Shannon’s theorem and Poisson boundaries for locally ...4.The Shannon-McMillan-Breiman theorem...

Shannon’s theorem and Poisson boundariesfor locally compact groups

Giulio TiozzoUniversity of Toronto

June 17, 2020

Summary

1. The Poisson representation formula - classical case

2. Poisson representation for groups3. The ray (and strip) criterion4. The Shannon-McMillan-Breiman theorem5. Proofs

joint with Behrang Forghani

Summary

1. The Poisson representation formula - classical case2. Poisson representation for groups

3. The ray (and strip) criterion4. The Shannon-McMillan-Breiman theorem5. Proofs


Summary

1. The Poisson representation formula - classical case2. Poisson representation for groups3. The ray (and strip) criterion

4. The Shannon-McMillan-Breiman theorem5. Proofs


Summary

1. The Poisson representation formula - classical case2. Poisson representation for groups3. The ray (and strip) criterion4. The Shannon-McMillan-Breiman theorem

5. Proofs


Summary

1. The Poisson representation formula - classical case2. Poisson representation for groups3. The ray (and strip) criterion4. The Shannon-McMillan-Breiman theorem5. Proofs


The Poisson representation formula

h∞(D) := u : D→ R bounded : ∆u = 0

Theorem (Poisson representation)There is a bijection

h∞(D) ↔ L∞(S1, λ)


TheoremGiven an integrable function f : S1 → R,

the function u : D→ R

u(reiθ) =1

2π

∫ π

−πPr (θ − t)f (eit ) dt

is harmonic on D, i.e. ∆u = 0. Here,

Pr (θ) :=1− r2

1 + r2 − 2r cos θ

is the Poisson kernel.


TheoremGiven an integrable function f : S1 → R, the function u : D→ R

u(reiθ) =1

2π

∫ π



Pr (θ) :=1− r2

1 + r2 − 2r cos θ




u(reiθ) =1

2π

∫ π


is harmonic on D, i.e. ∆u = 0.

Here,

Pr (θ) :=1− r2

1 + r2 − 2r cos θ




u(reiθ) =1

2π

∫ π



Pr (θ) :=1− r2

1 + r2 − 2r cos θ


The Poisson representation formula - II

h∞(D) := u : D→ R bounded : ∆u = 0

Theorem (Poisson)There is a bijection

h∞(D) ↔ L∞(∂D, λ)

I → boundary values

f (ξ) := limz→ξ

u(z)

I ← Poisson integral

u(reiθ) =1

2π

∫ π


The Poisson representation formula - II

h∞(D) := u : D→ R bounded : ∆u = 0

Theorem (Poisson)There is a bijection

h∞(D) ↔ L∞(∂D, λ)

I → boundary values (probabilistic interpretation)

f (ξ) := E( limzt→ξ

u(zt ))

where zt is Brownian motionI ← Poisson integral

u(reiθ) =1

2π

∫ π


The Poisson representation formula - IITheorem (Poisson)There is a bijection

h∞(D) ↔ L∞(∂D, λ)

I → boundary values (probabilistic interpretation)

f (ξ) := E( limzt→ξ

u(zt ))

where zt is Brownian motion

The Poisson representation formula - III

Recall PSL(2,R) = Isom+(H).

Given a = reiθ ∈ D, consider

ga(z) :=z − a1− az

so that ga(a) = 0, hence

g′a(z) :=1− |a|2

(1− az)2

and setting z = eit we obtain

dgaλ

dλ(t) =

∣∣∣g′a(eit )∣∣∣ =

1− r2

|1− rei(t−θ)|2= Pr (θ − t)

RN derivative of boundary action = Poisson kernel


Recall PSL(2,R) = Isom+(H). Given a = reiθ ∈ D,

consider

ga(z) :=z − a1− az


g′a(z) :=1− |a|2

(1− az)2


dgaλ

dλ(t) =

∣∣∣g′a(eit )∣∣∣ =

1− r2

|1− rei(t−θ)|2= Pr (θ − t)



Recall PSL(2,R) = Isom+(H). Given a = reiθ ∈ D, consider

ga(z) :=z − a1− az

so that ga(a) = 0,

hence

g′a(z) :=1− |a|2

(1− az)2


dgaλ

dλ(t) =

∣∣∣g′a(eit )∣∣∣ =

1− r2

|1− rei(t−θ)|2= Pr (θ − t)




ga(z) :=z − a1− az


g′a(z) :=1− |a|2

(1− az)2


dgaλ

dλ(t) =

∣∣∣g′a(eit )∣∣∣ =

1− r2

|1− rei(t−θ)|2= Pr (θ − t)




ga(z) :=z − a1− az


g′a(z) :=1− |a|2

(1− az)2

and setting z = eit

we obtain

dgaλ

dλ(t) =

∣∣∣g′a(eit )∣∣∣ =

1− r2

|1− rei(t−θ)|2= Pr (θ − t)




ga(z) :=z − a1− az


g′a(z) :=1− |a|2

(1− az)2


dgaλ

dλ(t)

