Stationary Dynamical Systems and the Furstenberg-Poisson ...hartmany/pdfs/lecture-notes.pdf · Stationary theory is a a theory that was developed by Furstenberg in the 60’s and

Stationary Dynamical Systemsand the Furstenberg-Poisson boundary

Graduate course, Northwestern University

Yair HartmanLast Update: Wednesday 14th March, 2018

1. Introduction

Lecture 1Stationary theory is a a theory that was developed by Furstenberg in the 60’s and the 70’s whichincludes the study of the Furstenberg-Poisson boundary.

It is a branch of ergodic theory that deals with measurable groups action which has severalsides. It is closely related to random walks on groups. Hence, different people find it interesting fordifferent reasons.

Before starting from the very basics, explaining and defining every object, I want to spendsometime over viewing the theory. Hence, if we mention terms that you are not familiar with, that’sfine - we will define everything when we will get there. So now just sit back and enjoy the view!

In general I want to present 3 perspectives.

TODO: discuss first the probability approach

Group theory - Application to rigidity theory

I’m mostly interested in this theory as a tool to study groups. This theory is especially relevant anduseful for the study of large infinite groups. Most of it is not relevant to abelian, or nilpotent groups.And while some of it relevant to solvable groups, the main focus is on non-amenable groups.

Here is a example:

Theorem 1.0.1 — Margulis’ normal subgroup theorem 67’. Let G be a connected simpleLie group with finite center and rkR ≥ 2 and let Γ be an irreducible lattice. Then any normalsubgroup N in Γ is either finite or is of finite index.

The first example of such Γ is SL3(Z). This theorem classifies the normal subgroups of SL3(Z)and a-priory seems to be totally unrelated to Poisson boundary and random walks. In some way,the abstract group SL3(Z) remembers that it came from the group SL3(R). A natural question is inwhat sense are they similar.

In what sense Z or Zn are similar to R or R3. The lattices are countable discrete groups and theambient groups are connected - so they seem very different. Yet, they are similar in a way. The

4 Chapter 1. Introduction

basic idea is that if we look at Z from very far away, we might think that we are looking at R.But, “far away” in what sense? One approach, is to define far away using a random walk. For

that, one fixes a some random walk on Z3, or SL3(Z) say, and then far a way would mean after along time. Indeed, in some way, these processes will look like random walks on R3 and SL3(R).

A stronger application, is the following.

Theorem 1.0.2 — Stuck-Zimmer theorem 94’. Let G as before. Then any faithful measurepreserving action of G is either transitive, or essentially free.

This theorem classifies the stabilizers of measure preserving actions of such groups, which canbe thought of as a generalization of normal subgroups, and indeed, this theorem implies the normalsubgroup theorem.

Again, while this result deals with measure preserving actions, the main tool here is a structuretheorem of certain stationary actions.

These two results, are stated here in the context of Lie groups, but in some sense, the mainmachinery in these proofs relay of abstract stationary theory more than working with the Lie groupstructure. And indeed, these results were proved by Bader-Shalom (’06) in a more abstract setup.In this course we will prove these results.

Probability - Random walks on groupsLoosely speaking, the Furstenberg-Poisson boundary is an objects that captures the asymptoticbehavior of the random walk. Although it is a measurable space, one can think of it as sort ofcompactification of the group that is constructed out of a random walk. In the discussion above,about the relation between lattices and their ambient groups, we said that they are similar in thesense that they share the same compactification.

There are many questions in probability that are related to random walks on groups, and theseare very active research fields. We will prove the existence of the Furstenberg-Poisson boundary,for every random walk on every group. In fact, the whole theory works perfectly fine for locallycompact second countable groups. However, we will develop the theory in the context of discretecountable groups, for simplicity.

In any case, a lot of research is concern with concrete realizations of the Furstenberg-Poissonboundary as measures on natural topological spaces that the group is question is acting on. Formany classes of groups there are natural boundaries: Gromov boundary of hyperbolic groups, flagvarieties of linear groups, Thurston’s boundary of mapping class groups, and so on. In all these case,using an entropy theory that we will develop, one can realize the Furstenberg-Poisson boundary onthese topological boundaries.

Theorem 1.0.3 Consider the simple random walk on the free group F2. Show that the boundaryof the tree, has a measure which is the Furstenberg-Poisson boundary.

Ergodic Theory - A good generalization of measure preserving actionsThe main focus of ergodic theory is the study of measure preserving actions. For convenience, wecan think of an action on a compact space where the measure doesn’t move when we hit it withgroup’s elements.

The theory behind these kind of actions is quite developed. For example, the celebrated ergodictheorem holds:

Theorem 1.0.4 — Birkhoff. Let (X ,m,T ) be an ergodic measure preserving system, and let

5

f ∈ L1(X ,m). The for m-almost every x,

limn

n

∑k=0

f (T k(x)) =∫

Xf (x)dm(x)

When T as an invertible transformation, so we are actually dealing with a Z-action. A naturalgeneralization holds for amenable groups:

Theorem 1.0.5 Let G be a discrete countable group, G y (X ,m) be an ergodic measure pre-serving action, f ∈ L1(X ,m) and Fn be a tempered Folner sequence. Then m-almost everyx,

limn

1|Fn| ∑

g∈Fn

f (gx) =∫

Xf (x)dm(x)

The existence of Folner sequence allows to take averages. So in the classical setup, there aretwo ingridients: a natural averaging scheme and the measure, being invariant.

1. Can we say anything about measure preserving actions of non amenable groups?2. Can we do anything with measures which are not invariant?These two questions are related in some sense: whenever an amenable groups acts on a compact

space, there is always invariant measure. Hence leaving the world of amenable groups, bothquestions are relevant: sometimes we will have invariant measures, but we won’t know how to takeaverages, and sometimes, there will be no invariant measures at all and in that case, we might wantto consider something weaker that invariant measure.

The nice feature of invariant measures, is that they are invariant. When considering measureswhich are not invariant, they changes as we apply sequence of group’s elements. Consider forexample the situation of the ergodic theorem. The fact that the measure is invariant, means thatwhen we hit it with these growing Folner sequences, we see different points, be we always see thesame measure. If the measure is not invariant, then things are starting to move and we loose control.

Here again the idea of a random walk becomes useful, addressing both issues. Given a randomwalk on a group we will consider measures which are stationary, with respect to this randomwalk. These will be the generalization of invariant measures that are always exist. At the sametime, having a random walk, gives a natural averaging schemes, and indeed there is a nice ergodictheorem in this context, called Kakutani’s random ergodic theorem.

Some terms

Groups

1. nilpotent groups2. amenable groups3. Cayley graph4. growth rate of a group5. the free group6. lamplighter groups, affine groups

Group actions

1. measurable actions2. ergodic measures, ergodic theorem3. non-singular actions4. factors5. universal objects

6 Chapter 1. Introduction

Analysis1. L1,L2,L∞ spaces2. Radon-Nikodym3. Gelfand’s theory

Probability1. Random walks2. Harmonic functions3. Marginals4. Shannon entropy

2. Background on measurable actions

Lecture 22.1 Measurable and topological actions

2.1.1 Borel spacesDefinition 1 1. A Borel space is a pair (X ,B) where X is a set and B is a σ -algebra on X ,

that is, a collection of subsets of X which contains /0 and X and is closed under takingcomplements and countable unions. The elements E ∈B are called Borel sets or measurablesets.

2. If (X1,B1) and (X2,B2) are Borel spaces, a map ϕ : X1→ X2 is a Borel map (or measurable)if ϕ−1 (B) ∈B1 for all B ∈B2.

3. ϕ is an isomorphism if it is a bijection between X1 and X2 and its inverse is also measurable.4. (Y,A ) is a Borel subspace of (X ,B) if Y ⊂ X and there exists some B ∈ B such that

A = B∩B.

Example 2.1 Let X be a topological space, and consider B to be the σ -algebra generated bythe open subsets.

Definition 2 (X ,B) is a standard Borel space if it is isomorphic to a Borel subspace of a completemetric space (in particular, it is separable).

All the spaces that we will meet along the way will be standard Borel spaces. But before that,let’s say some things on the level of Borel spaces.

Facts 1 • The cardinality of a standard Borel space is either finite, countable or continuum.• Two standard Borel spaces are isomorphic if and only if they have the same cardinality. It

says that is a sense, there is not enough structure in a Borel space. Therefore, we will addanother ingredient to the story - measures.

• In particular, any uncountable standard Borel space is isomorphic to the unit interval.• Most importantly: Any standard Borel space is separable, meaning there there exists

a countable collection of measurable sets Bn such that B = σ(Bn) and for any x 6= y,

8 Chapter 2. Background on measurable actions

there exists n such that x ∈ Bn and y 6∈ Bn).

2.1.2 Borel G-spaces

From now on, G will always stand for a discrete countable group. The theory that we will developholds to locally compact second countable groups, such as Lie groups, without to much adaptations.Since it is rich and interesting already for discrete countable groups, for simplicity, we won’t presentthis generality here.

Given a Borel space (X ,B), Let Aut(X ,B) denote the group of all Borel-automorphisms of X .

Definition 3 X is a Borel G-space, if G y (X ,B) in a measurable way, that is:There exists an action map G×X a−→ X which is measurable. We will simply write gx = a(g,x),

so any g is an isomorphism X → X . The map a is an action in the sense that and g1g2x = g1 (g2x).Another way do describe it is by a group homomorphism : G→ Aut(X ,B).

We will also consider topological actions. All of the topological spaces are assumed to beHausdorff, and usually we consider compact spaces.

Definition 4 A compact space X is a G-space, if G y X continuously, that is:There exists an action map G×X a−→ X which is continuous.If Homeo(X) denotes the group of all homeomorphisms from X to itself, then a continuous

action is a group homomorphism G→ Homeo(X).

Remark 1 The view of an action as a group homomorphism is useful to define these action formore general groups.

Since every continuous function is Borel (as usual, when we already have a topology, the Borelstructure is the one generated by the topology), we get that Homeo(X) is a subgroup of Aut(X ,B).In particular, any continuous action is a Borel action.

Let (X ,B) and (Y,A ) be two G-spaces. We say that a Borel map π : X →Y is a G-map if it isequivariant, namely π(gx) = gπ(x) for every x ∈ X and g ∈ G.

We will usually require maps between compact spaces to be continuous.

Definition 5 Let (X ,B) be a Borel G-space. A compact metric G-space Y is a compact model ofX if they are G-isomorphic as measurable spaces.

Remark 2 The unit interval is a compact model of every uncountable Borel space in the level ofspace, but not as G-spaces. Given an abstract Borel G-space we get an action on the unit interval,but this action is only Borel, and not continuous in general.

Theorem 1 — The compact model theorem. Any standard Borel G-space admits a compactmodel.

If the Borel space is not standard, then one can still find compact models, but there is noguaranteed that this model would be metrizable.

In some way, a topological model is similar to providing coordinate system to an abstractspace. A given Borel space admits many different compact models. However, sometimes there areproperties of the abstract space that are naturally reflected in a topological realizations. We will seean example of such a property of abstract space, that is equivalent to a some topological property,for any compact model.

One important corollary of the compact model theorem is that for any a Borel G-space, thestabilizer subgroups, Stab(x) = g|gx = x are closed subgroup of G (although it is not relevant forour setup of discrete groups).

2.2 Measured Borel Spaces 9

2.2 Measured Borel Spaces

Definition 6 A standard measured space is a triple (X ,B,ν) where• (X ,B) is a standard Borel space.• ν : B→ [0,∞] is a measure, namely, it is σ -additive linear function: if En ⊂F is pairwise

disjoint then ν (⋃

En) = ∑ν (En).We will mainly focus on probability measures, that is ν such that ν(X) = 1.

Once we have a measure in the picture, we want to ignore things that the measure doesn’t see.Measurable function on a measured space are only defined up to null sets. Hence functions arenot really functions on the set but only equivalence class of functions. Similarly, if (X ,ν) and(Y,η) are two measured spaces, then when we say that π : X → Y is a measurable map, we meanthat there exists X ′ ⊂ X and Y ′ ⊂ Y , with ν(X ′) = η(Y ′) = 1 and the map π : X ′→ Y ′ is a Borelisomorphism.

On one hand, it becomes a bit harder to work in this category - there are no points in the space,but only sets. No functions, only equivalence class of functions. On the other hand, the exact spaceabstractness makes this category very flexible. We will talk soon about factors and will see thingsthat are possible and very useful, that we cannot get in the topological setup.

Let π : X → X be a Borel map. Given a measure ν on X , the push-forward measure π∗ν isthe measure on Y defined by π∗ν(E) = ν(π−1(E)).

Two measures ν1,ν2 on the the same Borel space X are said to be equivalent, or in the samemeasure-class if ν1(E) = 0 ⇐⇒ ν2(E) = 0. In that case both ν1 is absolutely continuous withrespect to ν2 and also in the other direction and we denoted it by ν1 ∼ ν2.

Given a measured Borel space, let Aut∗(X ,ν) denote the group all the isomorphism φ : X →such that ν ∼ φ∗ν . So Aut∗(X ,ν)≤ Aut(X ,B).

We will only consider non-singular actions on standard measure spaces, meaning that g∗ν =gν and ν are equivalent for any g. Such measures are called quasi-invariant measures. In otherwords, the action map is a measurable group homomorphism G→ Aut*(X ,ν).

Finally, an action said to be a measure preserving action if ν = gν for all g. In that case wesay that ν is an invariant measure.

Again, if Aut(X ,ν) denote the subgroup of Aut∗(X ,ν) of isomorphisms φ : X → X such thatφ∗ν = ν then a measure preserving action is just such where the image of the action map is inAut(X ,ν).

The space of probabilitiesWe now mix topological and measurable actions.

Let X be a compact space. We denote by C (X) the algebra of all continuous functionsX → C. Riesz representation theorem states that measures on X (that are defined on the Borelsigma-algebra) are the same as positive linear functional on C (X). In particular, the space of allprobability measures on X , that we denote by Prob(X) is a subset of the dual space C (X).

The space Prob(X) is naturally equipped with the weak*-topology, namely νn→ ν if for anyf ∈ C (X), we have that νn( f )→ ν( f ). When X is compact we get that Prob(X) is a compactconvex space.

When G y X is a compact G-space, then we get a continuous action G y Prob(X), via thepush-forwarding the measures. Fixed points in this action are exactly invariant measures.

Lecture 3ErgodicityDefinition 7 A non-singular action G y (X ,ν) is said to be ergodic if there are no non-trivialinvariant subsets, that is if gA = A for all g ∈ G then ν(A) ∈ 0,1.


Exercise 2.1 Prove that m is ergodic if and only if any invariant function f : X → R (that is,f (x) = f (gx) for all g ∈ G) is constant.

Solution

Let X be a compact G-space. We denote by ProbG(X)⊂ Prob(X) to be the subset of invariantmeasures on X . In general, it might be empty. However, ProbG(X) is a convex compact space aswell.

A point ν in a convex space is extremal if the only solution to ν = cν1 +(1− c)ν2 is when cis either 0 or 1.

Lemma 1 Let X be a compact G-space. The following are equivalent for an invariant measurem ∈ ProbG(X):

1. m is ergodic, that is, if A is invariant (gA = A for all g ∈ G), then m(A) ∈ 0,1.2. m(A) ∈ 0,1 for all m-invariant A (m(gA∆A)) = 0 for all g ∈ G).3. m is extremal in ProbG(X).

