexity of classification problems in ergodic theory€¦ · T has discrete spectrum , i.e., there is a basis ... (countable) subgroup of T. It turns out that up to isomorphism](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0438a67e708231d40ce90f/html5/thumbnails/10.jpg)
Measure preserving transformations
Examples
X = T with the usual measure; T (z) = az, where a ∈ T, i.e.,T is a rotation.
X = 2Z, T (x)(n) = x(n− 1), i.e., the shift transformation.
Definition
A mpt T is ergodic if every T -invariant set has measure 0 or 1.
Any irrational, modulo π, rotation and the shift are ergodic.
The ergodic decomposition theorem shows that every mpt can becanonically decomposed into a (generally continuous) direct sum ofergodic mpt’s.
The complexity of classification problems in ergodic theory
![Page 11: The complexity of classification problems in ergodic theory€¦ · T has discrete spectrum , i.e., there is a basis ... (countable) subgroup of T. It turns out that up to isomorphism](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0438a67e708231d40ce90f/html5/thumbnails/11.jpg)
Measure preserving transformations
Examples
X = T with the usual measure; T (z) = az, where a ∈ T, i.e.,T is a rotation.
X = 2Z, T (x)(n) = x(n− 1), i.e., the shift transformation.
Definition
A mpt T is ergodic if every T -invariant set has measure 0 or 1.
Any irrational, modulo π, rotation and the shift are ergodic.
The ergodic decomposition theorem shows that every mpt can becanonically decomposed into a (generally continuous) direct sum ofergodic mpt’s.
The complexity of classification problems in ergodic theory
![Page 12: The complexity of classification problems in ergodic theory€¦ · T has discrete spectrum , i.e., there is a basis ... (countable) subgroup of T. It turns out that up to isomorphism](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0438a67e708231d40ce90f/html5/thumbnails/12.jpg)
Classifying measure preserving transformations
In ergodic theory one is interested in classifying ergodic mpt up tovarious notions of equivalence. I will consider below two suchstandard notions.
Isomorphism or conjugacy: A mpt S on (X,µ) is isomorphicto a mpt T on (Y, ν), in symbols S ∼= T , if there is anisomorphism ϕ of (X,µ) to (Y, ν) that sends S to T , i.e.,S = ϕ−1Tϕ.
Unitary isomorphism: To each mpt T on (X,µ) we can assignthe unitary operator UT : L2(X,µ)→ L2(X,µ) given byUT (f)(x) = f(T−1(x)). Then S, T are unitarily isomorphic,in symbols S ∼=u T , if US , UT are isomorphic.
Clearly ∼= implies ∼=u but the converse fails.
The complexity of classification problems in ergodic theory
![Page 13: The complexity of classification problems in ergodic theory€¦ · T has discrete spectrum , i.e., there is a basis ... (countable) subgroup of T. It turns out that up to isomorphism](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0438a67e708231d40ce90f/html5/thumbnails/13.jpg)
Classifying measure preserving transformations
In ergodic theory one is interested in classifying ergodic mpt up tovarious notions of equivalence. I will consider below two suchstandard notions.
Isomorphism or conjugacy: A mpt S on (X,µ) is isomorphicto a mpt T on (Y, ν), in symbols S ∼= T , if there is anisomorphism ϕ of (X,µ) to (Y, ν) that sends S to T , i.e.,S = ϕ−1Tϕ.
Unitary isomorphism: To each mpt T on (X,µ) we can assignthe unitary operator UT : L2(X,µ)→ L2(X,µ) given byUT (f)(x) = f(T−1(x)). Then S, T are unitarily isomorphic,in symbols S ∼=u T , if US , UT are isomorphic.
Clearly ∼= implies ∼=u but the converse fails.
The complexity of classification problems in ergodic theory
![Page 14: The complexity of classification problems in ergodic theory€¦ · T has discrete spectrum , i.e., there is a basis ... (countable) subgroup of T. It turns out that up to isomorphism](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0438a67e708231d40ce90f/html5/thumbnails/14.jpg)
Classifying measure preserving transformations
In ergodic theory one is interested in classifying ergodic mpt up tovarious notions of equivalence. I will consider below two suchstandard notions.
Isomorphism or conjugacy: A mpt S on (X,µ) is isomorphicto a mpt T on (Y, ν), in symbols S ∼= T , if there is anisomorphism ϕ of (X,µ) to (Y, ν) that sends S to T , i.e.,S = ϕ−1Tϕ.
Unitary isomorphism: To each mpt T on (X,µ) we can assignthe unitary operator UT : L2(X,µ)→ L2(X,µ) given byUT (f)(x) = f(T−1(x)). Then S, T are unitarily isomorphic,in symbols S ∼=u T , if US , UT are isomorphic.
Clearly ∼= implies ∼=u but the converse fails.
The complexity of classification problems in ergodic theory
![Page 15: The complexity of classification problems in ergodic theory€¦ · T has discrete spectrum , i.e., there is a basis ... (countable) subgroup of T. It turns out that up to isomorphism](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0438a67e708231d40ce90f/html5/thumbnails/15.jpg)
Classifying measure preserving transformations
In ergodic theory one is interested in classifying ergodic mpt up tovarious notions of equivalence. I will consider below two suchstandard notions.
Isomorphism or conjugacy: A mpt S on (X,µ) is isomorphicto a mpt T on (Y, ν), in symbols S ∼= T , if there is anisomorphism ϕ of (X,µ) to (Y, ν) that sends S to T , i.e.,S = ϕ−1Tϕ.
Unitary isomorphism: To each mpt T on (X,µ) we can assignthe unitary operator UT : L2(X,µ)→ L2(X,µ) given byUT (f)(x) = f(T−1(x)). Then S, T are unitarily isomorphic,in symbols S ∼=u T , if US , UT are isomorphic.
Clearly ∼= implies ∼=u but the converse fails.
The complexity of classification problems in ergodic theory
![Page 16: The complexity of classification problems in ergodic theory€¦ · T has discrete spectrum , i.e., there is a basis ... (countable) subgroup of T. It turns out that up to isomorphism](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0438a67e708231d40ce90f/html5/thumbnails/16.jpg)
Classifying measure preserving transformations
Two classical classification theorems:
(Halmos-von Neumann) An ergodic mpt has discretespectrum if UT has discrete spectrum , i.e., there is a basisconsisting of eigenvectors. In this case the eigenvalues aresimple and form a (countable) subgroup of T. It turns outthat up to isomorphism these are exactly the ergodic rotationsin compact metric groups G : T (g) = ag, where a ∈ G issuch that an : n ∈ Z is dense in G. For such T , let ΓT ≤ Tbe its group of eigenvalues. Then we have:
S ∼= T ⇔ S ∼=u T ⇔ ΓS = ΓT .
The complexity of classification problems in ergodic theory
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Classifying measure preserving transformations
Two classical classification theorems:
(Halmos-von Neumann) An ergodic mpt has discretespectrum if UT has discrete spectrum , i.e., there is a basisconsisting of eigenvectors. In this case the eigenvalues aresimple and form a (countable) subgroup of T. It turns outthat up to isomorphism these are exactly the ergodic rotationsin compact metric groups G : T (g) = ag, where a ∈ G issuch that an : n ∈ Z is dense in G. For such T , let ΓT ≤ Tbe its group of eigenvalues. Then we have:
S ∼= T ⇔ S ∼=u T ⇔ ΓS = ΓT .
The complexity of classification problems in ergodic theory
![Page 18: The complexity of classification problems in ergodic theory€¦ · T has discrete spectrum , i.e., there is a basis ... (countable) subgroup of T. It turns out that up to isomorphism](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0438a67e708231d40ce90f/html5/thumbnails/18.jpg)
Classifying measure preserving transformations
(Ornstein) Let Y = 1, . . . , n, p = (p1, · · · , pn) a probabilitydistribution on Y and form the product space X = Y Z withthe product measure µ. Consider the Bernoulli shift Tp on X.Its entropy is the real number H(p) = −
∑i pi log pi. Then
we have:Tp ∼= Tq ⇔ H(p) = H(q)
(but all the shifts are unitarily isomorphic).
The complexity of classification problems in ergodic theory
![Page 19: The complexity of classification problems in ergodic theory€¦ · T has discrete spectrum , i.e., there is a basis ... (countable) subgroup of T. It turns out that up to isomorphism](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0438a67e708231d40ce90f/html5/thumbnails/19.jpg)
Classifying measure preserving transformations
(Ornstein) Let Y = 1, . . . , n, p = (p1, · · · , pn) a probabilitydistribution on Y and form the product space X = Y Z withthe product measure µ. Consider the Bernoulli shift Tp on X.Its entropy is the real number H(p) = −
∑i pi log pi. Then
we have:Tp ∼= Tq ⇔ H(p) = H(q)
(but all the shifts are unitarily isomorphic).
The complexity of classification problems in ergodic theory
![Page 20: The complexity of classification problems in ergodic theory€¦ · T has discrete spectrum , i.e., there is a basis ... (countable) subgroup of T. It turns out that up to isomorphism](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0438a67e708231d40ce90f/html5/thumbnails/20.jpg)
Classifying measure preserving transformations
We will now consider the following question: Is it possible toclassify, in any reasonable way, general ergodic mpt?
We will see how ideas from descriptive set theory can throw somelight on this question.
The complexity of classification problems in ergodic theory
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Classifying measure preserving transformations
We will now consider the following question: Is it possible toclassify, in any reasonable way, general ergodic mpt?
We will see how ideas from descriptive set theory can throw somelight on this question.
The complexity of classification problems in ergodic theory
![Page 22: The complexity of classification problems in ergodic theory€¦ · T has discrete spectrum , i.e., there is a basis ... (countable) subgroup of T. It turns out that up to isomorphism](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0438a67e708231d40ce90f/html5/thumbnails/22.jpg)
Classification problems
I will next give an introduction to recent work in set theory,developed primarily over the last two decades, concerning a theoryof complexity of classification problems in mathematics, and thendiscuss its implications to the above problems.
The complexity of classification problems in ergodic theory
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Classification problems
A classification problem is given by:
A collection of objects X.
An equivalence relation E on X.
A complete classification of X up to E consists of:
A set of invariants I.
A map c : X → I such that xEy ⇔ c(x) = c(y).
For this to be of any interest both I, c must be as explicit andconcrete as possible.
The complexity of classification problems in ergodic theory
![Page 24: The complexity of classification problems in ergodic theory€¦ · T has discrete spectrum , i.e., there is a basis ... (countable) subgroup of T. It turns out that up to isomorphism](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0438a67e708231d40ce90f/html5/thumbnails/24.jpg)
Classification problems
A classification problem is given by:
A collection of objects X.
An equivalence relation E on X.
A complete classification of X up to E consists of:
A set of invariants I.
A map c : X → I such that xEy ⇔ c(x) = c(y).
For this to be of any interest both I, c must be as explicit andconcrete as possible.
The complexity of classification problems in ergodic theory
![Page 25: The complexity of classification problems in ergodic theory€¦ · T has discrete spectrum , i.e., there is a basis ... (countable) subgroup of T. It turns out that up to isomorphism](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0438a67e708231d40ce90f/html5/thumbnails/25.jpg)
Classification problems
A classification problem is given by:
A collection of objects X.
An equivalence relation E on X.
A complete classification of X up to E consists of:
A set of invariants I.
A map c : X → I such that xEy ⇔ c(x) = c(y).
For this to be of any interest both I, c must be as explicit andconcrete as possible.
The complexity of classification problems in ergodic theory
![Page 26: The complexity of classification problems in ergodic theory€¦ · T has discrete spectrum , i.e., there is a basis ... (countable) subgroup of T. It turns out that up to isomorphism](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0438a67e708231d40ce90f/html5/thumbnails/26.jpg)
Classification problems
Example
Classification of Bernoulli shifts up to isomorphism (Ornstein).Invariants: Reals
Example
Classification of ergodic measure-preserving transformations withdiscrete spectrum up to isomorphism (Halmos-von Neumann).Invariants: Countable subsets of T.
Example
Classification of unitary operators on a separable Hilbert space upto isomorphism (Spectral Theorem).Invariants: Measure classes, i.e., probability Borel measures ona Polish space up to measure equivalence.
The complexity of classification problems in ergodic theory
![Page 27: The complexity of classification problems in ergodic theory€¦ · T has discrete spectrum , i.e., there is a basis ... (countable) subgroup of T. It turns out that up to isomorphism](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0438a67e708231d40ce90f/html5/thumbnails/27.jpg)
Classification problems
Example
Classification of Bernoulli shifts up to isomorphism (Ornstein).Invariants: Reals
Example
Classification of ergodic measure-preserving transformations withdiscrete spectrum up to isomorphism (Halmos-von Neumann).Invariants: Countable subsets of T.
Example
Classification of unitary operators on a separable Hilbert space upto isomorphism (Spectral Theorem).Invariants: Measure classes, i.e., probability Borel measures ona Polish space up to measure equivalence.
The complexity of classification problems in ergodic theory
![Page 28: The complexity of classification problems in ergodic theory€¦ · T has discrete spectrum , i.e., there is a basis ... (countable) subgroup of T. It turns out that up to isomorphism](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0438a67e708231d40ce90f/html5/thumbnails/28.jpg)
Classification problems
Example
Classification of Bernoulli shifts up to isomorphism (Ornstein).Invariants: Reals
Example
Classification of ergodic measure-preserving transformations withdiscrete spectrum up to isomorphism (Halmos-von Neumann).Invariants: Countable subsets of T.
Example
Classification of unitary operators on a separable Hilbert space upto isomorphism (Spectral Theorem).Invariants: Measure classes, i.e., probability Borel measures ona Polish space up to measure equivalence.
The complexity of classification problems in ergodic theory
![Page 29: The complexity of classification problems in ergodic theory€¦ · T has discrete spectrum , i.e., there is a basis ... (countable) subgroup of T. It turns out that up to isomorphism](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0438a67e708231d40ce90f/html5/thumbnails/29.jpg)
Classification problems
Most often the collection of objects we try to classify can beviewed as forming a “nice” space, namely a Polish space, and theequivalence relation E turns out to be Borel or analytic (as asubset of X2).
For example, in studying mpt the appropriate space is the Polishgroup of mpt of a fixed (X,µ), with the so-called weak topology.Isomorphism then corresponds to conjugacy in that group, which isan analytic equivalence relation. Similarly unitary isomorphism isan analytic equivalence relation.
The complexity of classification problems in ergodic theory
![Page 30: The complexity of classification problems in ergodic theory€¦ · T has discrete spectrum , i.e., there is a basis ... (countable) subgroup of T. It turns out that up to isomorphism](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0438a67e708231d40ce90f/html5/thumbnails/30.jpg)
Classification problems
Most often the collection of objects we try to classify can beviewed as forming a “nice” space, namely a Polish space, and theequivalence relation E turns out to be Borel or analytic (as asubset of X2).
For example, in studying mpt the appropriate space is the Polishgroup of mpt of a fixed (X,µ), with the so-called weak topology.Isomorphism then corresponds to conjugacy in that group, which isan analytic equivalence relation. Similarly unitary isomorphism isan analytic equivalence relation.
The complexity of classification problems in ergodic theory
![Page 31: The complexity of classification problems in ergodic theory€¦ · T has discrete spectrum , i.e., there is a basis ... (countable) subgroup of T. It turns out that up to isomorphism](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0438a67e708231d40ce90f/html5/thumbnails/31.jpg)
Equivalence relations and reducibility
The theory of equivalence relations studies the set-theoretic natureof possible (complete) invariants and develops a mathematicalframework for measuring the complexity of classification problems.
The following simple concept is basic in organizing this study.
The complexity of classification problems in ergodic theory
![Page 32: The complexity of classification problems in ergodic theory€¦ · T has discrete spectrum , i.e., there is a basis ... (countable) subgroup of T. It turns out that up to isomorphism](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0438a67e708231d40ce90f/html5/thumbnails/32.jpg)
Equivalence relations and reducibility
The theory of equivalence relations studies the set-theoretic natureof possible (complete) invariants and develops a mathematicalframework for measuring the complexity of classification problems.
The following simple concept is basic in organizing this study.
