A pointwise ergodic theorem for quasi-pmp graphs Anush Tserunyan - - PowerPoint PPT Presentation

A pointwise ergodic theorem for quasi-pmp graphs

Anush Tserunyan

University of Illinois at Urbana-Champaign

pmp actions of groups and ergodicity

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

◮ A pmp action of Γ is ergodic if the only invariant measurable subsets are

null or conull.

◮ Dating back to Birkhoff, the pointwise ergodic theorem for pmp

actions of Z is a bridge between the global condition of the ergodicity of the action and the a.e. local combinatorics of the Schreier graph of the action as an f -labeled graph for each f ∈ L1(X, µ).

◮ The localizing windows Fn for testing this are taken from the group, e.g.,

the intervals Fn .

.= [0, n) ⊆ Z, and hence they are uniform throughout the

action space and are used at all x ∈ X at once.

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 mean 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. ◮ Analogous statements for free groups have also been proven by Nevo and

Stein, Grigorchuk, Bufetov, and Bowen and Nevo.

◮ Bowen and Nevo also have pointwise ergodic theorems for other

nonamenable groups, but for special kinds of actions.

The Schreier graph and uniform test windows

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

countable pmp graph GS on X — its Schreier graph: xGSy .

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

for a.e. x ∈ X, assert that the local averages Af [Fn · x] of an f ∈ L1(X, µ) over the test windows determined by the Fn at x converge to the global mean

• X f dµ.

◮ Let’s consider graphs in general.

More generally: pmp graphs

◮ G — a locally countable Borel graph on (X, µ). ◮ EG — the connectedness equivalence relation of G. ◮ G is ergodic if the only EG-invariant measurable subsets are null and

conull.

◮ G is pmp if every Borel automorphism γ of X that fixes G-connected

components setwise (i.e. γ(x) EG x for all x ∈ X) is measure preserving.

◮ Think: 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). ◮ G is quasi-pmp if every Borel automorphism γ of X that fixes

G-connected components setwise (i.e. γ(x) EG x for all x ∈ X) null preserving, i.e. maps null sets to null sets.

◮ 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.

Special case of quasi-pmp: ratio ergodic theorems

◮ Quasi-pmp graphs most often arise from quasi-pmp actions of groups. ◮ But why would one consider quasi-pmp actions?

Theorem (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.

◮ The quasi-pmp version of the pointwise ergodic theorem for Z would

imply Hopf’s theorem by rescaling the measure: dµg .

.= gdµ. ◮ I don’t know if the quasi-pmp pointwise ergodic theorem is true for Z,

but Hochman showed (2012) that the ratio ergodic theorem is false for

• n∈N Z along any sequence (Fn) of windows, and 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.

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.

◮ Since G may not come from a group, the testing windows can no longer

be uniform and will depend on the point x.

◮ Even if G came from a group action, its vertices have different weights

(quasi-pmp), so uniform testing windows may not yield correct averages (and they don’t for some groups by Hochman’s result).

◮ 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 (Ts. 2017)

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.

Credits and applications

In the pmp case, this was first proven by Tucker-Drob:

Theorem (Tucker-Drob 2016)

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

Applications of the pmp version

Answer to Bowen’s question: Every pmp ergodic countable treeable Borel equivalence relation admits an ergodic hyperfinite free factor. Ergodic 1% lemma (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–Ts. 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.

Applications of the quasi-pmp theorem

Corollary (General answer to Bowen’s question)

Every (not necessarily pmp or quasi-pmp) ergodic countable treeable Borel equivalence relation admits an ergodic hyperfinite free factor.

Corollary (Ratio ergodic theorem for quasi-pmp graphs, Ts. 2018)

Let G be a locally countable quasi-pmp ergodic Borel graph on (X, µ), let ρ : EG → R+ be the Radon–Nikodym cocycle corresponding to µ. There is an increasing sequence (Fn) of ●-connected finite Borel equivalence relations such that for any f , g ∈ L1(X, µ) with g > 0, for a.e. x ∈ X, lim

n→∞

Aρ

f [x]Fn

Aρ

g[x]Fn

=

• X f dµ
• X gdµ .

Two different proofs

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

probability 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 nontrivial 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. ◮ For our result of ergodic Hjorth’s lemma on cost attained, Miller and I

found a descriptive set theoretic argument to prove a weaker version of Tucker-Drob’s theorem, which was enough for our application.

◮ This left the bug in my head... resulting (a year later) in a constructive

and purely descriptive set theoretic proof of Tucker-Drob’s theorem that is applicable to quasi-pmp graphs.

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 some 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 some of the tools in the pmp case.

Asymptotic averages along a graph G

◮ Suffices to fix an f ∈ L∞(X, µ) and assume

• X f dµ = 0.