Proof. (1 =⇒ 2) Clear.(2 =⇒ 3) Assume that m is not extremal and m = cm1 +(1− c)m2 where 0 < c < 1. Then

both m1 and m2 are absolutely continuous w.r.t m. Let f1 (x) = dm1dm (x) be the Radon-Nikodym

derivative. f1 is the unique function that satisfies∫B

f (x)dm = m1 (A). Since m 6= m1, f1 is not

constant w.r.t. m, and so we can find a > 0 with A =

x| f (x)< a

such that 0 > m(A)> 1.Since both m and m1 are invariant, f1 is G-invariant w.r.t. m. In particular, m(gA∆A) = 0 for

all g ∈ G.(3 =⇒ 1) Assume that there is an invariant set A with m(A) = c for 0 < c < 1. Let m1 (B) =

1c m(B∩A) and m2 (B) = 1

1−c m(B∩ (X\A)

), for any B ∈B. Note that m1,m2 are invariant mea-

sures.Then(

cm1 +(1− c)m2)(B) = m(B∩A)+m

(B∩ (X\A)

)= m(B)

and so m is not extremal.

The fact that extremal measures are ergodic is very general. For the other direction we used thefact that if an invariant measure is absolutely continuous w.r.t. another invariant measure, then theRadon-Nikodym derivative is invariant function.

2.3 Factors

Definition 8 Let (X ,ν) and (Y,η) be two measured G-spaces.We say that (X ,ν) is an extension of (Y,η), and (Y,η) is a factor of (X ,ν) if there exists

Borel G-map (factor map) π : X → Y such that π∗ν = η .When X and Y are compact spaces, we require the continuous equivariant map π to be onto.

There are several ways to think of a factor, we list here the ones that we will use. First a picture:

TODO: Draw a square over a line

2.3 Factors 11

Algebras EmbeddingsA factor map π induces a natural embedding of algebras, via the “pull-back” map π∗ f = f (π(x)).

In the topological category, we get an embedding of the algebra C (Y )⊂ C (X). Functions inC (X) that are coming from Y are constant on the fibers.

A factor in the measurable category π : (X ,ν)→ (Y,η) gives a natural embedding of L∞(Y,η)⊂L∞(X ,ν).

So factors yields to G-invariant closed sub-algebra. In fact, the other direction is also true. Ifwe find say in C (X) a G-invariant closed sub-algebra, then it must be coming from a G-equivariantfactor.

2.3.1 Measurable factorsAs sub-sigma-algebrasLet (X ,B) be a Borel space and let ν a measure on X . A sub-sigma-algebra A is called ν-completeif any subset of a null set, A⊂ N where ν(N) = 0, belongs to A ∈A . It is not a heavy requirement- given a measure ν , one can always complete any sigma-algebra by adding to it all the measurablesubsets of the null sets.

Let (X ,B,ν)π−→ (Y,A ,η) be a factor map of measurable spaces. Consider the following sigma

algebra on X consists of all the sets of the form BY =

π−1(E)

E∈A. In other words, we are

pulling back the sigma-algebra from the factor.By that we get a G-invariant sub-sigma algebra BY ⊂B which is complete. Note that it is

invariant as a collection of subsets, not every set is invariant!

Theorem 2 — Macky’s point realization. Let (X ,B,ν) be a standard Borel space and let A ′ ⊂B be a sub-sigma algebra which is ν-complete. Then there exists a standard space (Y,A ,η) suchthat (X ,B,ν)

π−→ (Y,A ,η) and A ′ = π∗A .Now assume that (X ,B,ν) is a G-space (non-singular action). The space Y admits a natural

G-action (and the map is equivariant) if and only if A ′ is a G-invariant sigma-algebra.

Given (X ,B,ν), we can consider the space L∞(X ,B,ν). There is a natural correspondencebetween sub-sigma algebras A ⊂B and sub spaces L∞(X ,A ,ν)⊂ L∞(X ,B,ν). The embeddingis given by the sub space of A -measurable functions. A B-measurable function is A -measurableif and only if it is of the form π∗ f for a function f on the Mackey realization of A .

So, we can either work algebras or with sigma-algebras, and it is really matter of taste.

Example 2.2 Let G y (X ,B,ν), and let K ≤ G. Consider

BK =

E ∈B|kE = E∀k ∈ K.

Then BK is a sub-sigma algebra and we denote by K\\(X ,ν) the Mackey realization of BK . Weequipped K\\(X ,ν) with the pushed-forward ν measure on the Mackey space of BK .

We now note that for a normal subgroup N /G, N\\(X ,ν) is a G-space. Indeed, if E is Ninvariant then gE is also: ngE = gg−1ngE = gn′E = gE.

Theorem 2.3.1 — Disintegration. Let X be a compact space, ν ∈ Prob(X), and (X ,ν)π−→ (Y,η)

be a measurable factor map. Then there exist a Borel map Y → Prob(X), y 7→ νy such that

1. [Fibered measures] For ν-almost every y ∈ Y , νy

(π−1 (y)

)= 1.


2. [Disintegration]

ν =∫Y

νydη (y)

That is, for any Borel map f : X → C,

∫X

f (x)dν(x) =∫Y

∫X

f (x)dνy(x)

dη(y).

3. [Uniqueness] If there is another map which sends y 7→ ν ′y then ν ′y = νy for η-almost everyy.

Example 2.3 — Ergodic decomposition. The space Y = G\\(X ,ν) called the space ofG-ergodic components. Consider π : X → Y and denote by η the pushed forward measure onY .

First note that if ν is an ergodic measure then Y is the trivial action. There might be invariantmeasures, but all of them get ν-measure which is 0 or 1. It follows that Y is the trivial, one pointspace, with Dirac measure.

We claim that in general, νy is an ergodic measure for η-a.e. y. First, note that G actstrivially on G\\X - since each set in the sigma algebra is G-invariant. It follows that almost everyfiber π−1y is G-invariant set in X . Indeed, if π (x) = y then π (gx) = gπ (x) = gy = y.

Now let E ⊂ X be G-invariant. By definition, it is already measurable in Y = G\\X , and wecan consider π (E)⊂ Y . If y ∈ π (E) then νy (E) = 1 and otherwise, νy (E) = 0.

Remark 3 Now assume that the measure ν is an invariant measure. Since the measures νy are allergodic, this decomposition presents ν as an integral extremal measures.

In finite dimensional convex spaces, it is clear that any point can be written as a convex combi-nation of the extremal points. This holds also for infinite dimensional and it calls Krein–Milmantheorem. So once we know that ergodic measures are extremal, then we can get that one can presentany measure as an integral of ergodic.

Moreover, as this decomposition is unique, we get that ProbG(X) is a Choquet simplex (whichmeans that any point is being represented in a unique way as integral of the extremal).

Lecture 4Disintegration and conditional expectationsAs we saw, we can always pull back functions from a factor to its extension. However, thedisintegration theorem implies that in the measurable category, one can also push forward functionsfrom the extension to the factor.

Theorem 3 — Conditional expectation. Let (X ,B,ν) and let A ⊂B be a sub-sigma algebra.For a function f ∈ L1(X ,B,ν), there exists a unique function E( f |B) ∈ L1(X ,A ,ν), such

that for any set A ∈A ,∫A

f (x)dν (x) =∫A

E( f |B)(x)dν (x) .

Consider Y to be the Mackey space of A . Then we can think of E( f |B) = π∗ f as a functionon Y . How the function π∗ f decides what to give to y? There are many values in the fiber π−1(y).So the conditional expectation integrates the fiber according to νy.

2.3 Factors 13

Note that for any function f on Y we get π∗(π∗ f ) = f , but not the other way around!

In fact, π∗(π∗ f ) = E( f |A ) where A is the pull back of the σ -algebra from the factor.

Relative compact modelIf we have a measurable factor π : (X ,ν)→ (Y,η) we can build a compact model for each of them.So we get two compact G-spaces, X and Y . Also, we get a measurable factor map ϕ : (X ,ν)→ (Y,η)such that ϕ = π , ν-almost everywhere. But, we cannot guaranteed ϕ to be continuous with respectto the topologies that we got. One may hope to replace it by another factor map that will becontinuous. However, such a map do not exists in general. The best we can get are two compactspaces with a measurable map between them.

Exercise 2.2 The properties non-singular, invariant and ergodic are preserved under factors.

AddWhy we can push measures forward?

Why considering measures is natural, even if the space is a topological space?

2.3.2 MartingalesLet us recall a fundamental theorem from probability theory.

Bn is increasing σ -algebras (called filtration) in some probability space (X ,B,ν), that tendsto B. Meaning that B is the sigma algebra generated by the sets in

⋃n Bn.

Definition 9 A sequence of random variables Mn is a bounded martingale w.r.t. Bn if• E(|Mn|)< ∞

• Mn is Bn-measurable• E(Mn+1|Bn) = Mn

If we realize each sigma algebra Bn as a space (Xn,νn) so we have as sequence of factors (thereis no G action here)

(X ,ν)→ ··· → (Xn,νn)πn−→ (Xn−1,νn−1)→ ··· → (X1,ν1).

Now we can form the

L∞(X ,ν)→ ·· · → L∞(Xn,νn)πn∗−−→ L∞(Xn−1,νn−1)→ ··· → L∞(X1,ν1).

where the maps πn∗ are the conditional expectation. Given a function f ∈ L∞(X ,ν) we canconsider the sequence of functions fn ∈ L∞(Xn,νn) which are the conditional expectations. So thesequence fn forms a bounded martingale (since f , and hence all the rest, are bounded).

Since Bn →B it is natural to expect that fn → f is some sense. It is not hard to see that‖ f − fn‖2 → 0, when we consider fn as functions on X , by pulling them back. But in fact, fn

convergence to f is a stronger sense which is almost surely. The proof of that, which is classicalresult in probability is called the Martingale convergence theorem.

Theorem 4 — Martingale Convergence Theorem. Let Xn be a uniformly bounded martingale.Then Xn converges a.s. to a limit X with E(|X |)< ∞.

In other words, for any f ∈ L1(X ,ν) and ν-almost every x, fn(x)→ f (x).

3. Stationary theory

3.1 Random Walks on GroupsFor us, a random walk is just a measure µ ∈ Prob(G). We say that µ is generating, if the semigroupgenerated by the support of µ is the whole G. In terms of the random walk, it says that any elementin G has a positive probability to be visited sometime by the random walk.

In other words, if µ is not generating, then the random walk will always miss some part of G.So we will always assume that µ is generating.

The distribution of the random walk ofter 2 steps is µ ∗µ = µ2, and the nth step is distributedaccording to µn. Formally, we cam consider the multiplication map G×G→ G given by (g,h) 7→g · · ·h. Then µ ∗µ is the push forward of the measure µ×µ on G×G to G.

The generalization to µn is clear. Now µ is generating if and only if

⋃n∈N

Supp(µn) = G

which is equivalent to: Supp(µ) is a generating set of G.

Example 3.1 — Simple Random Walk on Zd .

Example 3.2 — Simple random walk on F2.

Example 3.3 — Lamplighter group. Let G = ⊕ZZ/2ZoZ. In other words, consider thegroup⊕ZZ/2Z. The group Z naturally acts on this group by automorphisms which is just actingon the index set: z f (n) = f (n− z)

The group ⊕ZZ/2Z is call the lamps group.Composition in this group is given by ( f1,z1)( f2,z2) = ( f1 + z1 f2,z1 + z2).

16 Chapter 3. Stationary theory

Example 3.4 — SL2(Z). Let S =

(0 −11 0

)and T =

(1 10 1

)then < S,T >= SL2(Z).

Consider the simple random walk. Can you imagine where a typical random walk is goingto?

3.2 Stationary measuresDefinition 10 A measured G-space (X ,ν) is a (G,µ)-stationary or just (G,µ)-space, if it is aG-space and ν = µ ∗ν . That is, for any f ∈ L∞(X ,ν),

∫X f (x)dν(x) = ∑g µ(g)

∫X f (gx)dν(x).

Example 3.5 If G y (X ,m) is a measure preserving action then m is µ-stationary for any µ .

Example 3.6 — F2 acting on ∂F2. Let ∂F2 be the space of all infinte reduced words in theletters a,b,a−1,b−1. This space is a compact space when equipped with the product topology.This topology is metrizable and a commonly used metric is given by d(x,y) = 1

r where r is thefirst letter such that xr 6= yr.

The group F2 acts on ∂F2 by adding the finite word at the beginning, and canceling if thereis a need. It is not hard to check that this action is continuous and that ∂F2 has no invariantmeasure.

To define a measure on ∂F2 it is enough to say what are the measures of cylinders. Let wbe is a finite reduced word. We denote by [w] the set of all points in ∂F2 which start with w.Consider the following measure ν Prob(∂F2). ν([w]) = 1

4·3n−1 where n is the length of the wordw.

Then, for any [w],

µ ∗ν([w]) =34

14 ·3n +

14

14 ·3n−2 =

14

(1

4 ·3n−1 +3

4 ·3n−1

)=

14 ·3n−1 = ν([w])

Lecture 5

3.2.1 The Markov Operator and properties of stationary measuresFix a measure µ ∈ Prob(G), and let G y X be a continuous action. The Markov operatorMµ : C (X)→ C (X) is define by Mµ( f )(x) = ∑g µ(g) f (gx). The dual operator, M∗µ : Prob(X)→Prob(X) is M∗µ(θ) = ∑g µ(g)gθ .

Stationary measures are exactly the fixed points of the dual operator M∗µ .

Remark 4 If ν is µ-stationary then it is also µn-stationary:

(µ ∗µ)∗ν = µ ∗ (µ ∗ν) = µ ∗ν = ν

With less symbols - let (X ,ν) be a stationary space, and find a compact model of X . Now ν is afixed point of M∗µ , and hence also it is a fixed point of any power of it.

Lemma 2 Any ergodic µ-stationary measure which is atomic-free, unless the space is finite and isinvariant.

Proof. Assume that there is an atom x with maximal ν measure for a stationary measure ν . Theorbit Gx is G-invariant with positive measure and by ergodicity ν(Gx) = 1. So we deal with acountable space X equipped with a transitive G-action.

Since ν is stationary, ν (x) = ∑g µ (g)ν (gx) but since ν (x) is the maximal value, ν (x) = ν (gx)for every g ∈ Supp(µ). Since ν is µn-stationary for any n, it holds for any g ∈

⋃n Supp(µn) = G.

3.2 Stationary measures 17

It follows that the measure ν is the uniform measure on the finite set Gx, hence, in particular it isinvariant.

Lemma 3 Every µ-stationary action is a non-singular action (where as usual, µ is a generatingmeasures).

Proof. We need to show that ν (A) = 0 if and only if gν (A) = 0 for all g ∈ G. Assume thatν (A) = 0. Then

0 = ν (A) = µ ∗ν (A) = ∑g

µ(g)ν (A)

so gν(A) = 0 for every g ∈ Supp(µ).For other g’s apply this argument on µn such that g ∈ Supp(µn). Note that for any g such n

exists by the generating assumption.

Corollary 3.2.1 The support of a stationary measure is an invariant set.

Lemma 4 Every compact G-space admits a stationary measure.

Proof. This follows by the amenability of N, when thinking of the action of the dual Markovoperator as an N-action.

Take some arbitrary θ ∈Prob(X) and consider the sequence νn =1n

(θ +µ ∗θ + · · ·+µn−1 ∗θ

).

The compactness assets that there exists a limiting measure ν (for a subsequence).

µ ∗νn−νn =1n(µ ∗θ + · · ·+µ

n ∗θ)− 1n

(θ +µ ∗θ + · · ·+µ

n−1 ∗θ

)=

1n(µn ∗θ −θ)→ 0

which shows that any accumulation point of νn is µ-stationary.Note: if X is metrizable, so the topology is separable. It means that the Borel structure is

separable and then measure, in particular, the stationary that we found is standard. If X is crazy, ν

is not necessarily standard.

A topological action G y X is a minimal if X has no G-invariant subset, or equivalently, everyorbit is dense.

Corollary 3.2.2 Let G y X be a continuous action, and assume that X has a unique stationarymeasure. Then X has at most one minimal component.

Proof. The support of the stationary measure is the unique minimal component, since in eachminimal component there is a stationary measure.

This shows how the existence of stationary measures can be used to study the topological action.Note that for amenable groups, the same is true for invariant measures, when the group is amenable.