The complexity of classification problems in ergodic theory
![Page 33: The complexity of classification problems in ergodic theory€¦ · T has discrete spectrum , i.e., there is a basis ... (countable) subgroup of T. It turns out that up to isomorphism](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0438a67e708231d40ce90f/html5/thumbnails/33.jpg)
Equivalence relations and reducibility
Definition
Let (X,E), (Y, F ) be equivalence relations. E is (Borel) reducibleto F , in symbols
E ≤B F,
if there is Borel map f : X → Y such that
x E y ⇔ f(x) F f(y).
Intuitive meaning:
The classification problem represented by E is at most ascomplicated as that of F .
F -classes are complete invariants for E.
The complexity of classification problems in ergodic theory
![Page 34: The complexity of classification problems in ergodic theory€¦ · T has discrete spectrum , i.e., there is a basis ... (countable) subgroup of T. It turns out that up to isomorphism](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0438a67e708231d40ce90f/html5/thumbnails/34.jpg)
Equivalence relations and reducibility
Definition
Let (X,E), (Y, F ) be equivalence relations. E is (Borel) reducibleto F , in symbols
E ≤B F,
if there is Borel map f : X → Y such that
x E y ⇔ f(x) F f(y).
Intuitive meaning:
The classification problem represented by E is at most ascomplicated as that of F .
F -classes are complete invariants for E.
The complexity of classification problems in ergodic theory
![Page 35: The complexity of classification problems in ergodic theory€¦ · T has discrete spectrum , i.e., there is a basis ... (countable) subgroup of T. It turns out that up to isomorphism](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0438a67e708231d40ce90f/html5/thumbnails/35.jpg)
Equivalence relations and reducibility
Definition
Let (X,E), (Y, F ) be equivalence relations. E is (Borel) reducibleto F , in symbols
E ≤B F,
if there is Borel map f : X → Y such that
x E y ⇔ f(x) F f(y).
Intuitive meaning:
The classification problem represented by E is at most ascomplicated as that of F .
F -classes are complete invariants for E.
The complexity of classification problems in ergodic theory
![Page 36: The complexity of classification problems in ergodic theory€¦ · T has discrete spectrum , i.e., there is a basis ... (countable) subgroup of T. It turns out that up to isomorphism](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0438a67e708231d40ce90f/html5/thumbnails/36.jpg)
Equivalence relations and reducibility
Definition
Let (X,E), (Y, F ) be equivalence relations. E is (Borel) reducibleto F , in symbols
E ≤B F,
if there is Borel map f : X → Y such that
x E y ⇔ f(x) F f(y).
Intuitive meaning:
The classification problem represented by E is at most ascomplicated as that of F .
F -classes are complete invariants for E.
The complexity of classification problems in ergodic theory
![Page 37: The complexity of classification problems in ergodic theory€¦ · T has discrete spectrum , i.e., there is a basis ... (countable) subgroup of T. It turns out that up to isomorphism](https://reader033.vdocument.in/reader033/viewer/2022060208/5f0438a67e708231d40ce90f/html5/thumbnails/37.jpg)
Equivalence relations and reducibility
Definition
E is bi-reducible to F if E is reducible to F and vice versa.
E ∼B F ⇔ E ≤B F and F ≤B E.
We also put:
Definition
E <B F ⇔ E ≤B F and F 6≤B E.
The complexity of classification problems in ergodic theory
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Equivalence relations and reducibility
Definition
E is bi-reducible to F if E is reducible to F and vice versa.
E ∼B F ⇔ E ≤B F and F ≤B E.
We also put:
Definition
E <B F ⇔ E ≤B F and F 6≤B E.
The complexity of classification problems in ergodic theory
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Equivalence relations and reducibility
Example
(Isomorphism of Bernoulli shifts) ∼B (=R)
Example
(Isomorphism of ergodic discrete spectrum mpt) ∼B Ec,where Ec is the equivalence relation on TN given by
(xn) Ec (yn)⇔ xn : n ∈ N = yn : n ∈ N
Example
(Isomorphism of unitary operators) ∼B ME,where ME is the equivalence relation on the Polish space ofprobability Borel measures on T given by
µ ME ν ⇔ µ ν and µ ν
The complexity of classification problems in ergodic theory
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Equivalence relations and reducibility
Example
(Isomorphism of Bernoulli shifts) ∼B (=R)
Example
(Isomorphism of ergodic discrete spectrum mpt) ∼B Ec,where Ec is the equivalence relation on TN given by
(xn) Ec (yn)⇔ xn : n ∈ N = yn : n ∈ N
Example
(Isomorphism of unitary operators) ∼B ME,where ME is the equivalence relation on the Polish space ofprobability Borel measures on T given by
µ ME ν ⇔ µ ν and µ ν
The complexity of classification problems in ergodic theory
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Equivalence relations and reducibility
Example
(Isomorphism of Bernoulli shifts) ∼B (=R)
Example
(Isomorphism of ergodic discrete spectrum mpt) ∼B Ec,where Ec is the equivalence relation on TN given by
(xn) Ec (yn)⇔ xn : n ∈ N = yn : n ∈ N
Example
(Isomorphism of unitary operators) ∼B ME,where ME is the equivalence relation on the Polish space ofprobability Borel measures on T given by
µ ME ν ⇔ µ ν and µ ν
The complexity of classification problems in ergodic theory
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Borel cardinality theory
The preceding concepts can be also interpreted as the basis of a“definable” or Borel cardinality theory for quotient spaces.
E ≤B F means that there is a Borel injection of X/E intoY/F , i.e., X/E has Borel cardinality less than or equal tothat of Y/F , in symbols
|X/E|B ≤ |Y/F |B
E ∼B F means that X/E and Y/F have the same Borelcardinality, in symbols
|X/E|B = |Y/F |B
E <B F means that X/E has strictly smaller Borelcardinality than Y/F , in symbols
|X/E|B < |Y/F |BThe complexity of classification problems in ergodic theory
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Borel cardinality theory
The preceding concepts can be also interpreted as the basis of a“definable” or Borel cardinality theory for quotient spaces.
E ≤B F means that there is a Borel injection of X/E intoY/F , i.e., X/E has Borel cardinality less than or equal tothat of Y/F , in symbols
|X/E|B ≤ |Y/F |B
E ∼B F means that X/E and Y/F have the same Borelcardinality, in symbols
|X/E|B = |Y/F |B
E <B F means that X/E has strictly smaller Borelcardinality than Y/F , in symbols
|X/E|B < |Y/F |BThe complexity of classification problems in ergodic theory
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Borel cardinality theory
The preceding concepts can be also interpreted as the basis of a“definable” or Borel cardinality theory for quotient spaces.
E ≤B F means that there is a Borel injection of X/E intoY/F , i.e., X/E has Borel cardinality less than or equal tothat of Y/F , in symbols
|X/E|B ≤ |Y/F |B
E ∼B F means that X/E and Y/F have the same Borelcardinality, in symbols
|X/E|B = |Y/F |B
E <B F means that X/E has strictly smaller Borelcardinality than Y/F , in symbols
|X/E|B < |Y/F |BThe complexity of classification problems in ergodic theory
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Borel cardinality theory
The preceding concepts can be also interpreted as the basis of a“definable” or Borel cardinality theory for quotient spaces.
E ≤B F means that there is a Borel injection of X/E intoY/F , i.e., X/E has Borel cardinality less than or equal tothat of Y/F , in symbols
|X/E|B ≤ |Y/F |B
E ∼B F means that X/E and Y/F have the same Borelcardinality, in symbols
|X/E|B = |Y/F |B
E <B F means that X/E has strictly smaller Borelcardinality than Y/F , in symbols
|X/E|B < |Y/F |BThe complexity of classification problems in ergodic theory
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Types of classification and mpt
An equivalence relation E on X is called concretely classifiable ifE ≤B (=Y ), for some Polish space Y , i.e., there is a Borel mapf : X → Y such that xEy ⇔ f(x) = f(y).
Thus isomorphism of Bernoulli shifts is concretely classifiable.However in the 1970’s Feldman showed that this fails for arbitrarympt (in fact even for the so-called K-automorphisms, a moregeneral class of mpt than Bernoulli shifts).
Theorem (Feldman)
Isomorphism of ergodic mpt is not concretely classifiable.
One can also see this using the fact that isomorphism of ergodicdiscrete spectrum mpt is not concretely classifiable.
The complexity of classification problems in ergodic theory
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Types of classification and mpt
An equivalence relation E on X is called concretely classifiable ifE ≤B (=Y ), for some Polish space Y , i.e., there is a Borel mapf : X → Y such that xEy ⇔ f(x) = f(y).
Thus isomorphism of Bernoulli shifts is concretely classifiable.However in the 1970’s Feldman showed that this fails for arbitrarympt (in fact even for the so-called K-automorphisms, a moregeneral class of mpt than Bernoulli shifts).
Theorem (Feldman)
Isomorphism of ergodic mpt is not concretely classifiable.
One can also see this using the fact that isomorphism of ergodicdiscrete spectrum mpt is not concretely classifiable.
The complexity of classification problems in ergodic theory
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Types of classification and mpt
An equivalence relation E on X is called concretely classifiable ifE ≤B (=Y ), for some Polish space Y , i.e., there is a Borel mapf : X → Y such that xEy ⇔ f(x) = f(y).
Thus isomorphism of Bernoulli shifts is concretely classifiable.However in the 1970’s Feldman showed that this fails for arbitrarympt (in fact even for the so-called K-automorphisms, a moregeneral class of mpt than Bernoulli shifts).
Theorem (Feldman)
Isomorphism of ergodic mpt is not concretely classifiable.
One can also see this using the fact that isomorphism of ergodicdiscrete spectrum mpt is not concretely classifiable.
The complexity of classification problems in ergodic theory
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Types of classification and mpt
An equivalence relation E on X is called concretely classifiable ifE ≤B (=Y ), for some Polish space Y , i.e., there is a Borel mapf : X → Y such that xEy ⇔ f(x) = f(y).
Thus isomorphism of Bernoulli shifts is concretely classifiable.However in the 1970’s Feldman showed that this fails for arbitrarympt (in fact even for the so-called K-automorphisms, a moregeneral class of mpt than Bernoulli shifts).
Theorem (Feldman)
Isomorphism of ergodic mpt is not concretely classifiable.
One can also see this using the fact that isomorphism of ergodicdiscrete spectrum mpt is not concretely classifiable.
The complexity of classification problems in ergodic theory
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Types of classification and mpt
An equivalence relation is called classifiable by countable structuresif it can be Borel reduced to isomorphism of countable structures(of some given type, e.g., groups, graphs, linear orderings, etc.).More precisely, given a countable signature L, denote by XL thespace of L-structures with universe N. This is a Polish space.Denote by ∼= the equivalence relation of isomorphism in XL. Wesay that an equivalence relation is classifiable by countablestructures if it is Borel reducible to isomorphism on XL, for someL.
Such types of classification occur often, for example, in operatoralgebras, topological dynamics, etc.
The complexity of classification problems in ergodic theory
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Types of classification and mpt
It follows from the Halmos-von Neumann theorem thatisomorphism (and unitary isomorphism) of ergodic discretespectrum mpt is classifiable by countable structures. On the otherhand we have:
Theorem (K-Sofronidis, 2001)
ME and thus isomorphism of unitary operators is not classifiable bycountable structures.
The complexity of classification problems in ergodic theory
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Types of classification and mpt
It follows from the Halmos-von Neumann theorem thatisomorphism (and unitary isomorphism) of ergodic discretespectrum mpt is classifiable by countable structures. On the otherhand we have:
Theorem (K-Sofronidis, 2001)
ME and thus isomorphism of unitary operators is not classifiable bycountable structures.
The complexity of classification problems in ergodic theory
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Types of classification and mpt
Theorem (Hjorth, 2001)
Isomorphism and unitary isomorphism of ergodic mpt cannot beclassified by countable structures.
This has more recently been strengthened as follows:
Theorem (Foreman-Weiss, 2004)
Isomorphism and unitary isomorphism of ergodic mpt cannot beclassified by countable structures on any generic class of ergodicmpt.
The complexity of classification problems in ergodic theory
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Types of classification and mpt
Theorem (Hjorth, 2001)
Isomorphism and unitary isomorphism of ergodic mpt cannot beclassified by countable structures.
This has more recently been strengthened as follows:
Theorem (Foreman-Weiss, 2004)
Isomorphism and unitary isomorphism of ergodic mpt cannot beclassified by countable structures on any generic class of ergodicmpt.
The complexity of classification problems in ergodic theory
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Types of classification and mpt
Theorem (Hjorth, 2001)
Isomorphism and unitary isomorphism of ergodic mpt cannot beclassified by countable structures.
This has more recently been strengthened as follows:
Theorem (Foreman-Weiss, 2004)
Isomorphism and unitary isomorphism of ergodic mpt cannot beclassified by countable structures on any generic class of ergodicmpt.
The complexity of classification problems in ergodic theory
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Types of classification and mpt
One can now in fact calculate the exact complexity of unitaryisomorphism.
Theorem (K, 2007)
i) Unitary isomorphism of ergodic mpt is Borel bireducible , i.e.,has exactly the same complexity, as measure equivalence.ii) Measure equivalence is Borel reducible to isomorphism ofergodic mpt.
More recently, Foreman-Rudolph-Weiss also showed the following:
Theorem (Foreman-Rudolph-Weiss, 2008)
Isomorphism of ergodic mpt is not Borel (it is clearly analytic).
However note that unitary isomorphism of mpt is Borel.
The complexity of classification problems in ergodic theory
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Types of classification and mpt
One can now in fact calculate the exact complexity of unitaryisomorphism.
Theorem (K, 2007)
i) Unitary isomorphism of ergodic mpt is Borel bireducible , i.e.,has exactly the same complexity, as measure equivalence.ii) Measure equivalence is Borel reducible to isomorphism ofergodic mpt.
More recently, Foreman-Rudolph-Weiss also showed the following:
Theorem (Foreman-Rudolph-Weiss, 2008)
Isomorphism of ergodic mpt is not Borel (it is clearly analytic).
However note that unitary isomorphism of mpt is Borel.
The complexity of classification problems in ergodic theory
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Types of classification and mpt
One can now in fact calculate the exact complexity of unitaryisomorphism.
Theorem (K, 2007)
i) Unitary isomorphism of ergodic mpt is Borel bireducible , i.e.,has exactly the same complexity, as measure equivalence.ii) Measure equivalence is Borel reducible to isomorphism ofergodic mpt.
More recently, Foreman-Rudolph-Weiss also showed the following:
Theorem (Foreman-Rudolph-Weiss, 2008)
Isomorphism of ergodic mpt is not Borel (it is clearly analytic).
However note that unitary isomorphism of mpt is Borel.
The complexity of classification problems in ergodic theory
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Types of classification and mpt
One can now in fact calculate the exact complexity of unitaryisomorphism.
Theorem (K, 2007)
i) Unitary isomorphism of ergodic mpt is Borel bireducible , i.e.,has exactly the same complexity, as measure equivalence.ii) Measure equivalence is Borel reducible to isomorphism ofergodic mpt.
More recently, Foreman-Rudolph-Weiss also showed the following:
Theorem (Foreman-Rudolph-Weiss, 2008)
Isomorphism of ergodic mpt is not Borel (it is clearly analytic).
However note that unitary isomorphism of mpt is Borel.
The complexity of classification problems in ergodic theory
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Types of classification and mpt
One can now in fact calculate the exact complexity of unitaryisomorphism.
Theorem (K, 2007)
i) Unitary isomorphism of ergodic mpt is Borel bireducible , i.e.,has exactly the same complexity, as measure equivalence.ii) Measure equivalence is Borel reducible to isomorphism ofergodic mpt.
More recently, Foreman-Rudolph-Weiss also showed the following:
Theorem (Foreman-Rudolph-Weiss, 2008)
Isomorphism of ergodic mpt is not Borel (it is clearly analytic).
However note that unitary isomorphism of mpt is Borel.
The complexity of classification problems in ergodic theory
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Types of classification and mpt
It follows from the last two theorems that
(∼=u) <B (∼=),
i.e., isomorphism of ergodic mpt is strictly more complicated thanunitary isomorphism.We have now seen that the complexity of unitary isomorphism ofergodic mpt can be calculated exactly and there are very stronglower bounds for isomorphism but its exact complexity is unknown.An obvious upper bound is the universal equivalence relationinduced by a Borel action of the automorphism group of themeasure space.
Problem
Is isomorphism of ergodic mpt Borel bireducible to the universalequivalence relation induced by a Borel action of theautomorphism group of the measure space?
