Entropy Theory for Sofic Group Actions Lewis Bowen Workshop on II 1 - - PowerPoint PPT Presentation

Entropy Theory for Sofic Group Actions

Lewis Bowen Workshop on II1 factors, May 2011

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 1 / 48

Notation

Let (X, µ) be a standard probability space.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 2 / 48

Notation

Let (X, µ) be a standard probability space. Let G be a countable discrete group acting by measure-preserving transformations on (X, µ).

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 2 / 48

Notation

Let (X, µ) be a standard probability space. Let G be a countable discrete group acting by measure-preserving transformations on (X, µ). Two actions G(X1, µ1), G(X2, µ2) are isomorphic if there exists a measure-space isomorphism φ : X1 → X2 with φ(gx) = gφ(x) for a.e. x ∈ X1 and for all g ∈ G.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 2 / 48

Notation

Let (X, µ) be a standard probability space. Let G be a countable discrete group acting by measure-preserving transformations on (X, µ). Two actions G(X1, µ1), G(X2, µ2) are isomorphic if there exists a measure-space isomorphism φ : X1 → X2 with φ(gx) = gφ(x) for a.e. x ∈ X1 and for all g ∈ G. Main Problem: Classify systems up to isomorphism.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 2 / 48

Bernoulli shifts

Let (K, κ) be a standard probability space.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 3 / 48

Bernoulli shifts

Let (K, κ) be a standard probability space. K G = {x : G → K}.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 3 / 48

Bernoulli shifts

Let (K, κ) be a standard probability space. K G = {x : G → K}. κG is the product measure on K G.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 3 / 48

Bernoulli shifts

Let (K, κ) be a standard probability space. K G = {x : G → K}. κG is the product measure on K G. G acts on K G by shifting. (gx)(f) = x(g−1f) for all x ∈ K G, g, f ∈ G.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 3 / 48

Bernoulli shifts

Let (K, κ) be a standard probability space. K G = {x : G → K}. κG is the product measure on K G. G acts on K G by shifting. (gx)(f) = x(g−1f) for all x ∈ K G, g, f ∈ G. G(K G, κG) is the Bernoulli shift over G with base space (K, κ).

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 3 / 48

Kolmogorov-Sinai entropy

For a probability space (K, κ), define the base entropy by H(K, κ) := −

• k∈K

κ(k) log

• κ(k)
• .

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 4 / 48

Kolmogorov-Sinai entropy

For a probability space (K, κ), define the base entropy by H(K, κ) := −

• k∈K

κ(k) log

• κ(k)
• .

Theorem (Kolmogorov, 1958)

If Z(K Z, κZ) is isomorphic to Z(LZ, λZ) then H(K, κ) = H(L, λ).

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 4 / 48

The Converse

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 5 / 48

The Converse

[Ornstein, 1970] H(K, κ) = H(L, λ) ⇒ Z(K Z, κZ) ∼ = Z(LZ, λZ).

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 5 / 48

The Converse

[Ornstein, 1970] H(K, κ) = H(L, λ) ⇒ Z(K Z, κZ) ∼ = Z(LZ, λZ).

Definition (Stepin, 1975)

A group G is Ornstein if H(K, κ) = H(L, λ) ⇒ G(K G, κG) ∼ = G(LG, λG).

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 5 / 48

The Converse

[Ornstein, 1970] H(K, κ) = H(L, λ) ⇒ Z(K Z, κZ) ∼ = Z(LZ, λZ).

Definition (Stepin, 1975)

A group G is Ornstein if H(K, κ) = H(L, λ) ⇒ G(K G, κG) ∼ = G(LG, λG). No finite group is Ornstein.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 5 / 48

The Converse

[Ornstein, 1970] H(K, κ) = H(L, λ) ⇒ Z(K Z, κZ) ∼ = Z(LZ, λZ).

Definition (Stepin, 1975)

A group G is Ornstein if H(K, κ) = H(L, λ) ⇒ G(K G, κG) ∼ = G(LG, λG). No finite group is Ornstein. Infinite amenable groups are Ornstein [Ornstein-Weiss, 1987].

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 5 / 48

The Converse

[Ornstein, 1970] H(K, κ) = H(L, λ) ⇒ Z(K Z, κZ) ∼ = Z(LZ, λZ).

Definition (Stepin, 1975)

A group G is Ornstein if H(K, κ) = H(L, λ) ⇒ G(K G, κG) ∼ = G(LG, λG). No finite group is Ornstein. Infinite amenable groups are Ornstein [Ornstein-Weiss, 1987]. If G contains an Ornstein subgroup H then G is Ornstein [Stepin, 1975].

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 5 / 48

The Converse

[Ornstein, 1970] H(K, κ) = H(L, λ) ⇒ Z(K Z, κZ) ∼ = Z(LZ, λZ).

Definition (Stepin, 1975)

A group G is Ornstein if H(K, κ) = H(L, λ) ⇒ G(K G, κG) ∼ = G(LG, λG). No finite group is Ornstein. Infinite amenable groups are Ornstein [Ornstein-Weiss, 1987]. If G contains an Ornstein subgroup H then G is Ornstein [Stepin, 1975]. Is every countably infinite group Ornstein?

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 5 / 48

New results

Definition

A group G is almost Ornstein if whenever (K, κ), (L, λ) are not two-atom spaces and H(K, κ) = H(L, λ) then G(K G, κG) ∼ = G(LG, λG).

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 6 / 48

New results

Definition

A group G is almost Ornstein if whenever (K, κ), (L, λ) are not two-atom spaces and H(K, κ) = H(L, λ) then G(K G, κG) ∼ = G(LG, λG).

Theorem (L.B., 2011)

Every countable infinite group is almost Ornstein.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 6 / 48

New results

Definition

A group G is almost Ornstein if whenever (K, κ), (L, λ) are not two-atom spaces and H(K, κ) = H(L, λ) then G(K G, κG) ∼ = G(LG, λG).