Remark about amenabilityIn classical ergodic theory, the study of limiting behaviors of an action G goes though the under-standing of invariant measures. However, invariant measures are not always exist. In facts, a groupG that any compact G-space admits an invariant measure is called amenable. Hence, the classicaltheory of a single transformation (Z or R action) was generalized to the action of amenable groupsduring the 20th century.

Definition 11 A group G is amenable if there exist an invariant measure on any compact G-space.

By definition, a non amenable group admits a compact G-space the classical theory to theseactions. In some sense, stationary measures without any invariant measure, and it is not clear howto generalize are good generalizations of invariant measures: there always exist, and yet they aremeaningful.

Claim 1 Let G be a non-amenable group. Then for any generating µ , there are µ-stationarymeasures which are not invariant.

Remark 5 In fact, this characterizes amenability: a group is amenable if and only if there exists agenerating measure µ such that all µ-stationary measures are already invariant.

The question what groups have the property that every µ-stationary measure is invariant forevery µ is still open. We will see soon examples of such groups.

Lecture 6

3.2.2 Kakutani ergodic theoremGiven X be a measurable G-space, we can consider the space X ×GN, and the skew-productmap T : X ×GN→ X ×GN , defined by T (x,(g1,g2, . . .)) = (g1x,(g2,g3, . . .)). Then T is a non-invertible measurable map.

Given a probability measure ν on X we equipped with the probability measure µN×ν .

Lemma 5 The measure ν is µ-stationary if and only if µN×ν is T -invariant.

Proof. Consider the factor map (which is not a G-map!) ϕ : X ×GN→ X . So ϕ ∗ (ν×µN) = ν .For any f ∈ L∞(X×GN,ν×µN), let f ∈ L∞(X ,ν) be f = ϕ∗( f ). We get that

ν( f ) = ν(ϕ∗ f ) = ϕ∗(ν×µN)(ϕ∗ f )) = ν×µ

N( f ).

Also,

µ ∗ν( f ) = ∑µ(g)∫

f (gx)dν(x) = ∑µ(g)∫

X×GNf (gx,(g2, . . .))dν×µ

N(x,(g2, . . .)) =

= ∑µ(g)∫

X×GNf (T (x,(g,g2, . . .)))dν×µ

N(x,(g2,g3, . . .))

=∫

X×GNf (T (x,(g,g1,g2, . . .)))dν×µ

N(x,(g,g2,g3, . . .))

= T∗(ν×µN)( f ).

It follows that ν×µN is T -invariant if and only if ν is µ-stationary

Theorem 3.2.3 Let (X ,ν) be µ-stationary space. The following are equivalent:1. The measures ν is ergodic2. The Markov operator Mµ on Lp(X ,ν), for all 1 ≤ p ≤ ∞ is ergodic. That is, the only

Pµ -invariant functions are the constant functions.

3.2 Stationary measures 19

3. The skew-product (X×GN,ν×µN,T ) is ergodic.

Proof. (1) =⇒ (2): For a given Mµ -invariant function f ∈ Lp(X ,ν), we show that set on whichf (x)≥ 0 is a G-invariant set, hence has trivial measure. By repeating this argument on f − c forany c we get the function f is constant.

So let f be Mµ -invariant, and assume further, that f+(x) = max f ,0(x) is also Mµ -invariant.Let E ⊂ X be a set with ν(E) = 1 on which both f and f+ are defined. Since the group g iscountable, the set E ′ =

⋂g gE is still of measure 1 (here we use that ν is non-singular). Now

let P′ = x ∈ E ′| f (x) ≥ 0. Every x ∈ P′ satisfies two equations: f (x) = ∑ µ(g) f (gx) and alsof (x) = ∑ µ(g) f+(gx). So ∑ µ(g)( f+(gx)− f (gx)) = 0 and since these are non negative terms,f+(gx) = f (gx) for any x ∈ P′ and g ∈ Supp(µ). Meaning that P′ is g invariant for every g ∈Supp(µ). Since the support contains a generating set, P′ is actually G-invariant, and by ergodicityhas a trivial ν-measure.

Now we show that f+ in Mµ -invariant. Since max f (x),0= f (x)+| f (x)|2 , we only need to show

that | f | is also Mµ .Note that | f |(x) ≤ |Mµ( f )|(x) and so Mµ(| f |)− | f | is a non-negative function in Lp(X ,ν).

Now, ∫X

Mµ(| f |)−| f |dν =∫

XMµ |( f )|dν−

∫X| f |dν

Since ν is µ-stationary, that is (Mµ)∗-invariant, this expression equals 0. As it is a non negative

function with 0 integral, we conclude that for ν-almost every x, Mµ(| f |)(x) = | f |(x).(2) =⇒ (3): Assume that Mµ is ergodic and let f ∈ L∞(X ×GN) be T -invariant. For any n

consider Bn the sigma algebra on GN generated by the first n letters. So given f we can push it toget fn =E( f |Bn). As the σ -algebras are nested, we get that fn =E( f |Bn) =E(E( f |Bn+1)|Bn) =E( fn+1|Bn).

As the functions fn depend only on the X coordinate and the first n coordinate of GN, we candefine functions fn on X×Gn by

fn(x,(g1,g2, . . .gn)) = fn(x,(g1,g2, . . . ,gn)) =∫

f (x,g1,g2, . . .)dµN(gn+1,gn+2, . . .).

Note that if we feed fn with a sequence g1,g2, . . .gn which cannot happen in the random walk,say if g1 6∈ Supp(µ) then fn would give 0.

Since f is T -invariant, we get that for any n,

fn(x,(g1,g2, . . .gn)) =∫

f (x,(g1,g2, . . .))dµN(gn+1,gn+2, . . .)

=∫

f (g1x,(g2, . . .))dµN(gn+1,gn+2, . . .)

= fn−1(g1x,(g2,g3, . . .gn))

and sofn(x,(g1,g2, . . .gn)) = f0(gngn−1 · · ·g1x)

In particular, for n = 1 we get that

f0(x) = E( f1|B0)(x) = ∑µ(g1) f1(x,(g1)) = ∑µ(g1) f0(g1x) = Mµ( f0)(x).

Since Mµ is ergodic, f0 = c is constant. Since f1 = E( f2|B1) the same argument shows that f1is constant function. As f0(x) = E( f1|B0)(x) we conclude that in fact f1 = f0 = c. The applies toany n, so we get that fn = c for any n.


Now by the Martingale convergence theorem, f is also the constant function c. In fact, we don’tneed the almost surely convergence result, and the soft, L2 result is enough. Since we know that allthe projections are constant, so‖ f − fn‖2→ 0 says that‖ f − c‖= 0 hence f = c almost surely.

(3) =⇒ (2): Any Mµ -invariant function in Lp(X ,ν) can be lifted to an Lp-function on theskew product, which is T -invariant and constant on GN, which shows that the skew product is notergodic.

(2) =⇒ (1): Any G-invariant set is Mµ -invariant, hence the existence of a G-invariant ofnon-trivial measure, implies that Mµ is not ergodic.

Lecture 7Corollary 3.2.4 — Kakutani ergodic theorem. Let (X ,ν) be an ergodic µ-stationary action.Then for any f ∈ L1(X ,ν) and µN-almost every (g1,g2,g3, . . .),

limn→∞

1n

n

∑k=1

f (gkgk−1 · · ·g2g1x) =∫

Xf (x)dν(x)

Proof. By the previous theorem we get that GN × ν is T -ergodic measure. Take a functionf ∈ L1(X ,ν) and lift it to a function f (x,(g1,g2, . . .)) = f (x), and apply the classical Birkhoffergodic theorem to get∫

Xf (x)dν(x) = lim

n→∞

1n

n

∑k=1

f (T k(x,(g1,g2, . . .))

= limn→∞

1n

n

∑k=1

f (gkgk−1 · · ·g2g1x,(gk+1,gk+2, . . .)

= limn→∞

1n

n

∑k=1

f (gkgk−1 · · ·g2g1x)

Remark 6 As invariant measures are stationary, this works also for invariant measures!It is in particular interesting when the group is non-amenable, where there is a difficulty to have

a natural way to take averages.

Corollary 3.2.5 Let X be compact, and let ν ∈ Prob(X) be an ergodic stationary measure. If η

is stationary ergodic and η << ν then η = ν .

Proof. Let f ∈C(X) be some function. Assume that η << ν , and let E ⊂ X be such that η(E)> 0and hence also ν(E)> 0. So there are E1 ⊂ E with ν(E1) = ν(E) and Ω1 ⊂ GN with µN(Ω1) = 1such that for any x ∈ E1 and (g1,g2, . . .) ∈Ω1, the ergodic theorem holds for ν .

Find E2 ⊂ E with η(E2) = η(E) and Ω2 with µN(Ω2) = 1 for which the theorem holds for η .Now apply for any point in E ′ = E1∩E2 and Ω′ = Ω1∩Ω2 to get

limn→∞

1n

n

∑k=1

f (gkgk−1 · · ·g2g1x) =∫

Xf (x)dν(x) =

∫X

f (x)dη(x).

Let X be a compact G-space. Denote by Probµ(X) to be the set of all µ-stationary measures onX . Then ProbG(X)⊂ Probµ(X)⊂ Prob(X).

3.3 Harmonic Functions 21

Exercise 3.1 Probµ(X) is a compact convex space.

Solution Show that the convolution operator (the dual of the Markov operator) is a continuousfunction.

Corollary 3.2.6 A µ-stationary is ergodic if and only if it is an extremal point in Probµ(X).

Proof. If A is an invariant set of non-trivial ν-measure, then the we can express ν as a convexcombination of the (normalized) restriction to A and the (normalized) restriction to the complementof A (one should verify that these are indeed stationary measures).

Now assume that ν is ergodic, and assume that ν = cη +(1− c)θ where η and θ are ergodic,stationary and 0 < c < 1. So η << ν in contradiction.

Here is the topological analogue of that: a topological action G y X is said to be minimal ifevery x, has a dense orbit (that is, G · x= X). The content of the following theorem that walksfrom (GN,µN) can be taken to revel the density.

Theorem 3.2.7 Let G y X be a continuous action on a compact metric space, and let µ ∈Prob(G) be generating. Then for µN-almost every walk g1,g2, . . . , and every x, the set gngn−1 · · ·g2g1xis dense in X .

TODO: write a proofProof.

3.2.3 Random Walks from dynamical perspectiveIn probability theory it is common to this of a random walk as a sequence of random variables.Sometimes it is useful to consider it in this way, and sometimes we will prefer the The space ofwalks and the Markovian measure.

Consider GN, the space of all sequences in G equipped with the product sigma algebra. Considerthe multiplication map m : GN → GN defined by (m(w))n = w1 ·w2 · · ·wn. The push-forwardmeasure Pµ = m∗µN is called the Markovian measure.

When GN is equipped with the product measure µN then the coordinates represents the incre-ments of the random walk.

When equipped with Pµ , we think of GN is the space of walks of the random walk, as the wn isthe position of the random walk at time n. To distinguish between the two, we denote the space ofwalks by (Ω,Pµ).

For example, consider the set of all paths ω such that e appears in infinitely many coordinates.If the measure of this set is 1 then the random walk is recurrent.

TODO: add: (Ω,Pµ) is a G-space, while (GN,µN) is not

3.3 Harmonic Functions

Definition 12 A function h : G→ R is µ-harmonic if

h(g) = ∑γ

µ (γ)h(gγ) .


Exercise 3.2 Find all µ-harmonic functions (bounded and unbounded) on Z where µ is thesimple random walk on Z. Conclude that the only bounded harmonic functions are the constantfunctions.

Exercise 3.3 Find a harmonic function for the simple random walk on F2.

We will be only interested here in bounded harmonic functions, that is functions in `∞(G) whichare harmonic. Denote the Banach space of all bounded harmonic functions by H∞(G,µ). Note thatH∞(G,µ) is never empty - it at least contains the constant functions. So H∞(G,µ) is s sub vectorspace of the Banach space `∞(G). It is also topologically closed. In other words, the sup norm onH∞(G,µ) induces a complete topology, so it is a Banach space.

Note that `∞(G) has another structure - it is a algebra, when considering the pointwise mul-tiplication. Pointwise multiplication of harmonic functions is not harmonic, so H∞(G,µ) is asub-Banach space, but not a sub-algebra.

Let h ∈ H∞(G,µ) be a bounded harmonic function.

Lemma 6 The limit limn→∞

h(wn) exists for Pµ -a.e. w ∈ Ω. So any h ∈ H∞(G,µ) corresponds to

some h ∈ L∞(Ω,Pµ), and

h(e) =∫Ω

h(w)dPµ(w).

Moreover, for a fixed g the limit limn→∞

h(gwn) = h(g,w) exists and

h(g) =∫Ω

h(g,w)dPµ(w).

Proof. Let Bn be the sigma algebra generated by the first n letters in Ω, including B0 which is thetrivial σ -algebra. Then Bn→B. Now define fn(w) = h(wn) and f0 = h(e). Then fn forms abounded martingale:

E( fn+1|Bn)(w) = ∑G

µ(γ)h(wnγ) = h(wn) = fn(w).

By the martingale it converges for a.e. w ∈ Ω. Once we know that h exists, then h(e) is theconditional expectation of h, with respect hence the equality.

For the “moreover“, fix g and define fn(w) = h(gwn) (and f0 = h(g)).

Remark 7 The meaning of the preceding lemma, is that if one want to understand something aboutlimiting behavior of a random walk - it is useful to consider bounded harmonic functions as theyhave an opinion of almost every walk.

Some idea is to say that two walks convergence to the same “direction” if all the boundedharmonic functions cannot distinguish between them. That is, define an equivalence relation onwalks, w ∼ w′ if limh(wn) = limh(w′n) for all h. However, maybe there are other functions thathave an opinion? We will deal with this questions when we will construct the Furstenberg-Poissonboundary.

Where do harmonic functions are coming from?

Lecture 8

3.3 Harmonic Functions 23

3.3.1 The Poisson TransformOne of the most natural source of G-spaces, is spaces of functions over G. For f : G→ X (whereX is some set) we write g. f (γ) = f g (γ) = f (gγ). This is a right-action: f g1g2 (γ) = f (g1g2γ) =f g1 (g2γ) = ( f g1)g2 (γ) so as functions f g1g2 = ( f g1)g2 .

More generally, if G acts on some space X , then G acts naturally on spaces of function over X .For example, consider L∞ (X ,ν), where X is a G-space, by again g. f (x) = f g(x) = f (gx).

Definition 13 Let (X ,ν) be a G-space. The Poisson transform is the map Pν : L∞(X ,ν)→ `∞(G)given by Pν( f )(g) = gν( f ) =

∫X f (gx)dν(x).

Lemma 7 The Poisson transform Pν : L∞(X ,ν)→ `∞(G) is a unital (sends the constant function1 to 1), positive (sends non-negative function on X to non negative functions on G), linear andG-equivariant.

Proof. Linearity and positivity are clear from since it is defined by integral, and it is unital since ν

and all the gν are probability measures.G-equivariance:

Pν( f γ)(g) =∫

Xf γ(gx)dν(x) =

∫X

f (γgx)dν(x) = Pν( f )(γg).

We just saw that any ν ∈ Prob(X), where X is a compact G-sapce gives a unital positive linearmap C(X)→ `∞(G).

Lemma 8 Let G y X be a continuous action, and let P : C(X)→ `∞(G) be a unital, positive andlinear equivariant map. Then there exists some ν ∈ Prob(X) such that Pν = P .

Hence we identified between Prob(X) and the space of unital positive linear equivariant maps.

Proof. For any continuous function f ∈C(X) define ν( f ) =P( f )(e). Note that ν is linear positivemap on C(X) with ν(1) = 1 and hence is a probability measure.

It is clear that H∞(G,µ) is a sub vector space of `∞(G), and we now observe that it is G-invariant. The key here is that the hamonicity condition is on the right side of the argument, whilethe action affects the left side:

Let h ∈ H∞(G,µ) and fix some g0 ∈ G. Then

∑γ

µ(γ)hg0(gγ) = ∑γ

µ(γ)h(g0gγ) = h(g0g) = hg0(g)

so hg0 is also µ-harmonic, since it is clearly bounded, we get that so G y H∞(G,µ).

