Descriptive combinatorics and ergodic theorems Anush Tserunyan - - PowerPoint PPT Presentation

Descriptive combinatorics and ergodic theorems

Anush Tserunyan

University of Illinois at Urbana-Champaign

“Gee Professor Kechris, descriptive set theory sure is powerful, and beautiful too!”

Descriptive set theory and equivalence relations

◮ Descriptive set theory studies definable sets/functions in Polish spaces.

Descriptive set theory and equivalence relations

◮ Descriptive set theory studies definable sets/functions in Polish spaces. ◮ In the last 30 years: definable (Borel, analytic) equivalence relations.

— View an equivalence relation E on a Polish space X as E ⊆ X 2.

Descriptive set theory and equivalence relations

◮ Descriptive set theory studies definable sets/functions in Polish spaces. ◮ In the last 30 years: definable (Borel, analytic) equivalence relations.

— View an equivalence relation E on a Polish space X as E ⊆ X 2.

◮ Such equivalence relations arise naturally all over mathematics:

Descriptive set theory and equivalence relations

◮ Descriptive set theory studies definable sets/functions in Polish spaces. ◮ In the last 30 years: definable (Borel, analytic) equivalence relations.

— View an equivalence relation E on a Polish space X as E ⊆ X 2.

◮ Such equivalence relations arise naturally all over mathematics:

Orbit equivalence relations of Borel actions of Polish groups on Polish spaces are analytic.

Descriptive set theory and equivalence relations

◮ Descriptive set theory studies definable sets/functions in Polish spaces. ◮ In the last 30 years: definable (Borel, analytic) equivalence relations.

— View an equivalence relation E on a Polish space X as E ⊆ X 2.

◮ Such equivalence relations arise naturally all over mathematics:

Orbit equivalence relations of Borel actions of Polish groups on Polish spaces are analytic. Many mathematical objects (such as Riemann surfaces, Banach spaces, measure-preserving transformations, etc.) can be encoded as points in Polish spaces.

Descriptive set theory and equivalence relations

◮ Descriptive set theory studies definable sets/functions in Polish spaces. ◮ In the last 30 years: definable (Borel, analytic) equivalence relations.

— View an equivalence relation E on a Polish space X as E ⊆ X 2.

◮ Such equivalence relations arise naturally all over mathematics:

Orbit equivalence relations of Borel actions of Polish groups on Polish spaces are analytic. Many mathematical objects (such as Riemann surfaces, Banach spaces, measure-preserving transformations, etc.) can be encoded as points in Polish spaces. Thinking of classifying these points up to some notion of equivalence (e.g., conformal equivalence, isomorphism, conjugacy), we define what it means for one such classification problem to be no harder than another: the Borel reducibility relation B between two equivalence relations.

Descriptive set theory and equivalence relations

◮ Descriptive set theory studies definable sets/functions in Polish spaces. ◮ In the last 30 years: definable (Borel, analytic) equivalence relations.

— View an equivalence relation E on a Polish space X as E ⊆ X 2.

◮ Such equivalence relations arise naturally all over mathematics:

Orbit equivalence relations of Borel actions of Polish groups on Polish spaces are analytic. Many mathematical objects (such as Riemann surfaces, Banach spaces, measure-preserving transformations, etc.) can be encoded as points in Polish spaces. Thinking of classifying these points up to some notion of equivalence (e.g., conformal equivalence, isomorphism, conjugacy), we define what it means for one such classification problem to be no harder than another: the Borel reducibility relation B between two equivalence relations.

◮ We will focus on countable Borel equivalence relations (CBERs).

— Here, “countable” means each equivalence class is countable.

Countable Borel equivalence relations via group actions

◮ These come from countable group actions: orbit equivalence relations

induced by Borel actions of countable groups are countable Borel.

Countable Borel equivalence relations via group actions

◮ These come from countable group actions: orbit equivalence relations

induced by Borel actions of countable groups are countable Borel.

◮ Conversely, the Feldman–Moore theorem states that these are all of them!

Countable Borel equivalence relations via group actions

◮ These come from countable group actions: orbit equivalence relations

induced by Borel actions of countable groups are countable Borel.

◮ Conversely, the Feldman–Moore theorem states that these are all of them! ◮ Thus, taking {γn} =. . Γ X with E = EΓ, each x ∈ X can refer to other

points in its E-class by names: γ0x, γ1x, γ2x . . .

Countable Borel equivalence relations via group actions

◮ These come from countable group actions: orbit equivalence relations

induced by Borel actions of countable groups are countable Borel.

◮ Conversely, the Feldman–Moore theorem states that these are all of them! ◮ Thus, taking {γn} =. . Γ X with E = EΓ, each x ∈ X can refer to other

points in its E-class by names: γ0x, γ1x, γ2x . . .

Countable Borel equivalence relations via group actions

◮ These come from countable group actions: orbit equivalence relations

induced by Borel actions of countable groups are countable Borel.

◮ Conversely, the Feldman–Moore theorem states that these are all of them! ◮ Thus, taking {γn} =. . Γ X with E = EΓ, each x ∈ X can refer to other

points in its E-class by names: γ0x, γ1x, γ2x . . .

◮ The abundance of Borel actions of countable groups makes the class of

CBERs extremely rich and B on it very complicated:

Adams–Kechris (using Zimmer’s cocycle superrigidity): N ⊇ A → EA such that for all A, B ⊆ N, A ⊆ B ⇔ EA B EB.

Countable Borel equivalence relations via graphs

◮ CBERs also come from graphs as the connectedness relations EG of

locally countable Borel graphs G.

— By a graph G on a space X we just mean a symmetric irreflexive G ⊆ X 2.