Theorem (L.B., 2011)

Every countable infinite group is almost Ornstein. If G has a cyclic subgroup C < G of prime order and the normalizer N(C) < G has infinite index, then G is Ornstein.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 6 / 48

Classification

• D. Ornstein and B. Weiss. Entropy and isomorphism theorems for

actions of amenable groups. J. Analyse Math. 48 (1987), 1–141.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 7 / 48

Classification

• D. Ornstein and B. Weiss. Entropy and isomorphism theorems for

actions of amenable groups. J. Analyse Math. 48 (1987), 1–141.

Theorem

If G is infinite and amenable then Bernoulli shifts over G are completely classified by their entropy (which equals their base measure entropy).

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 7 / 48

Classification

• D. Ornstein and B. Weiss. Entropy and isomorphism theorems for

actions of amenable groups. J. Analyse Math. 48 (1987), 1–141.

Theorem

If G is infinite and amenable then Bernoulli shifts over G are completely classified by their entropy (which equals their base measure entropy). What if G is nonamenable?

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 7 / 48

Factor maps

Definition

Given G(X, µ), G(Y, ν), a map φ : X → Y is a factor map if φ∗µ = ν, φ(gx) = gφ(x) for a.e. x ∈ X and all g ∈ G.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 8 / 48

Factor maps

Definition

Given G(X, µ), G(Y, ν), a map φ : X → Y is a factor map if φ∗µ = ν, φ(gx) = gφ(x) for a.e. x ∈ X and all g ∈ G. Let G be amenable.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 8 / 48

Factor maps

Definition

Given G(X, µ), G(Y, ν), a map φ : X → Y is a factor map if φ∗µ = ν, φ(gx) = gφ(x) for a.e. x ∈ X and all g ∈ G. Let G be amenable. Entropy is nonincreasing under factor maps.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 8 / 48

Factor maps

Definition

Given G(X, µ), G(Y, ν), a map φ : X → Y is a factor map if φ∗µ = ν, φ(gx) = gφ(x) for a.e. x ∈ X and all g ∈ G. Let G be amenable. Entropy is nonincreasing under factor maps. The full n-shift over G has entropy log(n).

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 8 / 48

Factor maps

Definition

Given G(X, µ), G(Y, ν), a map φ : X → Y is a factor map if φ∗µ = ν, φ(gx) = gφ(x) for a.e. x ∈ X and all g ∈ G. Let G be amenable. Entropy is nonincreasing under factor maps. The full n-shift over G has entropy log(n). = ⇒ the full 2-shift over G cannot factor onto the full 4-shift over G.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 8 / 48

The Ornstein-Weiss Example

Theorem (Ornstein-Weiss, 1987)

If F = a, b is the rank 2 free group then the full 2-shift over F factors

• nto the full 4-shift over F.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 9 / 48

The Ornstein-Weiss Example

Theorem (Ornstein-Weiss, 1987)

If F = a, b is the rank 2 free group then the full 2-shift over F factors

• nto the full 4-shift over F.

Define φ : (Z/2Z)F → (Z/2Z × Z/2Z)F by

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 9 / 48

The Ornstein-Weiss Example

Theorem (Ornstein-Weiss, 1987)

If F = a, b is the rank 2 free group then the full 2-shift over F factors

• nto the full 4-shift over F.

Define φ : (Z/2Z)F → (Z/2Z × Z/2Z)F by φ(x)(g) =

• x(g) + x(ga), x(g) + x(gb)
• .

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 9 / 48

More on factors

Open : If G is non-amenable, does every Bernoulli shift factor onto every Bernoulli shift?

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 10 / 48

More on factors

Open : If G is non-amenable, does every Bernoulli shift factor onto every Bernoulli shift? Ball (2005): for every non-amenable group G there is some m = m(G) > 0 such that the m-shift factors onto every Bernoulli shift. Bowen (2009): if G contains a rank 2 free subgroup then ‘yes’.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 10 / 48

Recap

If G is non-amenable then, for some n, the n-shift factors onto all other Bernoulli shifts.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 11 / 48

Recap

If G is non-amenable then, for some n, the n-shift factors onto all other Bernoulli shifts. So if “entropy theory” requires the n-shift to have entropy log(n), and that entropy does not increase under factors then there is no entropy theory for non-amenable groups.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 11 / 48

New results

Theorem (L. B., 2009)

If G is a sofic group then Kolmogorov’s direction holds. I.e., if G(K G, κG) is isomorphic to G(LG, λG) then H(K, κ) = H(L, λ).

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 12 / 48

The case G = Z.

Let T : X → X be an automorphism of (X, µ).

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 13 / 48

The case G = Z.

Let T : X → X be an automorphism of (X, µ). Let φ : X → A be an observable (i.e., a measurable map into a finite set).

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 13 / 48

The case G = Z.

Let T : X → X be an automorphism of (X, µ). Let φ : X → A be an observable (i.e., a measurable map into a finite set). Let x ∈ X be a typical element and consider the sequence (. . . , φ(T −1x), φ(x), φ(Tx), . . .).

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 13 / 48

The case G = Z.

Let T : X → X be an automorphism of (X, µ). Let φ : X → A be an observable (i.e., a measurable map into a finite set). Let x ∈ X be a typical element and consider the sequence (. . . , φ(T −1x), φ(x), φ(Tx), . . .). The idea: For n > 0, count the number of sequences (a1, a2, . . . , an) with elements ai ∈ A that approximate the above sequence.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 13 / 48

Local statistics

Let W ⊂ Z be finite. (W stands for window )

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 14 / 48

Local statistics

Let W ⊂ Z be finite. (W stands for window ) Define φW : X → AW = A × A × . . . × A

• W

by φW(x) :=

• φ(T wx)
• w∈W.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 14 / 48

Local statistics

Let W ⊂ Z be finite. (W stands for window ) Define φW : X → AW = A × A × . . . × A

• W

by φW(x) :=

• φ(T wx)
• w∈W.

φW

∗ µ is a measure on AW that encodes the local statistics .

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 14 / 48

Sequences

Let ψ : {1, . . . , n} → A be a map.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 15 / 48

Sequences

Let ψ : {1, . . . , n} → A be a map. ψW : {1, . . . , n} → AW is defined by ψW(j) =

• ψ(j + w)
• w∈W.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 15 / 48

Sequences

Let ψ : {1, . . . , n} → A be a map. ψW : {1, . . . , n} → AW is defined by ψW(j) =

• ψ(j + w)
• w∈W.