Lemma 9 Let (X ,ν) be a G-space. The measure ν is µ-stationary if and only if the image of Pν

is in H∞(G,µ).The measure ν is invariant if and only if the image of Pν are the constant functions.

Proof. Let f ∈ L∞(X ,ν), and let h = Pν( f ). Then h(g) = Pν( f )(g) = gν( f ) and

∑γ

µ(γ) ·h(gγ) = ∑γ

µ(γ) ·Pν( f )(gγ)

= ∑γ

µ(γ) ·gγν( f )

= g

(∑γ

µ(γ) · γν

)( f )

= gµ ∗ν( f )


So h is harmonic if and only if µ ∗ν = ν .The statement about invariant measures is clear.

3.4 Choquet-Deny theorem

Theorem 5 — Choquet-Deny. Let G be an abelian group, and let µ ∈ Prob(G) be some generatingmeasure. Then H∞(G,µ) consists of only constant functions.

In particular, any µ-stationary measure, for any µ on an abelian group is invariant.

Proof. Let K be the set of all µ-harmonic function which are bounded between 0 and 1. K is closedin the space of all real functions on G so it is compact and convex. By Krein-Milman K is theclosure of the convex hull of its extremal points, so it enough to show that that all the extremalfunctions are constant.

Recall that G acts on H∞(G,µ) from the left. Since G is abelian, the right G-action hg(γ) =h(gγ) also preserves harmonicity, hence preserves K. Let h be extremal in K. Then

h(g) = ∑γ

µ(γ) ·h(gγ) = ∑γ

µ(γ) ·h(γg) = ∑γ

µ(γ) ·hγ(g)

and so as functions, h = ∑γ µ(γ)hγ but h is extremal - so hγ = h for all γ ∈ Supp(µ). Since ν is µn

stationary and µ is generating, h is a constant function.

Exercise 3.4 Let µ be a generating measure on a nilpotent group G. Show that any µ-stationarymeasure is invariant.

Hint: Show first that any bounded harmonic function must be constant on the center of G.

Solution Todo

3.5 Conditional Measures

Note that in the following we consider a compact, metrizable stationary space.

Lemma 10 Let X be a compact metrizable G-space, and let ν ∈ Prob(X) be a µ-stationary measure.Then the limit limn→∞ wnν = νw exists for a.e. w ∈Ω, ν =

∫Ω

νwdPµ(w).Moreover, for any fixed g and Pµ -a.e. w the limit limn→∞ gwnν = νgw exists, and gνw = νgw.

Proof. Since X is metrizable, C(X) is separable, and let f1, f2, . . . be a countable set of functionswith ‖ fk‖ = 1 such that their span is dense in C(X). For each fk, there exists a subset Ωk ⊂ Ω

with Pµ(Ωk) = 1 on which the harmonic function Pν( f ) has a limit Pν( f )(wn) = limwnν( f ).Let Ω0 =

⋂k Ωk so it is a set of Pµ -measure one on which we can define for any w ∈Ω0 a linear

functional on the set fk. These functionals are bounded, hence continuous and so they extend tolinear functionals νw on C(X).

Lecture 9

3.6 On the category of stationary actions

Lets discuss some functorial properties of stationary spaces. We saw that a factor of a stationarymeasure is stationary.

Lemma 11 A factor of a stationary space is a stationary space.

3.6 On the category of stationary actions 25

Proof. Let (X ,ν)π−→ (Y,η) and assume that ν is µ-stationary. Then

µ ∗η = µ ∗πη = π (µ ∗ν) = πν = η .

Theorem 6 — (Naturality of the conditional measures. Let π : (X ,ν)→ (Y,η) be a G-factormeasurable map between two compact metrizable space. Then for Pµ -almost every w, π∗νw = ηw.

In particular, the conditional measures are measurable object, and do not depend on the compactmodel, as one might think. Indeed, if (X ,ν) and (Y,η) are two compact model of the sameabstract stationary action, then the measurable isomorphism π : X → Y form an isomorphism ofπ : (X ,νw)→ (Y,ηw) for Pµ -almost every w.

Note that this isomorphims is not a G-isomorphism (G doesn’t act on a single conditionalmeasure, but rather, takes one conditional to another one). Hence this isomorphism is as measurespaces, which is not a very strong condition. It does imply, for example, that the number of atoms,is the same, and this is independent on the compact model.

Remark 8 — Stationary measures as boundary maps. The existence of conditional measures,shows that a stationary measure on a compact metrizable space X can be thought of as a measurablemaps b : (Ω,Pµ)→ Prob(X). It is clear that b is shift invariant as νw is defined by a limit.

Since we know that any compact space admits a stationary measure, we get that such a map bis defined to Prob(X) for any compact metrizable G-space X .

The naturality results shows that if we have a factor map X → Y and instead of thinking ofstationary measures ν and η , but rather as maps bX and bY , then bY (w) = π(bX(w)).

We will discuss it further but already now we can observe some connection with amenability.Note that G acts on (Ω,Pµ) by “starting from g instead of e”, and that b is equivariant. Also, on(Ω,Pµ) the shift acts, and b is actually shift invariant. Hence if (Ω,Pµ) is trivial after modding outby the shift, then we get an invariant measure on any compact space, which implies that the groupis amenable!

If not, one can think of it as a nice replacement.

What about products? In general, if we have G y X and G y Y then the diagonal actionG y X×Y is defined by g(x,y) = (gx,gy).

Definition 14 Let (X ,ν) and (Y,η) be two (G,µ)-stationary space. A joining of ν and η is aµ-stationary measure νgη ∈ Prob(X×Y ) which ν and η as the projection. In a diagram,

(X×Y,νgη)π1

xx

π2

&&(X ,ν) (Y,η)

This is closely related to the notion of coupling in probability theory.Usually in ergodic theory when discussing invariant measures, a joining of invariant measures,

is an invariant measure on the product which extends both measure. In the invariant measure setup,there is a natural joining which is the product measure ν×η . Indeed g(ν×η) = gν×gη = ν×η .

In general, if we have two stationary actions, the diagonal action on the product space is notstationary anymore. The reason is, after thinking of the Poisson transform, that multiplication ofharmonic functions is not harmonic. In any case, any two stationary actions can be joined. Choosea compact model for X and for Y , and consider the subspace of all measures in Prob(X×Y ) whichprojects on ν and η . This is a (weak*-)compact convex space hence the dual of the Markov operator


has a fixed point there. More explicitly, start from some measure there, θ and take a weak*-limit ofthe averages of the convolutions of µ with θ .

But this is still quite abstract, the following is a hands-on construction when the stationarymeasures are standard. Recall that the compact model theorem says that when dealing with abstractstandard space, one can always find a metrizable compact model, and hence to get conditionalmeasures to work with.

The canonical joiningAny two stationary actions on standard spaces can be naturally joined using the following measure:

λ =∫Ω

νw×ηwdPµ(w).

Indeed, for each g ∈ G we have

gλ =∫Ω

gνw×gηwdPµ(w)

=∫Ω

νgw×ηgwdPµ(w)

=∫Ω

νw×ηwdgPµ(w)

where (gw)n = gwn. Hence we get that

∑g

µ(g) ·gλ = ∑g

µ(g) · (∫Ω

νw×ηwdgPµ(w))

=∫Ω

νw×ηwdPµ(w) = λ .

This construction becomes even more natural when considering stationary measure as boundarymaps. Instead of thinking about the stationary measures on X and on Y , consider the mapsbX : (Ω,Pµ)→ X and bY : (Ω,Pµ)→ Y . The first boundary map to X ×Y that comes to mind isbX ×bY : (Ω,Pµ)→ X×Y , which gives the measure λ above.

Under some conditions on the stationary spaces, this natural joining is also the unique one, butin general, there are many joinings of given two stationary measures.

Lemma 12 If ν is measure preserving then νgη = ν×η .

Proof.

νgη =∫Ω

νw×ηwdPµ (ω)

=∫Ω

ν×ηwdPµ (w)

= ν×∫

ηwdPµ (w)

= ν×η .

4. Boundary theory

4.1 Compact µ-boundaries

Definition 15 A compact (G,µ)-stationary, (B,ν) is called a compact-(G,µ)-boundary if νw =δbnd(w) is a Dirac measure for a.e. w ∈Ω.

Recall that a given a stationary measure ν on metrizable compact spaces yields to a boundarymap bnd : (Ω,Pµ)→ Prob(X). If ν happen to be µ-boundary, then we get a map bnd : (Ω,Pµ)→X ,and bnd∗Pµ = ν .

Example The trivial space (one point space), is a µ-boundary for any µ on any G.

Example The free group.We need to show that for any set, and almost every w, wnν(E)→0,1 It is enough to show it

for cylinder sets. So let [u] be a cylinder, so u is a word of length m. Since the simple random walkis transient, for almost every w, there exists a time t(w) when the random walk won’t visit the ballof radius 2m anymore. We get a map, defined almost everywhere, t : (Ω,Pµ)→ N such that the2m-prefix in wn, for all n > t(w) are the same word, say v. Now

wnν([u]) = ν((wn)−1[u]) = ν(g−1

n · · ·g−1m+1v−1[u])

Recall that ν ∈ Prob

Lemma 13 Compact boundaries are ergodic.

Proof. Let E ⊂ B be an invariant set. Then

νw (E) = limn→∞

wnν (E) = limn→∞

ν

((wn)

−1 E)= ν (E)

And since νω is a Dirac measure, ν (E) is either 0 or 1.

Lemma 14 Let (B,ν) be a compact boundary. Then the Poisson transform Pν : L∞(B,ν)→ `∞(G)is an isometry onto its image.

28 Chapter 4. Boundary theory

Proof. Note that any non-trivial µ-boundary satisfies the following property (known as the SATproperty): For any E with ν(E)> 0 and ε > 0, there exists some g such that ν(gE)> 1− ε .

Indeed, since ν = bnd∗Pµ we get that the preimage of E is a positive measure set of walks forwhich every w, limwnν(E) = δbnd(w)(E) = 1.

Now for characterstic functions, it follows that∥∥Pν(1E)

∥∥∞= 1. As the chracteristic functions

span a dense set in L∞ (B,ν), we conclude that Pν is an isometry.

Lemma 15 A measurable G-factor of a compact boundary is a compact boundary.

Proof. Let (X ,ν)π−→ (Y,η) both are compact, where (X ,ν) is a boundary. Then ηw = π(νw) for

Pµ -almost every w. Since the image of Dirac is Dirac, ηw are Dirac measures almost surely.

The name boundary comes from the idea that we can attached (B,ν) to the group G: Considerthe space G∪B, with a topology, that a sequence ω in G convergences to b ∈ B if bnd(ω) = b.

Claim 2 If (X ,ν) is a (G,µ)-boundary and ν is an invariant measure then (X ,ν) is trivial.

Proof. Since ν is invariant, νω = ν , and since (X ,ν) is a boundary, ν = νω is a Dirac measure.

Any µ-boundary gives some information on the µ-random walk - if bnd(ω1) 6= bnd(ω2) thenwe think of the walks as converging to different points at infinity. The trivial boundary does notdistinguish between the walks - they all goes to infinity.

Thus, we will be interested in finding the largest boundary possible. This boundary is theFurstenberg-Poisson boundary.

4.1.1 Joining of compact boundariesLemma 16 If (X ,ν) is a compact (G,µ)-boundary then νgη is the unique joining of ν and η .

Proof. Let λ be a some joining of ν and η . Then the conditional measures λw have marginalsπ1∗ (λw) = δb(w) and π2∗ (λw) = ηw. Since π1 (λw) is a Dirac measure, we get that λw = δb(w)×ηw

and so

λ =∫Ω

λwdPµ(w) =∫Ω

δb(w)×ηwdPµ(w) = νgη .

We get now a strong ergodic property of boundaries:

Corollary 1 Let (X ,ν) be a µ-boundary, and let (Y,m) be an ergodic measure preserving action.Then (X×Y,ν×m) is ergodic.

Proof. Pick compact models and consider the subspace K in Prob(X×Y ) consists of all stationarymeasures which projects on ν and m respectively. Then K is compact and convex.

We claim that a measures in an extremal point in K if and only if it is ergodic.Now since ν is a compact µ-boundary, there is a unique stationary joining, hence K is a single

point, which must be extremal and hence ergodic.

If the measure ν was an invariant measure, rather than stationary, then such a measure is calledweakly mixing.

Corollary 2 The only joining of a measure preserving system and a boundary is the product. Itmeans that in a sense, these properties are orthogonal to each other.

4.2 The Furstenberg-Poisson boundary 29

Corollary 3 The joining of two compact (G,µ)-boundaries is a compact (G,µ)-boundary.

Corollary 4 There exists a universal (G,µ) boundary (Π,ν): every (G,µ)-boundary is a factor if(Π,ν). The universal boundary called the Furstenberg-Poisson boundary.

4.2 The Furstenberg-Poisson boundary

Definition 16 An abstract (G,µ)-space, (B,ν) is called a (G,µ)-boundary if there exist a compactmodel of (B,ν) which is a compact boundary.

Theorem [Furstenberg ’73] Let (G,µ) be a locally compact second countable group with admis-sible measure. There exists a uniquely defined universal boundary called the Poisson boundary,(Π,ν), in the sense that any other boundary is a G-factor of (Π,ν).

By uniqueness we mean: any two universal boundaries are G-isomorphic.

4.2.1 Construction via harmonic functions

Real commutative C∗-algebrasLet A be a unital commutative real C∗-algebra is a commutative Banach algebra. That is,

A is a normed linear space over R with a commutative multiplication and a unit element and∥∥∥x2∥∥∥= ‖x‖2 for x ∈A .

An example of such an object is C (K) where K is compact Hausdorff space (note that notnecessarily metrizable).

This is indeed the only the source for examples: Gelfand-Naimark theorem relates these twocategories, of real C∗-algebra and compact spaces. More precisely, it says that for every suchalgebra A there exists a compact space K such that C (K) isometrically ∗-isomorphic to A (whenC (K) equipped with the sup norm: ‖ f‖

∞= supk∈K

∣∣ f (k)∣∣). The algebra A is separable if and onlyif K is metrizable. K is called the spectrum or the Gelfand space of A .

This correspondence between the categories of C∗-algebras and compact spaces, goes throughas follows:

There is a correspondence between closed sub algebra S ≤ A (containing the unit) andcontinuous onto maps K1→ K2 where K1 and K2 are the spectrums of A and S respectively.

Stone-Banach theoremAnother theorem that we will use is Stone-Banach. Let X ,Y compact spaces. If π : X → Y is a

homemorphism then it defines a function π∗ : C (Y )→ C (X) by π∗ ( f )(x) = f(π (x)

). Note that

π∗ is an isomerty and π∗ (1Y ) = 1X .Stone-Banach gives the other direction: if ϕ : C (Y )→ C (X) is an surjective linear isomerty

ϕ (1Y ) = 1X then it is induced by a homeomorphism π : X → Y .We will use it as follows: if G acts on C (X) by isometries then G acts on X .

4.2.2 The construction of Πtop

To avoid some technicalities, we will prove it to discrete G.Recall that the Martingale convergence theorem asserts that any bounded harmonic function

has an opinion on almost every walk ω ∈ (G,µ)N = (Ω,P) and we asked whether there are morefunctions with this property.

Definition Denote by B(G) the Banach the space of all bounded functions on G.

Let A be the algebra of all f ∈ B(G) such that the limit limn→∞

f (gwn) = f (g,w) exists for everyg ∈ G and P-a.e. ω ∈Ω.• We saw that H = H ∞ (G,µ)⊆A .