The complexity of classification problems in ergodic theory
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Types of classification and mpt
It follows from the last two theorems that
(∼=u) <B (∼=),
i.e., isomorphism of ergodic mpt is strictly more complicated thanunitary isomorphism.We have now seen that the complexity of unitary isomorphism ofergodic mpt can be calculated exactly and there are very stronglower bounds for isomorphism but its exact complexity is unknown.An obvious upper bound is the universal equivalence relationinduced by a Borel action of the automorphism group of themeasure space.
Problem
Is isomorphism of ergodic mpt Borel bireducible to the universalequivalence relation induced by a Borel action of theautomorphism group of the measure space?
The complexity of classification problems in ergodic theory
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Types of classification and mpt
It follows from the last two theorems that
(∼=u) <B (∼=),
i.e., isomorphism of ergodic mpt is strictly more complicated thanunitary isomorphism.We have now seen that the complexity of unitary isomorphism ofergodic mpt can be calculated exactly and there are very stronglower bounds for isomorphism but its exact complexity is unknown.An obvious upper bound is the universal equivalence relationinduced by a Borel action of the automorphism group of themeasure space.
Problem
Is isomorphism of ergodic mpt Borel bireducible to the universalequivalence relation induced by a Borel action of theautomorphism group of the measure space?
The complexity of classification problems in ergodic theory
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Actions of countable groups
More generally one also considers in ergodic theory the problem ofclassifying measure preserving actions of countable (discrete)groups Γ on standard measure spaces. The case Γ = Zcorresponds to the case of single transformations. We will nowlook at this problem from the point of view of the preceding theory.
The complexity of classification problems in ergodic theory
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Actions of countable groups
We will consider again isomorphism (also called conjugacy) andunitary isomorphism of actions. Two actions of the group Γ areisomorphic if there is a measure-preserving isomorphism of theunderlying spaces that conjugates the actions. They are unitarilyisomorphic if the corresponding unitary representations (theKoopman representations) are isomorphic.
We can form again in a canonical way a Polish space A(Γ, X, µ) ofall measure-preserving actions of Γ on (X,µ) and thenisomorphism and unitary isomorphism become analytic equivalencerelations on this space. We can therefore study their complexityusing the concepts introduced earlier.
The complexity of classification problems in ergodic theory
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Actions of countable groups
We will consider again isomorphism (also called conjugacy) andunitary isomorphism of actions. Two actions of the group Γ areisomorphic if there is a measure-preserving isomorphism of theunderlying spaces that conjugates the actions. They are unitarilyisomorphic if the corresponding unitary representations (theKoopman representations) are isomorphic.
We can form again in a canonical way a Polish space A(Γ, X, µ) ofall measure-preserving actions of Γ on (X,µ) and thenisomorphism and unitary isomorphism become analytic equivalencerelations on this space. We can therefore study their complexityusing the concepts introduced earlier.
The complexity of classification problems in ergodic theory
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Actions of countable groups
Theorem (Foreman - Weiss, Hjorth, 2004)
For any infinite countable group Γ, isomorphism of free, ergodic,measure-preserving actions of Γ is not classifiable by countablestructures.
Theorem (K, 2007)
For any infinite countable group Γ, unitary isomorphism of free,ergodic, measure-preserving actions of Γ is not classifiable bycountable structures.
The complexity of classification problems in ergodic theory
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Actions of countable groups
Theorem (Foreman - Weiss, Hjorth, 2004)
For any infinite countable group Γ, isomorphism of free, ergodic,measure-preserving actions of Γ is not classifiable by countablestructures.
Theorem (K, 2007)
For any infinite countable group Γ, unitary isomorphism of free,ergodic, measure-preserving actions of Γ is not classifiable bycountable structures.
The complexity of classification problems in ergodic theory
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Actions of countable groups
Except for abelian Γ, where we have the same picture as for Z, itis unknown however how isomorphism and unitary isomorphismrelations relate to ME. Hjorth and Tornquist have recently shownthat unitary isomorphism is at least a Borel equivalence relation.Also it is again not known what is the precise complexity of thesetwo equivalence relations. Is isomorphism Borel bireducible to theuniversal equivalence relation induced by a Borel action of theautomorphism group of the measure space?
The complexity of classification problems in ergodic theory
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Orbit equivalence
There is an additional important concept of equivalence betweenactions, called orbit equivalence. The study of orbit equivalence isa very active area today that has its origins in the connectionsbetween ergodic theory and operator algebras and the pioneeringwork of Dye.
Definition
Given an action of the group Γ on X we associate to it the orbitequivalence relation EXΓ , whose classes are the orbits of the action.Given measure-preserving actions of two groups Γ and ∆ on spaces(X,µ) and (Y, ν), resp., we say that they are orbit equivalent ifthere is an isomorphism of the underlying measure spaces thatsends EXΓ to EY∆.
Thus isomorphism clearly implies orbit equivalence but not viceversa.
The complexity of classification problems in ergodic theory
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Orbit equivalence
There is an additional important concept of equivalence betweenactions, called orbit equivalence. The study of orbit equivalence isa very active area today that has its origins in the connectionsbetween ergodic theory and operator algebras and the pioneeringwork of Dye.
Definition
Given an action of the group Γ on X we associate to it the orbitequivalence relation EXΓ , whose classes are the orbits of the action.Given measure-preserving actions of two groups Γ and ∆ on spaces(X,µ) and (Y, ν), resp., we say that they are orbit equivalent ifthere is an isomorphism of the underlying measure spaces thatsends EXΓ to EY∆.
Thus isomorphism clearly implies orbit equivalence but not viceversa.
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Orbit equivalence
There is an additional important concept of equivalence betweenactions, called orbit equivalence. The study of orbit equivalence isa very active area today that has its origins in the connectionsbetween ergodic theory and operator algebras and the pioneeringwork of Dye.
Definition
Given an action of the group Γ on X we associate to it the orbitequivalence relation EXΓ , whose classes are the orbits of the action.Given measure-preserving actions of two groups Γ and ∆ on spaces(X,µ) and (Y, ν), resp., we say that they are orbit equivalent ifthere is an isomorphism of the underlying measure spaces thatsends EXΓ to EY∆.
Thus isomorphism clearly implies orbit equivalence but not viceversa.
The complexity of classification problems in ergodic theory
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Orbit equivalence
Here we have the following classical result.
Theorem (Dye, 1959; Ornstein - Weiss, 1980)
Every two free, ergodic, measure-preserving actions of amenablegroups are orbit equivalent.
Thus there is a single orbit equivalence class in the space of free,ergodic, measure-preserving actions of an amenable group Γ.
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Orbit equivalence
Here we have the following classical result.
Theorem (Dye, 1959; Ornstein - Weiss, 1980)
Every two free, ergodic, measure-preserving actions of amenablegroups are orbit equivalent.
Thus there is a single orbit equivalence class in the space of free,ergodic, measure-preserving actions of an amenable group Γ.
The complexity of classification problems in ergodic theory
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Orbit equivalence
Here we have the following classical result.
Theorem (Dye, 1959; Ornstein - Weiss, 1980)
Every two free, ergodic, measure-preserving actions of amenablegroups are orbit equivalent.
Thus there is a single orbit equivalence class in the space of free,ergodic, measure-preserving actions of an amenable group Γ.
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Orbit equivalence
The situation for non-amenable groups has taken much longer tountangle. For simplicity, below “action” will mean “free, ergodic,measure-preserving action”. Schmidt, 1981, showed that everynon-amenable group which does not have Kazhdan’s property (T)admits at least two non-orbit equivalent actions and Hjorth, 2005,showed that every non-amenable group with property (T) hascontinuum many non-orbit equivalent actions. So everynon-amenable group has at least two non-orbit equivalent actions.
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Orbit equivalence
For general non-amenable groups though very little was knownabout the question of how many non-orbit equivalent actions theymight have. For example, until recently only finitely many distinctexamples of non-orbit equivalent actions of the free (non-abelian)groups were known. Gaboriau – Popa, 2005, finally showed thatthe free groups have continuum many non-orbit equivalent actions.In an important extension, Ioana, 2007, showed that every groupthat contains a free subgroup has continuum many such actions.However there are examples of non-amenable groups that containno free subgroups (Olshanski).
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Orbit equivalence
Finally, the question was completely resolved by Epstein.
Theorem (Epstein, 2007)
Every non-amenable group admits continuum many non-orbitequivalent free, ergodic, measure-preserving actions.
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Orbit equivalence
Finally, the question was completely resolved by Epstein.
Theorem (Epstein, 2007)
Every non-amenable group admits continuum many non-orbitequivalent free, ergodic, measure-preserving actions.
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Orbit equivalence
This still leaves open the possibility that there may be a concreteclassification of actions of some non-amenable groups up to orbitequivalence. However the following has been now proved bycombining very recent work of Ioana-K-Tsankov and the work ofEpstein.
Theorem (Epstein-Ioana-K-Tsankov)
Orbit equivalence of free, ergodic, measure preserving actions ofany non-amenable group is not classifiable by countable structures.
Thus we have a very strong dichotomy:
If a group is amenable, it has exactly one action up to orbitequivalence.
If it non-amenable, then orbit equivalence of its actions isunclassifiable in a strong sense.
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Orbit equivalence
This still leaves open the possibility that there may be a concreteclassification of actions of some non-amenable groups up to orbitequivalence. However the following has been now proved bycombining very recent work of Ioana-K-Tsankov and the work ofEpstein.
Theorem (Epstein-Ioana-K-Tsankov)
Orbit equivalence of free, ergodic, measure preserving actions ofany non-amenable group is not classifiable by countable structures.
Thus we have a very strong dichotomy:
If a group is amenable, it has exactly one action up to orbitequivalence.
If it non-amenable, then orbit equivalence of its actions isunclassifiable in a strong sense.
The complexity of classification problems in ergodic theory
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Orbit equivalence
This still leaves open the possibility that there may be a concreteclassification of actions of some non-amenable groups up to orbitequivalence. However the following has been now proved bycombining very recent work of Ioana-K-Tsankov and the work ofEpstein.
Theorem (Epstein-Ioana-K-Tsankov)
Orbit equivalence of free, ergodic, measure preserving actions ofany non-amenable group is not classifiable by countable structures.
Thus we have a very strong dichotomy:
If a group is amenable, it has exactly one action up to orbitequivalence.
If it non-amenable, then orbit equivalence of its actions isunclassifiable in a strong sense.
The complexity of classification problems in ergodic theory
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Orbit equivalence
This still leaves open the possibility that there may be a concreteclassification of actions of some non-amenable groups up to orbitequivalence. However the following has been now proved bycombining very recent work of Ioana-K-Tsankov and the work ofEpstein.
Theorem (Epstein-Ioana-K-Tsankov)
Orbit equivalence of free, ergodic, measure preserving actions ofany non-amenable group is not classifiable by countable structures.
Thus we have a very strong dichotomy:
If a group is amenable, it has exactly one action up to orbitequivalence.
If it non-amenable, then orbit equivalence of its actions isunclassifiable in a strong sense.
The complexity of classification problems in ergodic theory
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Orbit equivalence
This still leaves open the possibility that there may be a concreteclassification of actions of some non-amenable groups up to orbitequivalence. However the following has been now proved bycombining very recent work of Ioana-K-Tsankov and the work ofEpstein.
Theorem (Epstein-Ioana-K-Tsankov)
Orbit equivalence of free, ergodic, measure preserving actions ofany non-amenable group is not classifiable by countable structures.
Thus we have a very strong dichotomy:
If a group is amenable, it has exactly one action up to orbitequivalence.
If it non-amenable, then orbit equivalence of its actions isunclassifiable in a strong sense.
The complexity of classification problems in ergodic theory
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II. TURBULENCE AND CLASSIFICATION BYCOUNTABLE STRUCTURES
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The space of countable structures
A countable signature is a countable familyL = fii∈I ∪ Rjj∈J of function symbols fi, with fi of arityni ≥ 0, and relation symbols Rj , with Rj of arity mj ≥ 1. Astructure for L or L-structure has the form
A = 〈A, fAi i∈I , RAj j∈J〉,
where A is a nonempty set, fAi : Ani → A, and RAj ⊆ Amj .
Example
If L = ·, 1, where · and 1 are binary and nullary function symbolsrespectively, then a group is any L-structure G = 〈G, ·G , 1G〉 thatsatisfies the group axioms. Similarly, using various signatures, wecan study structures that correspond to fields, graphs, etc.
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The space of countable structures
A countable signature is a countable familyL = fii∈I ∪ Rjj∈J of function symbols fi, with fi of arityni ≥ 0, and relation symbols Rj , with Rj of arity mj ≥ 1. Astructure for L or L-structure has the form
A = 〈A, fAi i∈I , RAj j∈J〉,
where A is a nonempty set, fAi : Ani → A, and RAj ⊆ Amj .
Example
If L = ·, 1, where · and 1 are binary and nullary function symbolsrespectively, then a group is any L-structure G = 〈G, ·G , 1G〉 thatsatisfies the group axioms. Similarly, using various signatures, wecan study structures that correspond to fields, graphs, etc.
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The space of countable structures
We are interested in countably infinite structures here, so we canalways take (up to isomorphism) A = N.
Definition
Denote by XL the space of (countable) L-structures, i.e.,
XL =∏i∈I
N(Nni ) ×∏j∈J
2(Nmj ).
With the product topology (N and 2 being discrete) this is a Polishspace.
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The space of countable structures
We are interested in countably infinite structures here, so we canalways take (up to isomorphism) A = N.
Definition
Denote by XL the space of (countable) L-structures, i.e.,
XL =∏i∈I
N(Nni ) ×∏j∈J
2(Nmj ).
With the product topology (N and 2 being discrete) this is a Polishspace.
The complexity of classification problems in ergodic theory
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The space of countable structures
We are interested in countably infinite structures here, so we canalways take (up to isomorphism) A = N.
Definition
Denote by XL the space of (countable) L-structures, i.e.,
XL =∏i∈I
N(Nni ) ×∏j∈J
2(Nmj ).
With the product topology (N and 2 being discrete) this is a Polishspace.
The complexity of classification problems in ergodic theory
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The space of countable structures
Definition
Let S∞ be the infinite symmetric group of all permutations of N.It is a Polish group with the pointwise convergence topology. Itacts continuously on XL: If A ∈ XL, g ∈ S∞, then g · A is theisomorphic copy of A obtained by applying g. For example, ifI = ∅, Rjj∈J consists of a single binary relation symbol R, andA = 〈N, RA〉, then g · A = B, where(x, y) ∈ RB ⇔ (g−1(x), g−1(y)) ∈ RA. This is called the logicaction of S∞ on XL.
Clearly, ∃g(g · A = B)⇔ A ∼= B, i.e., the equivalence relationinduced by this action is isomorphism.
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The space of countable structures
Definition
Let S∞ be the infinite symmetric group of all permutations of N.It is a Polish group with the pointwise convergence topology. Itacts continuously on XL: If A ∈ XL, g ∈ S∞, then g · A is theisomorphic copy of A obtained by applying g. For example, ifI = ∅, Rjj∈J consists of a single binary relation symbol R, andA = 〈N, RA〉, then g · A = B, where(x, y) ∈ RB ⇔ (g−1(x), g−1(y)) ∈ RA. This is called the logicaction of S∞ on XL.
Clearly, ∃g(g · A = B)⇔ A ∼= B, i.e., the equivalence relationinduced by this action is isomorphism.
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The space of countable structures
Logic actions are universal among S∞-actions.
Theorem (Becker-K)
There is a countable signature L such that for every Borel actionof S∞ on a Polish space X there is a Borel equivariant injectionπ : X → XL (i.e. π(g · x) = g · π(x)). Thus every Borel S∞-spaceis Borel isomorphic to the logic action on an isomorphism-invariantBorel class of L-structures.
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The space of countable structures
Logic actions are universal among S∞-actions.
Theorem (Becker-K)
There is a countable signature L such that for every Borel actionof S∞ on a Polish space X there is a Borel equivariant injectionπ : X → XL (i.e. π(g · x) = g · π(x)). Thus every Borel S∞-spaceis Borel isomorphic to the logic action on an isomorphism-invariantBorel class of L-structures.