Countable Borel equivalence relations via graphs

◮ CBERs also come from graphs as the connectedness relations EG of

locally countable Borel graphs G.

— By a graph G on a space X we just mean a symmetric irreflexive G ⊆ X 2.

Countable Borel equivalence relations via graphs

◮ CBERs also come from graphs as the connectedness relations EG of

locally countable Borel graphs G.

— By a graph G on a space X we just mean a symmetric irreflexive G ⊆ X 2.

◮ Every CBER E on X comes from such a graph:

Countable Borel equivalence relations via graphs

◮ CBERs also come from graphs as the connectedness relations EG of

locally countable Borel graphs G.

— By a graph G on a space X we just mean a symmetric irreflexive G ⊆ X 2.

◮ Every CBER E on X comes from such a graph:

Take G .

.= E \ {(x, x) : x ∈ X}, the complete graph for E.

Countable Borel equivalence relations via graphs

◮ CBERs also come from graphs as the connectedness relations EG of

locally countable Borel graphs G.

— By a graph G on a space X we just mean a symmetric irreflexive G ⊆ X 2.

◮ Every CBER E on X comes from such a graph:

Take G .

.= E \ {(x, x) : x ∈ X}, the complete graph for E.

Even better: let Γ α X such that E = EΓ, take a symmetric generating set Γ = S, and define the Cayley–Schreier graph: xGSy .

.⇔ σ ·α x = y for some σ ∈ S.

Countable Borel equivalence relations via graphs

◮ CBERs also come from graphs as the connectedness relations EG of

locally countable Borel graphs G.

— By a graph G on a space X we just mean a symmetric irreflexive G ⊆ X 2.

◮ Every CBER E on X comes from such a graph:

Take G .

.= E \ {(x, x) : x ∈ X}, the complete graph for E.

Even better: let Γ α X such that E = EΓ, take a symmetric generating set Γ = S, and define the Cayley–Schreier graph: xGSy .

.⇔ σ ·α x = y for some σ ∈ S.

Then E = EGS.

Interplay with other subjects

◮ Any countable Borel group action Γ X can be turned into a

continuous one by replacing the Polish topology on X.

— Enables topological tools such as Baire category.

Interplay with other subjects

◮ Any countable Borel group action Γ X can be turned into a

continuous one by replacing the Polish topology on X.

— Enables topological tools such as Baire category.

◮ Borel group actions and graphs naturally occur on measure spaces (X, µ).

— Enables measure-theoretic tools such as the Borel–Cantelli lemma and much much more.

Interplay with other subjects

◮ Any countable Borel group action Γ X can be turned into a

continuous one by replacing the Polish topology on X.

— Enables topological tools such as Baire category.

◮ Borel group actions and graphs naturally occur on measure spaces (X, µ).

— Enables measure-theoretic tools such as the Borel–Cantelli lemma and much much more.

◮ All these together has created extremely active two-way traffic between

the study of CBERs and

ergodic theory measured group theory graph combinatorics geometric group theory percolation theory probabilistic combinatorics topological dynamics von Neumann algebras

Interplay with other subjects

◮ Any countable Borel group action Γ X can be turned into a

continuous one by replacing the Polish topology on X.

— Enables topological tools such as Baire category.

◮ Borel group actions and graphs naturally occur on measure spaces (X, µ).

— Enables measure-theoretic tools such as the Borel–Cantelli lemma and much much more.

◮ All these together has created extremely active two-way traffic between

the study of CBERs and

ergodic theory (ergodic theorems, mixing, entropy, . . . ) measured group theory (measure equivalence, cost, ℓ2-(co)homology, . . . ) graph combinatorics (colorings, matchings, flows, . . . ) geometric group theory (tree constructions, growth,. . . ) percolation theory (graph/action constructions, amenability barriers, . . . ) probabilistic combinatorics (Lov´ asz Local Lemma, concentration bounds, . . . ) topological dynamics (subshifts, topological entropy, compact actions, . . . ) von Neumann algebras (spectral gap, superrigidity, vN-dimension, . . . )

Our secret weapon

But what’s our contribution? What’s the difference between descriptive set theoretic and analytic thinking?

Our secret weapon

But what’s our contribution? What’s the difference between descriptive set theoretic and analytic thinking?

◮ We think pointwise, analyzing the local combinatorics at a point, whereas

analysts analyze the space through the prism of functions on it.

Our secret weapon

But what’s our contribution? What’s the difference between descriptive set theoretic and analytic thinking?

◮ We think pointwise, analyzing the local combinatorics at a point, whereas

analysts analyze the space through the prism of functions on it.

◮ What allows us to do this is the Luzin–Novikov uniformization theorem!

Our secret weapon

But what’s our contribution? What’s the difference between descriptive set theoretic and analytic thinking?

◮ We think pointwise, analyzing the local combinatorics at a point, whereas

analysts analyze the space through the prism of functions on it.

◮ What allows us to do this is the Luzin–Novikov uniformization theorem!

— Every Borel set B ⊆ X × Y with countable X-fibers is a countable union

• f Borel partial functions X ⇀ Y .

Our secret weapon

But what’s our contribution? What’s the difference between descriptive set theoretic and analytic thinking?

◮ We think pointwise, analyzing the local combinatorics at a point, whereas

analysts analyze the space through the prism of functions on it.

◮ What allows us to do this is the Luzin–Novikov uniformization theorem!

— Every Borel set B ⊆ X × Y with countable X-fibers is a countable union

• f Borel partial functions X ⇀ Y .

— Allows each x ∈ X to quantify over its (countable) equivalence class

Our secret weapon

But what’s our contribution? What’s the difference between descriptive set theoretic and analytic thinking?

◮ We think pointwise, analyzing the local combinatorics at a point, whereas

analysts analyze the space through the prism of functions on it.