=∣∣∣g′a(eit )

∣∣∣ =1− r2

|1− rei(t−θ)|2= Pr (θ − t)




ga(z) :=z − a1− az


g′a(z) :=1− |a|2

(1− az)2


dgaλ

dλ(t) =

∣∣∣g′a(eit )∣∣∣

=1− r2

|1− rei(t−θ)|2= Pr (θ − t)




ga(z) :=z − a1− az


g′a(z) :=1− |a|2

(1− az)2


dgaλ

dλ(t) =

∣∣∣g′a(eit )∣∣∣ =

1− r2

|1− rei(t−θ)|2

= Pr (θ − t)




ga(z) :=z − a1− az


g′a(z) :=1− |a|2

(1− az)2


dgaλ

dλ(t) =

∣∣∣g′a(eit )∣∣∣ =

1− r2

|1− rei(t−θ)|2= Pr (θ − t)




ga(z) :=z − a1− az


g′a(z) :=1− |a|2

(1− az)2


dgaλ

dλ(t) =

∣∣∣g′a(eit )∣∣∣ =

1− r2

|1− rei(t−θ)|2= Pr (θ − t)

RN derivative of boundary action

= Poisson kernel



ga(z) :=z − a1− az


g′a(z) :=1− |a|2

(1− az)2


dgaλ

dλ(t) =

∣∣∣g′a(eit )∣∣∣ =

1− r2

|1− rei(t−θ)|2= Pr (θ − t)


The Poisson representation formula - IV

Recall the Poisson formula

u(reiθ) =1

2π

∫ π

−πf (eit )Pr (θ − t) dt

Setting a = reiθ, dλ = dt2π

u(a) =

∫∂D

f (eit )dgaλ

dλ(t) dλ(t)

Finally if ξ = eit

u(a) =

∫∂D

f (ξ) dgaλ(ξ)

The Poisson representation formula - IV

Theorem (Poisson representation)If f : L∞(∂D, λ)→ R, then

u(a) =

∫∂D

f (ξ) dgaλ(ξ)

is harmonic on D.

Question. Can we generalize this to other groupsG 6= PSL2(R)?

General Poisson representation (Furstenberg)

Let G be a (lcsc) group, µ a probability measure on G.

DefinitionA function u : G→ R is µ-harmonic if

u(g) =

∫G

u(gh) dµ(h)

for all g ∈ G.Let B be a space on which G acts (measurably). A measure νon B is µ-stationary if

ν =

∫G

gν dµ(g).




u(g) =

∫G

u(gh) dµ(h)

for all g ∈ G.

Let B be a space on which G acts (measurably). A measure νon B is µ-stationary if

ν =

∫G

gν dµ(g).




u(g) =

∫G

u(gh) dµ(h)

for all g ∈ G.Let B be a space on which G acts (measurably).

A measure νon B is µ-stationary if

ν =

∫G

gν dµ(g).




u(g) =

∫G

u(gh) dµ(h)

for all g ∈ G.Let B be a space on which G acts (measurably). A measure νon B is µ-stationary if

ν =

∫G

gν dµ(g).

Random walks and µ-boundaries

Let µ be a prob. measure on G.

Consider the random walk

wn := g1g2 . . . gn

where (gi) are i.i.d. with distribution µ.

We denote Ω := (wn) the space of sample paths, andT ((wn)) := (wn+1) is the shift on Ω.

DefinitionA space (B, ν) is a µ-boundary if there exists a measurable map

bnd : Ω→ B

such that bnd = bnd T .

Random walks and µ-boundaries

Let µ be a prob. measure on G.Consider the random walk

wn := g1g2 . . . gn

where (gi) are i.i.d. with distribution µ.

We denote Ω := (wn) the space of sample paths, andT ((wn)) := (wn+1) is the shift on Ω.

DefinitionA space (B, ν) is a µ-boundary if there exists a measurable map

bnd : Ω→ B

such that bnd = bnd T .

Boundary convergenceSuppose that G < Isom(X ,d) a metric space, and that X has a“bordification” X = X ∪ ∂X .

Let o ∈ X a base point.

ow1o w2o

ξ

In most situations,lim

n→∞wno ∈ ∂X

exists a.s.

Boundary convergenceSuppose that G < Isom(X ,d) a metric space, and that X has a“bordification” X = X ∪ ∂X . Let o ∈ X a base point.

ow1o w2o

ξ


n→∞wno ∈ ∂X

exists a.s.


o

w1o w2o

ξ


n→∞wno ∈ ∂X

exists a.s.


ow1o

w2o

ξ


n→∞wno ∈ ∂X

exists a.s.


ow1o w2o

ξ


n→∞wno ∈ ∂X

exists a.s.