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Classification by countable structures
Definition
Let E be an equivalence relation on a Polish space X. We say theE admits classification by countable structures if there is acountable signature L and a Borel map f : X → XL such thatxEy ⇔ f(x) ∼= f(y), i.e., E ≤B∼=L
(=∼= |XL
).
This is equivalent to the following: There is a Borel S∞-space Ysuch that E ≤B EYS∞ , where xEYS∞y ⇔ ∃g ∈ S∞(g · x = y) is theequivalence relation induced by the S∞-action on Y .
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Classification by countable structures
Definition
Let E be an equivalence relation on a Polish space X. We say theE admits classification by countable structures if there is acountable signature L and a Borel map f : X → XL such thatxEy ⇔ f(x) ∼= f(y), i.e., E ≤B∼=L
(=∼= |XL
).
This is equivalent to the following: There is a Borel S∞-space Ysuch that E ≤B EYS∞ , where xEYS∞y ⇔ ∃g ∈ S∞(g · x = y) is theequivalence relation induced by the S∞-action on Y .
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Classification by countable structures
Examples
If E is concretely classifiable, then E admits classification bycountable structures.
Let X be an uncountable Polish space. Define Ec on XN by
(xn)Ec(yn)⇔ xn : n ∈ N = yn : n ∈ N.
Then Ec is classifiable by countable structures. (Ec is, up toBorel isomorphism, independent of X.)
(Giordano-Putnam-Skau) Topological orbit equivalence ofminimal homeomorphisms of the Cantor space is classifiableby countable structures.
(K) If G is Polish locally compact and X is a Borel G-space,then EXG admits classification by countable structures.
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Classification by countable structures
Examples
If E is concretely classifiable, then E admits classification bycountable structures.
Let X be an uncountable Polish space. Define Ec on XN by
(xn)Ec(yn)⇔ xn : n ∈ N = yn : n ∈ N.
Then Ec is classifiable by countable structures. (Ec is, up toBorel isomorphism, independent of X.)
(Giordano-Putnam-Skau) Topological orbit equivalence ofminimal homeomorphisms of the Cantor space is classifiableby countable structures.
(K) If G is Polish locally compact and X is a Borel G-space,then EXG admits classification by countable structures.
The complexity of classification problems in ergodic theory
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Classification by countable structures
Examples
If E is concretely classifiable, then E admits classification bycountable structures.
Let X be an uncountable Polish space. Define Ec on XN by
(xn)Ec(yn)⇔ xn : n ∈ N = yn : n ∈ N.
Then Ec is classifiable by countable structures. (Ec is, up toBorel isomorphism, independent of X.)
(Giordano-Putnam-Skau) Topological orbit equivalence ofminimal homeomorphisms of the Cantor space is classifiableby countable structures.
(K) If G is Polish locally compact and X is a Borel G-space,then EXG admits classification by countable structures.
The complexity of classification problems in ergodic theory
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Classification by countable structures
Examples
If E is concretely classifiable, then E admits classification bycountable structures.
Let X be an uncountable Polish space. Define Ec on XN by
(xn)Ec(yn)⇔ xn : n ∈ N = yn : n ∈ N.
Then Ec is classifiable by countable structures. (Ec is, up toBorel isomorphism, independent of X.)
(Giordano-Putnam-Skau) Topological orbit equivalence ofminimal homeomorphisms of the Cantor space is classifiableby countable structures.
(K) If G is Polish locally compact and X is a Borel G-space,then EXG admits classification by countable structures.
The complexity of classification problems in ergodic theory
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Classification by countable structures
Hjorth developed a theory called turbulence that provides the basicmethod for showing that equivalence relations do not admitclassification by countable structures.
Theorem (Hjorth)
Let G be a Polish group acting continuously on a Polish space X.If the action is turbulent, then EXG cannot be classified bycountable structures. In particular if E is an equivalence relationand EXG ≤B E for some turbulent action of a Polish group G onX, then E cannot be classified by countable structures.
In the next lecture I will explain the concept of turbulence andsketch some ideas in the proof of this theorem.
The complexity of classification problems in ergodic theory
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Classification by countable structures
Hjorth developed a theory called turbulence that provides the basicmethod for showing that equivalence relations do not admitclassification by countable structures.
Theorem (Hjorth)
Let G be a Polish group acting continuously on a Polish space X.If the action is turbulent, then EXG cannot be classified bycountable structures. In particular if E is an equivalence relationand EXG ≤B E for some turbulent action of a Polish group G onX, then E cannot be classified by countable structures.
In the next lecture I will explain the concept of turbulence andsketch some ideas in the proof of this theorem.
The complexity of classification problems in ergodic theory
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Classification by countable structures
Hjorth developed a theory called turbulence that provides the basicmethod for showing that equivalence relations do not admitclassification by countable structures.
Theorem (Hjorth)
Let G be a Polish group acting continuously on a Polish space X.If the action is turbulent, then EXG cannot be classified bycountable structures. In particular if E is an equivalence relationand EXG ≤B E for some turbulent action of a Polish group G onX, then E cannot be classified by countable structures.
In the next lecture I will explain the concept of turbulence andsketch some ideas in the proof of this theorem.
The complexity of classification problems in ergodic theory
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Turbulence
Let G be a Polish group acting continuously on a Polish space X.Below U (with various embellishments) is a typical nonempty openset in X and V (with various embellishments) is a typical opensymmetric nbhd of 1 ∈ G.
Definition
The (U, V )-local graph is given by
xRU,V y ⇔ x, y ∈ U &∃g ∈ V (g · x = y).
The (U, V )-local orbit of x ∈ U , denoted O(x, U, V ), is theconnected component of x in this graph. (If U = X and V = G,then O(x, U, V ) = G · x = the orbit of x)
Definition
A point x ∈ X is turbulent if ∀U 3 x ∀V(O(x, U, V ) has
nonempty interior). (Depends only on the orbit of x.)
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Turbulence
Let G be a Polish group acting continuously on a Polish space X.Below U (with various embellishments) is a typical nonempty openset in X and V (with various embellishments) is a typical opensymmetric nbhd of 1 ∈ G.
Definition
The (U, V )-local graph is given by
xRU,V y ⇔ x, y ∈ U &∃g ∈ V (g · x = y).
The (U, V )-local orbit of x ∈ U , denoted O(x, U, V ), is theconnected component of x in this graph. (If U = X and V = G,then O(x, U, V ) = G · x = the orbit of x)
Definition
A point x ∈ X is turbulent if ∀U 3 x ∀V(O(x, U, V ) has
nonempty interior). (Depends only on the orbit of x.)
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Turbulence
Let G be a Polish group acting continuously on a Polish space X.Below U (with various embellishments) is a typical nonempty openset in X and V (with various embellishments) is a typical opensymmetric nbhd of 1 ∈ G.
Definition
The (U, V )-local graph is given by
xRU,V y ⇔ x, y ∈ U &∃g ∈ V (g · x = y).
The (U, V )-local orbit of x ∈ U , denoted O(x, U, V ), is theconnected component of x in this graph. (If U = X and V = G,then O(x, U, V ) = G · x = the orbit of x)
Definition
A point x ∈ X is turbulent if ∀U 3 x ∀V(O(x, U, V ) has
nonempty interior). (Depends only on the orbit of x.)
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Turbulence
Definition
The action is called (generically) turbulent if
(i) every orbit is meager,
(ii) there is a dense, turbulent orbit.
This implies that the set of dense, turbulent orbits is comeager.
Proposition
Let G be a Polish group acting continuously on a Polish space Xand let x ∈ X. Suppose there is a neighborhood basis B(x) for xsuch that for all U ∈ B(x) and any open nonempty set W ⊆ U ,there is a continuous path (gt)0≤t≤1 in G with g0 = 1, g1 · x ∈Wand gt · x ∈ U for each t. Then x is turbulent.
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Turbulence
Definition
The action is called (generically) turbulent if
(i) every orbit is meager,
(ii) there is a dense, turbulent orbit.
This implies that the set of dense, turbulent orbits is comeager.
Proposition
Let G be a Polish group acting continuously on a Polish space Xand let x ∈ X. Suppose there is a neighborhood basis B(x) for xsuch that for all U ∈ B(x) and any open nonempty set W ⊆ U ,there is a continuous path (gt)0≤t≤1 in G with g0 = 1, g1 · x ∈Wand gt · x ∈ U for each t. Then x is turbulent.
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Turbulence
Definition
The action is called (generically) turbulent if
(i) every orbit is meager,
(ii) there is a dense, turbulent orbit.
This implies that the set of dense, turbulent orbits is comeager.
Proposition
Let G be a Polish group acting continuously on a Polish space Xand let x ∈ X. Suppose there is a neighborhood basis B(x) for xsuch that for all U ∈ B(x) and any open nonempty set W ⊆ U ,there is a continuous path (gt)0≤t≤1 in G with g0 = 1, g1 · x ∈Wand gt · x ∈ U for each t. Then x is turbulent.
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Turbulence
g1
gt
1
U
W
gt·x
x
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Turbulence
Example
The action of the Polish group (c0,+) (with the supnormtopology) on (RN,+) (with the product topology) by translation isturbulent. The orbits are the cosets of c0 in (RN,+), so they aredense and meager. Also, any x ∈ RN is turbulent. Clearly the setsof the form x+ C, with C ⊆ RN a convex open nbhd of 0, form anbhd basis for x ∈ RN. Now c0 is dense in any such C, sox+ (c0 ∩ C) is dense in x+ C. Let g ∈ c0 ∩ C. We will find acontinuous path (gt)0≤t≤1 from 1 to g in c0 such thatx+ gt ∈ x+ C for each t. Clearly gt = tg ∈ c0 ∩ C works.
There are groups that admit no turbulent action. For examplePolish locally compact groups and S∞.
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Turbulence
Example
The action of the Polish group (c0,+) (with the supnormtopology) on (RN,+) (with the product topology) by translation isturbulent. The orbits are the cosets of c0 in (RN,+), so they aredense and meager. Also, any x ∈ RN is turbulent. Clearly the setsof the form x+ C, with C ⊆ RN a convex open nbhd of 0, form anbhd basis for x ∈ RN. Now c0 is dense in any such C, sox+ (c0 ∩ C) is dense in x+ C. Let g ∈ c0 ∩ C. We will find acontinuous path (gt)0≤t≤1 from 1 to g in c0 such thatx+ gt ∈ x+ C for each t. Clearly gt = tg ∈ c0 ∩ C works.
There are groups that admit no turbulent action. For examplePolish locally compact groups and S∞.
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Generic ergodicity
Definition
Let E be an equivalence relation on a Polish space X and F anequivalence relation on a Polish space Y . A homomorphism fromE to F is a map f : X → Y such that xEy ⇒ f(x)Ff(y).We say that E is generically F -ergodic if for every Bairemeasurable homomorphism f there is a comeager set A ⊆ Xwhich f maps into a single F -class.
Example
Assume E = EXG is induced by a continuous G-action of a Polishgroup G on a Polish space X with a dense orbit. Then E isgenerically =Y -ergodic, for any Polish space Y .In particular, if every G-orbit is also meager, then E is notconcretely classifiable.
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Generic ergodicity
Definition
Let E be an equivalence relation on a Polish space X and F anequivalence relation on a Polish space Y . A homomorphism fromE to F is a map f : X → Y such that xEy ⇒ f(x)Ff(y).We say that E is generically F -ergodic if for every Bairemeasurable homomorphism f there is a comeager set A ⊆ Xwhich f maps into a single F -class.
Example
Assume E = EXG is induced by a continuous G-action of a Polishgroup G on a Polish space X with a dense orbit. Then E isgenerically =Y -ergodic, for any Polish space Y .In particular, if every G-orbit is also meager, then E is notconcretely classifiable.
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Generic ergodicity
Let as before Ec be the equivalence relation on (2N)N given by
(xn)Ec(yn)⇔ xn : n ∈ N = yn : n ∈ N.
Theorem (Hjorth)
The following are equivalent:
(i) E is generically Ec-ergodic.
(ii) E is generically EYS∞-ergodic for any Borel S∞-space Y .
It follows that if E satisfies these properties and if every E-class ismeager, then E cannot be classified by countable structures.
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Generic ergodicity
Let as before Ec be the equivalence relation on (2N)N given by
(xn)Ec(yn)⇔ xn : n ∈ N = yn : n ∈ N.
Theorem (Hjorth)
The following are equivalent:
(i) E is generically Ec-ergodic.
(ii) E is generically EYS∞-ergodic for any Borel S∞-space Y .
It follows that if E satisfies these properties and if every E-class ismeager, then E cannot be classified by countable structures.
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Generic ergodicity
Let as before Ec be the equivalence relation on (2N)N given by
(xn)Ec(yn)⇔ xn : n ∈ N = yn : n ∈ N.
Theorem (Hjorth)
The following are equivalent:
(i) E is generically Ec-ergodic.
(ii) E is generically EYS∞-ergodic for any Borel S∞-space Y .
It follows that if E satisfies these properties and if every E-class ismeager, then E cannot be classified by countable structures.
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Generic ergodicity
Theorem (Hjorth)
If a Polish group G acts continuously on a Polish space X and theaction is turbulent, then EXG is generically Ec-ergodic, so cannotbe classified by countable structures.
Idea of the proof:
Assume f : X → (2N)N is a Baire measurable homomorphism fromEXG to Ec. Let A(x) = f(x)n : n ∈ N, so that
xEXG y ⇒ A(x) = A(y).
Step 1: Let A = a ∈ 2N : ∀∗x (a ∈ A(x)), where ∀∗x means“on a comeager set of x.” Then A is countable.
Step 2: ∀∗x (A(x) = A), which completes the proof.
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Generic ergodicity
Theorem (Hjorth)
If a Polish group G acts continuously on a Polish space X and theaction is turbulent, then EXG is generically Ec-ergodic, so cannotbe classified by countable structures.
Idea of the proof:
Assume f : X → (2N)N is a Baire measurable homomorphism fromEXG to Ec. Let A(x) = f(x)n : n ∈ N, so that
xEXG y ⇒ A(x) = A(y).
Step 1: Let A = a ∈ 2N : ∀∗x (a ∈ A(x)), where ∀∗x means“on a comeager set of x.” Then A is countable.
Step 2: ∀∗x (A(x) = A), which completes the proof.
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Turbulence in the irreducible representations
Let H be a separable complex Hilbert space. We denote by U(H)the unitary group of H, i.e., the group of Hilbert spaceautomorphisms of H. The strong topology on U(H) is generatedby the maps T ∈ U(H) 7→ T (x) ∈ H (x ∈ H), and it is the sameas the weak topology generated by the mapsT ∈ U(H) 7→ 〈T (x), y〉 ∈ C (x, y ∈ H). With this topology U(H)is a Polish group.
If now Γ is a countable (discrete) group, Rep(Γ, H) is the space ofunitary representations of Γ on H, i.e., homomorphisms of Γ intoU(H) or equivalently actions of Γ on H by unitarytransformations. It is a closed subspace of U(H)Γ, equipped withthe product topology, so it is a Polish space. The group U(H) actscontinuously on Rep(Γ, H) via conjugacy T · π = TπT−1 (whereTπT−1(γ) = T π(γ) T−1) and the equivalence relation inducedby this action is isomorphism of unitary representations: π ∼= ρ.
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Turbulence in the irreducible representations
Let H be a separable complex Hilbert space. We denote by U(H)the unitary group of H, i.e., the group of Hilbert spaceautomorphisms of H. The strong topology on U(H) is generatedby the maps T ∈ U(H) 7→ T (x) ∈ H (x ∈ H), and it is the sameas the weak topology generated by the mapsT ∈ U(H) 7→ 〈T (x), y〉 ∈ C (x, y ∈ H). With this topology U(H)is a Polish group.
If now Γ is a countable (discrete) group, Rep(Γ, H) is the space ofunitary representations of Γ on H, i.e., homomorphisms of Γ intoU(H) or equivalently actions of Γ on H by unitarytransformations. It is a closed subspace of U(H)Γ, equipped withthe product topology, so it is a Polish space. The group U(H) actscontinuously on Rep(Γ, H) via conjugacy T · π = TπT−1 (whereTπT−1(γ) = T π(γ) T−1) and the equivalence relation inducedby this action is isomorphism of unitary representations: π ∼= ρ.