◮ What allows us to do this is the Luzin–Novikov uniformization theorem!

— Every Borel set B ⊆ X × Y with countable X-fibers is a countable union

• f Borel partial functions X ⇀ Y .

— Allows each x ∈ X to quantify over its (countable) equivalence class and give (Borel) names to the other guys in its class.

Our secret weapon

But what’s our contribution? What’s the difference between descriptive set theoretic and analytic thinking?

◮ We think pointwise, analyzing the local combinatorics at a point, whereas

analysts analyze the space through the prism of functions on it.

◮ What allows us to do this is the Luzin–Novikov uniformization theorem!

— Every Borel set B ⊆ X × Y with countable X-fibers is a countable union

• f Borel partial functions X ⇀ Y .

— Allows each x ∈ X to quantify over its (countable) equivalence class and give (Borel) names to the other guys in its class.

◮ We are not allowed to choose a point from each class! (Measure Theory 101)

Our secret weapon

But what’s our contribution? What’s the difference between descriptive set theoretic and analytic thinking?

◮ We think pointwise, analyzing the local combinatorics at a point, whereas

analysts analyze the space through the prism of functions on it.

◮ What allows us to do this is the Luzin–Novikov uniformization theorem!

— Every Borel set B ⊆ X × Y with countable X-fibers is a countable union

• f Borel partial functions X ⇀ Y .

— Allows each x ∈ X to quantify over its (countable) equivalence class and give (Borel) names to the other guys in its class.

◮ We are not allowed to choose a point from each class! (Measure Theory 101) ◮ Thus, our way of thinking is best described as originless combinatorics.

pmp actions of groups and ergodicity

As an example, today we’ll discuss ergodic theorems.

pmp actions of groups and ergodicity

As an example, today we’ll discuss ergodic theorems.

◮ (X, µ) — a standard probability space, e.g. [0, 1] with Lebesgue measure

pmp actions of groups and ergodicity

As an example, today we’ll discuss ergodic theorems.

◮ (X, µ) — a standard probability space, e.g. [0, 1] with Lebesgue measure ◮ Γ — a countable (discrete) group

pmp actions of groups and ergodicity

As an example, today we’ll discuss ergodic theorems.

◮ (X, µ) — a standard probability space, e.g. [0, 1] with Lebesgue measure ◮ Γ — a countable (discrete) group ◮ An action Γ (X, µ) is ergodic if every invariant measurable subset is

null or conull.

pmp actions of groups and ergodicity

As an example, today we’ll discuss ergodic theorems.

◮ (X, µ) — a standard probability space, e.g. [0, 1] with Lebesgue measure ◮ Γ — a countable (discrete) group ◮ An action Γ (X, µ) is ergodic if every invariant measurable subset is

null or conull.

◮ A probability measure preserving (pmp) action of Γ is a Borel action

Γ (X, µ) such that µ(γ · A) = µ(A), for each γ ∈ Γ and A ⊆ X.

pmp actions of groups and ergodicity

As an example, today we’ll discuss ergodic theorems.

◮ (X, µ) — a standard probability space, e.g. [0, 1] with Lebesgue measure ◮ Γ — a countable (discrete) group ◮ An action Γ (X, µ) is ergodic if every invariant measurable subset is

null or conull.

◮ A probability measure preserving (pmp) action of Γ is a Borel action

Γ (X, µ) such that µ(γ · A) = µ(A), for each γ ∈ Γ and A ⊆ X.

◮ Dating back to Birkhoff, pointwise ergodic theorems for pmp actions

Γ (X, µ) are bridges between the global condition of ergodicity and the a.e. local combinatorics of the action.

Pointwise ergodic theorems for pmp actions

Theorem (Pointwise ergodic, Birkhoff 1931)

A pmp action Z α (X, µ) is ergodic if and only if for each f ∈ L1(X, µ) and for a.e. x ∈ X, limit of local averages lim

n→∞ Af [Fn ·α x] =

• X

f dµ global average where Fn .

.= [0, n] ⊆ Z and Af [Fn ·α x] is the average of f over Fn ·α x.

Pointwise ergodic theorems for pmp actions

Theorem (Pointwise ergodic, Birkhoff 1931)

A pmp action Z α (X, µ) is ergodic if and only if for each f ∈ L1(X, µ) and for a.e. x ∈ X, limit of local averages lim

n→∞ Af [Fn ·α x] =

• X

f dµ global average where Fn .

.= [0, n] ⊆ Z and Af [Fn ·α x] is the average of f over Fn ·α x. ◮ This was generalized to amenable groups by Lindenstrauss (2001).

Pointwise ergodic theorems for pmp actions

Theorem (Pointwise ergodic, Birkhoff 1931)

A pmp action Z α (X, µ) is ergodic if and only if for each f ∈ L1(X, µ) and for a.e. x ∈ X, limit of local averages lim

n→∞ Af [Fn ·α x] =

• X

f dµ global average where Fn .

.= [0, n] ⊆ Z and Af [Fn ·α x] is the average of f over Fn ·α x. ◮ This was generalized to amenable groups by Lindenstrauss (2001). ◮ Analogous statements for free groups have been proven by Grigorchuk

(1987), Nevo (1994), Nevo–Stein (1994), Bufetov (2002), and Bowen–Nevo (2013).

Pointwise ergodic theorems for pmp actions

Theorem (Pointwise ergodic, Birkhoff 1931)

A pmp action Z α (X, µ) is ergodic if and only if for each f ∈ L1(X, µ) and for a.e. x ∈ X, limit of local averages lim

n→∞ Af [Fn ·α x] =

• X

f dµ global average where Fn .

