Sofic entropy theory Lewis Bowen Logic, Dynamics and their - - PowerPoint PPT Presentation

Sofic entropy theory

Lewis Bowen Logic, Dynamics and their Interactions with a celebration of the work of Dan Mauldin University of North Texas June 2012

Lewis Bowen (Texas A&M) Sofic entropy theory 1 / 29

Group actions

Let G be a countable discrete group.

Lewis Bowen (Texas A&M) Sofic entropy theory 2 / 29

Group actions

Let G be a countable discrete group. Let X be a set with a measure µ satisfying µ(X) = 1.

Lewis Bowen (Texas A&M) Sofic entropy theory 2 / 29

Group actions

Let G be a countable discrete group. Let X be a set with a measure µ satisfying µ(X) = 1. Let T : G → Aut(X, µ) be a homomorphism.

Lewis Bowen (Texas A&M) Sofic entropy theory 2 / 29

Group actions

Let G be a countable discrete group. Let X be a set with a measure µ satisfying µ(X) = 1. Let T : G → Aut(X, µ) be a homomorphism. T is a (probability-measure-preserving) action of G.

Lewis Bowen (Texas A&M) Sofic entropy theory 2 / 29

Group actions

Let G be a countable discrete group. Let X be a set with a measure µ satisfying µ(X) = 1. Let T : G → Aut(X, µ) be a homomorphism. T is a (probability-measure-preserving) action of G. Two actions T1 : G → Aut(X1, µ1), T2 : G → Aut(X2, µ2) are isomorphic if there exists a measure-space isomorphism φ : X1 → X2 with φ(T1(g)x) = T2(g)φ(x) for a.e. x ∈ X1, for every g ∈ G.

Lewis Bowen (Texas A&M) Sofic entropy theory 2 / 29

Group actions

Let G be a countable discrete group. Let X be a set with a measure µ satisfying µ(X) = 1. Let T : G → Aut(X, µ) be a homomorphism. T is a (probability-measure-preserving) action of G. Two actions T1 : G → Aut(X1, µ1), T2 : G → Aut(X2, µ2) are isomorphic if there exists a measure-space isomorphism φ : X1 → X2 with φ(T1(g)x) = T2(g)φ(x) for a.e. x ∈ X1, for every g ∈ G. Main Problem: Classify actions of G up to isomorphism.

Lewis Bowen (Texas A&M) Sofic entropy theory 2 / 29

Bernoulli shifts

Let (K, κ) be a probability space.

Lewis Bowen (Texas A&M) Sofic entropy theory 3 / 29

Bernoulli shifts

Let (K, κ) be a probability space. G acts on K G := {x : G → K} by shifting: (gx)(f) = x(g−1f) ∀x ∈ K G, g, f ∈ G.

Lewis Bowen (Texas A&M) Sofic entropy theory 3 / 29

Bernoulli shifts

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

Lewis Bowen (Texas A&M) Sofic entropy theory 3 / 29

Bernoulli shifts

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

k∈K ′ κ(k) log

• κ(k)
• if κ is supported on a

countable set K ′ ⊂ K. Otherwise, H(K, κ) := +∞.

Lewis Bowen (Texas A&M) Sofic entropy theory 3 / 29

Classification of Bernoulli Shifts

G(K, κ)G ∼ = G(L, λ)G

?

⇐ ⇒ H(K, κ) = H(L, λ)

Lewis Bowen (Texas A&M) Sofic entropy theory 4 / 29

Classification of Bernoulli Shifts

G(K, κ)G ∼ = G(L, λ)G

?

⇐ ⇒ H(K, κ) = H(L, λ) = ⇒ G = Z, Kolmogorov (1958)

Lewis Bowen (Texas A&M) Sofic entropy theory 4 / 29

Classification of Bernoulli Shifts

G(K, κ)G ∼ = G(L, λ)G

?

⇐ ⇒ H(K, κ) = H(L, λ) = ⇒ G = Z, Kolmogorov (1958) = ⇒ G amenable, folk theorem (1970s)

Lewis Bowen (Texas A&M) Sofic entropy theory 4 / 29

Classification of Bernoulli Shifts

G(K, κ)G ∼ = G(L, λ)G

?

⇐ ⇒ H(K, κ) = H(L, λ) = ⇒ G = Z, Kolmogorov (1958) = ⇒ G amenable, folk theorem (1970s)

Lewis Bowen (Texas A&M) Sofic entropy theory 4 / 29

Classification of Bernoulli Shifts

G(K, κ)G ∼ = G(L, λ)G

?