(define it arbitrarily if j + w / ∈ {1, . . . , n})

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 15 / 48

Sequences

Let ψ : {1, . . . , n} → A be a map. ψW : {1, . . . , n} → AW is defined by ψW(j) =

• ψ(j + w)
• w∈W.

(define it arbitrarily if j + w / ∈ {1, . . . , n}) Let u be the uniform measure on {1, . . . , n}. ψW

∗ u is a measure on AW

that encodes the local statistics of the sequence (ψ(1), . . . , ψ(n)) ∈ An.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 15 / 48

Entropy as a growth rate

Let dW(φ, ψ) be the l1-distance between φW

∗ µ and ψW ∗ u:

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 16 / 48

Entropy as a growth rate

Let dW(φ, ψ) be the l1-distance between φW

∗ µ and ψW ∗ u:

dW(φ, ψ) :=

• α∈AW
• φW

∗ µ(α) − ψW ∗ u(α)

• .

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 16 / 48

Entropy as a growth rate

Let dW(φ, ψ) be the l1-distance between φW

∗ µ and ψW ∗ u:

dW(φ, ψ) :=

• α∈AW
• φW

∗ µ(α) − ψW ∗ u(α)

• .

Theorem

h(T, φ) = inf

W⊂Z inf ǫ>0 lim n→∞

1 n log

• ψ : {1, . . . , n} → A : dW(φ, ψ) < ǫ
• .

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 16 / 48

Sofic Groups

A sofic approximation to G is a sequence Σ = {σi}∞

i=1 of maps

σi : G → Sym(mi) such that

1

for every f, g ∈ G, lim

i→∞

1 mi |{1 ≤ p ≤ mi : σi(g)σi(f)p = σi(gf)p}| = 1

2

for every f = g ∈ G, lim

i→∞

1 mi |{1 ≤ p ≤ mi : σi(g)p = σi(f)p}| = 1

3

limi→∞ mi = +∞.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 17 / 48

Sofic Groups

A sofic approximation to G is a sequence Σ = {σi}∞

i=1 of maps

σi : G → Sym(mi) such that

1

for every f, g ∈ G, lim

i→∞

1 mi |{1 ≤ p ≤ mi : σi(g)σi(f)p = σi(gf)p}| = 1

2

for every f = g ∈ G, lim

i→∞

1 mi |{1 ≤ p ≤ mi : σi(g)p = σi(f)p}| = 1

3

limi→∞ mi = +∞. G is sofic if there exists a sofic approximation to G.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 17 / 48

Sofic Groups

A sofic approximation to G is a sequence Σ = {σi}∞

i=1 of maps

σi : G → Sym(mi) such that

1

for every f, g ∈ G, lim

i→∞

1 mi |{1 ≤ p ≤ mi : σi(g)σi(f)p = σi(gf)p}| = 1

2

for every f = g ∈ G, lim

i→∞

1 mi |{1 ≤ p ≤ mi : σi(g)p = σi(f)p}| = 1

3

limi→∞ mi = +∞. G is sofic if there exists a sofic approximation to G.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 17 / 48

Sofic Groups

Residually finite groups are sofic. Hence all linear groups are sofic.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 18 / 48

Sofic Groups

Residually finite groups are sofic. Hence all linear groups are sofic. Amenable groups are sofic.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 18 / 48

Sofic Groups

Residually finite groups are sofic. Hence all linear groups are sofic. Amenable groups are sofic. (Gromov 1999, Weiss 2000, Elek-Szabo 2005) If G is sofic then G satisfies Gottshalk’s surjunctivity conjecture, Connes embedding conjecture, the Determinant conjecture, Kaplansky’s direct finiteness conjecture.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 18 / 48

Sofic Groups

Residually finite groups are sofic. Hence all linear groups are sofic. Amenable groups are sofic. (Gromov 1999, Weiss 2000, Elek-Szabo 2005) If G is sofic then G satisfies Gottshalk’s surjunctivity conjecture, Connes embedding conjecture, the Determinant conjecture, Kaplansky’s direct finiteness conjecture. Open : Is every countable group sofic?

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 18 / 48

Entropy for Sofic Groups

Let G(X, µ) be a system,

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 19 / 48

Entropy for Sofic Groups

Let G(X, µ) be a system, Σ = {σi} be a sofic approximation to G where σi : G → Sym(mi),

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 19 / 48

Entropy for Sofic Groups

Let G(X, µ) be a system, Σ = {σi} be a sofic approximation to G where σi : G → Sym(mi), φ : X → A be a measurable map into a finite set.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 19 / 48

Entropy for Sofic Groups

Let G(X, µ) be a system, Σ = {σi} be a sofic approximation to G where σi : G → Sym(mi), φ : X → A be a measurable map into a finite set. The idea: Count the number of observables ψ : {1, . . . , mi} → A so that (G, [mi], ui, ψ) approximates (G, X, µ, φ).

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 19 / 48

Approximating

If W ⊂ G is finite, let φW: X → AW be the map φW(x) :=

• φ
• wx
• w∈W.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 20 / 48

Approximating

If W ⊂ G is finite, let φW: X → AW be the map φW(x) :=

• φ
• wx
• w∈W.

Given ψ : {1, . . . , mi} → A, ψW: {1, . . . , mi} → AW is the map ψW(j) :=

• ψ
• σ(w)j
• w∈W.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 20 / 48

Approximating

If W ⊂ G is finite, let φW: X → AW be the map φW(x) :=

• φ
• wx
• w∈W.

Given ψ : {1, . . . , mi} → A, ψW: {1, . . . , mi} → AW is the map ψW(j) :=

• ψ
• σ(w)j
• w∈W.

Let dW(φ, ψ) be the l1-distance between φW

∗ µ and ψW ∗ u.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 20 / 48

Entropy for sofic groups

h

• Σ, φ
• := inf

W⊂G inf ǫ>0 lim sup i→∞

log

• {ψ : {1, . . . , mi} → A : dW(φ, ψ) ≤ ǫ}
• mi

.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 21 / 48

Entropy for sofic groups

h

• Σ, φ
• := inf

W⊂G inf ǫ>0 lim sup i→∞

log

• {ψ : {1, . . . , mi} → A : dW(φ, ψ) ≤ ǫ}
• mi

.