Boundary convergenceThen we define the hitting measure

ν(A) := P( limn→∞

wno ∈ A)

which is µ-stationary.

ow1o w2o

ξ

Moreover, (∂X , ν) is a µ-boundary, given by the map

Ω 3 bnd(ω) := limn→∞

wno ∈ ∂X

.

General Poisson representation - II (Furstenberg)Let G be a (lcsc) group, µ a probability measure on G, let (B, ν)be a µ-boundary.

The Poisson transform Φ : L∞(B, ν)→ H∞(G, µ) is

Φf (g) :=

∫B

f dgν

DefinitionA µ-boundary (B, ν) is the Poisson boundary if Φ is anisomorphism

L∞(B, ν)→ H∞(G, µ)

CorollaryPoisson boundary is trivial (= 1 point)⇔ bounded harmonicfunctions are constantExamples. Abelian groups; nilpotent groups



Φf (g) :=

∫B

f dgν


L∞(B, ν)→ H∞(G, µ)

CorollaryPoisson boundary is trivial (= 1 point)⇔ bounded harmonicfunctions are constant

Examples. Abelian groups; nilpotent groups



Φf (g) :=

∫B

f dgν


L∞(B, ν)→ H∞(G, µ)

CorollaryPoisson boundary is trivial (= 1 point)⇔ bounded harmonicfunctions are constantExamples. Abelian groups; nilpotent groups

Identification of the Poisson boundary

Let G < Isom(X ,d) be a group, µ a measure on G, ν the hittingmeasure on ∂X .

Question. Is (∂X , ν) the Poisson boundary for (G, µ)?

(Some) History.

I Furstenberg ’63: semisimple Lie groupsI Kaimanovich ’94: hyperbolic groupsI Karlsson-Margulis ’99: isometries of CAT(0) spacesI Kaimanovich-Masur ’99: mapping class groupI Bader-Shalom ’06: isometries of affine buildingsI Gautero-Matheus ’12: relatively hyperbolic groupsI Horbez ’16 : Out(Fn)I Maher-T. ’18: Cremona group




(Some) History.

I Furstenberg ’63: semisimple Lie groups

I Kaimanovich ’94: hyperbolic groupsI Karlsson-Margulis ’99: isometries of CAT(0) spacesI Kaimanovich-Masur ’99: mapping class groupI Bader-Shalom ’06: isometries of affine buildingsI Gautero-Matheus ’12: relatively hyperbolic groupsI Horbez ’16 : Out(Fn)I Maher-T. ’18: Cremona group




(Some) History.

I Furstenberg ’63: semisimple Lie groupsI Kaimanovich ’94: hyperbolic groups

I Karlsson-Margulis ’99: isometries of CAT(0) spacesI Kaimanovich-Masur ’99: mapping class groupI Bader-Shalom ’06: isometries of affine buildingsI Gautero-Matheus ’12: relatively hyperbolic groupsI Horbez ’16 : Out(Fn)I Maher-T. ’18: Cremona group




(Some) History.

I Furstenberg ’63: semisimple Lie groupsI Kaimanovich ’94: hyperbolic groupsI Karlsson-Margulis ’99: isometries of CAT(0) spaces

I Kaimanovich-Masur ’99: mapping class groupI Bader-Shalom ’06: isometries of affine buildingsI Gautero-Matheus ’12: relatively hyperbolic groupsI Horbez ’16 : Out(Fn)I Maher-T. ’18: Cremona group




(Some) History.

I Furstenberg ’63: semisimple Lie groupsI Kaimanovich ’94: hyperbolic groupsI Karlsson-Margulis ’99: isometries of CAT(0) spacesI Kaimanovich-Masur ’99: mapping class group

I Bader-Shalom ’06: isometries of affine buildingsI Gautero-Matheus ’12: relatively hyperbolic groupsI Horbez ’16 : Out(Fn)I Maher-T. ’18: Cremona group




(Some) History.

I Furstenberg ’63: semisimple Lie groupsI Kaimanovich ’94: hyperbolic groupsI Karlsson-Margulis ’99: isometries of CAT(0) spacesI Kaimanovich-Masur ’99: mapping class groupI Bader-Shalom ’06: isometries of affine buildings

I Gautero-Matheus ’12: relatively hyperbolic groupsI Horbez ’16 : Out(Fn)I Maher-T. ’18: Cremona group




(Some) History.

I Furstenberg ’63: semisimple Lie groupsI Kaimanovich ’94: hyperbolic groupsI Karlsson-Margulis ’99: isometries of CAT(0) spacesI Kaimanovich-Masur ’99: mapping class groupI Bader-Shalom ’06: isometries of affine buildingsI Gautero-Matheus ’12: relatively hyperbolic groups

I Horbez ’16 : Out(Fn)I Maher-T. ’18: Cremona group




(Some) History.

I Furstenberg ’63: semisimple Lie groupsI Kaimanovich ’94: hyperbolic groupsI Karlsson-Margulis ’99: isometries of CAT(0) spacesI Kaimanovich-Masur ’99: mapping class groupI Bader-Shalom ’06: isometries of affine buildingsI Gautero-Matheus ’12: relatively hyperbolic groupsI Horbez ’16 : Out(Fn)

I Maher-T. ’18: Cremona group




(Some) History.