.= [0, n] ⊆ Z and Af [Fn ·α x] is the average of f over Fn ·α x. ◮ This was generalized to amenable groups by Lindenstrauss (2001). ◮ Analogous statements for free groups have been proven by Grigorchuk

(1987), Nevo (1994), Nevo–Stein (1994), Bufetov (2002), and Bowen–Nevo (2013).

◮ Bowen–Nevo (2013) also have pointwise ergodic theorems for other

nonamenable groups, but for special kinds of actions.

Pointwise ergodic theorems for quasi-pmp actions

◮ An action Γ (X, µ) is quasi-pmp if every γ ∈ Γ is nonsingular, i.e.

maps nonnull sets to nonnull sets (possibly changing their measure).

Pointwise ergodic theorems for quasi-pmp actions

◮ An action Γ (X, µ) is quasi-pmp if every γ ∈ Γ is nonsingular, i.e.

maps nonnull sets to nonnull sets (possibly changing their measure).

◮ Why would one consider quasi-pmp actions?

Pointwise ergodic theorems for quasi-pmp actions

◮ An action Γ (X, µ) is quasi-pmp if every γ ∈ Γ is nonsingular, i.e.

maps nonnull sets to nonnull sets (possibly changing their measure).

◮ Why would one consider quasi-pmp actions?

Theorem (Ratio Pointwise Ergodic, Hopf 1937)

Let Z α (X, µ) be an ergodic pmp action. For each f , g ∈ L1(X, µ) with g > 0 and for a.e. x ∈ X, lim

n→∞

Af [Fn ·α x] Ag[Fn ·α x] =

• X f dµ
• X gdµ.

Pointwise ergodic theorems for quasi-pmp actions

◮ An action Γ (X, µ) is quasi-pmp if every γ ∈ Γ is nonsingular, i.e.

maps nonnull sets to nonnull sets (possibly changing their measure).

◮ Why would one consider quasi-pmp actions?

Theorem (Ratio Pointwise Ergodic, Hopf 1937)

Let Z α (X, µ) be an ergodic pmp action. For each f , g ∈ L1(X, µ) with g > 0 and for a.e. x ∈ X, lim

n→∞

Af [Fn ·α x] Ag[Fn ·α x] =

• X f dµ
• X gdµ.

◮ Replacing the measure µ with dµg . .= gdµ makes the action Γ (X, µg)

quasi-pmp and turns the ratio ergodic theorem into a usual one.

Pointwise ergodic theorems for quasi-pmp actions

◮ An action Γ (X, µ) is quasi-pmp if every γ ∈ Γ is nonsingular, i.e.

maps nonnull sets to nonnull sets (possibly changing their measure).

◮ Why would one consider quasi-pmp actions?

Theorem (Ratio Pointwise Ergodic, Hopf 1937)

Let Z α (X, µ) be an ergodic pmp action. For each f , g ∈ L1(X, µ) with g > 0 and for a.e. x ∈ X, lim

n→∞

Af [Fn ·α x] Ag[Fn ·α x] =

• X f dµ
• X gdµ.

◮ Replacing the measure µ with dµg . .= gdµ makes the action Γ (X, µg)

quasi-pmp and turns the ratio ergodic theorem into a usual one.

◮ A quasi-pmp ergodic theorem for Zd was proven by Feldman (2007) and

generalized by Hochman (2010) and Dooley–Jarrett (2016).

Pointwise ergodic theorems for quasi-pmp actions

◮ An action Γ (X, µ) is quasi-pmp if every γ ∈ Γ is nonsingular, i.e.

maps nonnull sets to nonnull sets (possibly changing their measure).

◮ Why would one consider quasi-pmp actions?

Theorem (Ratio Pointwise Ergodic, Hopf 1937)

Let Z α (X, µ) be an ergodic pmp action. For each f , g ∈ L1(X, µ) with g > 0 and for a.e. x ∈ X, lim

n→∞

Af [Fn ·α x] Ag[Fn ·α x] =

• X f dµ
• X gdµ.

◮ Replacing the measure µ with dµg . .= gdµ makes the action Γ (X, µg)

quasi-pmp and turns the ratio ergodic theorem into a usual one.

◮ A quasi-pmp ergodic theorem for Zd was proven by Feldman (2007) and

generalized by Hochman (2010) and Dooley–Jarrett (2016).

◮ Also known for lattice actions on homogeneous spaces (Nevo and co.), a

weaker form for groups of polynomial growth (Hochman 2013), an indirect version for free groups (Bowen–Nevo 2014).

Failures of pointwise ergodic for some quasi-pmp actions

◮ However, Hochman also showed (2012) that the ratio ergodic theorem is

false for Γ .

.= n∈N Z along any sequence (Fn) of window frames Fn ⊆ Γ.

Failures of pointwise ergodic for some quasi-pmp actions

◮ However, Hochman also showed (2012) that the ratio ergodic theorem is

false for Γ .

.= n∈N Z along any sequence (Fn) of window frames Fn ⊆ Γ. ◮ Moreover, it is also false for the nonabelian free groups along any

subsequence of balls.

Failures of pointwise ergodic for some quasi-pmp actions

◮ However, Hochman also showed (2012) that the ratio ergodic theorem is

false for Γ .

.= n∈N Z along any sequence (Fn) of window frames Fn ⊆ Γ. ◮ Moreover, it is also false for the nonabelian free groups along any

subsequence of balls.

◮ Thus, the pointwise ergodic theorem is false for the quasi-pmp actions of

• n∈N Z and the nonablian free groups.

Failures of pointwise ergodic for some quasi-pmp actions

◮ However, Hochman also showed (2012) that the ratio ergodic theorem is

false for Γ .