Theorem (L.B. ’09)

If φ1 and φ2 are generating then h

• Σ, φ1) = h(Σ, φ2). So let

h(Σ, G, X, µ) be this common number.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 21 / 48

Entropy for sofic groups

h

• Σ, φ
• := inf

W⊂G inf ǫ>0 lim sup i→∞

log

• {ψ : {1, . . . , mi} → A : dW(φ, ψ) ≤ ǫ}
• mi

.

Theorem (L.B. ’09)

If φ1 and φ2 are generating then h

• Σ, φ1) = h(Σ, φ2). So let

h(Σ, G, X, µ) be this common number.

Theorem (L.B. ’10, Kerr-Li ’10)

If G is amenable then h

• Σ, G, X, µ
• is the classical entropy of (G, X, µ).

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 21 / 48

Entropy for sofic groups

h

• Σ, φ
• := inf

W⊂G inf ǫ>0 lim sup i→∞

log

• {ψ : {1, . . . , mi} → A : dW(φ, ψ) ≤ ǫ}
• mi

.

Theorem (L.B. ’09)

If φ1 and φ2 are generating then h

• Σ, φ1) = h(Σ, φ2). So let

h(Σ, G, X, µ) be this common number.

Theorem (L.B. ’10, Kerr-Li ’10)

If G is amenable then h

• Σ, G, X, µ
• is the classical entropy of (G, X, µ).

Theorem (L.B. ’09)

h

• Σ, G, K G, κG

= H(K, κ).

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 21 / 48

Proof sketch

Theorem

If φ1 and φ2 are generating then h

• Σ, φ1) = h(Σ, φ2). So let

h(Σ, G, X, µ) be this common number.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 22 / 48

Proof sketch

Theorem

If φ1 and φ2 are generating then h

• Σ, φ1) = h(Σ, φ2). So let

h(Σ, G, X, µ) be this common number. Two observables φ : X → A, ψ : X → B are equivalent if the partitions {φ−1(a) : a ∈ A}, {ψ−1(b) : b ∈ B} agree up to measure zero.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 22 / 48

Proof sketch

Theorem

If φ1 and φ2 are generating then h

• Σ, φ1) = h(Σ, φ2). So let

h(Σ, G, X, µ) be this common number. Two observables φ : X → A, ψ : X → B are equivalent if the partitions {φ−1(a) : a ∈ A}, {ψ−1(b) : b ∈ B} agree up to measure zero. Let P be the set of all equivalence classes of observables φ with H(φ) < ∞.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 22 / 48

Proof sketch

Theorem

If φ1 and φ2 are generating then h

• Σ, φ1) = h(Σ, φ2). So let

h(Σ, G, X, µ) be this common number. Two observables φ : X → A, ψ : X → B are equivalent if the partitions {φ−1(a) : a ∈ A}, {ψ−1(b) : b ∈ B} agree up to measure zero. Let P be the set of all equivalence classes of observables φ with H(φ) < ∞.

Definition (Rohlin distance)

d(φ, ψ) := 2H(φ ∨ ψ) − H(ψ) − H(φ) = H(φ|ψ) + H(ψ|φ).

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 22 / 48

Proof sketch

Theorem

If φ1 and φ2 are generating then h

• Σ, φ1) = h(Σ, φ2). So let

h(Σ, G, X, µ) be this common number. Two observables φ : X → A, ψ : X → B are equivalent if the partitions {φ−1(a) : a ∈ A}, {ψ−1(b) : b ∈ B} agree up to measure zero. Let P be the set of all equivalence classes of observables φ with H(φ) < ∞.

Definition (Rohlin distance)

d(φ, ψ) := 2H(φ ∨ ψ) − H(ψ) − H(φ) = H(φ|ψ) + H(ψ|φ).

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 22 / 48

Proof sketch

Definition

φ refines ψ if H(ψ ∨ φ) = H(φ).

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 23 / 48

Proof sketch

Definition

φ refines ψ if H(ψ ∨ φ) = H(φ).

Definition

φ and ψ are combinatorially equivalent if there exists finite subsets K, L ⊂ G such that φK refines ψ and ψL refines φ.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 23 / 48

Proof sketch

Theorem

If φ is a generator then its combinatorial equivalence class is dense in the space of all generating observables.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 24 / 48

Proof sketch

Theorem

If φ is a generator then its combinatorial equivalence class is dense in the space of all generating observables.

Lemma

h(Σ, φ) is upper semi-continuous in φ.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 24 / 48

Proof sketch

Theorem

If φ is a generator then its combinatorial equivalence class is dense in the space of all generating observables.

Lemma

h(Σ, φ) is upper semi-continuous in φ.

Theorem

If φ and ψ are combinatorially equivalent then h(Σ, φ) = h(Σ, ψ).

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 24 / 48

Proof sketch

Definition

φ is a simple splitting of ψ if there exists f ∈ G and an observable ω refined by ψ such that φ = ψ ∨ ω ◦ f. φ is a splitting of ψ if it can be obtained from ψ by a sequence of simple splittings.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 25 / 48

Proof sketch

Definition

φ is a simple splitting of ψ if there exists f ∈ G and an observable ω refined by ψ such that φ = ψ ∨ ω ◦ f. φ is a splitting of ψ if it can be obtained from ψ by a sequence of simple splittings.

Lemma

If φ and ψ are equivalent then there exists an observable ω that is a splitting of both φ and ψ.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 25 / 48

Proof sketch

Definition

φ is a simple splitting of ψ if there exists f ∈ G and an observable ω refined by ψ such that φ = ψ ∨ ω ◦ f. φ is a splitting of ψ if it can be obtained from ψ by a sequence of simple splittings.

Lemma

If φ and ψ are equivalent then there exists an observable ω that is a splitting of both φ and ψ.

Proposition

If φ is a simple splitting of ψ then h(Σ, φ) = h(Σ, ψ).