I Furstenberg ’63: semisimple Lie groupsI Kaimanovich ’94: hyperbolic groupsI Karlsson-Margulis ’99: isometries of CAT(0) spacesI Kaimanovich-Masur ’99: mapping class groupI Bader-Shalom ’06: isometries of affine buildingsI Gautero-Matheus ’12: relatively hyperbolic groupsI Horbez ’16 : Out(Fn)I Maher-T. ’18: Cremona group

Ray approximation

Let G be a countable group and µ a measure on G.

A metric don G is temperate if ∃C s.t.

#g ∈ G : d(o,go) ≤ R ≤ CeCR

The measure has finite first moment if∫

G d(o,go) dµ(g) < +∞.

Theorem (Kaimanovich ∼ ’94)Let d be a temperate metric on G, suppose that µ has finite firstmoment w.r.t. d, and let (B, λ) be a µ–boundary. If there aremeasurable functions πn : B → G such that

d(wn, πn(bnd(ω)))

n→ 0

in probability, then (B, λ) is the Poisson boundary.

Ray approximation

Let G be a countable group and µ a measure on G. A metric don G is temperate if ∃C s.t.



G d(o,go) dµ(g) < +∞.


d(wn, πn(bnd(ω)))

n→ 0


Ray approximation




G d(o,go) dµ(g) < +∞.

Theorem (Kaimanovich ∼ ’94)Let d be a temperate metric on G,

suppose that µ has finite firstmoment w.r.t. d, and let (B, λ) be a µ–boundary. If there aremeasurable functions πn : B → G such that

d(wn, πn(bnd(ω)))

n→ 0


Ray approximation




G d(o,go) dµ(g) < +∞.

Theorem (Kaimanovich ∼ ’94)Let d be a temperate metric on G, suppose that µ has finite firstmoment w.r.t. d,

and let (B, λ) be a µ–boundary. If there aremeasurable functions πn : B → G such that

d(wn, πn(bnd(ω)))

n→ 0


Ray approximation




G d(o,go) dµ(g) < +∞.

Theorem (Kaimanovich ∼ ’94)Let d be a temperate metric on G, suppose that µ has finite firstmoment w.r.t. d, and let (B, λ) be a µ–boundary.

If there aremeasurable functions πn : B → G such that

d(wn, πn(bnd(ω)))

n→ 0


Ray approximation




G d(o,go) dµ(g) < +∞.

Theorem (Kaimanovich ∼ ’94)Let d be a temperate metric on G, suppose that µ has finite firstmoment w.r.t. d, and let (B, λ) be a µ–boundary. If there aremeasurable functions πn : B → G

such that

d(wn, πn(bnd(ω)))

n→ 0


Ray approximation




G d(o,go) dµ(g) < +∞.


d(wn, πn(bnd(ω)))

n→ 0

in probability,

then (B, λ) is the Poisson boundary.

Ray approximation




G d(o,go) dµ(g) < +∞.


d(wn, πn(bnd(ω)))

n→ 0


Ray approximation

Theorem (Kaimanovich ∼ ’94)If there exists a sequence πn : B → G such that

d(wn, πn(bnd(ω)))

n→ 0

in probability,


CorollaryIf random walk tracks geodesics sublinearly→ geometricboundary is Poisson boundary.Pick πn(ω) := γω(`n).

Ray approximation


d(wn, πn(bnd(ω)))

n→ 0



Ray approximation


d(wn, πn(bnd(ω)))

n→ 0


CorollaryIf random walk tracks geodesics sublinearly

→ geometricboundary is Poisson boundary.Pick πn(ω) := γω(`n).

Ray approximation


d(wn, πn(bnd(ω)))

n→ 0


CorollaryIf random walk tracks geodesics sublinearly→ geometricboundary is Poisson boundary.

Pick πn(ω) := γω(`n).

Ray approximation


d(wn, πn(bnd(ω)))

n→ 0



Ray approximationIf

d(wn, πn(bnd(ω)))

n→ 0


Ray approximationIf

d(wn, πn(bnd(ω)))

n→ 0

then (B, λ) is the Poisson boundary. Pick πn(ω) := γω(`n).

Ray approximationIf

d(wn, γω(`n))

n→ 0

then (B, λ) is the Poisson boundary. Pick πn(ω) := γω(`n).

Sublinear tracking and entropy

Sublinear tracking and entropy

Hn(Pξ) . log #B(γ(`n), rn)

since rn = o(n)

h(Pξ) = limn→∞

Hn(Pξ)

n. lim

1n

log #B(γ(`n), rn)→ 0

Ray approximation


d(wn, πn(bnd(ω)))

n→ 0


Question. (Kaimanovich ∼ ’96) Does the same criterion workfor locally compact groups?(Kaimanovich-Woess ’02) Ray criterion for invariant Markovoperators (intermediate between discrete and lcsc groups)

Ray approximation


d(wn, πn(bnd(ω)))

n→ 0


Question. (Kaimanovich ∼ ’96) Does the same criterion workfor locally compact groups?