⇐ ⇒ H(K, κ) = H(L, λ) = ⇒ G = Z, Kolmogorov (1958) = ⇒ G amenable, folk theorem (1970s) ⇐ = G = Z, Ornstein (1970)

Lewis Bowen (Texas A&M) Sofic entropy theory 4 / 29

Classification of Bernoulli Shifts

G(K, κ)G ∼ = G(L, λ)G

?

⇐ ⇒ H(K, κ) = H(L, λ) = ⇒ G = Z, Kolmogorov (1958) = ⇒ G amenable, folk theorem (1970s) ⇐ = G = Z, Ornstein (1970) ⇐ = G ⊃ Ornstein subgroup, Stepin (1975)

Lewis Bowen (Texas A&M) Sofic entropy theory 4 / 29

Classification of Bernoulli Shifts

G(K, κ)G ∼ = G(L, λ)G

?

⇐ ⇒ H(K, κ) = H(L, λ) = ⇒ G = Z, Kolmogorov (1958) = ⇒ G amenable, folk theorem (1970s) ⇐ = G = Z, Ornstein (1970) ⇐ = G ⊃ Ornstein subgroup, Stepin (1975) ⇐ = |G| = ∞ & G amenable, Ornstein-Weiss (1987)

Lewis Bowen (Texas A&M) Sofic entropy theory 4 / 29

Classification of Bernoulli Shifts

G(K, κ)G ∼ = G(L, λ)G

?

⇐ ⇒ H(K, κ) = H(L, λ) = ⇒ G = Z, Kolmogorov (1958) = ⇒ G amenable, folk theorem (1970s) ⇐ = G = Z, Ornstein (1970) ⇐ = G ⊃ Ornstein subgroup, Stepin (1975) ⇐ = |G| = ∞ & G amenable, Ornstein-Weiss (1987) ⇐ = |G| = ∞, (K, κ), (L, λ) not 2-atom spaces,

• B. (2011)

Lewis Bowen (Texas A&M) Sofic entropy theory 4 / 29

Classification of Bernoulli Shifts

G(K, κ)G ∼ = G(L, λ)G

?

⇐ ⇒ H(K, κ) = H(L, λ) = ⇒ G = Z, Kolmogorov (1958) = ⇒ G amenable, folk theorem (1970s) = ⇒ G sofic, B. (2010) ⇐ = G = Z, Ornstein (1970) ⇐ = G ⊃ Ornstein subgroup, Stepin (1975) ⇐ = |G| = ∞ & G amenable, Ornstein-Weiss (1987) ⇐ = |G| = ∞, (K, κ), (L, λ) not 2-atom spaces,

• B. (2011)

Lewis Bowen (Texas A&M) Sofic entropy theory 5 / 29

Factor maps

Lewis Bowen (Texas A&M) Sofic entropy theory 6 / 29

Factor maps

Definition

Let G(X, µ), G(Y, ν) be two systems and φ : X → Y a measurable map with φ∗µ = ν, φ(gx) = gφ(x) for a.e. x ∈ X and all g ∈ G. Then φ is a factor map from G(X, µ) to G(Y, ν).

Lewis Bowen (Texas A&M) Sofic entropy theory 6 / 29

Factor maps

Definition

Let G(X, µ), G(Y, ν) be two systems and φ : X → Y a measurable map with φ∗µ = ν, φ(gx) = gφ(x) for a.e. x ∈ X and all g ∈ G. Then φ is a factor map from G(X, µ) to G(Y, ν). Question: which Bernoulli shifts factor onto which Bernoulli shifts?

Lewis Bowen (Texas A&M) Sofic entropy theory 6 / 29

Factor maps

Definition

Let G(X, µ), G(Y, ν) be two systems and φ : X → Y a measurable map with φ∗µ = ν, φ(gx) = gφ(x) for a.e. x ∈ X and all g ∈ G. Then φ is a factor map from G(X, µ) to G(Y, ν). Question: which Bernoulli shifts factor onto which Bernoulli shifts?

Theorem

(Sinai 1960s, Ornstein-Weiss 1980) If G is amenable then G(K, κ)G factors onto G(L, λ)G if and only if H(K, κ) ≥ H(L, λ).

Lewis Bowen (Texas A&M) Sofic entropy theory 6 / 29

Factor maps

Definition

Let G(X, µ), G(Y, ν) be two systems and φ : X → Y a measurable map with φ∗µ = ν, φ(gx) = gφ(x) for a.e. x ∈ X and all g ∈ G. Then φ is a factor map from G(X, µ) to G(Y, ν). Question: which Bernoulli shifts factor onto which Bernoulli shifts?