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 25 / 48

Applications: von Neumann algebras

Definition

G1(X1, µ1) and G2(X2, µ2) are von Neumann equivalent (vNE) if L∞(X1, µ1) ⋊ G1 ∼ = L∞(X2, µ2) ⋊ G2.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 26 / 48

Applications: von Neumann algebras

Definition

G1(X1, µ1) and G2(X2, µ2) are von Neumann equivalent (vNE) if L∞(X1, µ1) ⋊ G1 ∼ = L∞(X2, µ2) ⋊ G2.

Theorem (Popa 2006)

If G is an ICC property (T) group then any two von Neumann equivalent Bernoulli shifts over G are isomorphic.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 26 / 48

Applications: von Neumann algebras

Definition

G1(X1, µ1) and G2(X2, µ2) are von Neumann equivalent (vNE) if L∞(X1, µ1) ⋊ G1 ∼ = L∞(X2, µ2) ⋊ G2.

Theorem (Popa 2006)

If G is an ICC property (T) group then any two von Neumann equivalent Bernoulli shifts over G are isomorphic.

Corollary

If, in addition, G is sofic and Ornstein then Bernoulli shifts over G are classified up to vNE by base measure entropy. E.g., this occurs when G = PSLn(Z) for n > 2.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 26 / 48

Applications: orbit equivalence

Definition

G1(X1, µ1) is orbit equivalent (OE) to G2(X2, µ2) if there exists a measure-space isomorphism φ : X1 → X2 such that φ(G1x) = G2φ(x) for a.e. x ∈ X1.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 27 / 48

Applications: orbit equivalence

Definition

G1(X1, µ1) is orbit equivalent (OE) to G2(X2, µ2) if there exists a measure-space isomorphism φ : X1 → X2 such that φ(G1x) = G2φ(x) for a.e. x ∈ X1.

Theorem (Dye 1959, Ornstein-Weiss 1980)

If G1 and G2 are amenable and infinite and their respective actions are ergodic and free then G1(X1, µ1) is OE to G2(X2, µ2).

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 27 / 48

OE rigidity

Theorem

For the following groups, OE of Bernoulli shift implies conjugacy of Bernoulli shifts: property (T) groups with ICC (Popa 2007), mapping class groups with 3g + n − 4 > 0, (g, n) / ∈ {(1, 2), (2, 0)} (Kida, 2008), direct products of infinite non-amenable groups with no finite normal subgroups (Popa, 2008).

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 28 / 48

OE rigidity

Theorem

For the following groups, OE of Bernoulli shift implies conjugacy of Bernoulli shifts: property (T) groups with ICC (Popa 2007), mapping class groups with 3g + n − 4 > 0, (g, n) / ∈ {(1, 2), (2, 0)} (Kida, 2008), direct products of infinite non-amenable groups with no finite normal subgroups (Popa, 2008).

Corollary

If G is as above then Bernoulli shifts over G are classified up to OE by base measure entropy.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 28 / 48

Free Groups: a special case

Let F = s1, . . . , sr. Let F act on (X, µ).

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 29 / 48

Free Groups: a special case

Let F = s1, . . . , sr. Let F act on (X, µ). Given an observable φ : X → A, define F(φ) := −(2r − 1)H(φ) +

r

• i=1

H(φ ∨ φ ◦ si); f(φ) := inf

n F

• φB(e,n)

.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 29 / 48

Free Groups: a special case

Let F = s1, . . . , sr. Let F act on (X, µ). Given an observable φ : X → A, define F(φ) := −(2r − 1)H(φ) +

r

• i=1

H(φ ∨ φ ◦ si); f(φ) := inf

n F

• φB(e,n)

.

Theorem

If φ1 and φ2 are generating then f(φ1) = f(φ2). So we may define f(F, X, µ) = f(φ1). Moreover, f(F, K F, κF) = H(K, κ).

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 29 / 48

Free Groups: a special case

For each n ≥ 1, let σn : F = s1, . . . , sr → Sym(n) be chosen uniformly at random.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 30 / 48

Free Groups: a special case

For each n ≥ 1, let σn : F = s1, . . . , sr → Sym(n) be chosen uniformly at random. Define h∗

• φ
• := inf

W inf ǫ>0 lim sup n→∞

log E

• {ψ : {1, . . . , n} → A : dW(φ, ψ) ≤ ǫ}
• n

.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 30 / 48

Free Groups: a special case

For each n ≥ 1, let σn : F = s1, . . . , sr → Sym(n) be chosen uniformly at random. Define h∗

• φ
• := inf

W inf ǫ>0 lim sup n→∞

log E

• {ψ : {1, . . . , n} → A : dW(φ, ψ) ≤ ǫ}
• n

.

Theorem

h∗

• φ
• = f(φ).

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 30 / 48

Some strange phenomena

f is not monotone under factor maps.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 31 / 48

Some strange phenomena

f is not monotone under factor maps. f is not well defined if the system does not have a finite entropy generating observable.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 31 / 48

Some strange phenomena

f is not monotone under factor maps. f is not well defined if the system does not have a finite entropy generating observable. If µ = tµ1 + (1 − t)µ2 where µ1 and µ2 are invariant and mutually singular then f(µ, φ) = tf(µ1, φ) + (1 − t)f(µ2, φ) − (r − 1)H(t) where H(t) = −t log(t) − (1 − t) log(1 − t).

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 31 / 48

Some strange phenomena

f is not monotone under factor maps. f is not well defined if the system does not have a finite entropy generating observable. If µ = tµ1 + (1 − t)µ2 where µ1 and µ2 are invariant and mutually singular then f(µ, φ) = tf(µ1, φ) + (1 − t)f(µ2, φ) − (r − 1)H(t) where H(t) = −t log(t) − (1 − t) log(1 − t). f can take negative values.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 31 / 48

A stationary Markov chain

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 32 / 48

A shift-invariant probability measure

We obtain an invariant probability measure on AZ where A is the state space.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 33 / 48

A different set of transition probabilities for the same stationary vector

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 34 / 48

The Cayley graph of the free group

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 35 / 48

A Markov chain over the free group

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 36 / 48

Markov chains and f

Theorem

A shift invariant probability measure µ on AF comes from a Markov chain if and only if f(µ, φ) = F(µ, φ) where φ is the canonical

• bservable, φ(x) := x(e).