(Kaimanovich-Woess ’02) Ray criterion for invariant Markovoperators (intermediate between discrete and lcsc groups)

Ray approximation


d(wn, πn(bnd(ω)))

n→ 0


Question. (Kaimanovich ∼ ’96) Does the same criterion workfor locally compact groups?(Kaimanovich-Woess ’02) Ray criterion for invariant Markovoperators (intermediate between discrete and lcsc groups)

Ray approximation for locally compact groups

Let G be a locally compact, second countable group,

and µ ameasure on G. Let m be Haar measure, and ρ := dµ

dm .A metric d on G is temperate if ∃C s.t.

m(g ∈ G : d(o,go) ≤ R) ≤ CeCR

Theorem (Forghani-T. ’19)Let d be a temperate metric on G, suppose that µ has finite firstmoment w.r.t. d, and let (B, λ) be a µ–boundary. If there aremeasurable functions πn : B → G such that

d(wn, πn(bnd(ω)))

n→ 0



Let G be a locally compact, second countable group, and µ ameasure on G.

Let m be Haar measure, and ρ := dµdm .

A metric d on G is temperate if ∃C s.t.



d(wn, πn(bnd(ω)))

n→ 0



Let G be a locally compact, second countable group, and µ ameasure on G. Let m be Haar measure, and ρ := dµ

dm .

A metric d on G is temperate if ∃C s.t.



d(wn, πn(bnd(ω)))

n→ 0







d(wn, πn(bnd(ω)))

n→ 0






Theorem (Forghani-T. ’19)Let d be a temperate metric on G, suppose that µ has finite firstmoment w.r.t. d,

and let (B, λ) be a µ–boundary. If there aremeasurable functions πn : B → G such that

d(wn, πn(bnd(ω)))

n→ 0






Theorem (Forghani-T. ’19)Let d be a temperate metric on G, suppose that µ has finite firstmoment w.r.t. d, and let (B, λ) be a µ–boundary.

If there aremeasurable functions πn : B → G such that

d(wn, πn(bnd(ω)))

n→ 0







d(wn, πn(bnd(ω)))

n→ 0







d(wn, πn(bnd(ω)))

n→ 0

in probability,







d(wn, πn(bnd(ω)))

n→ 0


Applications - Isometries of hyperbolic spaces

Recall the drift` := lim

n→∞

d(o,wno)

n

Theorem (Forghani-T. ’19)Let X be a proper δ-hyperbolic space and G < Isom(X ) be aclosed subgroup with bounded exponential growth. Let µ be a(spread-out) measure on G with finite first moment. Then:

I if ` = 0, then the Poisson boundary of (G, µ) is trivial;I if ` > 0, then the Poisson boundary of (G, µ) is (∂X , ν),

where ∂X is the Gromov boundary.

I For countable G: [Kaimanovich ’94] .



n→∞

d(o,wno)

n

Theorem (Forghani-T. ’19)Let X be a proper δ-hyperbolic space

and G < Isom(X ) be aclosed subgroup with bounded exponential growth. Let µ be a(spread-out) measure on G with finite first moment. Then:






n→∞

d(o,wno)

n

Theorem (Forghani-T. ’19)Let X be a proper δ-hyperbolic space and G < Isom(X ) be aclosed subgroup with bounded exponential growth.

Let µ be a(spread-out) measure on G with finite first moment. Then:






n→∞

d(o,wno)

n

Theorem (Forghani-T. ’19)Let X be a proper δ-hyperbolic space and G < Isom(X ) be aclosed subgroup with bounded exponential growth. Let µ be a(spread-out) measure on G with finite first moment.

Then:I if ` = 0, then the Poisson boundary of (G, µ) is trivial;I if ` > 0, then the Poisson boundary of (G, µ) is (∂X , ν),





n→∞

d(o,wno)

n


I if ` = 0, then the Poisson boundary of (G, µ) is trivial;

I if ` > 0, then the Poisson boundary of (G, µ) is (∂X , ν),where ∂X is the Gromov boundary.




n→∞

d(o,wno)

n





Applications - Isometries of CAT(0) spaces

Theorem (Forghani-T. ’19)Let X be a proper CAT(0) space

and G < Isom(X ) be a closedsubgroup with bounded exponential growth. Let µ be a(spread-out) measure on G with finite first moment. Then:


where ∂X is the visual boundary.

I For countable G: [Karlsson-Margulis ’99].


Theorem (Forghani-T. ’19)Let X be a proper CAT(0) space and G < Isom(X ) be a closedsubgroup with bounded exponential growth.