.= n∈N Z along any sequence (Fn) of window frames Fn ⊆ Γ. ◮ Moreover, it is also false for the nonabelian free groups along any

subsequence of balls.

◮ Thus, the pointwise ergodic theorem is false for the quasi-pmp actions of

• n∈N Z and the nonablian free groups.

◮ Still, is there any pointwise ergodic statement for these actions?

Failures of pointwise ergodic for some quasi-pmp actions

◮ However, Hochman also showed (2012) that the ratio ergodic theorem is

false for Γ .

.= n∈N Z along any sequence (Fn) of window frames Fn ⊆ Γ. ◮ Moreover, it is also false for the nonabelian free groups along any

subsequence of balls.

◮ Thus, the pointwise ergodic theorem is false for the quasi-pmp actions of

• n∈N Z and the nonablian free groups.

◮ Still, is there any pointwise ergodic statement for these actions? ◮ Let’s abandon the action and look at the induced Cayley–Schreier graph.

The Cayley–Schreier graph and uniform test windows

◮ A Borel action of a countable group Γ = S on (X, µ) induces a locally

countable Borel graph GS on X — its Cayley–Schreier graph: xGSy .

.⇔ σ · x = y for some σ ∈ S.

The Cayley–Schreier graph and uniform test windows

◮ A Borel action of a countable group Γ = S on (X, µ) induces a locally

countable Borel graph GS on X — its Cayley–Schreier graph: xGSy .

.⇔ σ · x = y for some σ ∈ S. ◮ Pointwise ergodic theorems extract a sequence (Fn) of subsets of Γ

(window frames)

The Cayley–Schreier graph and uniform test windows

◮ A Borel action of a countable group Γ = S on (X, µ) induces a locally

countable Borel graph GS on X — its Cayley–Schreier graph: xGSy .

.⇔ σ · x = y for some σ ∈ S. ◮ Pointwise ergodic theorems extract a sequence (Fn) of subsets of Γ

(window frames) and for a.e. x ∈ X, assert that the local averages Af [Fn · x] of an f ∈ L1(X, µ) over the test windows Fn · x converge to the global average

• X f dµ.

The Cayley–Schreier graph and uniform test windows

◮ A Borel action of a countable group Γ = S on (X, µ) induces a locally

countable Borel graph GS on X — its Cayley–Schreier graph: xGSy .

.⇔ σ · x = y for some σ ∈ S. ◮ Pointwise ergodic theorems extract a sequence (Fn) of subsets of Γ

(window frames) and for a.e. x ∈ X, assert that the local averages Af [Fn · x] of an f ∈ L1(X, µ) over the test windows Fn · x converge to the global average

• X f dµ.

◮ Let’s consider graphs in general.

More generally: pmp graphs

◮ G — a locally countable Borel graph on (X, µ).

More generally: pmp graphs

◮ G — a locally countable Borel graph on (X, µ). ◮ EG — the connectedness equivalence relation of G.

More generally: pmp graphs

◮ G — a locally countable Borel graph on (X, µ). ◮ EG — the connectedness equivalence relation of G. ◮ G is ergodic if each EG-invariant measurable subset is null or conull.

More generally: pmp graphs

◮ G — a locally countable Borel graph on (X, µ). ◮ EG — the connectedness equivalence relation of G. ◮ G is ergodic if each EG-invariant measurable subset is null or conull. ◮ G is pmp if every Borel automorphism γ of X that permutes every

G-connected component is measure preserving.

More generally: pmp graphs

◮ G — a locally countable Borel graph on (X, µ). ◮ EG — the connectedness equivalence relation of G. ◮ G is ergodic if each EG-invariant measurable subset is null or conull. ◮ G is pmp if every Borel automorphism γ of X that permutes every

G-connected component is measure preserving.

◮ In other words, the points in the same G-connected component have

equal mass.

Even more generally: quasi-pmp graphs

◮ G — a locally countable Borel graph on (X, µ).

Even more generally: quasi-pmp graphs

◮ G — a locally countable Borel graph on (X, µ). ◮ G is quasi-pmp if every Borel automorphism γ of X that permutes every

G-connected component is nonsingular (i.e. null preserving).

Even more generally: quasi-pmp graphs

◮ G — a locally countable Borel graph on (X, µ). ◮ G is quasi-pmp if every Borel automorphism γ of X that permutes every

G-connected component is nonsingular (i.e. null preserving).

◮ Think: the points in the same G-connected component have possibly

different nonzero masses.

Even more generally: quasi-pmp graphs

◮ G — a locally countable Borel graph on (X, µ). ◮ G is quasi-pmp if every Borel automorphism γ of X that permutes every

G-connected component is nonsingular (i.e. null preserving).

◮ Think: the points in the same G-connected component have possibly

different nonzero masses.

◮ This difference is given by a Borel cocycle ρ : EG → R+, which satisfies

ρ(x, y)ρ(y, z) = ρ(x, z).

Even more generally: quasi-pmp graphs

◮ G — a locally countable Borel graph on (X, µ). ◮ G is quasi-pmp if every Borel automorphism γ of X that permutes every

G-connected component is nonsingular (i.e. null preserving).

◮ Think: the points in the same G-connected component have possibly

different nonzero masses.

◮ This difference is given by a Borel cocycle ρ : EG → R+, which satisfies

ρ(x, y)ρ(y, z) = ρ(x, z).

◮ Thinking of ρ(x, y) as mass(x)/mass(y), write ρy(x) instead.

Even more generally: quasi-pmp graphs

◮ G — a locally countable Borel graph on (X, µ). ◮ G is quasi-pmp if every Borel automorphism γ of X that permutes every

G-connected component is nonsingular (i.e. null preserving).

◮ Think: the points in the same G-connected component have possibly

different nonzero masses.