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 37 / 48

Markov chains and f

Theorem

A shift invariant probability measure µ on AF comes from a Markov chain if and only if f(µ, φ) = F(µ, φ) where φ is the canonical

• bservable, φ(x) := x(e).

F varies continuously with transition probabilities.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 37 / 48

Markov chains and f

Theorem

A shift invariant probability measure µ on AF comes from a Markov chain if and only if f(µ, φ) = F(µ, φ) where φ is the canonical

• bservable, φ(x) := x(e).

F varies continuously with transition probabilities. If µ is uniformly distributed on a set of cardinality n then F(µ, φ) = −(r − 1) log(n).

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 37 / 48

Markov chains and f

Theorem

A shift invariant probability measure µ on AF comes from a Markov chain if and only if f(µ, φ) = F(µ, φ) where φ is the canonical

• bservable, φ(x) := x(e).

F varies continuously with transition probabilities. If µ is uniformly distributed on a set of cardinality n then F(µ, φ) = −(r − 1) log(n). There exist mixing Markov chains with negative f-invariant.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 37 / 48

Markov chains and f

Theorem

A shift invariant probability measure µ on AF comes from a Markov chain if and only if f(µ, φ) = F(µ, φ) where φ is the canonical

• bservable, φ(x) := x(e).

F varies continuously with transition probabilities. If µ is uniformly distributed on a set of cardinality n then F(µ, φ) = −(r − 1) log(n). There exist mixing Markov chains with negative f-invariant. Problem : Classify Markov chains over a free group.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 37 / 48

The space of actions

Let (X, µ) be an atomless standard probability space.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 38 / 48

The space of actions

Let (X, µ) be an atomless standard probability space. Let A denote the set of all m.p. actions of F = a, b on (X, µ) with the weak topology.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 38 / 48

The space of actions

Let (X, µ) be an atomless standard probability space. Let A denote the set of all m.p. actions of F = a, b on (X, µ) with the weak topology.

Theorem

The set of actions measurably conjugate to a Markov chain is dense in

• A. Moreover, for every action T there exists a sequence of Markov

actions T1, T2, . . . that converges to T such that f(Ti) ց f(T) as i → ∞.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 38 / 48

An alternative formula

Define F ′(φ) by F ′(φ) := ha(φ) + hb(φ) − H(φ) where ha is the entropy rate of φ with respect to the transformation x → a · x.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 39 / 48

An alternative formula

Define F ′(φ) by F ′(φ) := ha(φ) + hb(φ) − H(φ) where ha is the entropy rate of φ with respect to the transformation x → a · x. Define f ′(φ) by f ′(φ) := inf

n F ′(φB(e,n)).

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 39 / 48

An alternative formula

Define F ′(φ) by F ′(φ) := ha(φ) + hb(φ) − H(φ) where ha is the entropy rate of φ with respect to the transformation x → a · x. Define f ′(φ) by f ′(φ) := inf

n F ′(φB(e,n)).

Theorem

f ′ = f.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 39 / 48

An alternative formula

Define F ′(φ) by F ′(φ) := ha(φ) + hb(φ) − H(φ) where ha is the entropy rate of φ with respect to the transformation x → a · x. Define f ′(φ) by f ′(φ) := inf

n F ′(φB(e,n)).

Theorem

f ′ = f. If the system is Markov, then ha(φ) = H(φ ∨ φa) − H(φ). So, F ′ = F = f = f ′.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 39 / 48

Side remark: other groups

Suppose Γ = G ∗A H where G, H and A are amenable. Define FΓ(φ) := hG(φ) + hH(φ) − hA(φ) fΓ(φ) := inf

n FΓ(φn).

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 40 / 48

Side remark: other groups

Suppose Γ = G ∗A H where G, H and A are amenable. Define FΓ(φ) := hG(φ) + hH(φ) − hA(φ) fΓ(φ) := inf

n FΓ(φn).

Theorem

If φ1 and φ2 are generating then fΓ(φ1) = fΓ(φ2). So we may define fΓ(Γ, X, µ) = fΓ(φ1). Moreover, fΓ(Γ, K F, κF) = H(K, κ).

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 40 / 48

Applications

Theorem (Abramov-Rohlin)

If φ, ψ are observables of a Z-dynamical system then h(φ ∨ ψ) = h(φ) + h(ψ|φ) where h(ψ|φ) := lim

n→∞

1 2n + 1H

• n
• i=−n

ψT i|

• j∈Z

φT j .

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 41 / 48

Applications

Theorem (Abramov-Rohlin)

If φ, ψ are observables of a Z-dynamical system then h(φ ∨ ψ) = h(φ) + h(ψ|φ) where h(ψ|φ) := lim

n→∞

1 2n + 1H

• n
• i=−n

ψT i|

• j∈Z

φT j .

Theorem

If φ, ψ are observables of an F-dynamical system then f(φ ∨ ψ) = f(φ) + f(ψ|φ) where φF is the smallest F-invariant σ-algebra containing φ, F(ψ|φ) := −(2r − 1)H(ψ|φF) +

r

• i=1

H(ψ ∨ ψsi|φF) f(ψ|φ) := inf

n F(ψn|φ).

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 41 / 48

Finite to one maps

If (Z, ζ) is finite and ζ is uniformly distributed then f(F, Z, ζ) = −(r − 1) log(|Z|).

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 42 / 48

Finite to one maps

If (Z, ζ) is finite and ζ is uniformly distributed then f(F, Z, ζ) = −(r − 1) log(|Z|).

Theorem

If (F, X, µ) admits an n-to-1 factor map onto (F, Y, ν) then f(F, X, µ) = f(F, Y, ν) − (r − 1) log(n).

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 42 / 48

Systems of algebraic origin

Let G be a compact separable group and let T : G → G be a group automorphism fixing a closed normal subgroup N.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 43 / 48

Systems of algebraic origin

Let G be a compact separable group and let T : G → G be a group automorphism fixing a closed normal subgroup N.

Theorem (Yuzvinskii, 1965)

h(T, G, HaarG) = h(T, N, HaarN ) + h(T, G/N, HaarG/N ).

