descriptive combinatorics and ergodic theoremsanush tserunyan university of illinois at...

Descriptive combinatorics and ergodictheorems

Anush Tserunyan

University of Illinois at Urbana-Champaign

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

Descriptive set theory and equivalence relations

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

I In the last 30 years: definable (Borel, analytic) equivalence relations.

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

I Such equivalence relations arise naturally all over mathematics:

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

Many mathematical objects (such as Riemann surfaces, Banach spaces,measure-preserving transformations, etc.) can be encoded as points inPolish spaces.

Thinking of classifying these points up to some notion of equivalence (e.g.,conformal equivalence, isomorphism, conjugacy), we define what it meansfor one such classification problem to be no harder than another: the Borelreducibility relation 6B between two equivalence relations.

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

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

Countable Borel equivalence relations via group actions

I These come from countable group actions: orbit equivalence relationsinduced by Borel actions of countable groups are countable Borel.

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

I Thus, taking {γn} =.. Γ y X with E = EΓ, each x ∈ X can refer to otherpoints in its E -class by names: γ0x , γ1x , γ2x . . .

I The abundance of Borel actions of countable groups makes the class ofCBERs extremely rich and 6B on it very complicated:

Adams–Kechris (using Zimmer’s cocycle superrigidity): N ⊇ A 7→ EA suchthat for all A,B ⊆ N, A ⊆ B ⇔ EA 6B EB .

Countable Borel equivalence relations via graphs

I CBERs also come from graphs as the connectedness relations EG oflocally countable Borel graphs G .

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

I 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 yα X such that E = EΓ, take a symmetric generatingset Γ = 〈S〉, and define the Cayley–Schreier graph:

xGSy ..⇔ σ ·α x = y for some σ ∈ S .

Then E = EGS.

Interplay with other subjects

I Any countable Borel group action Γ y X can be turned into acontinuous one by replacing the Polish topology on X .

— Enables topological tools such as Baire category.

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

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

I All these together has created extremely active two-way traffic betweenthe study of CBERs and

ergodic theory

(ergodic theorems, mixing, entropy, . . . )

measured group theory

(measure equivalence, cost, `2-(co)homology, . . . )

graph combinatorics

(colorings, matchings, flows, . . . )

geometric group theory

(tree constructions, growth,. . . )

percolation theory

(graph/action constructions, amenability barriers, . . . )

probabilistic combinatorics

(Lovasz Local Lemma, concentration bounds, . . . )

topological dynamics

(subshifts, topological entropy, compact actions, . . . )

von Neumann algebras

(spectral gap, superrigidity, vN-dimension, . . . )

ergodic theory

graph combinatorics

percolation theory

ergodic theory

graph combinatorics

percolation theory

ergodic theory (ergodic theorems, mixing, entropy, . . . )

measured group theory (measure equivalence, cost, `2-(co)homology, . . . )

graph combinatorics (colorings, matchings, flows, . . . )

geometric group theory (tree constructions, growth,. . . )

percolation theory (graph/action constructions, amenability barriers, . . . )

probabilistic combinatorics (Lovasz 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 settheoretic and analytic thinking?

I We think pointwise, analyzing the local combinatorics at a point, whereasanalysts analyze the space through the prism of functions on it.

I 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 unionof Borel partial functions X ⇀ Y .

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

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

I Thus, our way of thinking is best described as originless combinatorics.

Our secret weapon

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

andgive (Borel) names to the other guys in its class.

Our secret weapon

pmp actions of groups and ergodicity

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

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

I Γ — a countable (discrete) group

I An action Γ y (X , µ) is ergodic if every invariant measurable subset isnull or conull.

I A probability measure preserving (pmp) action of Γ is a Borel actionΓ y (X , µ) such that µ(γ · A) = µ(A), for each γ ∈ Γ and A ⊆ X .

I Dating back to Birkhoff, pointwise ergodic theorems for pmp actionsΓ y (X , µ) are bridges between the global condition of ergodicity and thea.e. local combinatorics of the action.

Pointwise ergodic theorems for pmp actions

Theorem (Pointwise ergodic, Birkhoff 1931)

A pmp action Z yα (X , µ) is ergodic if and only if for each f ∈ L1(X , µ)and for a.e. x ∈ X ,

limit of local averages limn→∞

Af [Fn ·α x ] =

∫Xf dµ global average

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

I This was generalized to amenable groups by Lindenstrauss (2001).I Analogous statements for free groups have been proven by Grigorchuk

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

I Bowen–Nevo (2013) also have pointwise ergodic theorems for othernonamenable groups, but for special kinds of actions.

Af [Fn ·α x ] =

I This was generalized to amenable groups by Lindenstrauss (2001).