Theorem

(Sinai 1960s, Ornstein-Weiss 1980) If G is amenable then G(K, κ)G factors onto G(L, λ)G if and only if H(K, κ) ≥ H(L, λ). G amenable ⇒ h(G(Z/nZ, un)G) = log(n) (the full n-shift). So the full 2-shift over G cannot factor onto the full 4-shift over G.

Lewis Bowen (Texas A&M) Sofic entropy theory 6 / 29

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) Sofic entropy theory 7 / 29

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) Sofic entropy theory 7 / 29

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)
• .

Theorem (B. 2011)

If G contains a subgroup isomorphic to F then every nontrivial Bernoulli shift over G factors onto every other Bernoulli shift over G.

Lewis Bowen (Texas A&M) Sofic entropy theory 7 / 29

Factors between Bernoulli shifts

Theorem (Gaboriau-Lyons, 2009)

For every non-amenable group G there is some space (K, κ) with H(K, κ) < ∞ and an essentially free action F(K, κ)G with F-orbits contained in the G-orbits.

Lewis Bowen (Texas A&M) Sofic entropy theory 8 / 29

Factors between Bernoulli shifts

Theorem (Gaboriau-Lyons, 2009)

For every non-amenable group G there is some space (K, κ) with H(K, κ) < ∞ and an essentially free action F(K, κ)G with F-orbits contained in the G-orbits.

Theorem (Ball, 2005)

For every non-amenable group G there is some (K, κ) with H(K, κ) < 0 such that G(K, κ)G factors onto every Bernoulli shift.

Lewis Bowen (Texas A&M) Sofic entropy theory 8 / 29

New Results: Sofic Groups

Theorem (B., 2010)

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) Sofic entropy theory 9 / 29

New Results: Sofic Groups

Theorem (B., 2010)

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). History

1

L.B. 2010: free groups, sofic groups, finitely generated actions;

2

Kerr-Li 2011: topological and measure-theoretic sofic entropy; removed finite generation hypothesis

3

New equivalent definitions have been found by Kerr-Li, Kerr and Zhang.

Lewis Bowen (Texas A&M) Sofic entropy theory 9 / 29

Pseudo-metrics and separating sets

Let ρ be a pseudo-metric on a set Y. A (ρ, ǫ)-separated subset S ⊂ Y satisfies ρ(s, t) ≥ ǫ for every s = t ∈ S.

Lewis Bowen (Texas A&M) Sofic entropy theory 10 / 29

Pseudo-metrics and separating sets

Let ρ be a pseudo-metric on a set Y. A (ρ, ǫ)-separated subset S ⊂ Y satisfies ρ(s, t) ≥ ǫ for every s = t ∈ S. Nǫ(Z, ρ) is the max. cardinality of a (ρ, ǫ)-separated subset of Z ⊂ Y.

Lewis Bowen (Texas A&M) Sofic entropy theory 10 / 29

Pseudo-metrics and separating sets

Let ρ be a pseudo-metric on a set Y. A (ρ, ǫ)-separated subset S ⊂ Y satisfies ρ(s, t) ≥ ǫ for every s = t ∈ S. Nǫ(Z, ρ) is the max. cardinality of a (ρ, ǫ)-separated subset of Z ⊂ Y. Define pseudo-metrics ρ2 and ρ∞ on Y n by: ρ2

• (x1, . . . , xn), (y1, . . . , yn)
• =
• 1

n

n

• i=1

ρ(xi, yi)2, ρ∞

• (x1, . . . , xn), (y1, . . . , yn)
• :=

max

i

ρ(xi, yi).

Lewis Bowen (Texas A&M) Sofic entropy theory 10 / 29

Topological entropy for Z-actions

Let T : X → X be a homeomorphism of a compact topological space.

Lewis Bowen (Texas A&M) Sofic entropy theory 11 / 29

Topological entropy for Z-actions

Let T : X → X be a homeomorphism of a compact topological space. Let Orb(T, n) := {(x, Tx, . . . , T n−1x) ∈ X n : x ∈ X}.

Lewis Bowen (Texas A&M) Sofic entropy theory 11 / 29

Topological entropy for Z-actions

Let T : X → X be a homeomorphism of a compact topological space. Let Orb(T, n) := {(x, Tx, . . . , T n−1x) ∈ X n : x ∈ X}. htop(T, ρ) := sup

ǫ>0

lim sup

n→∞

n−1 log Nǫ(Orb(T, n), ρ∞) ρ is dynamically generating if for any x = y ∈ X there is some n ∈ Z such that ρ(T nx, T ny) > 0.