• Unlike H , A is an algebra: the pointwise multiplication preserves the existence of the limita.e., the norm is the sup norm.• A This is also a G-space: the action f g (γ) = f (gγ) preserves the condition that the limit

exists for all g and a.e. ω .Let I be the set of all functions in A that the limit exists and equal to zero for every g ∈ G andP-a.e. ω ∈Ω.• I is an ideal.• The G-action preserves I .

Thus, G acts on the C∗-algebra A /I !

Lemma A = H ⊕I .

Proof. First, H ∩I = 0 since we saw that h(g) =∫

Ωf (g,ω)dPµ(w) so if f (g,w) = 0 for

almost every w then h(g) = 0.Given f ∈A define h f (g) =

∫f (g,w)dPµ(w). So h f is harmonic:

∑γ∈G

µ (γ)h f (gγ) = ∑γ∈G

µ(γ)∫

Ω

f (gγ,w)dPµ(w)

= ∑γ∈G

µ(γ)∫Ω

f (g,γw)dPµ(w)

=∫Ω

f (g,w)dPµ(w) = h f (g)

f −h f ∈I :

h f (gwn) =∫Ω

f (gwn,w′)dPµ(w′)

= E(

f |g,w1,w2, . . .wn

)Hence limh f (gwn) = f (g,w) hence f −h f indeed in I .

Remark 9 This lemma can be consider as an if-and-only-if version of the martingale convergencetheorem. Being bounded harmonic is not just a good reason for a.s. convergence but essentially theonly reason for a bounded function to converge.

Corollary 5 H is not just a Banach space - there is a multiplication that makes it an algebra, witha G-action on it.

This multiplication is the pointwise multiplication modulo I . That is, h1 · h2 = h whereh1h2 = h+ i.

Explicitly,Thus, H admits a spectrum Πtop: this is a compact space such that H ≈ C

(Πtop

)(Gelfand

representation). For h ∈H , denote by h the image by Gelfand.

Πtop is a G-space

Since H is a G-space, we can push the G-action forward to C(Πtop

). That is, any g ∈ G defines

an automorphism of C(Πtop

). Such an automorphism always induced (Stone-Banach) by some

homeomorphisms of Πtop. This defined a G-action on Πtop.

4.2 The Furstenberg-Poisson boundary 31

The measure ν

Until now we have a G-sapce. We are looking for a measure on the space.Define for each g ∈ G, Lg : H ∞→ R by

Lg (h) = h(g) =∫Ω

h(g,ω)dP(ω) .

In words, given h, and an element g integrate the limiting values of h along all the future conditionedthat we start to walk from g.

It induces a positive linear function Lg on C(Πtop

)and by Riesz, we have a νg ∈P

(Πtop

)such that for h ∈ C

(Πtop

),

νg(h)=∫

Πtop

h(x)dνg (x) = Lg(h)= Lg (h) =

∫Ω

h(g,ω)dP(ω) = h(g)

Properties of these measures:• The G-action is gνe = νg

gνe(h)= νe

(hg)= νe

(hg)= hg (e) = h(g) = νg

(h)

• ν = νe is stationary.ν(h)= h(e) = ∑

γ∈Gµ (γ)h(γ) = ∑

γ∈Gµ (γ)νγ

(h)= µ ∗ν

(h)

• Gelfand=PoissonWith this measure, the Gelfand representation h↔ h and the Poisson transform coincide: Forh we need to show that h = Pν(h).

Pν(h)(g) = gν(h)= νg

(h)= h(g)

•(Πtop,ν

)is a boundary.

Fix w such that νw exists, and let h ∈ C(Πtop

).∫

Πtop

h(x)dνw(x) = limn→∞

∫h(x)d(wnν)(x)

= limn→∞

Lwn

(h)= lim

n→∞Lwn (h)

= limn→∞

h(wn)

Apply the last equations on h2 to get

limn→∞

h2(wn) =∫

Πtop

h2(x)dνw(x).

Recall that the Gelfand map h↔ h is multiplicative. This is actually a tautology: we definedthe multiplication on H by this map. Thus,

limn→∞

h2(wn) = ( limn→∞

h(wn))2 =

∫Πtop

h(x)dνw (x)

2

But from Cauchy-Schwartz we get


∫Πtop

(h(x)

)2 dνω (x)≥

∫Πtop

h(x)dνω (x)

2

So we have an equality in Cauchy-Schwartz and so h must be constant on Supp(νω). But it holdsfor any h ∈ C

(Πtop

)so νω is a Dirac measure.

Universality and Uniqueness

Let (B,η) be a compact boundary. Then we showed that the Poisson transform

Pν : L∞(B,η)→ H∞(G,µ)

form an isometry. We now further claim:

Lemma 17 If (B,ν) is a compact boundary then Pν is multiplicative Pν : L∞ (B,ν)→H ∞ (G,µ).

Proof. We need to show that for any f1, f2 ∈ L∞ (B,ν), Pν( f1 · f2) = Pν( f1) ·Pν( f2), where thefirst multiplication is the pointwise multiplication on X and the second is as the multiplication ofharmonic functions (modulo I ).

So we need to show that for a.e. w ∈ (Ω,Pµ),

limn

Pν( f1 · f2)(wn)−Pν( f1)(wn) ·Pν( f2)(wn) = 0.

Let w be such that νw exists and denote by b ∈ B the point such that νw = δb.

limn

∫B

f1(x) f2(x)dwnν(x) =∫B

f1(x) f2(x)dνw(x)

= f1(b) f2(b)

And since

limn

∫B

fi(x)dwnν(x) = fi(b)

we conclude that Pν is multiplicative.

Corollary 6 For a compact µ-boundary, the image Im(Pν) is a subalgebra S ≤ H∞(G,µ).Moreover, the Poisson transform is a C∗-isomorphism L∞ (B,ν)≈S .

By correspondence between sub C∗ algebras and compact factors, we get that B is continuesfactor of Πtop.

In particular, any bounded harmonic function is given as a Poisson transform of some functionon the Furstenberg-Poisson boundary.

4.3 Zimmer’s Construction (shift ergodic components) 33

4.2.3 Disadvantage of the topological Poisson boundaryUp to here we constructed a compact topological stationary space Πtop such that every compactboundary is a continuous factor of Πtop. In general this space Πtop is too large. In particular, thereis no reason that it will be metrizable, so it is unnatural to work with.

For example, in order to prove weak* converges (which plays an important role in this theory),one must show that for every continuous function the limit exists, when we ignore a zero measureset. Thus if C (X) is separable it is enough to show it for each function from the dense set (and theintersection of the correlated full measure sets will still be of full measure).

Moreover, for discrete groups, Πtop is not metrazible, although we know nice and intuitivemodels for the Poisson boundary of F2 or lamplighter groups.

The deep point here is that factor (not continuous one) of a boundary is a boundary. Hence,instead of talking about “continuous-universal space” we can ask for just universal space.

That is, we can take a different compact model for Πtop. Any boundary will be a factor (again,not continuous anymore) of it, and then this space can be chosen to be separable (in this procedureone can take the advantage of G being separable). However, We won’t describe this constructionhere, but we will consider a whole different prespective: We will construct an abstract space, andthen one can take any compact model of it.

As topological space, neither of these compact models won’t be unique in the topologicalcategory - and indeed for many groups, there are different (G-isomorphic but non homeomorphic)models for the Poisson boundary which are natural with respect to the group.

The advantage of this topological construction is the intuition: we understand what is themeaning of the measure ν - this is just the hitting distribution on the boundary. Moreover, for anyg ∈ G the measure gν represents the hitting distribution for a random walk that starts from g. Thusthe G-action can be though as a re-rooting.

4.3 Zimmer’s Construction (shift ergodic components)4.3.1 Arrows and stars

Let (X ,A ,ν)π−→ (Y,B,η).

Recall that the measure ν can be disintegrated ν =∫Y

νydη (y) with supp(νy)⊂ π−1 (y)

There two natural maps:• π∗ : L∞ (Y,B,η)→ L∞ (X ,A ,ν) which is composition: (π∗ f )(x) = f

(π (x)

).

Note that

ν(π∗ f)=∫X

π∗ f (x)dν (x) =

∫X

f(π (x)

)dν (x) =

∫Y

f (y)dπ∗ν (y) = π∗ν ( f )

This is the easy direction, and it works also as a function π∗ : C (Y )→ C (X).• π∗ : L∞ (X ,A ,ν)→ L∞ (Y,B,η). Given f ∈ L∞ (X ,A ,ν), (π∗ f )(y) =

∫f (x)dνy (x) =

νy ( f ).

Claim 3 π∗π∗ = idL∞(Y ) and π∗π∗ = E

(·|π−1B

)Proof. Observe that(

π∗π∗ f)(y) =

∫X

(π∗ f)(x)dνy (x) =

∫X

f(π (x)

)dνy (x) = f (y)

where the last equality follows from the facts that supp(νy)⊂ π−1 (y).

In the other direction, for f ∈ L∞ (X ,A ,ν) we claim that π∗π∗ f = E(

f |π−1B)

.


It is clear that it is measurable in π−1B. And for any B ∈B,

∫π−1B

f (x)dν (x) =∫X

1π−1B (x) · f (x)dν (x)

=∫Y

νy (1π−1B · f )dη (y)

=∫B

νy ( f )dη (y)

=∫B

(π∗ f )dη

=∫B

(π∗ f )dπ∗ν

=∫

π−1B

(π∗π∗ f)

dν

So the function π∗π∗ f satisfies the properties of the conditional expectation, so by the unique-ness we get equality.

4.3.2 Construction by Shift ergodic componentsLet (Ω,B,P) = (G,µ)N. Define T : Ω→Ω by T (ω) = (ω1 ·ω2,ω3, . . .) and consider the G actiongω = (gω1,ω2,ω3, . . .).

These two actions, T and G commute:

T (gω) = T (gω1,ω2, . . .) = (g ·ω2,ω3, . . .) = g(ω2,ω3, . . .) = gT (ω)

Now consider IT to be the σ -algebra on Ω of T -invariant sets, that is IT =

B ∈B|T−1B = B

.

The Mackey corresponding factor, (Π,ν) = T\\(Ω,P) is the Poisson-Furstenberg boundary.The following will give us characterization of (Π,ν), in terms of the measurable functions from

it. It is a general fact about Mackey spaces, when we apply it to our case.

Lemma 18 Let (X ,η) be some probability space. Every measurable f : Π→ X defines a T -invariant function F : Ω→X . Moreover, every T -invariant F : Ω→X is defined by some f : Π→X .

(Ω,B,P) F //

π

&&

(X ,C ,η)

(Π,A ,ν)

f88

Proof. Let F : Ω→ X be a measurable function. We want to show that F is T -invariant if and onlyif there exists such an f .

First we observe that• F is T -invariant if and only if F is measurable w.r.t. IT : if F T = F then for any measurable

C ∈ C ,F−1 (C) = (F T )−1 (C)

= T−1(

F−1 (C))


so F−1 (C) ∈IT . For the other direction, assume F is IT -measurable we show that F−1 (C) =

(F T )−1 (C) for any C∈C . (F T )−1 (C)=T−1(

F−1 (C))

but F is IT -measurable so F−1 (C)∈

IT so indeed T−1(

F−1 (C))= F−1 (C).

• There exists f if and only if F is IT -measurable: if there exists such f , such that F = π∗ fso F is π−1A = IT measurable, by definition. If F is IT -measurable then we can definef = π∗F . Now

π∗ f = π

∗π∗F = E

(F |π−1A

)= E(F |IT ) = F.

Lemma 19 (Π,ν) is a stationary space.

Proof. Note that (Ω,P) is not a stationary space: µ ∗P= µ ∗µ×µ×·· · 6= P: for F ∈ L∞ (Ω,P),

∫F (ω1,ω2,ω3 . . .)dT∗P =

∫F(T (ω1,ω2,ω3 . . .)

)dP

=∫

F(ω1ω2,ω3,...

)dP

=∫

Fd(

µ ∗µ×µN)

so we have µ ∗P= T∗P. Since Π is the factor of T -invariant sets, both P and T∗P project to thesame measure ν , so it is a stationary measure.

More formally, observe that if F : Ω→ X comes from a function on Π then it is T -invariant,hence T∗P(F) = P(F). So for any f ∈ L∞ (Π,ν) we have

µ ∗ν ( f ) = µ ∗ (π∗P)( f ) = π∗ (µ ∗P)( f ) = µ ∗P(π∗ f)

= T∗P(π∗ f)= P

(π∗ f)= π∗P( f ) = ν ( f )

so µ ∗ν = ν .

Lemma 20 The Furstenberg transform F : L∞ (Π,ν)→H ∞ (G,µ) is a bijection.In other words: there is a bijection between T -invariant functions on Ω and G-harmonic

functions.

Proof. Recall that for f ∈ L∞ (Π,ν), F ( f )(g) =∫

f dgν in this proof we write f = F ( f ).On the other hand, for h ∈H ∞ (G,µ) recall that by the martingale convergence theorem the

limit h(e,ω) = limn→∞

h(ω1 · · ·ωn) exists for a.e. ω ∈Ω. That is, h defines some function on Ω which

is T -invariant. Hence it defines a function h ∈ L∞ (Π,ν).

We now claim that these are inverse transforms of each other, that is f = f and ˆh = h:Let f ∈ L∞ (Π,ν) and denote by f = π∗ f ∈ L∞ (Ω,P) so f is T -invariant. Note that f (g) =∫

f dgν =∫

π∗ f dgπ∗P=∫

f dgP. Denote ω = π (ω).


¯f (ω) = limn→∞

f (ω1 · · ·ωn)

limn→∞

∫Ω

f d (ω1 · · ·ωnP)

= limn→∞

∫Ω

f(ω1 · · ·ωn,ω

′)dP(ω′)

= limn→∞

∫Ω

f T n (ω1, . . . ,ωn,ω

′)dP(ω′)

= limn→∞

∫Ω

f(ω1, . . . ,ωn,ω

′)dP(ω′)

= f (ω) = f (ω)

For h ∈H ∞ (G,µ),

ˆh(g) = gν(h)= gπ∗P

(h)

= gP(

limn→∞

h(ω1 · · ·ωn)

)= lim

n→∞

∫Ω

h(ω1 · · ·ωn)dgP(ω)

= limn→∞

∫Ω

h(gω1 · · ·ωn)dP(ω)

= limn→∞

∫G

· · ·∫G

h(gω1 · · ·ωn)dµ (ω1) · · ·dµ (ωn)

= limn→∞

h(g) = h(g)

Corollary 7 The G-action on (Π,ν) is ergodic, that is every G-invariant set A ⊂ B (that isν (gA∆g) = 0 for all g ∈ G) is of trivial measure (ν (A) ∈ 0,1)

Proof. Let A be a G-invariant set and define f = χA. So f is G-invariant: g f = g(χA) = χg−1A =χA = f .

By the equivariance of the Furstenberg transorm, if h ∈H ∞ is F ( f ) then

h(g) =∫

f dgν =∫

g f dν =∫

f dν = h(e)

so h is a constant function h≡ c. Since constant functions on L∞ (Π,ν) yield constant functions, bythe injectivity of F we conclude that f is constant, that is, ν (A) ∈ 0,1.


4.3.3 Compact and abstract boundariesWe originally defined a boundary as a compact stationary space with Dirac measures as limitingmeasures.

Then we say that being a boundary is equivalent to being a factor of the Poisson boundary.As such we can think of boundaries as G-invariant sub σ -algebras of the Poisson boundary.The Furstenberg transform of a boundary is an isomorphism onto a G-invariant sub algebra of

the algebra of H ∞ (G,µ), and actually any such sub algebra corresponded to some boundary.

Theorem 7 Let (B,η) be an abstract (G,µ)-stationary space. Then the following are equivalent:1. (B,η) is a factor of the Poisson boundary2. Any compact model of (B,η) is a compact (G,µ)-boundary.3. There exists a compact model of (B,η) which is a compact (G,µ)-boundary.

In this case we call (B,η) an abstract (G,µ)-boundary (or just boundary).