I Analogous statements for free groups have been proven by Grigorchuk(1987), Nevo (1994), Nevo–Stein (1994), Bufetov (2002), andBowen–Nevo (2013).

Af [Fn ·α x ] =

Pointwise ergodic theorems for quasi-pmp actions

I An action Γ y (X , µ) is quasi-pmp if every γ ∈ Γ is nonsingular, i.e.maps nonnull sets to nonnull sets (possibly changing their measure).

I Why would one consider quasi-pmp actions?

Theorem (Ratio Pointwise Ergodic, Hopf 1937)

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

limn→∞

Af [Fn ·α x ]

Ag [Fn ·α x ]=

∫X f dµ∫X gdµ

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

quasi-pmp and turns the ratio ergodic theorem into a usual one.I A quasi-pmp ergodic theorem for Zd was proven by Feldman (2007) and

generalized by Hochman (2010) and Dooley–Jarrett (2016).I Also known for lattice actions on homogeneous spaces (Nevo and co.), a

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

limn→∞

Af [Fn ·α x ]

Ag [Fn ·α x ]=

∫X f dµ∫X gdµ

limn→∞

Af [Fn ·α x ]

Ag [Fn ·α x ]=

∫X f dµ∫X gdµ

limn→∞

Af [Fn ·α x ]

Ag [Fn ·α x ]=

∫X f dµ∫X gdµ

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

I A quasi-pmp ergodic theorem for Zd was proven by Feldman (2007) andgeneralized by Hochman (2010) and Dooley–Jarrett (2016).

I Also known for lattice actions on homogeneous spaces (Nevo and co.), aweaker form for groups of polynomial growth (Hochman 2013), anindirect version for free groups (Bowen–Nevo 2014).

limn→∞

Af [Fn ·α x ]

Ag [Fn ·α x ]=

∫X f dµ∫X gdµ

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

I Also known for lattice actions on homogeneous spaces (Nevo and co.), aweaker form for groups of polynomial growth (Hochman 2013), anindirect version for free groups (Bowen–Nevo 2014).

limn→∞

Af [Fn ·α x ]

Ag [Fn ·α x ]=

∫X f dµ∫X gdµ

Failures of pointwise ergodic for some quasi-pmp actions

I However, Hochman also showed (2012) that the ratio ergodic theorem isfalse for Γ ..=

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

I Moreover, it is also false for the nonabelian free groups along anysubsequence of balls.

I Thus, the pointwise ergodic theorem is false for the quasi-pmp actions of⊕n∈N Z and the nonablian free groups.

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

I Let’s abandon the action and look at the induced Cayley–Schreier graph.

The Cayley–Schreier graph and uniform test windows

I A Borel action of a countable group Γ = 〈S〉 on (X , µ) induces a locallycountable Borel graph GS on X — its Cayley–Schreier graph:

xGSy ..⇔ σ · x = y for some σ ∈ S .

I Pointwise ergodic theorems extract a sequence (Fn) of subsets of Γ(window frames) and for a.e. x ∈ X , assert that the local averagesAf [Fn · x ] of an f ∈ L1(X , µ) over the test windows Fn · x converge tothe global average

∫X f dµ.

I Let’s consider graphs in general.

I Pointwise ergodic theorems extract a sequence (Fn) of subsets of Γ(window frames)

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

∫X f dµ.

More generally: pmp graphs

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

I EG — the connectedness equivalence relation of G .

I G is ergodic if each EG -invariant measurable subset is null or conull.

I G is pmp if every Borel automorphism γ of X that permutes everyG -connected component is measure preserving.

I In other words, the points in the same G -connected component haveequal mass.

Even more generally: quasi-pmp graphs

I G is quasi-pmp if every Borel automorphism γ of X that permutes everyG -connected component is nonsingular (i.e. null preserving).

I Think: the points in the same G -connected component have possiblydifferent nonzero masses.

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

ρ(x , y)ρ(y , z) = ρ(x , z).

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

µ(γB) =

∫Bρx(γx) dµ(x).

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

Aρf [U] ..=

∑u∈U f (u)ρx(u)∑

u∈U ρx(u),

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

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

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

I Think: the points in the same G -connected component have possiblydifferent nonzero masses.

ρ(x , y)ρ(y , z) = ρ(x , z).

µ(γB) =

Aρf [U] ..=

u∈U ρx(u),

G -connected component is nonsingular (i.e. null preserving).I Think: the points in the same G -connected component have possibly

different nonzero masses.

ρ(x , y)ρ(y , z) = ρ(x , z).