◮ This difference is given by a Borel cocycle ρ : EG → R+, which satisfies

ρ(x, y)ρ(y, z) = ρ(x, z).

◮ Thinking of ρ(x, y) as mass(x)/mass(y), write ρy(x) instead. ◮ It makes µ ρ-invariant,

Even more generally: quasi-pmp graphs

◮ G — a locally countable Borel graph on (X, µ). ◮ G is quasi-pmp if every Borel automorphism γ of X that permutes every

G-connected component is nonsingular (i.e. null preserving).

◮ Think: the points in the same G-connected component have possibly

different nonzero masses.

◮ This difference is given by a Borel cocycle ρ : EG → R+, which satisfies

ρ(x, y)ρ(y, z) = ρ(x, z).

◮ Thinking of ρ(x, y) as mass(x)/mass(y), write ρy(x) instead. ◮ It makes µ ρ-invariant, i.e. for any γ as above,

µ(γB) =

• B

ρx(γx) dµ(x).

Even more generally: quasi-pmp graphs

◮ G — a locally countable Borel graph on (X, µ). ◮ G is quasi-pmp if every Borel automorphism γ of X that permutes every

G-connected component is nonsingular (i.e. null preserving).

◮ Think: the points in the same G-connected component have possibly

different nonzero masses.

◮ This difference is given by a Borel cocycle ρ : EG → R+, which satisfies

ρ(x, y)ρ(y, z) = ρ(x, z).

◮ Thinking of ρ(x, y) as mass(x)/mass(y), write ρy(x) instead. ◮ It makes µ ρ-invariant, i.e. for any γ as above,

µ(γB) =

• B

ρx(γx) dµ(x).

◮ The ρ-weighted average of a function f over a finite G-connected U ⊆ X:

Aρ

f [U] . .=

• u∈U f (u)ρx(u)
• u∈U ρx(u)

, where x is any/some point in the G-connected component of U.

A pointwise ergodic theorem for quasi-pmp graphs

◮ Let G be a locally countable quasi-pmp ergodic graph on (X, µ) and let

ρ : EG → R+ be the corresponding cocycle.

A pointwise ergodic theorem for quasi-pmp graphs

◮ Let G be a locally countable quasi-pmp ergodic graph on (X, µ) and let

ρ : EG → R+ be the corresponding cocycle.

◮ Even if G came from a group action (it always does), Hochman’s results

show that uniform/deterministic window frames taken from the group may not yield correct averages.

A pointwise ergodic theorem for quasi-pmp graphs

◮ Let G be a locally countable quasi-pmp ergodic graph on (X, µ) and let

ρ : EG → R+ be the corresponding cocycle.

◮ Even if G came from a group action (it always does), Hochman’s results

show that uniform/deterministic window frames taken from the group may not yield correct averages.

◮ Nevertheless, we can obtain a Borel pairwise disjoint collection of such

windows in X (i.e., a finite Borel equivalence relation on X) ensuring that each window is G-connected — the main challenge.

A pointwise ergodic theorem for quasi-pmp graphs

◮ Let G be a locally countable quasi-pmp ergodic graph on (X, µ) and let

ρ : EG → R+ be the corresponding cocycle.

◮ Even if G came from a group action (it always does), Hochman’s results

show that uniform/deterministic window frames taken from the group may not yield correct averages.

◮ Nevertheless, we can obtain a Borel pairwise disjoint collection of such

windows in X (i.e., a finite Borel equivalence relation on X) ensuring that each window is G-connected — the main challenge. In other words,

Theorem (Ծ . 2018)

There is an increasing sequence (Fn) of G-connected finite Borel equivalence relations on X such that for every f ∈ L1(X, µ), lim

n→∞ Aρ f [x]Fn =

• X

f dµ, for a.e. x ∈ X. Here, Aρ

f [x]Fn is the ρ-weighted average of f over the Fn-class [x]Fn of x.

Deterministic vs. random test windows

Credits and applications

The main theorem in the pmp case was first proven by Tucker-Drob:

Theorem (Tucker-Drob 2016)

The pointwise ergodic theorem is true for locally countable ergodic pmp graphs.

Credits and applications

The main theorem in the pmp case was first proven by Tucker-Drob:

Theorem (Tucker-Drob 2016)

The pointwise ergodic theorem is true for locally countable ergodic pmp graphs.

Applications

Answer to Bowen’s question: Every pmp ergodic countable treeable Borel equivalence relation admits an ergodic hyperfinite free factor.

Ergodic 1% lemma (Tucker-Drob and Conley–Gaboriau–Marks–Tucker-Drob):

Every locally countable ergodic nonhyperfinite pmp Borel graph G admits Borel sets A ⊆ X of arbitrarily small measure such that G|A is still ergodic and nonhyperfinite.

Could have been used in Ergodic Hjorth’s lemma for cost attained (Miller–Ծ . 2017): If a countable pmp ergodic Borel equivalence relation E is treeable

and has cost n ∈ N ∪ {∞}, then it is induced by an a.e. free action of Fn such that each of the n standard generators of Fn alone acts ergodically.

Credits and applications

The main theorem in the pmp case was first proven by Tucker-Drob:

Theorem (Tucker-Drob 2016)

The pointwise ergodic theorem is true for locally countable ergodic pmp graphs.

Applications

Answer to Bowen’s question: Every ✘✘

✘ ❳❳ ❳

pmp ergodic countable treeable Borel equivalence relation admits an ergodic hyperfinite free factor.

Ergodic 1% lemma (Tucker-Drob and Conley–Gaboriau–Marks–Tucker-Drob):

Every locally countable ergodic nonhyperfinite pmp Borel graph G admits Borel sets A ⊆ X of arbitrarily small measure such that G|A is still ergodic and nonhyperfinite.