Let µ be a(spread-out) measure on G with finite first moment. Then:





Theorem (Forghani-T. ’19)Let X be a proper CAT(0) space and G < Isom(X ) be a closedsubgroup with bounded exponential growth. Let µ be a(spread-out) measure on G with finite first moment.

Then:I if ` = 0, then the Poisson boundary of (G, µ) is trivial;I if ` > 0, then the Poisson boundary of (G, µ) is (∂X , ν),




Theorem (Forghani-T. ’19)Let X be a proper CAT(0) space and G < Isom(X ) be a closedsubgroup with bounded exponential growth. Let µ be a(spread-out) measure on G with finite first moment. Then:

I if ` = 0, then the Poisson boundary of (G, µ) is trivial;

I if ` > 0, then the Poisson boundary of (G, µ) is (∂X , ν),where ∂X is the visual boundary.



Theorem (Forghani-T. ’19)Let X be a proper CAT(0) space and G < Isom(X ) be a closedsubgroup with bounded exponential growth. Let µ be a(spread-out) measure on G with finite first moment. Then:




Applications - Diestel-Leader graphs

Consider the Diestel-Leader graph

DLq,r := (o1,o2) ∈ Tq × Tr : h1(o1) + h2(o2) = 0.

and edges (z1, z2) ∼ (y1, y2) if z1 ∼ y1 and z2 ∼ y2.



DLq,r := (o1,o2) ∈ Tq × Tr : h1(o1) + h2(o2) = 0.


Theorem (Forghani-T. ’19)Let G < Aut(DLq,r ) be a closed subgroup, let µ be a measureon G with finite first moment, and let V be its vertical drift.Then:

1. if V = 0, then the Poisson boundary of (G, µ) is trivial;2. if V < 0, then the Poisson boundary of (G, µ) is ∂Tq ×ω2

with the hitting measure;3. if V > 0, then the Poisson boundary of (G, µ) is ω1 × ∂Tr

with the hitting measure.



DLq,r := (o1,o2) ∈ Tq × Tr : h1(o1) + h2(o2) = 0.


Theorem (Forghani-T. ’19)Let G < Aut(DLq,r ) be a closed subgroup,

let µ be a measureon G with finite first moment, and let V be its vertical drift.Then:






DLq,r := (o1,o2) ∈ Tq × Tr : h1(o1) + h2(o2) = 0.


Theorem (Forghani-T. ’19)Let G < Aut(DLq,r ) be a closed subgroup, let µ be a measureon G

with finite first moment, and let V be its vertical drift.Then:






DLq,r := (o1,o2) ∈ Tq × Tr : h1(o1) + h2(o2) = 0.


Theorem (Forghani-T. ’19)Let G < Aut(DLq,r ) be a closed subgroup, let µ be a measureon G with finite first moment,

and let V be its vertical drift.Then:






DLq,r := (o1,o2) ∈ Tq × Tr : h1(o1) + h2(o2) = 0.


Theorem (Forghani-T. ’19)Let G < Aut(DLq,r ) be a closed subgroup, let µ be a measureon G with finite first moment, and let V be its vertical drift.

Then:1. if V = 0, then the Poisson boundary of (G, µ) is trivial;2. if V < 0, then the Poisson boundary of (G, µ) is ∂Tq ×ω2





DLq,r := (o1,o2) ∈ Tq × Tr : h1(o1) + h2(o2) = 0.



1. if V = 0, then the Poisson boundary of (G, µ) is trivial;

2. if V < 0, then the Poisson boundary of (G, µ) is ∂Tq ×ω2with the hitting measure;

3. if V > 0, then the Poisson boundary of (G, µ) is ω1 × ∂Trwith the hitting measure.



DLq,r := (o1,o2) ∈ Tq × Tr : h1(o1) + h2(o2) = 0.




with the hitting measure;

3. if V > 0, then the Poisson boundary of (G, µ) is ω1 × ∂Trwith the hitting measure.



DLq,r := (o1,o2) ∈ Tq × Tr : h1(o1) + h2(o2) = 0.






The Shannon-McMillan-Breiman theoremThe (Avez) entropy is

H(µ) := −∑

g

µ(g) logµ(g)

h := limn→∞

H(µn)

nwhere µn(g) := P(wn = g).

Theorem (Kaimanovich-Vershik ∼ ’83, Derriennic ∼ ’85)Let G be a countable group and µ a measure on G, with finiteentropy h. Then almost surely

limn→∞

(−1

nlogµn(wn)

)= h.

Proof. Using subadditivity:

µn+m(wn+m) ≥ µn(wn)µm(w−1n wn+m)


H(µ) := −∑

g

µ(g) logµ(g)

h := limn→∞

H(µn)

n

where µn(g) := P(wn = g).


limn→∞

(−1

nlogµn(wn)

)= h.




H(µ) := −∑

g

µ(g) logµ(g)

h := limn→∞

H(µn)



limn→∞

(−1

nlogµn(wn)

)= h.