µ(γB) =

Aρf [U] ..=

u∈U ρx(u),

different nonzero masses.I This difference is given by a Borel cocycle ρ : EG → R+, which satisfies

ρ(x , y)ρ(y , z) = ρ(x , z).

µ(γB) =

Aρf [U] ..=

u∈U ρx(u),

ρ(x , y)ρ(y , z) = ρ(x , z).

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

I It makes µ ρ-invariant, i.e. for any γ as above,

µ(γB) =

Aρf [U] ..=

u∈U ρx(u),

ρ(x , y)ρ(y , z) = ρ(x , z).

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

i.e. for any γ as above,

µ(γB) =

Aρf [U] ..=

u∈U ρx(u),

ρ(x , y)ρ(y , z) = ρ(x , z).

µ(γB) =

Aρf [U] ..=

u∈U ρx(u),

ρ(x , y)ρ(y , z) = ρ(x , z).

µ(γB) =

Aρf [U] ..=

u∈U ρx(u),

A pointwise ergodic theorem for quasi-pmp graphs

I Let G be a locally countable quasi-pmp ergodic graph on (X , µ) and letρ : EG → R+ be the corresponding cocycle.

I Even if G came from a group action (it always does), Hochman’s resultsshow that uniform/deterministic window frames taken from the groupmay not yield correct averages.

I Nevertheless, we can obtain a Borel pairwise disjoint collection of suchwindows in X (i.e., a finite Borel equivalence relation on X ) ensuring thateach window is G -connected — the main challenge. In other words,

Theorem (Ծ. 2018)

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

limn→∞

Aρf [x ]Fn =

∫Xf dµ, for a.e. x ∈ X .

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

Theorem (Ծ. 2018)

limn→∞

Aρf [x ]Fn =

I Nevertheless, we can obtain a Borel pairwise disjoint collection of suchwindows in X (i.e., a finite Borel equivalence relation on X ) ensuring thateach window is G -connected — the main challenge.

In other words,

Theorem (Ծ. 2018)

limn→∞

Aρf [x ]Fn =

Theorem (Ծ. 2018)

limn→∞

Aρf [x ]Fn =

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 pmpgraphs.

Applications

Answer to Bowen’s question: Every pmp ergodic countable treeable Borelequivalence 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 admitsBorel sets A ⊆ X of arbitrarily small measure such that G |A is stillergodic 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 treeableand 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.

Applications

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

Applications

Answer to Bowen’s question: Every ��XXXpmp ergodic countable treeable Borelequivalence relation admits an ergodic hyperfinite free factor.

Two different proofs

I Tucker-Drob’s proof of his theorem is based on a rather deep result inpercolation theory by Hutchcroft and Nachmias (2015) onindistinguishability of the Wired Uniform Spanning Forest.

I Furthermore, to derive his theorem from this, Tucker-Drob makes use ofother probabilistic techniques:

(Gaboriau–Lyons) Wilson’s algorithm rooted at infinity

(essentially Hatami–Lovasz–Szegedy) an analogue for graphs of theAbert–Weiss theorem.

I All these techniques are inherently pmp.

I A year later, I found a combinatorial/purely descriptive set theoretic proofof Tucker-Drob’s theorem (for pmp graphs).

I Another half a year later, the argument became applicable to quasi-pmpgraphs, yielding a generalization.

The main theorem (again) and what’s involved

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

Theorem

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

Aρf [x ]F ≈ε

∫Xf 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 -classG -connected.

Theorem

Aρf [x ]F ≈ε

∫Xf dµ

Theorem

Aρf [x ]F ≈ε

∫Xf dµ

Borel G -prepartitions

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

I It suffices to fix an f ∈ L∞(X , µ) and assume∫X f dµ = 0.

I Building a G -connected finite Borel equivalence relation amounts toconstructing a Borel collection P of pairwise disjoint finite G -connectedsubsets of X — call it a G -prepartition.

— We do not require the domain dom(P) ..=⋃P to be all of X .

I To prove the theorem, we need to build a Borel G -prepartition P with1− ε domain whose cells U ∈ P satisfy Af [U] ≈ε 0 — call these U good.

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

I How much land does this maximal P conquer? That is: what’s themeasure of dom(P)?

I The first step in the proof is showing that dom(P) meets everyG -connected component.

I Thus, dom(P) has positive measure, but this measure could be arbitrarilysmall.

Maximal prepartition — not good enough

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

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

I However, dom(P) may leave out infinite G -connected components.

I How to get rid of these infinite clusters?

I Need a stronger notion of maximality — packed prepartitions!

I Going into these however is too much for a morning talk :)

Thanks!

descriptive combinatorics and ergodic theoremsanush tserunyan university of illinois at...

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