Could have been used in Ergodic Hjorth’s lemma for cost attained (Miller–Ծ . 2017): If a countable pmp ergodic Borel equivalence relation E is treeable

and has cost n ∈ N ∪ {∞}, then it is induced by an a.e. free action of Fn such that each of the n standard generators of Fn alone acts ergodically.

Two different proofs

◮ Tucker-Drob’s proof of his theorem is based on a rather deep result in

percolation theory by Hutchcroft and Nachmias (2015) on indistinguishability of the Wired Uniform Spanning Forest.

Two different proofs

◮ Tucker-Drob’s proof of his theorem is based on a rather deep result in

percolation theory by Hutchcroft and Nachmias (2015) on indistinguishability of the Wired Uniform Spanning Forest.

◮ Furthermore, to derive his theorem from this, Tucker-Drob makes use of

• ther probabilistic techniques:

(Gaboriau–Lyons) Wilson’s algorithm rooted at infinity (essentially Hatami–Lov´ asz–Szegedy) an analogue for graphs of the Abert–Weiss theorem.

Two different proofs

◮ Tucker-Drob’s proof of his theorem is based on a rather deep result in

percolation theory by Hutchcroft and Nachmias (2015) on indistinguishability of the Wired Uniform Spanning Forest.

◮ Furthermore, to derive his theorem from this, Tucker-Drob makes use of

• ther probabilistic techniques:

(Gaboriau–Lyons) Wilson’s algorithm rooted at infinity (essentially Hatami–Lov´ asz–Szegedy) an analogue for graphs of the Abert–Weiss theorem.

◮ All these techniques are inherently pmp.

Two different proofs

◮ Tucker-Drob’s proof of his theorem is based on a rather deep result in

percolation theory by Hutchcroft and Nachmias (2015) on indistinguishability of the Wired Uniform Spanning Forest.

◮ Furthermore, to derive his theorem from this, Tucker-Drob makes use of

• ther probabilistic techniques:

(Gaboriau–Lyons) Wilson’s algorithm rooted at infinity (essentially Hatami–Lov´ asz–Szegedy) an analogue for graphs of the Abert–Weiss theorem.

◮ All these techniques are inherently pmp. ◮ A year later, I found a combinatorial/purely descriptive set theoretic proof

• f Tucker-Drob’s theorem (for pmp graphs).

Two different proofs

◮ Tucker-Drob’s proof of his theorem is based on a rather deep result in

percolation theory by Hutchcroft and Nachmias (2015) on indistinguishability of the Wired Uniform Spanning Forest.

◮ Furthermore, to derive his theorem from this, Tucker-Drob makes use of

• ther probabilistic techniques:

(Gaboriau–Lyons) Wilson’s algorithm rooted at infinity (essentially Hatami–Lov´ asz–Szegedy) an analogue for graphs of the Abert–Weiss theorem.

◮ All these techniques are inherently pmp. ◮ A year later, I found a combinatorial/purely descriptive set theoretic proof

• f Tucker-Drob’s theorem (for pmp graphs).

◮ Another half a year later, the argument became applicable to quasi-pmp

graphs, yielding a generalization.

The main theorem (again) and what’s involved

Diagonalizing through a dense sequence in L1(X, µ) reduces the main theorem to:

Theorem

Let G be a locally countable quasi-pmp ergodic Borel graph on (X, µ) and let ρ : EG → R+ be the cocycle. For every f ∈ L1(X, µ) and ε > 0, there is a G-connected finite Borel equivalence relation F on X such that Aρ

f [x]F ≈ε

• X

f dµ for all but ε-measure-many x ∈ X.

The main theorem (again) and what’s involved

Diagonalizing through a dense sequence in L1(X, µ) reduces the main theorem to:

Theorem

Let G be a locally countable quasi-pmp ergodic Borel graph on (X, µ) and let ρ : EG → R+ be the cocycle. For every f ∈ L1(X, µ) and ε > 0, there is a G-connected finite Borel equivalence relation F on X such that Aρ

f [x]F ≈ε

• X

f dµ for all but ε-measure-many x ∈ X. The proof required new tools: asymptotic averages along a graph finitizing vertex-cuts (exploits nonhyperfiniteness) saturated and packed prepartitions an iteration technique via measure-compactness for a cocycle ρ on EG, notions of ρ-ratio and (G, ρ)-visibility.

The main theorem (again) and what’s involved

Diagonalizing through a dense sequence in L1(X, µ) reduces the main theorem to:

Theorem

Let G be a locally countable quasi-pmp ergodic Borel graph on (X, µ) and let ρ : EG → R+ be the cocycle. For every f ∈ L1(X, µ) and ε > 0, there is a G-connected finite Borel equivalence relation F on X such that Aρ

f [x]F ≈ε

• X

f dµ for all but ε-measure-many x ∈ X. The proof required new tools: asymptotic averages along a graph finitizing vertex-cuts (exploits nonhyperfiniteness) saturated and packed prepartitions an iteration technique via measure-compactness for a cocycle ρ on EG, notions of ρ-ratio and (G, ρ)-visibility. In the remaining time, I’ll discuss the difficulty of getting each F-class G-connected.

Borel G-prepartitions

◮ For the sake of the talk, assume G is pmp.

Borel G-prepartitions

◮ For the sake of the talk, assume G is pmp. ◮ It suffices to fix an f ∈ L∞(X, µ) and assume

• X f dµ = 0.

Borel G-prepartitions

◮ For the sake of the talk, assume G is pmp. ◮ It suffices to fix an f ∈ L∞(X, µ) and assume

• X f dµ = 0.