Proof. (1⇒ 2) Let π : (Π,ν)→ (B,η), and consider B as a compact metric space. Find C0⊂C (B)a countable dense set and let D = π∗ f f∈C0

⊂ C (Π).

For any π∗ f ∈ L∞ (Π,ν), π∗ f = π∗ f . For any f ∈C0 there exists full measure Ω f such thatthe equality holds for any ω ∈C f (where ω is the projection from Ω→ B). Intersect all these setsto get a full measure Ω′ ⊂Ω such that the equality holds for all ω ∈Ω′ and f ∈C0.

We get that for any ω ∈Ω′ and any π∗ f ∈ D we have

π∗ f (ω) = π∗ f (ω) = lim

n→∞π∗ f (ω1 · · ·ωn) = lim

n→∞

∫π∗ f d (ω1 · · ·ωn)ν

On the other hand,∫

π∗ f dω1 · · ·ωnν =∫

f dω1 · · ·ωnη so we get ω1 · · ·ωnη ( f )→ π∗ ( f )(ω)=f(π (ω)

)for any f ∈C0 and any ω ∈Ω′. By continuity, this convergence holds for any f ∈ C (B)

so by definition, ω1 · · ·ωnη → δπ(ω) for almost every ω ∈Ω′.(2⇒ 3) is clear.(3⇒ 1) We consider now a compact boundary (B,η). The function ω 7→ ηω = δb is a measur-

able function Ω→ B. This map is clearly T -invariant, hence it defines a function π : Π→ B.

Let µ be the SRW on F2. We claim that there is a stationary measure on the boundary of thetree which is µ-boundary.

For any r, let τr : GN→ N be such that τr(w) = maxn‖g1 · · ·gn‖ ≤ r Each τr is defined on afull measure set, and consider its intersection. Now define bnd : Ω→ ∂Tree by bnd(w)r = gτr(w).

Lamplighter group ⊕Z(Z/2Z)oZ. Elements are of the form ( f ,k) where f : Z→ Z/2Z withfinite support and k ∈ Z. Z acts on ⊕Z(Z/2Z) by shifting the index. Multiplication is given by

( f1,k1)( f2,k2) = ( f1 + k1 f2,k1 + k2)

generated by 3 elements. It follows that

( f ,k)−1 = (−k f ,−k)

here we used that f−1 = f .The commutator subgroup is the lamps, and the quotient is Z so it is amenable (solvable).Consider the space Ω and denote by wn = ( fn,kn). Project µ to µ on Z. If µ is not symmetric

then there is a map con f : Ω→Π(Z/2Z) which is given by con f (w)(z) = lim fn(z). This map isshift invariant so we get a µ-boundary.

5. Properties of the Furstenberg-Poisson boundary

Corollary 8 Since we know that the Poisson boundary of finite index subgroup with the hittingmeasure is the same as the original we get that for any vitrually nilpotent group any generatingmeasure, the Poisson boundary is trivial.

Open problem: It is an open question to describe the class of groups that have trivial Poissonboundary for any measure.

5.1 Operations of group, and the Poisson boundaryNo endomorphism: Barycenter map: Let M be a convex space, and consider Prob(M). As always,we can embed M in Prob(M) via m 7→ δm.

Since M is convex, we have also a map in the other direction, called the barycenter map.bar : Prob(M)→M which is the continues extension of the map bar(∑n

i=1 piδmi) = ∑ni=1 pimi. The

composition M→ Prob(M)→M is the identity on M.The canonical example is when M = Prob(X). When G acts on X , we get an action on M and

the map bar is equivariant.

Lemma 21 Let (X ,ν) be a µ-boundary. Then the only endomorphism is the identity. Namely, ifα : X → X is a measurable map that commutes with the G-action, then α = id.

Proof. Let η = 12(ν +α∗ν). So η is a µ-stationary and it’s conditional measures are given by

ηw =12(νw +α∗νw) =

12(δbnd(w)+δα(bnd(w))).

We claim that η is a µ-boundary, which implies that α = id. Define α : X → Prob(X) by α(x) =12(δx +δα(x)).

The ν pushforward via Prob(X)α∗−→ Prob(Prob(X)) given by 1

2(δν +δα∗ν).Now we can use bar : Prob(Prob(X))→ Prob(X) to push this measure to get bar∗(α∗ν) =

12(ν +α∗ν) = η .

Hence η is a µ-boundary as an equivariant image of the µ-boundary ν .

40 Chapter 5. Properties of the Furstenberg-Poisson boundary

5.1.1 Quotient groupsLet (G,µ) and let G→ Q with kernel N. Denote by µ ∈P (Q) to be the projected random walk.It is just the push-forward ϕ∗µ where ϕ is the group homomorphism.

Let (X ,ν) be a (Q, µ)-stationary space. Then we can define a G-action on X that factorizesthrough ϕ , that is gx = ϕ (g)x.

Claim 4 (X ,ν) is (G,µ)-stationary if and only if it is (Q, µ)-stationary.Moreover, ν is µ-boundary if and only if it is µ-boundary.

Proof. ∑g∈G µ (g)gν = ∑g∈G µ (g)ϕ (g)ν = ∑q∈Q µ (q)qν .Now νw = limwnν = limϕ(wn)ν If (B,ν) is a (Q, µ)-compact boundary then it is (G,µ)-

compact boundary: which is a Dirac measure for a.e. µ-random walk.

Lemma 22 The Poisson boundary Π(Q, µ)≈ N\\Π(G,µ).

Proof. The space N\\Π(G,µ) is the maximal factor of Π(G,µ) on which N acts trivially. By theabove, Π(Q, µ), is a µ-boudnary, that is, a factor of Π(G,µ) on which N acts trivially. Hence weget that N\\Π(G,µ)→Π(Q, µ).

On the other hand, any G-space on which N acts trivially, can be considered as a Q-space (pickelements gq from ϕ−1 (q)). Hence we get a Q-action on N\\Π(G,µ). Once both Q and G act onN\\Π(G,µ), we get that it is (Q, µ)-boundary, hence Π(Q, µ)→ N\\Π(G,µ).

The composition of these two maps is an endomorphism hence these spaces are isomorphic.

A different approach: Consider the two action on (Ω,P), now P is the Markov measures, soit holds the positions of the random walk and not the increments. One action is the shift, andthe other is modding out each coordinate by N. These actions commute, and so we get thatΠ(Q, µ) = N\Π(G,µ).

5.1.2 Recurrent subgroupsOne interesting question is to relate the Poisson boundary of a group and its subgroups. In particular,if Γ is a “large” subgroup is G, can we identify the Poisson boundary of Γ with the one of G?

Remark 10 The historical motivation is coming from the study of Lie groups. A subgroup Γ≤ Gis a lattice if Γ is a discrete group and the space of cosets G/Γ admits a G-invariant measure. Thefirst example is Z≤ R.

Is many senses lattices are “approximations” of the enveloping groups. One aspect, for exampleis the following question: can a lattice in SL2 (R) be also a lattice in SL3 (R). Nowadays, one canprove that it cannot be using many tools. However, the first proof was given by Furstenberg and itwas the question that led him to define the Poisson boundary.

The philosophical idea that if one is very far, he cannot distinguish between the lattice and theenveloping group. The Poisson boundary is a way to consider groups from a far point of view - anylocal behavior of the random walk disappears in the Poisson boundary.

Let Γ≤ G be a subgroup and let µ ∈P (G) be a random walk. Denote (Ω,P) = (G,µ)N.

Definition 17 Γ is called µ-recurrent if the µ-random walk almost surely visits Γ infinitely manytimes.

Example 5.1 F2×Z2.

5.1 Operations of group, and the Poisson boundary 41

Example 5.2 Let Γ≤ F2 be the commutator group, that is, the group generated by elementsof the form [w1,w2] = w1w2w−1

1 w−12 . Algebraically, Γ is isomorphic to the free group F∞ with

finitely many generators. The quotient F2/Γ is isomorphic to Z2 and the projected random walkµ is the simple random walk on Z2. Since Z2 is recurrent w.r.t. the simple random walk weconclude that Γ is µ-recurrent.

Definition 18 Let Γ be a µ-recurrent subgroup. Denote by τ : Ω→ N , the first return time

τ (ω) = min

n≥ 1|ω1 · · ·ωn ∈ Γ.

Γ is µ-positive recurrent if E(τ)< ∞ and µ-null recurrent if E(τ) = ∞.

Example 5.3 — Two floors F2. Let G = F2×Z2 and let µ (e,1) = p and µ (g,0) = 1−p4 for

g ∈

a,a−1,b,b−2

. The subgroup F2×0 is a µ-positive recurrent subgroup.

What about the commutator group is F2?

Theorem 8 — Kac’s formula. If Γ is µ-recurrent then E(τ) = [G : Γ].

Proof. Consider the Markov chain where the states are Γ\G and perform the right random walk.That is, X0 = Γ, the trivial coset and Xn = ΓZn where Zn ∼ µn. Observe that the original randomwalk visits Γ exactly when the Markov chain visits the trivial coset. This Markov chain is irreducible(due to the generating assumption), doubly stochastic and recurrent.

Now, if the index is finite, so the Markov chain is finite and there exists a stationary probabilitydistribution η which is uniform. Thus by Kac’s formula, the expected return time to Γe is1/η (Γe) = [G : Γ].

If the index is infinite, this Markov chain admits no finite stationary measure, and hence, it isnull recurrent. In other words, E(τ) = ∞.

Corollary 9 The property µ-positive recurrent is independence on µ!

Definition 19 Let Φ : Ω→ Γ be the hitting map:

Φ(ω) = ω1 · · ·ωτ(ω).

Now we can push forward P to get a measure on Γ. Let θ ∈P (Γ) be θ = Φ∗P.

Example 5.4 Consider the two floors free group. θ has full support.

Definition 20 Now let (X ,ν) be a (G,µ)-stationary space. We can consider the restricted actionΓ y X . We will show that actually ν is a θ -stationary measure.

The proof relies on the optional stopping theorem from probability theory. We won’t get intothe details here and just rephrase it for our purposes.

Let Ψ be the Γ-restriction map. That is, for any f : G→ R, we let Ψ( f ) : Γ→ R defined byΨ( f )(γ) = f (γ).

Theorem 9 — Optional stopping theorem. The map Ψ : H ∞ (G,µ)→H ∞ (Γ,θ) is an isomet-ric bijection.

First, one should prove that the restriction of a µ-harmonic function is θ -harmonic. Next, theinteresting part is to observe that µ-harmonic functions are determined by their values on Γ.

42 Chapter 5. Properties of the Furstenberg-Poisson boundary

Theorem 10 (X ,ν) is (Γ,θ)-stationary. Moreover, (G,µ) and (Γ,θ) share the same Poissonboundary.

Proof. Let Fµ be the Furstenberg transform Fµ : L∞ (X ,ν)→H ∞ (G,µ) defined by Fµ ( f )(g) =gν ( f ). This map can be defined for non stationary actions as well. Note that ν is µ-stationary ifand only if the image contains only µ-harmonic functions.

Then Fθ =ΨFµ , the Furstenberg transform for the (Γ,θ)-action, maps L∞ (X ,ν)→H ∞ (Γ,θ).Since the image is inside the space of harmonic functions, ν is θ -stationary.

It follows that for (X ,ν) = Π(G,µ), Fθ = ΨFµ is an isometric bijection between L∞ (X ,ν)and H ∞ (Γ,θ) which implies that (X ,ν) is actually the Poisson boundary of (Γ,θ).

5.2 Relatively measure preserving factors

Let π : (X ,ν)→ (Y,η). Consider the disintegration ν =∫

νydη (y).

Definition 5.2.1 A factor map π : (X ,ν)→ (Y,η) is relatively measure preserving if the disin-tegration map X → Prob(Y ) is equivariant.

Fact 1 π : (X ,ν)→∗ is relatively measure preserving if and only if (X ,ν) is measure preserving.

Explicitly it means that if gνy = νgy for every g and η-a.e. y ∈ Y , which is equivalent todgνg−1y

dνy(x) = 1 for νy-a.e. x and η-a.e. y ∈ Y and every g ∈ G.

Lemma 23 The factor π : (X ,ν)→ (Y,η) is relatively measure preserving if and only if thefunctions dgν

dνare Y -measurable, for all g.

Remark 11 We may write these conditions just for g ∈ supp(µ). It is equivalent as for any g ∈ Gsince we consider µ to be a generating measure.

Proof. Write the disintegrations:ν =∫

νydη (y) and gν w.r.t. gη :

gν =∫

(gν)y dgη (y) =∫

(gν)gy dη .

By the linearity of the G action on P (X) we get that

g ·ν = g(∫

νydη

)=∫

g ·νydη .

By the uniqueness of the disintegration, we get that gνy = (gν)gy .

Now the factor is relatively measure preserving is equivalent to gνy = νgy if and only ifνgy = (gν)gy, which we can rewrite as dνgy

d(gν)gy(x) = 1 for a.e. x.

(4 =⇒ 3) Assume that dgν

dνis Y -measurable so π∗ dgη

dη= dgν

dν.

Now∫f dgν =

∫f

dgν

dνdν

=∫ ∫ (

fdgν

dν

)dνydη

But the assumption is that dgν

dνis constant on the fibers so

=∫ ∫

f dνyπ∗dgν

dνdη =

∫ ∫f dνy

dgη

dηdη .

5.2 Relatively measure preserving factors 43

Since it holds for any f , we have equality of measures:

gν =∫

νydgη

dηdη

If we just disintegrate gν we get

gνy = (gν)gy .

gν =∫

(gν)y dgη =∫ (gν)y

dgν

(4 ⇐⇒ 5) We already saw that in general ϕY ≤ ϕX and equality holds if and only if all theRadon-Nikudym derivatives in X are Y -measurable.

Consider the Furstenberg transforms from X and from Y (FX ,FY respectively).

Corollary 10 For any π : (X ,ν)→ (Y,η) , Im(FY ) ⊂ Im(FX). If π is measure preserving thenIm(FX) = Im(FY ).

Proof. Note that the Furstenberg transform FY generates harmonic function using the functionsdgη

dη:

FY ( f )(g) = gη ( f ) =∫

f dgη =∫

fdgη

dηdη .

We can think of the functions dgη

dηas a kernel for this transform: some kind of basis that using it FY

generates bounded harmonic functions. Thus, since we always can pull back dgη

dηto dgν

dνit is clear

that Im(FY )⊂ Im(FX).If π is measure preserving then all the “basis” that FX uses, are actually already in Y so they

share the same image.

Exercise 5.1 Prove the corollary without using RanonNikudym derivatives. For the equalitypart, use the definition of relatively measure preserving (gνy = νgy)

5.2.1 Product groupsLet us just mention now how the Poisson boundary behaves with products. We will give the prooflater on, when after developing some machinery.

Let (G1,µ1) and (G2,µ2) be two random walks on two different group. Then

Theorem 11 Π(G1×G2,µ1×µ2) = Π(G1,µ1)×Π(G2,µ2).

Note that G1 is a quotient group, and the projected measure is µ1. Hence Π(G1,µ1) is aµ-boudnary. The same for (G2,µ2). Hence we get that Π(G1,µ1)×Π(G2,µ2) is a µ-boundary.

To claim that they are actually isomorphic we need the following notion.We will discuss this notion later, but meanwhile, note that (X ,ν) is measure-preserving if on

only if Π : (X ,ν)→∗ is relatively measure preserving.

Lemma 24 If (X ,ν) is a µ-boundary and π(X ,ν)→ (Y,η) is relatively measure preserving thenπ is an isomorphism.

Proof. X → Y → Prob(X)→ X is equivariant hence should be the identity.

6. Entropy theory

6.1 Furstenberg entropy

Let (X ,ν) be a (G,µ)-stationary space.