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Turbulence in the irreducible representations
A representation π ∈ Rep(Γ, H) is irreducible if it has nonon-trivial invariant closed subspaces. Let Irr(Γ, H) ⊆ Rep(Γ, H)be the space of irreducible representations. Then it can be shownthat Irr(Γ, H) is a Gδ subset of Rep(Γ, H), and thus also a Polishspace in the relative topology.
From now on we assume that H is ∞-dimensional.
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Turbulence in the irreducible representations
A representation π ∈ Rep(Γ, H) is irreducible if it has nonon-trivial invariant closed subspaces. Let Irr(Γ, H) ⊆ Rep(Γ, H)be the space of irreducible representations. Then it can be shownthat Irr(Γ, H) is a Gδ subset of Rep(Γ, H), and thus also a Polishspace in the relative topology.
From now on we assume that H is ∞-dimensional.
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Turbulence in the irreducible representations
Thoma has shown that if Γ is abelian-by-finite then we haveIrr(Γ, H) = ∅, but if Γ is not abelian-by-finite then ∼= |Irr(Γ, H) isnot concretely classifiable. Hjorth extended this by showing that itis not even classifiable by countable structures. This is proved byshowing that the conjugacy action is turbulent on an appropriateconjugacy invariant closed subspace of Irr(Γ, H). We will needbelow this result for the case Γ = F2 = the free group with twogenerators, so we will state and sketch the proof of the followingstronger result for this case.
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Turbulence in the irreducible representations
Theorem (Hjorth)
The conjugacy action of U(H) on Irr(F2, H) is turbulent.
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Turbulence in the irreducible representations
Note that we can identify Rep(F2, H) with U(H)2 and the actionof U(H) on U(H)2 becomes
T · (U1, U2) = (TU1T−1, TU2T
−1).
I will now outline the proof of that theorem.
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Turbulence in the irreducible representations
Note that we can identify Rep(F2, H) with U(H)2 and the actionof U(H) on U(H)2 becomes
T · (U1, U2) = (TU1T−1, TU2T
−1).
I will now outline the proof of that theorem.
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Turbulence in the irreducible representations
Step 1: Irr(F2, H) is dense Gδ in Rep(F2, H) = U(H)2.
Note that if U(n) = U(Cn), then, with some canonicalidentifications, U(1) ⊆ U(2) ⊆ · · · ⊆ U(H) and
⋃n U(n) = U(H).
Now U(n) is compact, connected, so, by a result of Schreier-Ulam,the set of (g, h) ∈ U(n)2 such that 〈g, h〉 = U(n) is a dense Gδ inU(n)2. By Baire Category this shows that the set of(g, h) ∈ U(H)2 with 〈g, h〉 = U(H) is dense Gδ, since for eachnonempty open set N in U(H), the set of (g, h) ∈ U(H)2 thatgenerate a subgroup intersecting N is open dense. Thus thegeneric pair (g, h) ∈ U(H)2 generates a dense subgroup of U(H)so, viewing (g, h) as a representation, any (g, h)-invariant closedsubspace A is invariant for all of 〈g, h〉 = U(H), hence A = 0 orA = H and the generic pair is irreducible.
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Turbulence in the irreducible representations
Step 1: Irr(F2, H) is dense Gδ in Rep(F2, H) = U(H)2.
Note that if U(n) = U(Cn), then, with some canonicalidentifications, U(1) ⊆ U(2) ⊆ · · · ⊆ U(H) and
⋃n U(n) = U(H).
Now U(n) is compact, connected, so, by a result of Schreier-Ulam,the set of (g, h) ∈ U(n)2 such that 〈g, h〉 = U(n) is a dense Gδ inU(n)2. By Baire Category this shows that the set of(g, h) ∈ U(H)2 with 〈g, h〉 = U(H) is dense Gδ, since for eachnonempty open set N in U(H), the set of (g, h) ∈ U(H)2 thatgenerate a subgroup intersecting N is open dense. Thus thegeneric pair (g, h) ∈ U(H)2 generates a dense subgroup of U(H)so, viewing (g, h) as a representation, any (g, h)-invariant closedsubspace A is invariant for all of 〈g, h〉 = U(H), hence A = 0 orA = H and the generic pair is irreducible.
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Turbulence in the irreducible representations
Step 2: Every orbit is meager.
It is enough to show that every conjugacy class in U(H) is meager.This is a classical result but here is a simple proof recently foundby Rosendal.
For each infinite I ⊆ N, let
A(I) = T ∈ U(H) : ∃i ∈ I (T i = 1).
It is easy to check that A(I) is dense in U(H). (Use the fact that⋃n U(n) is dense in U(H). Then it is enough to approximate
elements of each U(n) by A(I) and this can be easily done usingthe fact that elements of U(n) are conjugate in U(n) to diagonalunitaries.)
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Turbulence in the irreducible representations
Step 2: Every orbit is meager.
It is enough to show that every conjugacy class in U(H) is meager.This is a classical result but here is a simple proof recently foundby Rosendal.
For each infinite I ⊆ N, let
A(I) = T ∈ U(H) : ∃i ∈ I (T i = 1).
It is easy to check that A(I) is dense in U(H). (Use the fact that⋃n U(n) is dense in U(H). Then it is enough to approximate
elements of each U(n) by A(I) and this can be easily done usingthe fact that elements of U(n) are conjugate in U(n) to diagonalunitaries.)
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Turbulence in the irreducible representations
Let now V0 ⊇ V1 ⊇ · · · be a basis of open nbhds of 1 in U(H) andput B(I, k) = T ∈ U(H) : ∃i ∈ I(i > k and T i ∈ Vk). Thiscontains A(I \ 0, . . . , k), so is open dense. Thus
C(I) =⋂k
B(I, k) = T : ∃(in) ∈ IN (T in → 1)
is comeager and conjugacy invariant. If a conjugacy class C isnonmeager, it will thus be contained in all C(I), I ⊆ N infinite.Thus T ∈ C ⇒ Tn → 1, so letting d be a left invariant metric forU(H) we have for T ∈ C, d(T, 1) = d(Tn+1, Tn)→ 0, whenceT = 1, a contradiction.
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Turbulence in the irreducible representations
Step 3: There is a dense conjugacy class in Irr(F2, H) (so the setof all π ∈ Irr(F2, H) with dense conjugacy class is dense Gδ inIrr(F2, H)).
As Irr(F2, H) is dense Gδ in Rep(F2, H) it is enough to findπ ∈ Rep(F2, H) with dense conjugacy class in Rep(F2, H) – thenthe set of all such π’s is dense Gδ so intersects Irr(F2, H). Let(πn) be dense in Rep(F2, H) and let π ∼=
⊕n πn, π ∈ Rep(F2, H).
This π easily works.
(This also gives an easy proof of a result of Yoshizawa: Thereexists an irreducible representation of F2 which weakly containsany representation of F2.)
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Turbulence in the irreducible representations
Step 3: There is a dense conjugacy class in Irr(F2, H) (so the setof all π ∈ Irr(F2, H) with dense conjugacy class is dense Gδ inIrr(F2, H)).
As Irr(F2, H) is dense Gδ in Rep(F2, H) it is enough to findπ ∈ Rep(F2, H) with dense conjugacy class in Rep(F2, H) – thenthe set of all such π’s is dense Gδ so intersects Irr(F2, H). Let(πn) be dense in Rep(F2, H) and let π ∼=
⊕n πn, π ∈ Rep(F2, H).
This π easily works.
(This also gives an easy proof of a result of Yoshizawa: Thereexists an irreducible representation of F2 which weakly containsany representation of F2.)
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Turbulence in the irreducible representations
Step 3: There is a dense conjugacy class in Irr(F2, H) (so the setof all π ∈ Irr(F2, H) with dense conjugacy class is dense Gδ inIrr(F2, H)).
As Irr(F2, H) is dense Gδ in Rep(F2, H) it is enough to findπ ∈ Rep(F2, H) with dense conjugacy class in Rep(F2, H) – thenthe set of all such π’s is dense Gδ so intersects Irr(F2, H). Let(πn) be dense in Rep(F2, H) and let π ∼=
⊕n πn, π ∈ Rep(F2, H).
This π easily works.
(This also gives an easy proof of a result of Yoshizawa: Thereexists an irreducible representation of F2 which weakly containsany representation of F2.)
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Turbulence in the irreducible representations
Step 4: Let π ∈ Irr(F2, H) have dense conjugacy class. Then π isturbulent.
Thus by Steps 2,3,4, the conjugacy action of U(H) on Irr(F2, H)is turbulent.
Lemma
For ρ, σ ∈ Irr(F2, H), the following are equivalent:
(i) ρ ∼= σ,
(ii) ∃(Tn) ∈ U(H)N such that Tn · ρ = TnρT−1n → σ and no
subsequence of (Tn) converges in the weak topology ofB1(H) = T ∈ B(H) : ||T || ≤ 1 (a compact metrizablespace) to 0.
(Here B(H) is the space of bounded linear operators on H. Prooffollows from Schur’s Lemma and the compactness of the unit ballof B(H).)
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Turbulence in the irreducible representations
Step 4: Let π ∈ Irr(F2, H) have dense conjugacy class. Then π isturbulent.
Thus by Steps 2,3,4, the conjugacy action of U(H) on Irr(F2, H)is turbulent.
Lemma
For ρ, σ ∈ Irr(F2, H), the following are equivalent:
(i) ρ ∼= σ,
(ii) ∃(Tn) ∈ U(H)N such that Tn · ρ = TnρT−1n → σ and no
subsequence of (Tn) converges in the weak topology ofB1(H) = T ∈ B(H) : ||T || ≤ 1 (a compact metrizablespace) to 0.
(Here B(H) is the space of bounded linear operators on H. Prooffollows from Schur’s Lemma and the compactness of the unit ballof B(H).)
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Turbulence in the irreducible representations
Step 4: Let π ∈ Irr(F2, H) have dense conjugacy class. Then π isturbulent.
Thus by Steps 2,3,4, the conjugacy action of U(H) on Irr(F2, H)is turbulent.
Lemma
For ρ, σ ∈ Irr(F2, H), the following are equivalent:
(i) ρ ∼= σ,
(ii) ∃(Tn) ∈ U(H)N such that Tn · ρ = TnρT−1n → σ and no
subsequence of (Tn) converges in the weak topology ofB1(H) = T ∈ B(H) : ||T || ≤ 1 (a compact metrizablespace) to 0.
(Here B(H) is the space of bounded linear operators on H. Prooffollows from Schur’s Lemma and the compactness of the unit ballof B(H).)
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Turbulence in the irreducible representations
Lemma
Given any nonempty open W ⊆ Irr(F2, H) (open in the relativetopology) and orthonormal e1, . . . , ep ∈ H, there are orthonormale1, . . . , ep, ep+1, . . . , eq and T ∈ U(H) such that
(i) T (ei) ⊥ ej , ∀i, j ≤ q;
(ii) T 2(ei) = −ei, ∀i ≤ q;
(iii) T = id on (H0 ⊕ T (H0))⊥, where H0 = 〈e1, . . . , eq〉;(iv) T · π ∈W .
(Use the fact that there is an irreducible σ in W which is not inthe conjugacy class of π. By the density of the conjugacy class ofπ, there is a sequence of unitaries Tn with Tn · π → σ, so by theprevious lemma Tn weakly converges to 0.)
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Turbulence in the irreducible representations
Lemma
Given any nonempty open W ⊆ Irr(F2, H) (open in the relativetopology) and orthonormal e1, . . . , ep ∈ H, there are orthonormale1, . . . , ep, ep+1, . . . , eq and T ∈ U(H) such that
(i) T (ei) ⊥ ej , ∀i, j ≤ q;
(ii) T 2(ei) = −ei, ∀i ≤ q;
(iii) T = id on (H0 ⊕ T (H0))⊥, where H0 = 〈e1, . . . , eq〉;(iv) T · π ∈W .
(Use the fact that there is an irreducible σ in W which is not inthe conjugacy class of π. By the density of the conjugacy class ofπ, there is a sequence of unitaries Tn with Tn · π → σ, so by theprevious lemma Tn weakly converges to 0.)
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Turbulence in the irreducible representations
We now show that π is turbulent. Fix a basic nbhd of π of the form
U = ρ ∈ Irr(F2, H) : ∀γ ∈ F ∀i, j ≤ k|〈ρ(γ)(ei), ej〉 − 〈π(γ)(ei), (ej)〉| < ε,
ε > 0, F ⊆ F2 finite, e1, . . . , ek orthonormal. Lete1, . . . , ek, ek+1, . . . , ep be an orthonormal basis for the span ofe1, . . . , ek ∪ π(γ)(ei) : γ ∈ F, 1 ≤ i ≤ k, and let W ⊆ U bean arbitrary nonempty open set. Then let e1, . . . , ep, ep+1, . . . , eqand T be as in the previous lemma (so that T · π ∈W ). It isenough to find a continuous path (Tθ)0≤θ≤π/2 in U(H) withT0 = 1, Tπ/2 = T , and Tθ · π ∈ U for all θ. Take
Tθ(ei) = (cos θ)ei + (sin θ)T (ei)Tθ(T (ei)) = (− sin θ)ei + (cos θ)T (ei),
for i = 1, . . . , q and let Tθ = id on (H0 ⊕ T (H0))⊥, whereH0 = 〈e1, . . . , eq〉. Then one can easily see that Tθ · π ∈ U , for allθ.
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Turbulence in the irreducible representations
θ
T (ei)
−ei ei
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III. THE PROOF OF THE NON-CLASSIFICATIONTHEOREM
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The non-classification theorem
My goal in the remaining two lectures is to prove the followingresult.
Theorem (Epstein-Ioana-K-Tsankov)
Let Γ be a countable non-amenable group. Then orbit equivalencefor measure-preserving, free, ergodic (in fact mixing) actions of Γis not classifiable by countable structures.
I will start be giving a very rough idea of the proof and thendiscussing the (rather extensive) set of results needed to implementit.
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The non-classification theorem
My goal in the remaining two lectures is to prove the followingresult.
Theorem (Epstein-Ioana-K-Tsankov)
Let Γ be a countable non-amenable group. Then orbit equivalencefor measure-preserving, free, ergodic (in fact mixing) actions of Γis not classifiable by countable structures.
I will start be giving a very rough idea of the proof and thendiscussing the (rather extensive) set of results needed to implementit.
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The non-classification theorem
My goal in the remaining two lectures is to prove the followingresult.
Theorem (Epstein-Ioana-K-Tsankov)
Let Γ be a countable non-amenable group. Then orbit equivalencefor measure-preserving, free, ergodic (in fact mixing) actions of Γis not classifiable by countable structures.
I will start be giving a very rough idea of the proof and thendiscussing the (rather extensive) set of results needed to implementit.
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Basic definitions
A standard measure space (X,µ) is a Polish space X with anon-atomic probability Borel measure µ. All such spaces areisomorphic to [0, 1] with Lebesgue measure on the Borel sets.The measure algebra MALGµ of µ is the algebra of Borel sets ofX, modulo null sets, with the topology induced by the metricd(A,B) = µ(A∆B). Let Aut(X,µ) be the group ofmeasure-preserving automorphisms of (X,µ) (again modulo nullsets) with the weak topology, i.e., the one generated by the mapsT 7→ T (A) (A ∈ MALGµ). It is a Polish group.
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Basic definitions
If Γ is a countable group, denote by A(Γ, X, µ) the space ofmeasure-preserving actions of Γ on (X,µ) or equivalently,homomorphisms of Γ into Aut(X,µ). It is a closed subspace ofAut(X,µ)Γ with the product topology, so also a Polish space. Wesay that a ∈ A(Γ, X, µ) is free if ∀γ 6= 1 (γ · x 6= x, a.e.), and it isergodic if every invariant Borel set A ⊆ X is either null or conull.We say that a ∈ A(Γ, X, µ), b ∈ A(Γ, Y, ν) are orbit equivalent,aŒ b, if, denoting by Ea, Eb the equivalence relations induced bya, b respectively, Ea is isomorphic to Eb, in the sense that there isa measure-preserving isomorphism of (X,µ) to (Y, ν) that sendsEa to Eb (modulo null sets). Thus Œ is an equivalence relation onA(Γ, X, µ) and previous theorem asserts that it cannot beclassified by countable structures if Γ is not amenable.
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Idea of the proof
We start with the following fact.