◮ Looking at a G-connected component Y , we need to somehow partition

it into (arbitrarily large) finite connected pieces such that each of them gets the average of f close to 0.

◮ But what if all values of f on Y are at least 97? ◮ More generally, what if there are no arbitrarily large finite G-connected

sets whose average is roughly 0?

Definition

Call a real a ∈ R a G-asymptotic average of f at x ∈ X if there are arbitrarily large finite G-connected sets V ∋ x with Af [V ] arbitrarily close to a. Denote the set of such a by Af [G](x).

◮ Invariance: The map x → Af [G](x) is EG-invariant; hence constant a.e.

(by ergodicity), so we just write Af [G] instead.

◮ Convexity: Af [G] is an interval (discrete intermediate value property).

Borel G-prepartitions

◮ 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.

Lemma (Local-global bridge)

There is a Borel G-prepartition U with conull domain such that the f -averages over its cells U ∈ U are (almost) in Af [G].

◮ If 0 /

∈ Af [G], then Af [G] lies on one side of 0 in R, say positive, so

◮ a prepartition U from the lemma yields a contradiction: the global mean

• f f (calculated over U) becomes strictly positive.

◮ So, it must be that 0 ∈ Af [G].

Maximal prepartition — not good enough

◮ Now we’d like 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 have we conquered? I.e., what’s the measure of dom(P)? ◮ Because 0 ∈ Af [G], it would have to meet every G-connected

component, so have positive measure, but it could be arbitrarily small.

◮ How to get rid of these infinite clusters? ◮ Need a stronger notion of maximality — packed prepartitions!

Packed prepartitions

◮ For r ∈ (0, 1], call a finite set V ⊆ X an r-pack over a prepartition P if

V ∩ dom(P) is a union of sets in P # of new points = |V \ dom(P)| r · |V ∩ dom(P)| = # of old points.

◮ A 1 3-pack V over P: |V \ dom(P)| = 13 1 3 · 36 = 1 3 · |V ∩ dom(P)|. ◮ Call a G-prepartition into “good” sets r-packed there is no “good”

r-pack V over P.

◮ Theorem (Ts.) Borel r-packed prepartitions exist modulo a null set.

time?

No infinite clusters left out

◮ Taking a packed P, we no longer have infinite clusters left since

• therwise there would be a pack over P.

◮ Great, dom(P) now looks combinatorially/locally large, but what about

its measure?

◮ If we had an absolute lower bound on µ

• dom(P)
• (depending only on

G), then perhaps we could iterate.

◮ But what would give us an absolute lower bound on µ

• dom(P)
• ?

◮ The nonhyperfiniteness of G and its exploitation via finitizing vertex-cuts!

Finitizing vertex-cuts and nonhyperfiniteness

◮ Philosophy: when a graph G is nonhyperfinite, one has to destroy a

nontrivial chunk of it to make it hyperfinite.

◮ More precisely, call a set C ⊆ X a finitizing vertex-cut if G−C . .= G|X\C

is component-finite.

◮ Define the finitizing vertex-price of G, noted fvpµ(G), as the infimum of

µ(C) over all Borel finitizing vertex-cuts of G.

◮ Easy to show: G nonhyperfinite =

⇒ fvpµ(G) > 0.

◮ Since the packed prepartition P we built doesn’t leave out infinite

clusters, dom(P) is a finitizing vertex-cut.

◮ Hence, the measure of dom(P) is bounded below by fvpµ(G) > 0 — an

absolute lower bound!

Iteration via measure-compactness

◮ We take a coherent sequence (Pn) of G-prepartitions that get more and

more packed and contain larger and larger sets whose f -averages are closer and closer to 0.

◮ We want to put these together, but coherence only ensures that

• k<n E(Pk) is an equivalence relation, yet

n dom(Pn) may still be

small.

◮ µ

• dom(Pn)
• fvpµ(G) for all n, so we take the lim sup...

Measure-compactness (Trivial)

In a probability space, if infinitely-many events each happen with probability α, then with probability α, infinitely-many of them happen all at once.

◮ Hence, lim supn dom(Pn) has positive measure. ◮ Once we know it has positive measure, the packedness condition implies

that lim supn dom(Pn) is actually conull, so we have conquered all of X!

◮ Thus, putting enough finitely-many Pn-s together, we get a desired

G-connected finite Borel equivalence relation F. QED

Grazie

No infinite clusters left out

◮ Taking a packed P, we no longer have infinite clusters left since

• therwise there would be a pack over P.

◮ One has to still argue that looking locally large implies that dom(P) is

also globally (measure-wise) large, but I’ll save this for another talk.

Grazie