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 ∈ L 1 ( X , µ ). ◮ The localizing windows F n for testing this are taken from the group, e.g., the intervals F n . . = [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 ∈ L 1 ( X , µ ) and for a.e. x ∈ X, � limit of local averages n →∞ A f [ F n · α x ] = lim f d µ global mean X where F n . . = [0 , n ) ⊆ Z and A f [ F n · α x ] is the average of f over F n · 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 G S on X — its Schreier graph: xG S y . . ⇔ σ · x = y for some σ ∈ S . ◮ Pointwise ergodic theorems extract a sequence ( F n ) of subsets of Γ and for a.e. x ∈ X , assert that the local averages A f [ F n · x ] of an f ∈ L 1 ( X , µ ) over the test windows determined by the F n at x converge � to the global mean X f d µ . ◮ Let’s consider graphs in general.
More generally: pmp graphs ◮ G — a locally countable Borel graph on ( X , µ ). ◮ E G — the connectedness equivalence relation of G . ◮ G is ergodic if the only E G -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 ) E G 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 ) E G 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 : E G → R + , which satisfies ρ ( x , y ) ρ ( y , z ) = ρ ( x , z ) . ◮ Thinking of ρ ( x , y ) as mass ( x ) / mass ( y ), write ρ y ( x ) instead. ◮ It makes µ ρ -invariant, i.e. for any γ as above, � ρ x ( γ x ) d µ ( x ) . µ ( γ B ) = B ◮ The ρ -weighted average of a function f over a finite G -connected U ⊆ X : u ∈ U f ( u ) ρ x ( u ) � A ρ f [ U ] . . = , u ∈ U ρ x ( 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 ∈ L 1 ( X , µ ) with g > 0 and for a.e. x ∈ X, � A f [ F n · α x ] X f d µ lim A g [ F n · α x ] = X gd µ. � n →∞ ◮ The quasi-pmp version of the pointwise ergodic theorem for Z would imply Hopf’s theorem by rescaling the measure: d µ g . . = gd µ . ◮ 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 ( F n ) 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 : E G → 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 ( F n ) of G-connected finite Borel equivalence relations on X such that � for every f ∈ L 1 ( X , µ ) , lim n →∞ A ρ f [ x ] F n = f d µ, for a.e. x ∈ X . X Here, A ρ f [ x ] F n is the ρ -weighted average of f over the F n -class [ x ] F n 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 F n such that each of the n standard generators of F n 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 : E G → R + be the Radon–Nikodym cocycle corresponding to µ . There is an increasing sequence ( F n ) of ● -connected finite Borel equivalence relations such that for any f , g ∈ L 1 ( X , µ ) with g > 0 , for a.e. x ∈ X, A ρ � X f d µ f [ x ] F n lim = X gd µ . A ρ � g [ x ] F n n →∞
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 other 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 L 1 ( 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 : E G → R + be the cocycle. For every f ∈ L 1 ( X , µ ) and ε > 0 , there is a G-connected finite Borel equivalence relation F on X such that � A ρ f [ x ] F ≈ ε f d µ X 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 E G , notions of ρ -ratio and ( G , ρ )-visibility. In the remaining time, I’ll discuss some of the tools in the pmp case.
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