Theorem (Gaussian actions)
To each π ∈ Rep(Γ, H), we can assign in a Borel way an actionaπ ∈ A(Γ, X, µ) (on some standard measure space (X,µ)), calledthe Gaussian action associated to π, such that
(i) π ∼= ρ⇒ aπ ∼= aρ,
(ii) If κaπ0 is the Koopman representation on L20(X,µ) associated
to aπ, then π ≤ κaπ0 .
Moreover if π is irreducible, aπ is weakly mixing.
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Idea of the proof
We start with the following fact.
Theorem (Gaussian actions)
To each π ∈ Rep(Γ, H), we can assign in a Borel way an actionaπ ∈ A(Γ, X, µ) (on some standard measure space (X,µ)), calledthe Gaussian action associated to π, such that
(i) π ∼= ρ⇒ aπ ∼= aρ,
(ii) If κaπ0 is the Koopman representation on L20(X,µ) associated
to aπ, then π ≤ κaπ0 .
Moreover if π is irreducible, aπ is weakly mixing.
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Idea of the proof
Explanations:
(i) a, b ∈ A(Γ, X, µ) are isomorphic, a ∼= b if there is aT ∈ Aut(X,µ) taking a to b,
TγaT−1 = γb, ∀γ ∈ Γ.
(Here we let γa = a(γ).)
(ii) Let a ∈ A(Γ, X, µ). Let
L20(X,µ) = f ∈ L2(X,µ) :
∫f = 0 = C⊥.
The Koopman representation κa0 of Γ on L20(X,µ) is given by
(γ · f)(x) = f(γ−1 · x).
(iii) If π ∈ Rep(Γ, H), ρ ∈ Rep(Γ, H ′), then π ≤ ρ iff π isisomorphic to a subrepresentation of ρ, i.e., the restriction ofρ to an invariant, closed subspace of H ′.
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Idea of the proof
Explanations:
(i) a, b ∈ A(Γ, X, µ) are isomorphic, a ∼= b if there is aT ∈ Aut(X,µ) taking a to b,
TγaT−1 = γb, ∀γ ∈ Γ.
(Here we let γa = a(γ).)
(ii) Let a ∈ A(Γ, X, µ). Let
L20(X,µ) = f ∈ L2(X,µ) :
∫f = 0 = C⊥.
The Koopman representation κa0 of Γ on L20(X,µ) is given by
(γ · f)(x) = f(γ−1 · x).
(iii) If π ∈ Rep(Γ, H), ρ ∈ Rep(Γ, H ′), then π ≤ ρ iff π isisomorphic to a subrepresentation of ρ, i.e., the restriction ofρ to an invariant, closed subspace of H ′.
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Idea of the proof
Explanations:
(i) a, b ∈ A(Γ, X, µ) are isomorphic, a ∼= b if there is aT ∈ Aut(X,µ) taking a to b,
TγaT−1 = γb, ∀γ ∈ Γ.
(Here we let γa = a(γ).)
(ii) Let a ∈ A(Γ, X, µ). Let
L20(X,µ) = f ∈ L2(X,µ) :
∫f = 0 = C⊥.
The Koopman representation κa0 of Γ on L20(X,µ) is given by
(γ · f)(x) = f(γ−1 · x).
(iii) If π ∈ Rep(Γ, H), ρ ∈ Rep(Γ, H ′), then π ≤ ρ iff π isisomorphic to a subrepresentation of ρ, i.e., the restriction ofρ to an invariant, closed subspace of H ′.
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Idea of the proof
So let π ∈ Irr(F2, H) and look at aπ ∈ A(F2, X, µ). We will thenmodify aπ, in a Borel and isomorphism preserving way, to anotheraction a(π) ∈ A(F2, Y, ν) (for reasons to be explained later) andfinally apply a construction of Epstein to “co-induce” appropriatelya(π), in a Borel and isomorphism preserving way, to an actionb(π) ∈ A(Γ, Z, ρ), which will turn out to be free and ergodic (infact mixing i.e., µ(γ ·A ∩B)→ µ(A)µ(B) as γ →∞, for everyBorel A,B). So we finally have a Borel function
π ∈ Irr(F2, H) 7→ b(π) ∈ A(Γ, Z, ρ).
PutπRρ⇔ b(π) Œ b(ρ).
Then R is an equivalence relation on Irr(F2, H), and π ∼= ρ⇒ πRρ(since π ∼= ρ⇒ aπ ∼= aρ ⇒ a(π) ∼= a(ρ)⇒ b(π) ∼= b(ρ)).
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Idea of the proof
Fact: R has countable index over ∼= (i.e., every R-class containsonly countably many ∼=-classes).
If now Œ on A(Γ, Z, ρ) admitted classification by countablestructures, so would R on Irr(F2, H). So let F : Irr(F2, H)→ XL
be Borel, where XL is the Polish space of countable structures fora signature L, with
πRρ⇔ F (π) ∼= F (ρ).
Therefore π ∼= ρ⇒ F (π) ∼= F (ρ). So there is a comeager setA ⊆ Irr(F2, H) and A0 ∈ XL such that F (π) ∼= A0, ∀π ∈ A. Butevery ∼=-class in Irr(F2, H) is meager, so by the previous fact everyR-class in Irr(F2, H) is meager, so there are R-inequivalentπ, ρ ∈ A, and thus F (π) 6∼= F (ρ), a contradiction.
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Idea of the proof
Fact: R has countable index over ∼= (i.e., every R-class containsonly countably many ∼=-classes).
If now Œ on A(Γ, Z, ρ) admitted classification by countablestructures, so would R on Irr(F2, H). So let F : Irr(F2, H)→ XL
be Borel, where XL is the Polish space of countable structures fora signature L, with
πRρ⇔ F (π) ∼= F (ρ).
Therefore π ∼= ρ⇒ F (π) ∼= F (ρ). So there is a comeager setA ⊆ Irr(F2, H) and A0 ∈ XL such that F (π) ∼= A0, ∀π ∈ A. Butevery ∼=-class in Irr(F2, H) is meager, so by the previous fact everyR-class in Irr(F2, H) is meager, so there are R-inequivalentπ, ρ ∈ A, and thus F (π) 6∼= F (ρ), a contradiction.
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Gaussian actions
Sketch of proof. For simplicity we will discuss the case of realHilbert spaces, the complex case being handled by appropriatecomplexifications.
Let H be an infinite-dimensional, separable, real Hilbert space.Consider the product space (RN, µN), where µ is the normalized,centered Gaussian measure on R with density 1√
2πe−x
2/2. Let
pi : RN → R, i ∈ N, be the projection functions. The closed linearspace 〈pi〉 ⊆ L2
0(RN, µN) (real valued) has countable infinitedimension, so we can assume that H = 〈pi〉 ⊆ L2
0(RN, µN).
Lemma
If S ∈ O(H) = the orthogonal group of H (i.e., the group ofHilbert space automorphisms of H), then we can extend uniquelyS to S ∈ Aut(RN, µN) in the sense that the Koopman operatorOS : L2
0(RN, µN)→ L20(RN, µN) defined by OS(f) = f (S)−1
extends S, i.e., OS |H = S.
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Gaussian actions
Sketch of proof. For simplicity we will discuss the case of realHilbert spaces, the complex case being handled by appropriatecomplexifications.
Let H be an infinite-dimensional, separable, real Hilbert space.Consider the product space (RN, µN), where µ is the normalized,centered Gaussian measure on R with density 1√
2πe−x
2/2. Let
pi : RN → R, i ∈ N, be the projection functions. The closed linearspace 〈pi〉 ⊆ L2
0(RN, µN) (real valued) has countable infinitedimension, so we can assume that H = 〈pi〉 ⊆ L2
0(RN, µN).
Lemma
If S ∈ O(H) = the orthogonal group of H (i.e., the group ofHilbert space automorphisms of H), then we can extend uniquelyS to S ∈ Aut(RN, µN) in the sense that the Koopman operatorOS : L2
0(RN, µN)→ L20(RN, µN) defined by OS(f) = f (S)−1
extends S, i.e., OS |H = S.
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Gaussian actions
Sketch of proof. For simplicity we will discuss the case of realHilbert spaces, the complex case being handled by appropriatecomplexifications.
Let H be an infinite-dimensional, separable, real Hilbert space.Consider the product space (RN, µN), where µ is the normalized,centered Gaussian measure on R with density 1√
2πe−x
2/2. Let
pi : RN → R, i ∈ N, be the projection functions. The closed linearspace 〈pi〉 ⊆ L2
0(RN, µN) (real valued) has countable infinitedimension, so we can assume that H = 〈pi〉 ⊆ L2
0(RN, µN).
Lemma
If S ∈ O(H) = the orthogonal group of H (i.e., the group ofHilbert space automorphisms of H), then we can extend uniquelyS to S ∈ Aut(RN, µN) in the sense that the Koopman operatorOS : L2
0(RN, µN)→ L20(RN, µN) defined by OS(f) = f (S)−1
extends S, i.e., OS |H = S.
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Gaussian actions
Lemma
If S ∈ O(H) = the orthogonal group of H (i.e., the group ofHilbert space automorphisms of H), then we can extend uniquelyS to S ∈ Aut(RN, µN) in the sense that the Koopman operatorOS : L2
0(RN, µN)→ L20(RN, µN) defined by OS(f) = f (S)−1
extends S, i.e., OS |H = S.
Thus if π ∈ Rep(Γ, H), we can extend each π(γ) ∈ O(H) toπ(γ) ∈ Aut(RN, µN). Let aπ ∈ A(Γ,RN, µN) be defined byaπ(γ) = π(γ). This clearly works.
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Gaussian actions
Lemma
If S ∈ O(H) = the orthogonal group of H (i.e., the group ofHilbert space automorphisms of H), then we can extend uniquelyS to S ∈ Aut(RN, µN) in the sense that the Koopman operatorOS : L2
0(RN, µN)→ L20(RN, µN) defined by OS(f) = f (S)−1
extends S, i.e., OS |H = S.
Thus if π ∈ Rep(Γ, H), we can extend each π(γ) ∈ O(H) toπ(γ) ∈ Aut(RN, µN). Let aπ ∈ A(Γ,RN, µN) be defined byaπ(γ) = π(γ). This clearly works.
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Gaussian actions
Lemma
If S ∈ O(H) = the orthogonal group of H (i.e., the group ofHilbert space automorphisms of H), then we can extend uniquelyS to S ∈ Aut(RN, µN) in the sense that the Koopman operatorOS : L2
0(RN, µN)→ L20(RN, µN) defined by OS(f) = f (S)−1
extends S, i.e., OS |H = S.
Proof of Lemma:
The pi’s form an orthonormal basis for H. Let S(pi) = qi ∈ H.Then let θ : RN → RN be defined by θ(x) = (q0(x), q1(x), . . . ).Then θ is 1-1, since the σ-algebra generated by (qi) is the Borelσ-algebra, modulo null sets, so (qi) separates points modulo nullsets. Moreover θ preserves µN. This follows from the fact thatevery f ∈ 〈pi〉 (including qi) has centered Gaussian distribution andthe qi are independent, since E(qiqj) = 〈qi, qj〉 = 〈pi, pj〉 = δij .Thus θ ∈ Aut(RN, µN). Now put S = θ−1.
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Gaussian actions
Lemma
If S ∈ O(H) = the orthogonal group of H (i.e., the group ofHilbert space automorphisms of H), then we can extend uniquelyS to S ∈ Aut(RN, µN) in the sense that the Koopman operatorOS : L2
0(RN, µN)→ L20(RN, µN) defined by OS(f) = f (S)−1
extends S, i.e., OS |H = S.
Proof of Lemma:
The pi’s form an orthonormal basis for H. Let S(pi) = qi ∈ H.Then let θ : RN → RN be defined by θ(x) = (q0(x), q1(x), . . . ).Then θ is 1-1, since the σ-algebra generated by (qi) is the Borelσ-algebra, modulo null sets, so (qi) separates points modulo nullsets. Moreover θ preserves µN. This follows from the fact thatevery f ∈ 〈pi〉 (including qi) has centered Gaussian distribution andthe qi are independent, since E(qiqj) = 〈qi, qj〉 = 〈pi, pj〉 = δij .Thus θ ∈ Aut(RN, µN). Now put S = θ−1.
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Review
Recall that our plan consists of the three steps
π(1)−−→ aπ
(2)−−→ a(π)(3)−−→ b(π),
where π is an irreducible unitary representation of F2, aπ is thecorresponding Gaussian action of F2, (2) is the “perturbation” to anew action of F2 and (3) is the co-inducing construction, whichfrom the F2-action a(π) produces a Γ-action b(π).
We already discussed step (1). We next discuss step (3).
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Review
Recall that our plan consists of the three steps
π(1)−−→ aπ
(2)−−→ a(π)(3)−−→ b(π),
where π is an irreducible unitary representation of F2, aπ is thecorresponding Gaussian action of F2, (2) is the “perturbation” to anew action of F2 and (3) is the co-inducing construction, whichfrom the F2-action a(π) produces a Γ-action b(π).
We already discussed step (1). We next discuss step (3).
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Properties of Epstein’s co-induced action
Theorem
Let Γ be non-amenable. Given a ∈ A(F2, Y, ν) we can construct in a Borel wayb ∈ A(Γ, Z, ρ) and a′ ∈ A(F2, Z, ρ) (Z, ρ independent of a) with the followingproperties:
(i) Ea′ ⊆ Eb,(ii) b is free and ergodic (in fact mixing),
(iii) a′ is free,
(iv) a is a factor of a′ via a map f : Z → Y (f independent of a), i.e.,
f(δa′ · z) = δa · f(z) (δ ∈ F2), and f∗ρ = ν, and in fact if a is ergodic, then for
every a′-invariant Borel set A ⊆ Z of positive measure, if ρA =ρ|Aρ(A)
, then a is
a factor of (a′|A, ρA) via f .
(v) If a free action a ∈ A(F2, Y , ν) is a factor of the action a via g : Y → Y
Yg // Y
Z
f
OO
then for γ ∈ Γ \ 1, gf(γb · z) 6= gf(z), ρ-a.e.
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Review
Recall that our plan consisted of the three steps
π(1)−−→ aπ
(2)−−→ a(π)(3)−−→ b(π),
where π is an irreducible unitary representation of F2, aπ is thecorresponding Gaussian action of F2, (2) is the “perturbation” to anew action of F2 and (3) is the co-inducing construction, whichfrom the F2-action a(π) produces a Γ-action b(π).
I discussed the first step and summarized the key properties of thethird step. I will discuss it in detail later but I will next describethe second step.
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Review
Recall that our plan consisted of the three steps
π(1)−−→ aπ
(2)−−→ a(π)(3)−−→ b(π),
where π is an irreducible unitary representation of F2, aπ is thecorresponding Gaussian action of F2, (2) is the “perturbation” to anew action of F2 and (3) is the co-inducing construction, whichfrom the F2-action a(π) produces a Γ-action b(π).
I discussed the first step and summarized the key properties of thethird step. I will discuss it in detail later but I will next describethe second step.
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An action of F2
We now deal with construction (2). The key here is a particularaction of F2 on T2 utilized first to a great effect by Popa,Popa-Gaboriau and then Ioana. The group SL2(Z) acts on (T2, λ)(λ is Lebesgue measure) in the usual way by matrix multiplication
A · (z1, z2) = (A−1)t(z1
z2
).
This is free, measure-preserving, and ergodic, in fact weak mixing.
Fix also a copy of F2 with finite index in SL2(Z) and denote therestriction of this action to F2 by α0. It is also free,measure-preserving and weak mixing. For any c ∈ A(F2, X, µ), welet a(c) ∈ A(F2,T2 ×X,λ× µ) be the product action
a(c) = α0 × c(i.e. γa(c) · (z, x) = (γα0 · z, γc · x)). Then in our case we take
a(π) = a(aπ) = α0 × aπ.Again a(π) is ergodic, in fact weak mixing.
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An action of F2
We now deal with construction (2). The key here is a particularaction of F2 on T2 utilized first to a great effect by Popa,Popa-Gaboriau and then Ioana. The group SL2(Z) acts on (T2, λ)(λ is Lebesgue measure) in the usual way by matrix multiplication
A · (z1, z2) = (A−1)t(z1
z2
).
This is free, measure-preserving, and ergodic, in fact weak mixing.