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 43 / 48

Systems of algebraic origin

Let G be a compact separable group and let T : G → G be a group automorphism fixing a closed normal subgroup N.

Theorem (Yuzvinskii, 1965)

h(T, G, HaarG) = h(T, N, HaarN ) + h(T, G/N, HaarG/N ).

Theorem (L.B. - Gutman, 2009)

If G is totally disconnected and F acts by automorphisms on G with closed normal subgroup N then f(F, G, HaarG) = f(F, N, HaarN ) + f(F, G/N, HaarG/N ).

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 43 / 48

Systems of algebraic origin

Let G be a compact separable group and let T : G → G be a group automorphism fixing a closed normal subgroup N.

Theorem (Yuzvinskii, 1965)

h(T, G, HaarG) = h(T, N, HaarN ) + h(T, G/N, HaarG/N ).

Theorem (L.B. - Gutman, 2009)

If G is totally disconnected and F acts by automorphisms on G with closed normal subgroup N then f(F, G, HaarG) = f(F, N, HaarN ) + f(F, G/N, HaarG/N ). Let G = (Z/2Z)F. Let N = {0, 1}. By Ornstein-Weiss’ example, G/N ∼ = G × G = (Z/2Z × Z/2Z)F.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 43 / 48

Systems of algebraic origin

Let G be a compact separable group and let T : G → G be a group automorphism fixing a closed normal subgroup N.

Theorem (Yuzvinskii, 1965)

h(T, G, HaarG) = h(T, N, HaarN ) + h(T, G/N, HaarG/N ).

Theorem (L.B. - Gutman, 2009)

If G is totally disconnected and F acts by automorphisms on G with closed normal subgroup N then f(F, G, HaarG) = f(F, N, HaarN ) + f(F, G/N, HaarG/N ). Let G = (Z/2Z)F. Let N = {0, 1}. By Ornstein-Weiss’ example, G/N ∼ = G × G = (Z/2Z × Z/2Z)F. f(F, G, HaarG) = f(F, N, HaarN ) + f(F, G/N, HaarG/N )

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 43 / 48

Systems of algebraic origin

Let G be a compact separable group and let T : G → G be a group automorphism fixing a closed normal subgroup N.

Theorem (Yuzvinskii, 1965)

h(T, G, HaarG) = h(T, N, HaarN ) + h(T, G/N, HaarG/N ).

Theorem (L.B. - Gutman, 2009)

If G is totally disconnected and F acts by automorphisms on G with closed normal subgroup N then f(F, G, HaarG) = f(F, N, HaarN ) + f(F, G/N, HaarG/N ). Let G = (Z/2Z)F. Let N = {0, 1}. By Ornstein-Weiss’ example, G/N ∼ = G × G = (Z/2Z × Z/2Z)F. f(F, G, HaarG) = f(F, N, HaarN ) + f(F, G/N, HaarG/N ) log(2) = − log(2) + log(4).

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 43 / 48

Systems of algebraic origin

Let f ∈ Z[G]. G acts on Z[G]/Z[G]f and therefore on the dual Xf =

• Z[G]/Z[G]f, a compact group. It preserves the Haar measure.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 44 / 48

Systems of algebraic origin

Let f ∈ Z[G]. G acts on Z[G]/Z[G]f and therefore on the dual Xf =

• Z[G]/Z[G]f, a compact group. It preserves the Haar measure.

Theorem

If G is residually finite and Σ is a sofic approximation determined by a normal chain Ni < G and f ∈ Z[G] is invertible in l1(G) then h(Σ, G(Xf, Haar)) = log(det(f)).

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 44 / 48

Systems of algebraic origin

Let f ∈ Z[G]. G acts on Z[G]/Z[G]f and therefore on the dual Xf =

• Z[G]/Z[G]f, a compact group. It preserves the Haar measure.

Theorem

If G is residually finite and Σ is a sofic approximation determined by a normal chain Ni < G and f ∈ Z[G] is invertible in l1(G) then h(Σ, G(Xf, Haar)) = log(det(f)). G = Z: Yuzvinskii 1967 G = Zd: Lind-Schmidt-Ward 1990 G amenable: Deninger 2006, Deninger-Schmidt 2007, Li 2010 G non-amenable: L. B. 2009 G non-amenable and topological entropy case: Kerr-Li 2010.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 44 / 48

The Kerr-Li approach

Let S = {pn}n∈N ⊂ L∞

R (X, µ), pn∞ ≤ 1, W ⊂ G, d > 0,

SW,d :=   

j

• i=1

fi ◦ wi : j ≤ d, fi ∈ {p1, . . . , pd}, wi ∈ W    .

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 45 / 48

The Kerr-Li approach

Let S = {pn}n∈N ⊂ L∞

R (X, µ), pn∞ ≤ 1, W ⊂ G, d > 0,

SW,d :=   

j

• i=1

fi ◦ wi : j ≤ d, fi ∈ {p1, . . . , pd}, wi ∈ W    . For σ : G → Sym(mi), let UPµ(S, W, d, δ, σ) to be the set of all unital positive linear maps ϕ : L∞(X, µ) → Cmi that are

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 45 / 48

The Kerr-Li approach

Let S = {pn}n∈N ⊂ L∞

R (X, µ), pn∞ ≤ 1, W ⊂ G, d > 0,

SW,d :=   

j

• i=1

fi ◦ wi : j ≤ d, fi ∈ {p1, . . . , pd}, wi ∈ W    . For σ : G → Sym(mi), let UPµ(S, W, d, δ, σ) to be the set of all unital positive linear maps ϕ : L∞(X, µ) → Cmi that are approximately multiplicative : ϕ(j

i=1 fi ◦ wi) − j i=1 ϕ(fi ◦ wi)2 < δ

for j ≤ d, fi ∈ {p1, . . . , pd}, wi ∈ W

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 45 / 48

The Kerr-Li approach

Let S = {pn}n∈N ⊂ L∞

R (X, µ), pn∞ ≤ 1, W ⊂ G, d > 0,

SW,d :=   

j

• i=1

fi ◦ wi : j ≤ d, fi ∈ {p1, . . . , pd}, wi ∈ W    . For σ : G → Sym(mi), let UPµ(S, W, d, δ, σ) to be the set of all unital positive linear maps ϕ : L∞(X, µ) → Cmi that are approximately multiplicative : ϕ(j

i=1 fi ◦ wi) − j i=1 ϕ(fi ◦ wi)2 < δ

for j ≤ d, fi ∈ {p1, . . . , pd}, wi ∈ W approximately measure-preserving : |ζ ◦ ϕ(f) − µ(f)| < δ for f ∈ SW,d