Definition 21 The Radon-Nikudym cocycle ρ : G×X → R is

ρ (g,x) =− logdg−1ν

dν(x)

This is an additive cocycle:

ρ (g1g2,x) = − logdg−1

2 g−11 ν

dν(x)

= − log

(dg−1

2 g−11 ν

dg−12 ν

(x)dg−1

2 ν

dν(x)

)

= − logdg−1

1 ν

dν(g2x)− log

dg−12 ν

dν(x)

= ρ (g1,g2x)+ρ (g2,x)

by the chain rule of the Radon-Nikudym derivative.

Definition 22 Define ϕ : G→ R by

ϕ (g) =∫X

ρ (g,x)dν (x)

ϕ measures the deformation each g makes on ν . Indeed, using the convexity of− log, by Jensenϕ is non-negative:

46 Chapter 6. Entropy theory

ϕ (g) =∫X

− logdg−1ν

dν(x)dν (x)

≥ − log∫X

dg−1ν

dν(x)dν (x)

= − log∫X

dg−1ν (x) = 0.

Equality holds if and only if dg−1ν

dνis constant a.e. that is, ν and g−1ν are the same.

In terms of information theory, ϕ (g) = DKL (gν‖ν) is the Kullback-Leibler divegence.

Definition 23 The Furstenberg entropy of a (G,µ) stationary space, (X ,ν) is

hµ (X ,ν) =∫

ϕ (g)dµ (g) .

This quantify the average deformation that G makes when it acts on ν . In particular, it followsfor the discussion above that hµ (X ,ν) = 0 if and only if ν is G-invariant (we also say in that casethat (X ,ν) is measure preserving).

Lemma 25 The entropy decreases with factors: if π : (X ,ν)→ (Y,η) then hµ (X ,ν)≥ hµ (Y,η).

Proof. Consider the conditional expectation operator π∗ : L∞ (X ,ν)→ L∞ (Y,η). Let f > 0 withlog f ∈ L1 (X ,ν). By Jensen inequality for conditional expectations, since − log is strictly convexwe get

− log(π∗ ( f )(y)

)≤ π∗ (− log f )(y)

for η-a.e. y ∈ Y and equality holds if and only if f is Y -measurable.Apply it to f = dg−1ν

ν. A general fact about Radon-Nikudym derivatives is that the projection

of a derivative is the derivative of the projected measures.In our case,

π∗dg−1ν

dν=

dπ∗

(g−1ν

)dπ∗ν

=dg−1η

dη

So the Jensen inequality says that

− logdg−1η

dη(y)≤ π∗

(− log

dg−1ν

dν

)(y)

After integrating against η we get

ϕY (g)≤∫

π∗

(− log

dg−1ν

dν

)(y)dπ∗ν =

∫− log

dg−1ν

dν(x)dν (x) = ϕX (g)

In particular hµ (Y,η) ≤ hµ (X ,ν), and equality hold if and only if each of the dg−1ν

dνare

Y -measurable.

Lemma hµn (X ,ν) = n ·hµ (X ,ν)

Proof. Exercise.

6.2 Random walk entropy 47

6.2 Random walk entropy

Recall the definition of usual entropy of an atomic measure:

H (µ) = ∑g∈G−µ (g) log µ (g) .

with the convention that 0 log(0) = 0.

H(µ) = log(|X |)−DKL(µ||u)

In this section we will assume that H (µ)< ∞.Simple information theoretical argument shows that given two measures on the group, µ and

µ ′,

H(µ ∗µ

′)≤ H (µ)+H(µ′) .

Thus, the sequence H (µn) is sub additive so the limit

hRW (G,µ) = lim1n

H (µn)

exists. This value (can be either 0 or ∞) called the random walk entropy (or Avez entropy, orasymptotic entropy).

Theorem 12 hµ (X ,ν)≤ hRW (G,µ)

Proof. ν = µ ∗ν = ∑g

µ (g) ·gν then

1 =dν

dν(x) =

d(∑ µ (g) ·gν

)dν

(x) = ∑µ (g)dgν

dν(x)≥ µ (g0)

dg0ν

dν(x)

so dgν

dν(x)≤ 1

µ(g) . Now, using the cocycle property,

ρ (g,x) =−ρ

(g−1,gx

)= log

dgν

dν(gx)≤− log µ (g)

So

hµ (X ,ν) = ∑g

µ(g)∫X

ρ (g,x)dν (x)≤∑g

µ (g)(− log µ (g)) = H (µ) .

Thus we get

hµ (X ,ν) =1n

hµn (X ,ν)≤ 1n

H (µn)→ hRW (G,µ)

Corollary 11 If hRW (G,µ) = 0 then the Poisson boundary is trivial.


Definition Let G be a finitely generated group and fix a generating set S.G has exponential growth if limn→∞ |Sn|

1n > 1 and subexponential otherwise.

G is said to have polynomial growth if limn→∞|Sn|nd ≤C for some fixed C and d.

One can prove that these properties are group properties - that is, independent of choice of S.

Remark 12 A highly non-trivial theorem of Gromov states that G has polynomial growth if andonly if it is virtually nilpotent.

It was open question whether there are intermediate growth groups - subexponential groupswhich growth faster then any polynomial.

The first such groups was constructed by Grigichuk.

Corollary 12 If G has subexponential growth then for any generating µ with finite support thenΠ(G,µ) is trivial.

Proof. H (µn)≤ log |Sn| and then 1n H (µn)≤ 1

n log |Sn|= log |Sn|1n → 0.

Note that for polynomial groups, we know much more - by Gromov - it is virtually nilpotentand so for any measure the Poisson boundary is trivial (any generating measure - with no furtherassumptions like finite support or finite moments). However, it applies also for finitely supportedmeasures, on intermediate growth.

6.2.1 Equality of the entropiesTheorem 13 [Kaimanovich-Vershik] For hRW (G,µ) = hµ (Π,ν) .

Let (Ω,Pµ), where Pµ is as usual the Markov measure and not the product.Consider the following partitions of the space. α1 is the partitions by the first cylinder, and

denote by α1 (ω) the element in the partition α1 that contains ω . Let τn to be the partition whereω ∼τn ω ′ if they agree after time n. Note that τn is a decreasing family of sigma algebras, andthe tail sigma algebra is the limit τ = limn→∞ τn.

Another construction of the Poisson boundary was given by Kaimanovich-Vershik when theyrealized the Poisson boundary as the space (Ω,Pµ) when equipped with the tail sigma algebra. It isvery similar to Zimmer’s construction where he considered the sigma algebra of invariant sets. Theinvariant sigma algebra and the tail sigma algebra are different.

w∼I w′ if there exists some N such that for all n < N, wn = w′n.w∼τ w′ if there exist some k and N such that for all n > N, wn+k = w′n.However, for generating µ , these two sigma algebras are equivalent mod Pµ . So let’s think of

Poisson boundary as the realization of the tail sigma algebra. In that case, bnd : (Ω,Pµ)→Π(G,µ)is given by w 7→ τ(w).

Given a space, say (Ω,P) and a sigma algebras, say α1, the entropy if α1 is defined to be

H (α1) =∫Ω

− logP(α1 (ω)

)dP(ω)

which coincide, in this case, with H (µ).Given another sigma algebra, say τ , the conditional entropy of α1 w.r.t. τ is

H (α1|τ) =∫Ω

− logP[α1 (ω) |τ (ω)

]dP(ω) .

We will prove the theorem by showing that the both quantities are equal to

hRW (G,µ) = H (µ)−H (α1|τ) = hµ (Π,ν) .


Lemma 26 hRW (G,µ) = H (µ)−H (α1|τ).

Proof. Let ω ∈Ω. write zn = ω1 · · ·ωn. Then,

P(α1 (ω) |τn (ω)

)=

µ (z1) ·µn−1(

z−11 zn

)µn (zn)

.

and so H (α1|τn) =∫Ω

− logP(α1 (ω) |τn (ω)

)dP(ω) = H (µ)+H

(µn−1

)−H (µn).

H (α1|τn) ≤ H (α1|τn+1) since τn+1 is a sub partition of τn, so knowing τn gives at least theamount of information about the first step, as τn+1. Thus,

H (µ)+H(

µn−1)−H (µn)≤ H (µ)+H (µn)−H

(µ

n+1)

and so

H(

µn−1)−H (µn)≤ H (µn)−H

(µ

n+1).

That is, the sequence Fn = H (µn)−H(

µn−1)

is a decreasing sequence of non-negativenumbers and hence it convergences, and so also its Cesaro sums. Now

hRW (G,µ) = lim1n

H (µn) = lim1n

n

∑k=1

Fk = limFn.

Therefore,

H (α1|τ) = limn→∞

H (α1|τn) = H (µ)− limn→∞

(H(

µn−1)−H (µn)

)= H (µ)−hRW (G,µ) .

Lemma 27 hµ (Π,ν) = H (µ)−H (α1|τ).

Proof. Let E ⊂Π with ν (E)> 0 and [g]⊂Ω be a cylinder. Then

P([g]∩bnd−1E

)= P

([g])Pg

(bnd−1E

)= P

([g])

gν (E) .

And so

P([g] |bnd−1E

)= P

([g]) gν (E)

ν (E).

Since it holds for any E, we can write the density of this measure for a single b ∈Π (for a.e. b):

P[α1(w)|τ (ω)

]= µ(w1)

dw1ν

dν

(bnd(w)

)


H (µ)−H (α1|τ) = −∑g

µ (g) log µ (g)+∫Ω

logP[α1 (ω) |τ (ω)

]dP(ω)

= −∑w1

µ(w1) log µ(w1)+∑w1

µ(w1)∫Ω

logP[α1 (ω) |τ(w)

]dPw1(w)

= −∑w1

µ(w1) log µ(w1)+∑w1

µ(w1)∫Ω

log(

µ(w1)dw1ν

dν

(bnd(w)

))dPw1(w)

= ∑w1

µ(w1)∫Ω

log(

dw1ν

dν

(bnd(w)

))dPw1(w)

= ∑w1

µ(w1)∫Ω

−ρ(w−11 ,bnd(w))dPw1(w)

By the cocycle property, ρ(w1,w−11 bnd(w)) = ρ(w1,bnd(θw)) where θ is the shift on Ω. Since

bnd is shift invariant, and ω1 = g

= ∑w1

µ(w1)∫Ω

ρ(w1,bnd(w))dPw1(w)

= ∑g

µ (g)∫Ω

ρ (g,b)dν (b)

= hµ (Π,ν) .

Corollary 13 The Poisson boundary is trivial if and only if hRW (G,µ) = 0.

6.2.2 A Shannon-type theorem for the random walk entropyRecall Kingman’s subadditive ergodic theorem:

Theorem 14 Let (X ,m) be a probability space and T : X→ X a measure preserving transformation.Suppose that fn ∈ L1 (X ,m) is a sub additive sequence, that is, fn+m ≤ fn + fm T n for all n,m.Then the limit f = lim 1

n fn ≥−∞ exists a.s. and f is T -invariant function.If, furthermore, m is T -ergodic, then the limit f is the constant number which is infn

∫ 1n fn (x)dm(x).

Let Cng be the cylinder of all ω such that ω1 · · ·ωn = g.

Theorem 15 Let (G,µ) with H (µ)< ∞. Then

limn→∞−1

nlog µ

n(wn) = hRW (G,µ) .

for a.e. ω , and as functions in L1.

Proof. Here we want to consider the shift σ : (GN,µN)→ (GN,µN). Define fn(w) =− log µn(wn).Write wn = g1 · · ·gn.

Note that µn+m(g1 · · ·gngn+1 · · ·gn+m)≥ µn(g1 · · ·gn)µm(gn+1 · · ·gn+m) and thus

fn+m(w) = − log µn+m(g1 · · ·gngn+1 · · ·gn+m)

≤ − log µn(g1 · · ·gn)− log µ

m(gn+1 · · ·gn+m)

= fn(w)+ fm(σnw)


where σ : Ω→ Ω is the shift. Since the entropy is finite, f1 ∈ L1 (G,µ)N, and thus by the thesubadditive ergodic theorem, the limit lim−1

n log µn(wn) exists and it is equal to

inf1n

∫− log µ

n(wn)dµN(w) = inf

1n

H (µn) = hRW (G,µ) .

Corollary 14 Assume that there exist δ > 0 and a sequence An ⊂ G such that µn (An) > δ andlog |An|= o(n). Then Π(G,µ) is trivial.

Proof. Let φn(w) =−1n log µn(wn). So φn(w)→ h for almost every w.

For any n let Cn ⊂ Ω be the set of all walks that at time n visited at An. Let ψn(w) = φn(w) ·1Cn(w). So we get∫

ψn(w)dPµ(w) = −1n

∫Cn

log µn(wn)dPµ(w)

=1n

H(µn|Zn ∈ An)

≤ 1n

log |An| → 0

That is ψn→ 0 in L1. Hence we can find nk such that ψnk(w)→ 0 almost surely. Let D be thefull measure set of w for which ψnk(w)→ 0 for all w ∈ D, and also φn(w)→ h.

As a subsequence of numbers φnk(w)→ h for all w ∈ D. Hence it is enough to find a positivemeasure set on which φnk(w)→ 0.

Now let C be the set of w such that w ∈Cn for infinitely many n’s. That is, C = supCn. Fatou’slemma says that Pµ(C)≥ limsupPµ(Cn)≥ δ .

Now for any w ∈C∩D we get that φnk(w) = ψnk(w)→ 0.

Remark 13 We didn’t use the fact that we are talking about iid random walk. In particular, if wehave some other measure Λ ∈ Prob(Ω) with a Shannon theorem, that is−1

n logΛ(Cnwn) convergence

a.s. and in L1 to some constant number, and we have a collection of sub exponentially family ofsets An that is visited for Λ-almost every walk, then this constant is 0.

We can think on the collection An as guessing where the random walk is, at time n. The gamehere is to find the walker at time n, where we allow our guesses to be of sub-exponential size. Wewill soon use this idea to have a criterion when a certain boundary is actually the Poisson boundary.

6.2.3 A Shannon type theorem for Furstenberg entropyLet (X ,ν) be a G-space. Consider the one sided Bernoulli shift (Ω,P) = (G,µ)N with the shiftθ : Ω→Ω θ (ω1,ω2, . . .) = (ω2,ω3, . . .). And let T : Ω×X →Ω×X be T (ω,x) = (θω,ω1x).

Claim 5 P×ν is T -invariant measure if and only if ν is µ-stationary. P×ν is ergodic if and onlyif ν is ergodic.

Proof. Since θ is ergodic, the only T -invariant sets are of the form E×Ω where E ⊂ X . ThusT−1 (Ω×E) =

(Ω× (µ ∗E)

), both parts are now follow

Theorem 16 Let (X ,ν) be an ergodic (G,µ)-stationary space with finite entropy. Then If

hµ (X ,ν) = limn→∞−1n log d(ωn···ω1)

−1ν

dν(x) for P×ν almost every (ω,x).


Proof. Recall the cocycle ρ (g,x) =− log dg−1ν

dν(x) and let ρ (ω,x) = ρ (ω1,x).

The cocycle property, ρ (ω2ω1,x) = ρ (ω2,ω1x)+ρ (ω1,x), implies that

ρ (ω,x)+ ρ(T (ω,x)

)= ρ (ω1,x)+ρ (ω2,ω1x)

= ρ (ω2ω1,x)

and more generally, ∑nk=1 ρ

(T k (ω,x)

)= ρ (ωn · · ·ω1,x) .

Since we assume that the entropy is finite, ρ (ω,x) = ρ (ω1,x) ∈ L1 (P×ν). Now the ergodictheorem says that

limn→∞

1n

n

∑k=1

ρ

(T k (ω,x)

)=∫

X×Ω

ρ (ω,x)dP×ν = hµ (X ,ν) .

Now,

1n

n

∑k=1

ρ

(T k (x,ω)

)=

1n

ρ (ωn · · ·ω1,x)

= −1n

logd (ωn · · ·ω1)

−1ν

dν(x) .