Fix also a copy of F2 with finite index in SL2(Z) and denote therestriction of this action to F2 by α0. It is also free,measure-preserving and weak mixing. For any c ∈ A(F2, X, µ), welet a(c) ∈ A(F2,T2 ×X,λ× µ) be the product action
a(c) = α0 × c(i.e. γa(c) · (z, x) = (γα0 · z, γc · x)). Then in our case we take
a(π) = a(aπ) = α0 × aπ.Again a(π) is ergodic, in fact weak mixing.
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A separability argument
The key property of the passage from c to a(c) is the followingseparability result established by Ioana (in a somewhat differentcontext – but his proof works as well here).
Below, for each c ∈ A(F2, X, µ), with a(c) = α0 × c ∈ A(F2, Y, ν),where Y = T2 ×X, ν = λ× µ, we let b(c) ∈ A(Γ, Z, ρ),a′(c) ∈ A(F2, Z, ρ) come from Epstein’s co-inducing construction.
Theorem (Ioana)
If (ci)i∈I is an uncountable family of actions in A(F2, X, µ) and(b(ci))i∈I are mutually orbit equivalent, then there is uncountableJ ⊆ I such that if i, j ∈ J , we can find Borel sets Ai, Aj ⊆ Z ofpositive measure which are respectively a′(ci), a′(cj)-invariant anda′(ci)|Ai ∼= a′(cj)|Aj with respect to the normalized measures ρAi ,ρAj .
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A separability argument
The key property of the passage from c to a(c) is the followingseparability result established by Ioana (in a somewhat differentcontext – but his proof works as well here).
Below, for each c ∈ A(F2, X, µ), with a(c) = α0 × c ∈ A(F2, Y, ν),where Y = T2 ×X, ν = λ× µ, we let b(c) ∈ A(Γ, Z, ρ),a′(c) ∈ A(F2, Z, ρ) come from Epstein’s co-inducing construction.
Theorem (Ioana)
If (ci)i∈I is an uncountable family of actions in A(F2, X, µ) and(b(ci))i∈I are mutually orbit equivalent, then there is uncountableJ ⊆ I such that if i, j ∈ J , we can find Borel sets Ai, Aj ⊆ Z ofpositive measure which are respectively a′(ci), a′(cj)-invariant anda′(ci)|Ai ∼= a′(cj)|Aj with respect to the normalized measures ρAi ,ρAj .
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A separability argument
The key property of the passage from c to a(c) is the followingseparability result established by Ioana (in a somewhat differentcontext – but his proof works as well here).
Below, for each c ∈ A(F2, X, µ), with a(c) = α0 × c ∈ A(F2, Y, ν),where Y = T2 ×X, ν = λ× µ, we let b(c) ∈ A(Γ, Z, ρ),a′(c) ∈ A(F2, Z, ρ) come from Epstein’s co-inducing construction.
Theorem (Ioana)
If (ci)i∈I is an uncountable family of actions in A(F2, X, µ) and(b(ci))i∈I are mutually orbit equivalent, then there is uncountableJ ⊆ I such that if i, j ∈ J , we can find Borel sets Ai, Aj ⊆ Z ofpositive measure which are respectively a′(ci), a′(cj)-invariant anda′(ci)|Ai ∼= a′(cj)|Aj with respect to the normalized measures ρAi ,ρAj .
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Completion of the proof
We now complete the proof of the main theorem.
Recall the chain
π(1)−−→ aπ
(2)−−→ a(π)(3)−−→ b(π),
where π is an irreducible unitary representation of F2, aπ is thecorresponding Gaussian action of F2, (2) is the “perturbation” to anew action of F2 and (3) is the co-inducing construction, whichfrom the F2-action a(π) produces a Γ-action b(π).
We also have a′(π) coming from a(π) in step (3). Recall that wedefined
πRρ⇔ b(π) Œ b(ρ).
To complete the proof, we only had to show that R has countableindex over ∼=. We now prove this:
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Completion of the proof
We now complete the proof of the main theorem.
Recall the chain
π(1)−−→ aπ
(2)−−→ a(π)(3)−−→ b(π),
where π is an irreducible unitary representation of F2, aπ is thecorresponding Gaussian action of F2, (2) is the “perturbation” to anew action of F2 and (3) is the co-inducing construction, whichfrom the F2-action a(π) produces a Γ-action b(π).
We also have a′(π) coming from a(π) in step (3). Recall that wedefined
πRρ⇔ b(π) Œ b(ρ).
To complete the proof, we only had to show that R has countableindex over ∼=. We now prove this:
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Completion of the proof
We now complete the proof of the main theorem.
Recall the chain
π(1)−−→ aπ
(2)−−→ a(π)(3)−−→ b(π),
where π is an irreducible unitary representation of F2, aπ is thecorresponding Gaussian action of F2, (2) is the “perturbation” to anew action of F2 and (3) is the co-inducing construction, whichfrom the F2-action a(π) produces a Γ-action b(π).
We also have a′(π) coming from a(π) in step (3). Recall that wedefined
πRρ⇔ b(π) Œ b(ρ).
To complete the proof, we only had to show that R has countableindex over ∼=. We now prove this:
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Completion of the proof
Assume, towards a contradiction, that (πi)i∈I is an uncountablefamily of pairwise non-isomorphic representations in Irr(F2, H)such that if bi = b(πi), then (bi) are pairwise orbit equivalent.Recall the chain:
πi → aπi = ci → a(πi) = a(ci) = α0×ci → b(ci) = bi, a′(ci) = a′i.
By Ioana’s theorem, we can find uncountable J ⊆ I such that ifi, j ∈ J , there are a′i, a
′j-resp. invariant Borel sets Ai, Aj of
positive measure, so that a′i|Ai ∼= a′j |Aj . Moreover by property (iv)of the co-inducing construction, a(ci) is a factor of a′i|Ai andsimilarly for a(cj), a′j |Aj .
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Completion of the proof
Fix i0 ∈ J . Then for any j ∈ J , fix Ai0 , Aj of positive measurewhich are respectively a′(ci0), a′(cj)-invariant anda′(ci0)|Ai ∼= a′(cj)|Aj . We have
πj ≤ κaπj0 ( = κ
cj0 ) ≤ κα0×cj
0 ( = κa(cj)0 ) ≤ κ
a′j |Aj0
∼= κa′i0|Ai0
0 ≤ κa′i00 .
Thus (πj) is (up to isomorphism) an uncountable family of
pairwise non-isomorphic irreducible subrepresentations of κa′i00 , a
contradiction.
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Review
We have finished the proof of the main theorem modulo Ioana’stheorem and defining and proving the properties of the co-inducedconstruction. We will discuss first the proof of Ioana’s theorem,assuming the properties of the co-induced action, and deal withthat at the end.
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Ioana’s Theorem
Recall that α0 is the canonical action of F2 on T2.
Below, for each c ∈ A(F2, X, µ), with a(c) = α0 × c ∈ A(F2, Y, ν),where Y = T2 ×X, ν = λ× µ, we let b(c) ∈ A(Γ, Z, ρ),a′(c) ∈ A(F2, Z, ρ) come from Epstein’s co-inducing construction.
Theorem (Ioana)
If (ci)i∈I is an uncountable family of actions in A(F2, X, µ) and(b(ci))i∈I are mutually orbit equivalent, then there is uncountableJ ⊆ I such that if i, j ∈ J , we can find Borel sets Ai, Aj ⊆ Z ofpositive measure which are respectively a′(ci), a′(cj)-invariant anda′(ci)|Ai ∼= a′(cj)|Aj with respect to the normalized measures ρAi ,ρAj .
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Ioana’s Theorem
Recall that α0 is the canonical action of F2 on T2.
Below, for each c ∈ A(F2, X, µ), with a(c) = α0 × c ∈ A(F2, Y, ν),where Y = T2 ×X, ν = λ× µ, we let b(c) ∈ A(Γ, Z, ρ),a′(c) ∈ A(F2, Z, ρ) come from Epstein’s co-inducing construction.
Theorem (Ioana)
If (ci)i∈I is an uncountable family of actions in A(F2, X, µ) and(b(ci))i∈I are mutually orbit equivalent, then there is uncountableJ ⊆ I such that if i, j ∈ J , we can find Borel sets Ai, Aj ⊆ Z ofpositive measure which are respectively a′(ci), a′(cj)-invariant anda′(ci)|Ai ∼= a′(cj)|Aj with respect to the normalized measures ρAi ,ρAj .
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Ioana’s Theorem
Recall that α0 is the canonical action of F2 on T2.
Below, for each c ∈ A(F2, X, µ), with a(c) = α0 × c ∈ A(F2, Y, ν),where Y = T2 ×X, ν = λ× µ, we let b(c) ∈ A(Γ, Z, ρ),a′(c) ∈ A(F2, Z, ρ) come from Epstein’s co-inducing construction.
Theorem (Ioana)
If (ci)i∈I is an uncountable family of actions in A(F2, X, µ) and(b(ci))i∈I are mutually orbit equivalent, then there is uncountableJ ⊆ I such that if i, j ∈ J , we can find Borel sets Ai, Aj ⊆ Z ofpositive measure which are respectively a′(ci), a′(cj)-invariant anda′(ci)|Ai ∼= a′(cj)|Aj with respect to the normalized measures ρAi ,ρAj .
The complexity of classification problems in ergodic theory
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Ioana’s Theorem
We now outline the proof of this theorem.
We have the following situation, using the properties of theco-induced action, and some simple manipulations:
Let bi = b(ci). Then (bi)i∈I is an uncountable family of free,ergodic actions in A(Γ, Z, ρ), a′i = a′(ci) are free in A(F2, Z, ρ),there is E such that Ebi = E for each i ∈ I, Ea′i ⊆ E, and α0 is a
factor of a′i via a map pi : Z → T2 such that
(∗) for γ ∈ Γ \ 1, i ∈ I, pi(γbi · z) 6= pi(z), ρ-a.e.
Z
pi
a′i_
T2 α0
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Ioana’s Theorem
We now outline the proof of this theorem.
We have the following situation, using the properties of theco-induced action, and some simple manipulations:
Let bi = b(ci). Then (bi)i∈I is an uncountable family of free,ergodic actions in A(Γ, Z, ρ), a′i = a′(ci) are free in A(F2, Z, ρ),there is E such that Ebi = E for each i ∈ I, Ea′i ⊆ E, and α0 is a
factor of a′i via a map pi : Z → T2 such that
(∗) for γ ∈ Γ \ 1, i ∈ I, pi(γbi · z) 6= pi(z), ρ-a.e.
Z
pi
a′i_
T2 α0
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Ioana’s Theorem
We now outline the proof of this theorem.
We have the following situation, using the properties of theco-induced action, and some simple manipulations:
Let bi = b(ci). Then (bi)i∈I is an uncountable family of free,ergodic actions in A(Γ, Z, ρ), a′i = a′(ci) are free in A(F2, Z, ρ),there is E such that Ebi = E for each i ∈ I, Ea′i ⊆ E, and α0 is a
factor of a′i via a map pi : Z → T2 such that
(∗) for γ ∈ Γ \ 1, i ∈ I, pi(γbi · z) 6= pi(z), ρ-a.e.
Z
pi
a′i_
T2 α0
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Ioana’s Theorem
The action of SL2(Z) on Z2 by matrix multiplication gives asemidirect product SL2(Z) n Z2, and similarly F2 n Z2 (as we viewF2 as a subgroup of SL2(Z)). The key point is that (F2 n Z2,Z2)has the so-called relative property (T):
∃ finite Q ⊆ F2 n Z2, ε > 0, such that for any unitaryrepresentation π ∈ Rep(F2 n Z2, H), if v is a (Q, ε)-invariant unitvector (i.e., ||π(q)(v)− v|| < ε, ∀q ∈ Q), then there is aZ2-invariant vector w with ||v − w|| < 1
We will next define a family of representations of this group on anappropriate Hilbert space.
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Ioana’s Theorem
The action of SL2(Z) on Z2 by matrix multiplication gives asemidirect product SL2(Z) n Z2, and similarly F2 n Z2 (as we viewF2 as a subgroup of SL2(Z)). The key point is that (F2 n Z2,Z2)has the so-called relative property (T):
∃ finite Q ⊆ F2 n Z2, ε > 0, such that for any unitaryrepresentation π ∈ Rep(F2 n Z2, H), if v is a (Q, ε)-invariant unitvector (i.e., ||π(q)(v)− v|| < ε, ∀q ∈ Q), then there is aZ2-invariant vector w with ||v − w|| < 1
We will next define a family of representations of this group on anappropriate Hilbert space.
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Ioana’s Theorem
Consider the σ-finite measure space (E,P ) where for each Borelset A ⊆ E, P (A) =
∫|Az|dρ(z).
Given i, j ∈ I, we will define a representation
πi,j ∈ Rep(F2 n Z2, L2(E,P )). Identify Z2 with T2 = the group ofcharacters of T2 so that m = (m1,m2) ∈ Z2 is identified withχm(z1, z2) = zm1
1 zm22 . Via this identification, the action of F2 on
Z2 by matrix multiplication is identified with the shift action of F2
on T2: δ · χ(t) = χ(δ−1 · t), δ ∈ F2, χ ∈ T2, t ∈ T2. Then the
semidirect product F2 n Z2 is identified with F2 n T2 andmultiplication is given by
(δ1, χ1)(δ2, χ2) = (δ1δ2, χ1(δ1 · χ2)).
If χ ∈ T2, let ηiχ = χ pi : Z → T. Then define πi,j as follows
πi,j(δ, χ)(f)(x, y) = ηiχ(x)ηjχ(y)f((δ−1)a′i · x, (δ−1)a
′j · y)
for δ ∈ F2, χ ∈ T2.The complexity of classification problems in ergodic theory
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Ioana’s Theorem
Consider the σ-finite measure space (E,P ) where for each Borelset A ⊆ E, P (A) =
∫|Az|dρ(z).
Given i, j ∈ I, we will define a representation
πi,j ∈ Rep(F2 n Z2, L2(E,P )). Identify Z2 with T2 = the group ofcharacters of T2 so that m = (m1,m2) ∈ Z2 is identified withχm(z1, z2) = zm1
1 zm22 . Via this identification, the action of F2 on
Z2 by matrix multiplication is identified with the shift action of F2
on T2: δ · χ(t) = χ(δ−1 · t), δ ∈ F2, χ ∈ T2, t ∈ T2. Then the
semidirect product F2 n Z2 is identified with F2 n T2 andmultiplication is given by
(δ1, χ1)(δ2, χ2) = (δ1δ2, χ1(δ1 · χ2)).
If χ ∈ T2, let ηiχ = χ pi : Z → T. Then define πi,j as follows
πi,j(δ, χ)(f)(x, y) = ηiχ(x)ηjχ(y)f((δ−1)a′i · x, (δ−1)a
′j · y)
for δ ∈ F2, χ ∈ T2.The complexity of classification problems in ergodic theory
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Ioana’s Theorem
Then if v = 1∆, where ∆ = (z, z) : z ∈ Z, using theseparability of L2(Z, ρ) and L2(E,P ) one can find J ⊆ Iuncountable such that if i, j ∈ J , then
||πi,j(q)(v)− v||L2(E,P ) < ε,
∀q ∈ Q.
So by relative property (T), there is f ∈ L2(E,P ) with||f − 1∆|| < 1 and f is Z2-invariant (for πi,j), i.e.,
f(x, y) = ηiχ(x)ηjχ(y)f(x, y), ∀χ ∈ T2.
Since f 6= 0 (as ||1∆|| = 1 and ||f − 1∆|| < 1) the set
S = (x, y) ∈ E : ηiχ(x) = ηjχ(y), ∀χ ∈ T2
has positive P -measure.
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Ioana’s Theorem
Then if v = 1∆, where ∆ = (z, z) : z ∈ Z, using theseparability of L2(Z, ρ) and L2(E,P ) one can find J ⊆ Iuncountable such that if i, j ∈ J , then
||πi,j(q)(v)− v||L2(E,P ) < ε,
∀q ∈ Q.
So by relative property (T), there is f ∈ L2(E,P ) with||f − 1∆|| < 1 and f is Z2-invariant (for πi,j), i.e.,
f(x, y) = ηiχ(x)ηjχ(y)f(x, y), ∀χ ∈ T2.
Since f 6= 0 (as ||1∆|| = 1 and ||f − 1∆|| < 1) the set
S = (x, y) ∈ E : ηiχ(x) = ηjχ(y), ∀χ ∈ T2
has positive P -measure.