H(µ) := −∑

g

µ(g) logµ(g)

h := limn→∞

H(µn)


Theorem (Kaimanovich-Vershik ∼ ’83, Derriennic ∼ ’85)Let G be a countable group and µ a measure on G,

with finiteentropy h. Then almost surely

limn→∞

(−1

nlogµn(wn)

)= h.




H(µ) := −∑

g

µ(g) logµ(g)

h := limn→∞

H(µn)


Theorem (Kaimanovich-Vershik ∼ ’83, Derriennic ∼ ’85)Let G be a countable group and µ a measure on G, with finiteentropy h.

Then almost surely

limn→∞

(−1

nlogµn(wn)

)= h.




H(µ) := −∑

g

µ(g) logµ(g)

h := limn→∞

H(µn)



limn→∞

(−1

nlogµn(wn)

)= h.



The Shannon-McMillan-Breiman theorem

Theorem (Kaimanovich-Vershik ’83, Derriennic ’85)Let G be a countable group and µ a measure on G, with finiteentropy h. Then almost surely

limn→∞

(−1

nlogµn(wn)

)= h.



The Shannon-McMillan-Breiman theorem

Theorem (Kaimanovich-Vershik ’83, Derriennic ’81)Let G be a countable group and µ a measure on G, with finiteentropy h. Then almost surely

limn→∞

(−1

nlogµn(wn)

)= h.


− logµn+m(wn+m) ≤ − logµn(wn)− logµm(w−1n wn+m)

and by Kingman’s theorem.

Question. (Derriennic) Can we generalize to locally compactgroups?

Entropies

Let µ be a measure on G, abs. cont. w.r.t. the Haar measure m

and let ρ := dµdm .

The differential entropy is

hdiff (µ) := −∫

Gρ(g) log ρ(g)dm(g)

Note: it can be negative!The Furstenberg entropy of the µ–boundary (B, λ) is

h(λ) :=

∫G×B

logdgλdλ

(γ) dgλ(γ)dµ(g)

Entropies

Let µ be a measure on G, abs. cont. w.r.t. the Haar measure mand let ρ := dµ

dm .

The differential entropy is




h(λ) :=

∫G×B

logdgλdλ

(γ) dgλ(γ)dµ(g)

Entropies


dm .The differential entropy is




h(λ) :=

∫G×B

logdgλdλ

(γ) dgλ(γ)dµ(g)

Entropies





Note: it can be negative!

The Furstenberg entropy of the µ–boundary (B, λ) is

h(λ) :=

∫G×B

logdgλdλ

(γ) dgλ(γ)dµ(g)

Entropies






h(λ) :=

∫G×B

logdgλdλ

(γ) dgλ(γ)dµ(g)

Poisson boundary

Let (∂G, ν) be the Poisson boundary of (G, µ).

Theorem (Derriennic)For any µ-boundary (B, λ), we have

h(λ) ≤ h(ν).

If h(ν) is finite, then

h(ν) = h(λ)⇔ (B, λ) is the Poisson boundary

Poisson boundary



h(λ) ≤ h(ν).

If h(ν) is finite,

then


Poisson boundary



h(λ) ≤ h(ν).


h(ν) = h(λ)

⇔ (B, λ) is the Poisson boundary

Poisson boundary



h(λ) ≤ h(ν).



The Shannon-McMillan-Breiman theorem - locallycompact case

Let G be a lcsc group,

µ a prob. measure on G, µ m,ρ := dµ

dm .

Theorem (Forghani-T. ’19)Let (∂G, ν) be the Poisson boundary of (G, µ). If ρ is boundedand the differential entropy Hn <∞ for any n, then

lim infn→∞

(−1

nlog

dµn

dm(wn)

)= h(ν)

almost surely.Recall h(ν) = Furstenberg entropy.


Let G be a lcsc group, µ a prob. measure on G,

µ m,ρ := dµ

dm .


lim infn→∞

(−1

nlog

dµn

dm(wn)

)= h(ν)



Let G be a lcsc group, µ a prob. measure on G, µ m,

ρ := dµdm .


lim infn→∞

(−1

nlog

dµn

dm(wn)

)= h(ν)



Let G be a lcsc group, µ a prob. measure on G, µ m,ρ := dµ

dm .


lim infn→∞

(−1

nlog

dµn

dm(wn)

)= h(ν)




dm .

Theorem (Forghani-T. ’19)Let (∂G, ν) be the Poisson boundary of (G, µ).

If ρ is boundedand the differential entropy Hn <∞ for any n, then

lim infn→∞

(−1

nlog

dµn

dm(wn)

)= h(ν)




dm .

Theorem (Forghani-T. ’19)Let (∂G, ν) be the Poisson boundary of (G, µ). If ρ is boundedand the differential entropy Hn <∞ for any n,

then

lim infn→∞

(−1

nlog

dµn

dm(wn)

)= h(ν)




dm .


lim infn→∞

(−1

nlog

dµn

dm(wn)

)= h(ν)

almost surely.