◮ Building a G-connected finite Borel equivalence relation amounts to

constructing a Borel collection P of pairwise disjoint finite G-connected subsets of X — call it a G-prepartition.

— We do not require the domain dom(P) .

.= P to be all of X.

Borel G-prepartitions

◮ For the sake of the talk, assume G is pmp. ◮ It suffices to fix an f ∈ L∞(X, µ) and assume

• X f dµ = 0.

◮ Building a G-connected finite Borel equivalence relation amounts to

constructing a Borel collection P of pairwise disjoint finite G-connected subsets of X — call it a G-prepartition.

— We do not require the domain dom(P) .

.= P to be all of X. ◮ To prove the theorem, we need to build a Borel G-prepartition P with

1 − ε domain whose cells U ∈ P satisfy Af [U] ≈ε 0 — call these U good.

Borel G-prepartitions

◮ For the sake of the talk, assume G is pmp. ◮ It suffices to fix an f ∈ L∞(X, µ) and assume

• X f dµ = 0.

◮ Building a G-connected finite Borel equivalence relation amounts to

constructing a Borel collection P of pairwise disjoint finite G-connected subsets of X — call it a G-prepartition.

— We do not require the domain dom(P) .

.= P to be all of X. ◮ To prove the theorem, we need to build a Borel G-prepartition P with

1 − ε domain whose cells U ∈ P satisfy Af [U] ≈ε 0 — call these U good.

◮ By Kechris–Miller, there is always a Borel maximal such prepartition P.

Borel G-prepartitions

◮ For the sake of the talk, assume G is pmp. ◮ It suffices to fix an f ∈ L∞(X, µ) and assume

• X f dµ = 0.

◮ Building a G-connected finite Borel equivalence relation amounts to

constructing a Borel collection P of pairwise disjoint finite G-connected subsets of X — call it a G-prepartition.

— We do not require the domain dom(P) .

.= P to be all of X. ◮ To prove the theorem, we need to build a Borel G-prepartition P with

1 − ε domain whose cells U ∈ P satisfy Af [U] ≈ε 0 — call these U good.

◮ By Kechris–Miller, there is always a Borel maximal such prepartition P. ◮ How much land does this maximal P conquer? That is: what’s the

measure of dom(P)?

Borel G-prepartitions

◮ For the sake of the talk, assume G is pmp. ◮ It suffices to fix an f ∈ L∞(X, µ) and assume

• X f dµ = 0.

◮ Building a G-connected finite Borel equivalence relation amounts to

constructing a Borel collection P of pairwise disjoint finite G-connected subsets of X — call it a G-prepartition.

— We do not require the domain dom(P) .

.= P to be all of X. ◮ To prove the theorem, we need to build a Borel G-prepartition P with

1 − ε domain whose cells U ∈ P satisfy Af [U] ≈ε 0 — call these U good.

◮ By Kechris–Miller, there is always a Borel maximal such prepartition P. ◮ How much land does this maximal P conquer? That is: what’s the

measure of dom(P)?

◮ The first step in the proof is showing that dom(P) meets every

G-connected component.

Borel G-prepartitions

◮ For the sake of the talk, assume G is pmp. ◮ It suffices to fix an f ∈ L∞(X, µ) and assume

• X f dµ = 0.

◮ Building a G-connected finite Borel equivalence relation amounts to

constructing a Borel collection P of pairwise disjoint finite G-connected subsets of X — call it a G-prepartition.

— We do not require the domain dom(P) .

.= P to be all of X. ◮ To prove the theorem, we need to build a Borel G-prepartition P with

1 − ε domain whose cells U ∈ P satisfy Af [U] ≈ε 0 — call these U good.

◮ By Kechris–Miller, there is always a Borel maximal such prepartition P. ◮ How much land does this maximal P conquer? That is: what’s the

measure of dom(P)?

◮ The first step in the proof is showing that dom(P) meets every

G-connected component.

◮ Thus, dom(P) has positive measure, but this measure could be arbitrarily

small.

Maximal prepartition — not good enough

P — a Borel maximal G-prepartition into good cells U, i.e. Af [U] ≈ε 0.

Maximal prepartition — not good enough

P — a Borel maximal G-prepartition into good cells U, i.e. Af [U] ≈ε 0.

◮ To prove the theorem, we need the measure of dom(P) to be 1 − ε.

Maximal prepartition — not good enough

P — a Borel maximal G-prepartition into good cells U, i.e. Af [U] ≈ε 0.

◮ To prove the theorem, we need the measure of dom(P) to be 1 − ε. ◮ However, dom(P) may leave out infinite G-connected components.

Maximal prepartition — not good enough

P — a Borel maximal G-prepartition into good cells U, i.e. Af [U] ≈ε 0.

◮ To prove the theorem, we need the measure of dom(P) to be 1 − ε. ◮ However, dom(P) may leave out infinite G-connected components. ◮ How to get rid of these infinite clusters?

Maximal prepartition — not good enough

P — a Borel maximal G-prepartition into good cells U, i.e. Af [U] ≈ε 0.

◮ To prove the theorem, we need the measure of dom(P) to be 1 − ε. ◮ However, dom(P) may leave out infinite G-connected components. ◮ How to get rid of these infinite clusters? ◮ Need a stronger notion of maximality — packed prepartitions!

Maximal prepartition — not good enough

P — a Borel maximal G-prepartition into good cells U, i.e. Af [U] ≈ε 0.

◮ To prove the theorem, we need the measure of dom(P) to be 1 − ε. ◮ However, dom(P) may leave out infinite G-connected components. ◮ How to get rid of these infinite clusters? ◮ Need a stronger notion of maximality — packed prepartitions! ◮ Going into these however is too much for a morning talk :)

Thanks!