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Ioana’s Theorem
Using the fact that E is generated by bj , and the fact thatcharacters separate points, it follows that for almost all x ∈ X,there is at most one y such that (x, y) ∈ S. Indeed, fix x such that
(x, y1) ∈ S, (x, y2) ∈ S with y1 6= y2. Then y1 = γbj1 · x,
y2 = γbj2 · x, γ1 6= γ2. But also
∀χ(ηiχ(x) = ηjχ(γbj1 · x) = ηjχ(γbj2 · x)
).
But T2 separates points, so
pj(γbj1 · x) = pj(γ
bj2 · x).
Let γ1γ−12 = γ 6= 1. Then
pj(γbj · γbj2 · x) = pj(γ
bj2 · x)
which happens only on a null set by (∗) for pj .
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Ioana’s Theorem
Let
Ai = x : ∃ unique y (x, y) ∈ S, Aj = y : ∃x ∈ Ai (x, y) ∈ S.
Then Ai and Aj are respectively a′i-invariant and a′j-invariant setsof positive measure. Let ϕ : Ai → Aj be defined byϕ(x) = y ⇔ (x, y) ∈ S. Then ϕ shows that a′i|Ai ∼= a′j |Aj .
This completes the proof of Ioana’s theorem.
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Ioana’s Theorem
Let
Ai = x : ∃ unique y (x, y) ∈ S, Aj = y : ∃x ∈ Ai (x, y) ∈ S.
Then Ai and Aj are respectively a′i-invariant and a′j-invariant setsof positive measure. Let ϕ : Ai → Aj be defined byϕ(x) = y ⇔ (x, y) ∈ S. Then ϕ shows that a′i|Ai ∼= a′j |Aj .
This completes the proof of Ioana’s theorem.
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Ioana’s Theorem
Let
Ai = x : ∃ unique y (x, y) ∈ S, Aj = y : ∃x ∈ Ai (x, y) ∈ S.
Then Ai and Aj are respectively a′i-invariant and a′j-invariant setsof positive measure. Let ϕ : Ai → Aj be defined byϕ(x) = y ⇔ (x, y) ∈ S. Then ϕ shows that a′i|Ai ∼= a′j |Aj .
This completes the proof of Ioana’s theorem.
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Co-induced actions
Theorem
Let Γ be non-amenable. Given a ∈ A(F2, Y, ν) we can construct in a Borel wayb ∈ A(Γ, Z, ρ) and a′ ∈ A(F2, Z, ρ) (Z, ρ independent of a) with the followingproperties:
(i) Ea′ ⊆ Eb,(ii) b is free and ergodic (in fact mixing),
(iii) a′ is free,
(iv) a is a factor of a′ via a map f : Z → Y (f independent of a), i.e.,
f(δa′ · z) = δa · f(z) (δ ∈ F2), and f∗ρ = ν, and in fact if a is ergodic, then for
every a′-invariant Borel set A ⊆ Z of positive measure, if ρA =ρ|Aρ(A)
, then a is
a factor of (a′|A, ρA) via f .
(v) If a free action a ∈ A(F2, Y , ν) is a factor of the action a via g : Y → Y
Yg // Y
Z
f
OO
then for γ ∈ Γ \ 1, gf(γb · z) 6= gf(z), ρ-a.e.
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Co-induced actions
We have two countable groups ∆ and Γ (in our case ∆ will be F2)and are given a free, ergodic, measure-preserving action a0 of ∆ on(Ω, ω) and a free, measure-preserving action b0 of Γ on (Ω, ω) withEa0 ⊆ Eb0 (note that b0 is also ergodic). LetN = [Eb0 : Ea0 ] = (the number of Ea0-classes in eachEb0-class) ∈ 1, 2, 3, . . . ,ℵ0. Work below with N = ℵ0 the othercases being similar.
Given these data we will describe Epstein’s co-inducingconstruction that, given any a ∈ A(∆, Y, ν), will produceb ∈ A(Γ, Z, ρ), where Z = Ω× Y N, ρ = ω × νN, called theco-induced action of a, modulo (a0, b0),
b = CInd(a0, b0)Γ∆(a),
which will satisfy the previous theorem.
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Co-induced actions
We have two countable groups ∆ and Γ (in our case ∆ will be F2)and are given a free, ergodic, measure-preserving action a0 of ∆ on(Ω, ω) and a free, measure-preserving action b0 of Γ on (Ω, ω) withEa0 ⊆ Eb0 (note that b0 is also ergodic). LetN = [Eb0 : Ea0 ] = (the number of Ea0-classes in eachEb0-class) ∈ 1, 2, 3, . . . ,ℵ0. Work below with N = ℵ0 the othercases being similar.
Given these data we will describe Epstein’s co-inducingconstruction that, given any a ∈ A(∆, Y, ν), will produceb ∈ A(Γ, Z, ρ), where Z = Ω× Y N, ρ = ω × νN, called theco-induced action of a, modulo (a0, b0),
b = CInd(a0, b0)Γ∆(a),
which will satisfy the previous theorem.
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Co-induced actions
Put E = Ea0 , F = Eb0 . We can then find a sequence(Cn) ∈ Aut(Ω, ω)N of choice functions, i.e., C0 = id and Cn(w)is a transversal for the E-classes contained in [w]F .
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Co-induced actions
[w]F
Cn(w)
w
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To prove this, define first a sequence (Dn) of Borel choicefunctions as follows: define the equivalence relation on Γγ ∼w δ ⇔ (γb0 · w)E(δb0 · w). Let γn,w be a transversal for ∼wwith γ0,w = 1 and put Dn(w) = γb0n,w · w.
We can then use the ergodicity of E to modify (Dn) to a sequence(Cn) of 1-1 choice functions (which are then in Aut(Ω, ω)) asfollows: Fix n ∈ N and consider Dn. As it is countable-to-1, letΩ =
⊔∞k=1 Yk be a Borel partition such that Dn|Yk is 1-1. Let then
Zk = Dn(Yk), so that µ(Zk) = µ(Yk). Since E is ergodic, there isTk ∈ Aut(Ω, ω) with Tk(w)Ew, a.e., such that Tk(Zk) = Yk. Letthen Cn(w) = Tk(Dn(w)), if w ∈ Yk. We have Cn(w)EDn(w)and Cn is 1-1. So Cn are choice functions and each Cn is 1-1.
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Co-induced actions
Using the Cn we can define the index cocycle
ϕE,F = ϕ : F → S∞ = the symmetric group of N
given by
ϕ(w1, w2)(k) = n⇔ [Ck(w1)]E = [Cn(w2)]E
(cocycle means: ϕ(w2, w3)ϕ(w1, w2) = ϕ(w1, w3) wheneverw1Fw2Fw3). Define also for each (w1, w2) ∈ F , ~δ(w1, w2) ∈ ∆N
by(~δ(w1, w2)n)a0 · Cϕ(w1,w2)−1(n)(w1) = Cn(w2).
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Co-induced actions
[w]F
w2
n δ
k
w1
φ(w1,w2)(k)=n
~δ(w1,w2)n=δ
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Co-induced actions
Now S∞ acts on the product group ∆N by shift:(σ · ~δ)n = ~δσ−1(n), where ~δ = (~δn) ∈ ∆N. So we can form the
semi-direct product S∞ n ∆N.Given a ∈ A(∆, Y, ν), we then have a measure-preserving action ofS∞ n ∆N on (Y N, νN) by
((σ,~δ) · ~y)n = ( ~δn)a · ~yσ−1(n).
Finally we have a cocycle for the action b0, ψ : Γ× Ω→ S∞ n ∆N
given by
(∗) ψ(γ,w) =(ϕ(w, γb0 · w), ~δ(w, γb0 · w)
)(cocycle means: ψ(γ1γ2, w) = ψ(γ1, γ2 · w)ψ(γ2, w)). Finally let
b = CInd(a0, b0)Γ∆(a)
be the skew product b = b0 nψ (Y N, µN), i.e., for γ ∈ Γ
γb · (w, ~y) = (γb0 · w,ψ(γ,w) · ~y)
=(γb0 · w, (n 7→ (~δ(w, γb0 · w)n)a · ~yϕ(w,γb0 ·w)−1(n))
).
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Co-induced actions
We also let a′ = a0 nψ′ (Y N, µN), where ψ′ is the cocycle for theaction a0 given by replacing b0 by a0 in (∗). Thus for δ ∈ ∆
δa′ · (w, ~y) =
(δa0 · w, (n 7→ (~δ(w, δa0 · w)n)a · ~yϕ(w,δa0 ·w)−1(n)
).
One can now verify without great difficulty the properties (i) to (v)of the co-induced construction stated earlier, except for theergodicity in part (ii). Moreover in the case we are interested in(∆ = F2 and Γ non-amenable), we need to find the seed actionsa0, b0.
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Co-induced actions
Theorem
Let Γ be non-amenable. Given a ∈ A(F2, Y, ν) we can construct in a Borel wayb ∈ A(Γ, Z, ρ) and a′ ∈ A(F2, Z, ρ) (Z, ρ independent of a) with the followingproperties:
(i) Ea′ ⊆ Eb,(ii) b is free and ergodic (in fact mixing),
(iii) a′ is free,
(iv) a is a factor of a′ via a map f : Z → Y (f independent of a), i.e.,
f(δa′ · z) = δa · f(z) (δ ∈ F2), and f∗ρ = ν, and in fact if a is ergodic, then for
every a′-invariant Borel set A ⊆ Z of positive measure, if ρA =ρ|Aρ(A)
, then a is
a factor of (a′|A, ρA) via f .
(v) If a free action a ∈ A(F2, Y , ν) is a factor of the action a via g : Y → Y
Yg // Y
Z
f
OO
then for γ ∈ Γ \ 1, gf(γb · z) 6= gf(z), ρ-a.e.
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Co-induced actions
We verify the properties of a′, b:
(i) Ea′ ⊆ Eb: trivial as Ea0 ⊆ Eb0 .
(ii) b is free: trivial as b0 is free.
(iii) a′ is free: trivial as a0 is free.
(iv) Let f : Z = Ω× Y N → Y be given by f(w, ~y) = ~y0. Then ais a factor of a′ via f (this follows from C0(w) = w).Next we show that if a is ergodic and A ⊆ Ω× Y N haspositive measure and is a′-invariant, then f∗ρA = ν. LetB ⊆ Y , ν(B) = 1 be a-invariant and such that ν|B is theunique a-invariant probability measure on B. Thenρ(f−1(B)) = 1, so f∗ρA lives on B and then f∗ρA = ν.
The complexity of classification problems in ergodic theory
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Co-induced actions
(v) If a ∈ A(∆, Y , ν) is a free action which is a factor of a viag : Y → Y , then for γ ∈ Γ \ 1, gf(γb · z) 6= gf(z), ρ-a.e:Fix γ ∈ Γ \ 1. We need to show that
(∗∗) g((~δ(w, γb0 · w)0)a · ~yϕ(w,γb0 ·w)−1(0)
)6= g(~y0)
for almost all w, ~y. Assume not, i.e. for positively many w, ~y(∗∗) fails. We can also assume that ϕ(w, γb0 · w)−1(0) = k isfixed for the w, ~y and ~δ(w, γb · w)0 = δ is also fixed. Thusg(δa · ~yk) = δa · g(~yk) = g(~y0) on a set of positive measure ofw, ~y. If k 6= 0 this is false using Fubini. If k = 0, then δ = 1by the freeness of a, so w = γb0 · w for a positive set of w,contradicting the freeness of b0.
The complexity of classification problems in ergodic theory
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Small subequivalence relations
In general it is not clear that the second part of (ii), i.e., “b isergodic” is true. However if E = Ea0 is “small” in F = Eb0 in thesense to be described below, then this will be the case and in fact bwill be mixing.
For γ ∈ Γ, let
|γ|E = ω(w : (w, γb0 · w) ∈ E) ∈ [0, 1].
We say that E = Ea0 is small in F = Eb0 , if |γ|E → 0 as γ →∞.
Theorem (Ioana-K-Tsankov)
In the above notation and assuming also that b0 is mixing, if E issmall in F , then for any a ∈ A(∆, Y, ν), b = CInd(a0, b0)Γ
∆(a) ismixing.
The complexity of classification problems in ergodic theory
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Small subequivalence relations
In general it is not clear that the second part of (ii), i.e., “b isergodic” is true. However if E = Ea0 is “small” in F = Eb0 in thesense to be described below, then this will be the case and in fact bwill be mixing.
For γ ∈ Γ, let
|γ|E = ω(w : (w, γb0 · w) ∈ E) ∈ [0, 1].
We say that E = Ea0 is small in F = Eb0 , if |γ|E → 0 as γ →∞.
Theorem (Ioana-K-Tsankov)
In the above notation and assuming also that b0 is mixing, if E issmall in F , then for any a ∈ A(∆, Y, ν), b = CInd(a0, b0)Γ
∆(a) ismixing.
The complexity of classification problems in ergodic theory
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Small subequivalence relations
In general it is not clear that the second part of (ii), i.e., “b isergodic” is true. However if E = Ea0 is “small” in F = Eb0 in thesense to be described below, then this will be the case and in fact bwill be mixing.
For γ ∈ Γ, let
|γ|E = ω(w : (w, γb0 · w) ∈ E) ∈ [0, 1].
We say that E = Ea0 is small in F = Eb0 , if |γ|E → 0 as γ →∞.
Theorem (Ioana-K-Tsankov)
In the above notation and assuming also that b0 is mixing, if E issmall in F , then for any a ∈ A(∆, Y, ν), b = CInd(a0, b0)Γ
∆(a) ismixing.
The complexity of classification problems in ergodic theory
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A measure theoretic version of the von NeumannConjecture
Thus to complete the proof of the properties of the co-inducedconstruction, we will need to show that for ∆ = F2, Γnon-amenable, there are free, ergodic a0 ∈ A(∆,Ω, ω), andb0 ∈ A(Γ,Ω, ω) free, mixing with Ea0 ⊆ Eb0 and Ea0 small in Eb0 .
The complexity of classification problems in ergodic theory
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A measure theoretic version of the von NeumannConjecture
This is based on a construction of Gaboriau-Lyons, using ideasfrom probability theory as well as the theory of costs, who provedthat there are such a0, b0 without considering the smallnesscondition, which was later established by Ioana-K-Tsankov. TheGaboriau-Lyons result provided an affirmative answer to a measuretheoretic version of von Neumann’s Conjecture.
To make sure now that Ea0 is small in Eb0 one can either chooseappropriately some parameters in the Gaboriau-Lyons constructionor else one starts with any a0, b0 as above and co-induces by(a0, b0) an appropriate Bernoulli percolation a of F2 to get b0 andthen shows that one can find a small subequivalence relationEa0 ⊆ Eb0 generated by a free, ergodic action a0 of F2.
The complexity of classification problems in ergodic theory
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A measure theoretic version of the von NeumannConjecture
This is based on a construction of Gaboriau-Lyons, using ideasfrom probability theory as well as the theory of costs, who provedthat there are such a0, b0 without considering the smallnesscondition, which was later established by Ioana-K-Tsankov. TheGaboriau-Lyons result provided an affirmative answer to a measuretheoretic version of von Neumann’s Conjecture.
To make sure now that Ea0 is small in Eb0 one can either chooseappropriately some parameters in the Gaboriau-Lyons constructionor else one starts with any a0, b0 as above and co-induces by(a0, b0) an appropriate Bernoulli percolation a of F2 to get b0 andthen shows that one can find a small subequivalence relationEa0 ⊆ Eb0 generated by a free, ergodic action a0 of F2.
The complexity of classification problems in ergodic theory
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A final comment
We finally note the following result:
Theorem (Ioana-K-Tsankov)
Let Γ be an arbitrary infinite countable group and let b0 be a free,measure preserving, mixing action of Γ on (Ω, ω). Then there is afree, measure preserving, mixing action a0 of Z on the same spacesuch that E = Ea0 ⊆ F = Eb0 and E is small in F .
This gives a method of producing, starting with arbitrary measurepreserving actions of Z, apparently new types of measurepreserving, mixing actions of any infinite countable group Γ.
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Reference
(A.K. and R. Tucker-Drob) The complexity of classificationproblems in ergodic theory, preprint.
http://www.math.caltech.edu/people/kechris.html
The complexity of classification problems in ergodic theory