For boundaries we can have another limit. Note that here we take the positive direction ofthe random walk, evaluate the Radon Nikudym derivative in a specific point and no take the limitwithout the minus.

Lemma 28 Let (B,ν) be a boundary with finite entropy, then for almost every w ∈ bnd−1(b)

1n

logdg1 · · ·gnν

dν(b)→ hµ (B,ν) .

Proof. We want to show that −1n ρ((g1 · · ·gn)

−1,b))→ hµ(B,ν) = ∑g∫

ρ(g,b)dν(b). Lets calcu-late the expression:

ρ

(g−1

n · · ·g−11 ,b

)= ρ

(g−1

n · · ·g−12 ,g−1

1 b)+ρ

(g−1

1 ,b)

= ρ

(g−1

n · · ·g−13 ,g−1

2 g−11 b)+ρ

(g−1

2 ,g−11 b)+ρ

(g−1

1 ,b)

= ρ

(g−1

n · · ·g−14 ,g−1

3 g−12 g−1

1 b)+ρ

(g−1

3 ,g−12 g−1

1 b)+ρ

(g−1

2 ,g−11 b)+ρ

(g−1

1 ,b)

=n

∑k=1

ρ

(g−1

k ,g−1k−1 · · ·g

−11 b)

Note that for w ∈ bnd−1(b) we have that g−1k−1 · · ·g

−11 bnd(w) = bnd(σ k−1w) = bnd(w), since

bnd is σ -invariant function. Thus we get

−1n

ρ(g−1n · · ·g−1

1 ,bnd(w)) =− 1n

n

∑k=1

ρ

(g−1

k ,bnd(w)).

So we need to show that the right term convergence to the Furstenberg entropy.Let ϕ ∈L1

(Ω,µN

)be the function ϕ(w)= ρ(g1,bnd(σ(w))= ρ

(g1,g−1

1 bnd(w))=−ρ

(g−1

1 ,bnd(w))

.

6.3 A criterion for the Poisson boundary 53

Note that ϕ(σw) = −ρ

(g−1

2 ,bnd(w))

, since bnd is θ -invariant function. Now the ergodicaverage is

1n

n

∑k=1

ϕ(σ kw) =−1n

n

∑k=1

ρ(g−1k ,bnd(w))

and by the ergodic theorem it convergence to∫

ϕ(w)dµN(w). We now verify that this is indeedthe Furstenberg entropy of (B,ν):

∫ϕ(w)dµ

N(w) =∫

ρ(g1,bnd(σw))dµN(w)

= ∑g

µ(g)∫

ρ(g,bnd(σw))dµN(w)

= ∑µ (g)∫B

ρ (g,b)dν (b)

= hµ (B,ν) .

6.3 A criterion for the Poisson boundaryLet (B,ν) be a boundary. We want to get a tool that tells us whether (B,ν) is the Poisson boundary.Out tool is based on the entropy and so we will assume that H (µ)< ∞.

Claim 6 If (B1,ν1)→ (B2,ν2) is a measure preserving map between two boundaries, then theyare equal.

Proof. There are several ways to see this fact. Recall that each boundary defines, via its Furstenbergtransform a G-invariant closed sub algebra of H ∞ (G,µ) and vice versa. Since the images of theFurstenbeg transforms coincide, they are isomorphic.

Corollary 15 A boundary (B,ν) is the Poisson boundary if and only if its entropy equals to theentropy of the Poisson boundary.

Recall that if we can guess the position of the random walk in a subexponential scale, thenhRW = 0. Now we apply this argument, for conditional entropy. That is, to show that B is thePoisson boundary we claim that it is enough to guess the position of the random walk in subexponential scale, given the information of B. In other words, given the information that the randomwalk hits (through the boundary map) the point b ∈ B we need to guess where the walker is, at timen.

Definition 24 A probability measure Λ ∈P (Ω) has an asymptotic entropy h(Λ) if the limit

−1n

logΛ

(Cn

g1···gn

)→ h(Λ)

exist for a.e. ω ∈Ω and in L1 (Λ).The Shannon-type theorem that we proved showed that the Markov measure of the µ-random

walk has an asymptotic entropy (the Markov measure is the measure on Ω when we think of it asthe space of walks and not increments).

Let (B,ν) be a µ-boundary. For any b ∈ B, we can take the Markov chain on G which isconditioned to hit b. We denote this measure on Ω by Pb.

Recall that given E ⊂ B with ν (E)> 0 and Ce,g1,...,gn ⊂Ω be a cylinder. Then


P(

Ce,g1,...,gn ∩bnd−1E)= P

(Ce,g1,...,gn

)Pg

(bnd−1E

)= P

(Ce,g1,...,gn

)gν (E) .

where Pg is the Markov chain that starts from g instead of e. Now

P(

Ce,g1,...,gn |bnd−1E)= P

(Ce,g1,...,gn

) gν (E)ν (E)

.

Since it holds for any E, we can write the density of this measure for a single b ∈ B (for a.e. b):

Pbµ(C

ng1,...,gn

) = Pµ(Cng1,...,gn

)dg1, . . . ,gnν

dν(b).

Lemma 29 Let (B,ν) be a (G,µ)-boundary. Then for a.e. b∈ B, the measure Pb has an asymptoticentropy and it is equal to

h(Pb)= hRW (G,µ)−hµ (B,ν) .

Proof. Apply log on Pbµ(C

ng1,...,gn

) = Pµ(Cng1,...,gn

)dg1···gnν

dν(b) and let w ∈ bnd−1(b). Then

−1n

logPbµ(C

ng1,...,gn

) = −1n

logPµ(Cng1···gn

)− 1n

logdg1 · · ·gnν

dν(b)

→ hRW (G,µ)−hµ (B,ν)

Corollary 16 A boundary (B,ν) is the Poisson boundary if and only if h(Pbµ) = 0 for almost every

b ∈ B.In particular, if (B,ν) is a µ-boundary, and there exist some δ > 0 and a sequence of maps

πn : B→ 2G such that Pbµ(C

nπn(b)

)> δ and log∣∣πn (b)

∣∣= o(n), then (B,ν) is the Poisson boundary.

6.4 Furstenberg’s entropy realization problemProblem 6.1 Given (G,µ) find all values in [0,hRW (G,µ)] obtained as the Furstenberg entropy ofergodic µ-stationary actions.

6.4.1 Entropy gapConsider the classical settings of ergodic theory, that is G y (X ,m) is a probability measurepreserving action.

Consider L2 (X ,m). We have a G-action on L2 (X ,m), by g f (x) = f(

g−1x)

. Note that this is aunitary representation of G:

‖g f‖22 =

∫ ∣∣∣ f (g−1x)∣∣∣2 dm(x) =

∫ ∣∣ f (x)∣∣2 dg−1m(x) = ‖ f‖22 .

It is a very useful tool. For example, one can define ergodicity as the only G-invariant subspaceis the one dimensional space of constant functions. Similarly weakly mixing can also be formulatedvia this Koompan representation.

6.4 Furstenberg’s entropy realization problem 55

For non measure preserving measures the action is no longer unitary. However, we can fixit. The following works for quasi invariant measures, but we can think only on stationary actionsG y (X ,ν).

Let π : G→U (L2(X ,ν)) be the unitary representation defined by:

π (g) f (x) = f(

g−1x)(dgν

dν(x)) 1

2

.

And then

∥∥π (g) f∥∥2

2 =∫ ∣∣∣ f (g−1x

)∣∣∣2 dgν

dν(x)dν (x)

=∫ ∣∣∣ f (g−1x

)∣∣∣2 dgν (x)

=∫ ∣∣ f (x)∣∣2 dν (x)

= ‖ f‖22

However, the situation becomes quite different than the one in the measure preserving setup.For example, even the one dimensional subspace of constant functions is no longer invariant.

Lemma 30 Let G y (X ,nu) be an ergodic action. If there exists a non-invariant vector in L2(X ,ν)then ν is equivalent to a G-invariant probability measure.

Proof. Let f be an invariant function. So f (x) = f (g−1x)dgν

dν(x)

12 for a.e. x. Let S = x| f (x) 6= 0

the support of f . So S is non invariant G-invariant set hence, by ergodicity ν(S) = 1. Now definedη(x) = f (x)2dν(x). Since S has full measure, the new measure, η , is equivalent to ν .

Now,

dη(g−1x) = f (g−1x)2dν(g−1x) = f (g−1x)2 dgν

dν(x)dν(x) = f (x)2dν(x) = dη(x)

that is, η is a G-invariant measure. It has finite mass since η(X) =∫

1dη(x) =∫

f (x)2dν(x) =‖ f‖2. By normalizing f we can make η a G-invariant probability measure.

Corollary 17 Let (X ,ν) be an ergodic non-invariant stationary action. Then L2(X ,ν) admits nonon-zero invariant vectors.

We now turn to relate this representation with the Furstenberg entropy. Note that

⟨π (g)1,1

⟩=∫ (dgν

dν

) 12

(x)dν (x)

and so

ϕ (g) = −2∫X

log

(dg−1ν

dν(x)

) 12

(x)dν (x)

≥ −2log∫X

(dg−1ν

dν(x)

) 12

(x)dν (x)

= −2log⟨

π

(g−1)

1,1⟩

= −2log⟨1,π(g)1

⟩


Let π (µ) : L2 (X ,ν)→ L2 (X ,ν) be the operator

π (µ)( f ) = ∑µ (g)π (g) f .

We get

hµ (X ,ν) = ∑g

µ(g)ϕ(g)

≥ ∑µ(g)−2log⟨1,π(g)1

⟩= −2log∑µ(g)

⟨1,π(g)1

⟩≥ −2log

∥∥π(µ∥∥

Now define

‖µ‖T = sup∥∥π (µ)

∥∥ : π unitary representation with no invariant vectors

Definition 25 If G has property (T) then ‖µ‖T < 1 for every fully supported measure measure µ .The original definition says that G has property (T) if any representation that admits almost

invariant vector, admits an invariant vector.Typical (discrete) examples are the groups SLn (Z) for n≥ 3. If G is amenable group that has

also property (T) then G is finite (compact).A quotient of property (T) groups has property (T) and hence the product of property (T) group

with non property (T) doesn’t have property (T).

Corollary 18 If G has property (T) then for any generating random walk µ , there exists an entropygap. That is, there exists Cµ > 0 such that any non invariant ergodic stationary space has entropygreater than Cµ .

Note that there exists groups without property (T) with an entropy gap - for example Z×Gwhere G has (T).

However, it is interesting to understand the class of groups with an entropy gap. For example -is it true that a simple group without (T) cannot have gap?

The only know group that are known to have no gap, for any measure with finite first momentare lamplighters over recurrent base group, free groups and virtually free groups, as SL2 (Z).

6.5 Poisson bundlesCosG = Hg. Note that Hg = gg−1Hg = gHg−1 The advantage is that we get a left action

6.6 Identifications of Poisson boundariesThe lamplighter group.

6.6.1 Group compactificationFor some classes of groups there are “natural” boundaries, which comes from the geometry of thegroup. Some of these spaces were proved to be the Poisson boundary for natural measures. Let usmentioned some examples.

Let M be a metric space. By a compactification of M we mean a compact Hausdorff spaceM ⊂ M which is second countable (there exists a countable basis for the topology) such that Mopen and dense in M. The boundary of M is ∂M = M\M.

6.6 Identifications of Poisson boundaries 57

Now we consider M to be a group. But then, we require the compactification to be compatiblewith the group action:

Let G be a countable group. We say that G = G∪∂G is a compatible compactification if theleft action of G on itself extends to an action of G on G by homeomorphisms.

Definition 26 A compactification is said to be non-elementary if there is no finite set in ∂G whichis fixed by G.

A (non recurrent) random walk will convergence to an element in ∂G. Thus we get a mapΠ : (Ω,P)→ ∂G. It is clear that ν = Π∗P is stationary and moreover, since Π is shift invariant, so(∂G,ν) is a boundary.

6.6.2 Ends of a groupFor a compact subset K is a locally compact space M we denote by EK (M) the set of the connectedcomponents of M\K. If K1 ⊂ K2 then there is a natural map EK2 (M)→ EK1 (M). The projectivelimit is the space of ends as K exhaust the whole space M is denoted by E (M). The compactificationM = M∪E (M) is called the ends compactification.

If G is a finitely generated group then we can play this game on the Cayley graph of G.Example: Z has 2 ends. Z2 has one. For the free group, E (F2) is the set of infinite reduced

words.

6.6.3 Gromov boundaryDefinition 27 Let M be a geodesic metric space. A geodesic triangle (or, three points) are δ -thinif the distance of any point on any of the edges to the two other edges is at most δ .

M is called δ -hyperbolic is every triangle is δ -thin.M is called hyperbolic if it is δ hyperbolic for some δ ≥ 0.

Pick some origin o ∈ M. A geodesic ray from o is an isometry γ : [0,∞)→ M such that,γ (0) = o and the path γ

([0, t]

)is a geodesic connected o and γ (t), for any t.

We say that γ1 and γ2 are equivalent if they stay within a bounded distance from each other.That is, there exist K such that d

(γ1 (t) ,γ2 (t)

)≤ K for all t.

The Gromov boundary of M is the set of equivalence classes of geodesic rays.Now play this game on a group. The Gromoc boundary is a compatible compactification.Example: The free group.

Theorem Let G be a hyperbolic group or a group with infinitely many ends, and let µ be agenerating random walk. Then if µ has finite first moment then (∂G,Π∗P) is the Poisson boundary.

6.6.4 Lamplighter groups6.6.5 The Affine group

Consider the affine group ax+b. That is, each element g is of the form(ag,bg

)and the multiplica-

tion is composition of maps.That is, gh is the map

ag (ahx+bh)+bg = agahx+(agbh +bg

).

A nice way to represent this group is by ax+b 7→

(a b0 1

)and the action is on the vectors

of the form

(x1

).


Now lets consider A f f(Z[

12

]), that is the set of elements (x, f ) =

(2x f0 1

)where f = m

2n .

We can see that

(x1, f1)(x2, f2) = (x1 + x2, f1 +2x1 f2)

and in general

(x1, f1) · · ·(xn, fn) = (x1 + · · ·+ xn, f1 +2y1 f2 + · · ·+2yn−1 fn) .

Consider the homomorphism A f f(Z[

12

])→ Z, (x, f ) 7→ x. Given µ on A f f

(Z[

12

])let

µZ ∈P (Z) be the projected random walk. Write α = EµZ for the drift.This group is finitely generated by the two elements

b =

(2 00 1

)= (1,0) , a =

(1 10 1

)= (0,1)

with the relation a2b = ba. Which falls into the Baumslag-Solitar groups format: BS (1,2). Ingeneral, BS (n,m) =

⟨a,b|anb = bam

⟩.

Let π : LL(Z,Z)→ A f f(Z[

12

])be the map

π (x, f ) =

(x,∑

k2k f (k)

).

We can see that

K = kerπ =

(x, f ) |x = 0,∑

k2k f (k) = 0

.

and so LL(Z,Z)/K = A f f(Z[

12

]). In particular, for a measure µ ∈P (L) and projected measure

µ ∈P (A) we have that Π(A, µ) = K\Π(L,µ).In particular, if Π(L,µ) is trivial, so do Π(A, µ).Let µ be the SRW. Then clearly, Π(L,µ) is trivial and so do Π(A, µ).

Theorem 17 Let µ ∈P

(A f f

(Z[

12

]))be generating with finite first moment. Let µZ be the

projected random walk on Z via the map (x, f ) 7→ x and let α = ∑zµZ (z). Then1. If α < 0 then for almost every path (yn,ϕn) the limit exists limϕn = ϕ∞ ∈ R. And R with

the distribution of ϕ∞ is the Poisson boundary.2. If α = 0 then the Poisson boundary is trivial.3. If α > 0 then the limit limϕn = ϕ∞ ∈Q2 (numbers of the form a

2b ) in the 2-adic topology.

|x|2 = 2−n where x = 2n ·q where q is odd, the metric induced by this norm.

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