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 45 / 48

The Kerr-Li approach

Let S = {pn}n∈N ⊂ L∞

R (X, µ), pn∞ ≤ 1, W ⊂ G, d > 0,

SW,d :=   

j

• i=1

fi ◦ wi : j ≤ d, fi ∈ {p1, . . . , pd}, wi ∈ W    . For σ : G → Sym(mi), let UPµ(S, W, d, δ, σ) to be the set of all unital positive linear maps ϕ : L∞(X, µ) → Cmi that are approximately multiplicative : ϕ(j

i=1 fi ◦ wi) − j i=1 ϕ(fi ◦ wi)2 < δ

for j ≤ d, fi ∈ {p1, . . . , pd}, wi ∈ W approximately measure-preserving : |ζ ◦ ϕ(f) − µ(f)| < δ for f ∈ SW,d approximately equivariant : ϕ(f ◦ w) − ϕ(f) ◦ σ(w)2 < δ for w ∈ W, f ∈ {p1, . . . , pd}.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 45 / 48

The Kerr-Li approach

For ϕ, ψ : L∞(X, µ) → Cm unital positive linear maps let ρS(ϕ, ψ) =

∞

• n=1

1 2n ϕ(pn) − ψ(pn)2.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 46 / 48

The Kerr-Li approach

For ϕ, ψ : L∞(X, µ) → Cm unital positive linear maps let ρS(ϕ, ψ) =

∞

• n=1

1 2n ϕ(pn) − ψ(pn)2. Let Nε(UPµ(S, W, d, δ, σi), ρS) be the maximum cardinality of an ε-separated subset of UPµ(S, W, d, δ, σi) w.r.t. ρS.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 46 / 48

The Kerr-Li approach

For ϕ, ψ : L∞(X, µ) → Cm unital positive linear maps let ρS(ϕ, ψ) =

∞

• n=1

1 2n ϕ(pn) − ψ(pn)2. Let Nε(UPµ(S, W, d, δ, σi), ρS) be the maximum cardinality of an ε-separated subset of UPµ(S, W, d, δ, σi) w.r.t. ρS. hµ(Σ, S) = sup

ε>0

inf

W inf d∈N inf δ>0 lim sup i→∞

1 mi log Nε(UPµ(S, W, d, δ, σi), ρS)

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 46 / 48

The Kerr-Li approach

For ϕ, ψ : L∞(X, µ) → Cm unital positive linear maps let ρS(ϕ, ψ) =

∞

• n=1

1 2n ϕ(pn) − ψ(pn)2. Let Nε(UPµ(S, W, d, δ, σi), ρS) be the maximum cardinality of an ε-separated subset of UPµ(S, W, d, δ, σi) w.r.t. ρS. hµ(Σ, S) = sup

ε>0

inf

W inf d∈N inf δ>0 lim sup i→∞

1 mi log Nε(UPµ(S, W, d, δ, σi), ρS)

Theorem (Kerr-Li, 2010)

If S, T are dynamically generating (i.e., not contained in a proper G-invariant von Neumann subalgebra) then hµ(Σ, S) = hµ(Σ, T ). So let h(Σ, G, X, µ) be this number.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 46 / 48

The Kerr-Li approach

Topological sofic entropy is defined similarly with C(X) in place of L∞

R (X, µ).

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 47 / 48

The Kerr-Li approach

Topological sofic entropy is defined similarly with C(X) in place of L∞

R (X, µ).

Theorem (Kerr-Li 2010, Variational principle)

If GX is continuous action on a compact metrizable space X then htop(Σ, G, X) = sup

µ∈MG(X)

h(Σ, G, X, µ). In particular, if MG(X) is empty then htop(Σ, G, X) = −∞.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 47 / 48

The Kerr-Li approach

Topological sofic entropy is defined similarly with C(X) in place of L∞

R (X, µ).

Theorem (Kerr-Li 2010, Variational principle)

If GX is continuous action on a compact metrizable space X then htop(Σ, G, X) = sup

µ∈MG(X)

h(Σ, G, X, µ). In particular, if MG(X) is empty then htop(Σ, G, X) = −∞.

Corollary (Gromov 1999, Weiss 2000, Kerr-Li 2010)

If A is a finite set and φ : AG → AG is a G-equivariant continuous injective map then φ is surjective.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 47 / 48

Further Results & Open Questions

Does Ornstein theory extend to non-abelian free groups?: factors

• f Bernoulli shifts, factors onto Bernoulli shifts, mixing Markov

chains, etc.

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 48 / 48

Further Results & Open Questions

Does Ornstein theory extend to non-abelian free groups?: factors

• f Bernoulli shifts, factors onto Bernoulli shifts, mixing Markov

chains, etc. Is sofic entropy an invariant of von Neumann equivalence or orbit equivalence for Property (T) groups?

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 48 / 48

Further Results & Open Questions

Does Ornstein theory extend to non-abelian free groups?: factors

• f Bernoulli shifts, factors onto Bernoulli shifts, mixing Markov

chains, etc. Is sofic entropy an invariant of von Neumann equivalence or orbit equivalence for Property (T) groups? Is there an entropy theory for actions of sofic groups by automorphisms on II1 factors?

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 48 / 48

Further Results & Open Questions

Does Ornstein theory extend to non-abelian free groups?: factors

• f Bernoulli shifts, factors onto Bernoulli shifts, mixing Markov

chains, etc. Is sofic entropy an invariant of von Neumann equivalence or orbit equivalence for Property (T) groups? Is there an entropy theory for actions of sofic groups by automorphisms on II1 factors? Does the f-invariant extend to more general groups?

Lewis Bowen (Texas A&M) Entropy Theory for Sofic Group Actions 48 / 48