Recall h(ν) = Furstenberg entropy.



dm .


lim infn→∞

(−1

nlog

dµn

dm(wn)

)= h(ν)


Proof of Shannon theorem

Let (B, λ) be a µ–boundary.

Suppose ρ ≤ C.Let ρn := dµn

dm , w∞ := bnd(ω).

1. By the ergodic theorem, for P–almost every ω = (wn)

limn

1n

logdwnλ

dλ(w∞) = h(λ).

2. Let fn : (M, κ)→ R≥0 with∫

fn dκ ≤ C. Then

lim infn

(−1

nlog fn(x)

)≥ 0 κ-a.e.

3. Check ∫Ω

dwn−1λ

dλ(w∞)ρn(wn) dP ≤ C.


Let (B, λ) be a µ–boundary. Suppose ρ ≤ C.

Let ρn := dµndm , w∞ := bnd(ω).


limn

1n

logdwnλ

dλ(w∞) = h(λ).


fn dκ ≤ C. Then

lim infn

(−1

nlog fn(x)

)≥ 0 κ-a.e.

3. Check ∫Ω

dwn−1λ



Let (B, λ) be a µ–boundary. Suppose ρ ≤ C.Let ρn := dµn

dm ,

w∞ := bnd(ω).


limn

1n

logdwnλ

dλ(w∞) = h(λ).


fn dκ ≤ C. Then

lim infn

(−1

nlog fn(x)

)≥ 0 κ-a.e.

3. Check ∫Ω

dwn−1λ




dm , w∞ := bnd(ω).


limn

1n

logdwnλ

dλ(w∞) = h(λ).


fn dκ ≤ C. Then

lim infn

(−1

nlog fn(x)

)≥ 0 κ-a.e.

3. Check ∫Ω

dwn−1λ




dm , w∞ := bnd(ω).

1. By the ergodic theorem,

for P–almost every ω = (wn)

limn

1n

logdwnλ

dλ(w∞) = h(λ).


fn dκ ≤ C. Then

lim infn

(−1

nlog fn(x)

)≥ 0 κ-a.e.

3. Check ∫Ω

dwn−1λ




dm , w∞ := bnd(ω).


limn

1n

logdwnλ

dλ(w∞) = h(λ).


fn dκ ≤ C. Then

lim infn

(−1

nlog fn(x)

)≥ 0 κ-a.e.

3. Check ∫Ω

dwn−1λ




dm , w∞ := bnd(ω).


limn

1n

logdwnλ

dλ(w∞) = h(λ).


fn dκ ≤ C.

Then

lim infn

(−1

nlog fn(x)

)≥ 0 κ-a.e.

3. Check ∫Ω

dwn−1λ




dm , w∞ := bnd(ω).


limn

1n

logdwnλ

dλ(w∞) = h(λ).


fn dκ ≤ C. Then

lim infn

(−1

nlog fn(x)

)≥ 0 κ-a.e.

3. Check ∫Ω

dwn−1λ


Proof of Shannon theorem - II4. Hence

lim infn

(−1

nlog ρn(wn)− 1

nlog

dwnλ

dλ(w∞)

)≥ 0.

lim infn

(−1

nlog ρn(wn)

)≥ h(λ)

5. (Derriennic)

limn→∞

∫Ω

−1n

log ρn(wn) dP = h(ν)

6. By Fatou’s lemma

h(λ) ≤∫

Ω

lim infn

(−1

nlog ρn(wn)

)dP ≤ lim

n

∫Ω

(−1

nlog ρn(wn)

)dP = h(ν)

7. Thus setting λ = ν, a.s.

lim infn

(−1

nlog ρn(wn)

)= h(ν)

QED


lim infn

(−1

nlog ρn(wn)− 1

nlog

dwnλ

dλ(w∞)

)≥ 0.

lim infn

(−1

nlog ρn(wn)

)≥ h(λ)

5. (Derriennic)

limn→∞

∫Ω

−1n



h(λ) ≤∫

Ω

lim infn

(−1

nlog ρn(wn)

)dP ≤ lim

n

∫Ω

(−1

nlog ρn(wn)

)dP = h(ν)

7. Thus setting λ = ν,

a.s.

lim infn

(−1

nlog ρn(wn)

)= h(ν)

QED


lim infn

(−1

nlog ρn(wn)− 1

nlog

dwnλ

dλ(w∞)

)≥ 0.

lim infn

(−1

nlog ρn(wn)

)≥ h(λ)

5. (Derriennic)

limn→∞

∫Ω

−1n



h(λ) ≤∫

Ω

lim infn

(−1

nlog ρn(wn)

)dP ≤ lim

n

∫Ω

(−1

nlog ρn(wn)

)dP = h(ν)

7. Thus setting λ = ν, a.s.

lim infn

(−1

nlog ρn(wn)

)= h(ν)

QED

The end

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Shannon’s theorem and Poisson boundaries for locally ...4.The Shannon-McMillan-Breiman theorem...

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