Lewis Bowen (Texas A&M) Sofic entropy theory 11 / 29

Topological entropy for Z-actions

Let T : X → X be a homeomorphism of a compact topological space. Let Orb(T, n) := {(x, Tx, . . . , T n−1x) ∈ X n : x ∈ X}. htop(T, ρ) := sup

ǫ>0

lim sup

n→∞

n−1 log Nǫ(Orb(T, n), ρ∞) ρ is dynamically generating if for any x = y ∈ X there is some n ∈ Z such that ρ(T nx, T ny) > 0.

Theorem (Rufus Bowen, 70’s)

If ρ1, ρ2 are dynamically generating, then htop(T, ρ1) = htop(T, ρ2). So htop(T) := htop(T, ρ1). Topological entropy was defined earlier, using open covers, by Adler, Konheim and McAndrew (1965).

Lewis Bowen (Texas A&M) Sofic entropy theory 11 / 29

Measure-theoretic entropy

Suppose T preserves a Borel probability measure µ on X. Idea: measure entropy is the exponential rate of growth of the number

• f ǫ-separated partial orbits of T that are statistically relevant.

Lewis Bowen (Texas A&M) Sofic entropy theory 12 / 29

Measure-theoretic entropy

Suppose T preserves a Borel probability measure µ on X. Idea: measure entropy is the exponential rate of growth of the number

• f ǫ-separated partial orbits of T that are statistically relevant.

Given a finite set L ⊂ C(X) and δ > 0, let Orb(T, L, δ, n) :=

• (x, Tx, . . . , T n−1x) ∈ X n : x ∈ X,
• f dµ − 1

n

n−1

• i=0

f(T ix)

• < δ ∀f ∈ L
• .

Lewis Bowen (Texas A&M) Sofic entropy theory 12 / 29

Measure-theoretic entropy

hµ(T, ρ) := sup

ǫ>0

inf

L⊂C(X) inf δ>0 lim sup n→∞

n−1 log Nǫ(Orb(T, L, δ, n), ρ∞).

Lewis Bowen (Texas A&M) Sofic entropy theory 13 / 29

Measure-theoretic entropy

hµ(T, ρ) := sup

ǫ>0

inf

L⊂C(X) inf δ>0 lim sup n→∞

n−1 log Nǫ(Orb(T, L, δ, n), ρ∞).

Theorem

If ρ is dynamically generating, and T is ergodic, then hµ(T, ρ) = hµ(T).

Lewis Bowen (Texas A&M) Sofic entropy theory 13 / 29

Sofic groups

Definition

G is sofic if there is a sequence σi : G → Sym(di) of maps (which need not be homomorphisms!) such that for every g, h ∈ G, lim

i→∞ d−1 i

#{p ∈ [di] : σi(g)σi(h)p = σi(gh)p} = 1 and if g = h then lim

i→∞ d−1 i

#{p ∈ [di] : σi(g)p = σi(h)p} = 1.

Lewis Bowen (Texas A&M) Sofic entropy theory 14 / 29

Fundamentals

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

Lewis Bowen (Texas A&M) Sofic entropy theory 15 / 29

Fundamentals

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

Lewis Bowen (Texas A&M) Sofic entropy theory 15 / 29

Fundamentals

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) Sofic entropy theory 15 / 29

Fundamentals

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) Sofic entropy theory 15 / 29

Topological entropy for sofic groups

Let GX be an action by homeomorphisms on a compact metrizable space,

Lewis Bowen (Texas A&M) Sofic entropy theory 16 / 29

Topological entropy for sofic groups

Let GX be an action by homeomorphisms on a compact metrizable space, Σ = {σi} be a sofic approximation to G where σi : G → Sym(di),

Lewis Bowen (Texas A&M) Sofic entropy theory 16 / 29

Topological entropy for sofic groups

Let GX be an action by homeomorphisms on a compact metrizable space, Σ = {σi} be a sofic approximation to G where σi : G → Sym(di), Given W ⊂ G finite, δ > 0, let Orb(W, δ, σi) be the set of all maps φ : [di] → X such that ρ2(φ ◦ σi(w), w ◦ φ) < δ, ∀w ∈ W. These are approximate partial orbits.

Lewis Bowen (Texas A&M) Sofic entropy theory 16 / 29

Topological entropy for sofic groups

hΣ(GX, ρ) := sup

ǫ>0

inf

W⊂G inf δ>0 lim sup i→∞

d−1

i

log Nǫ(Orb(W, δ, σi), ρ∞).

Lewis Bowen (Texas A&M) Sofic entropy theory 17 / 29

Topological entropy for sofic groups

hΣ(GX, ρ) := sup

ǫ>0

inf

W⊂G inf δ>0 lim sup i→∞

d−1

i

log Nǫ(Orb(W, δ, σi), ρ∞).

Theorem (Kerr-Li, 2011)

If ρ1, ρ2 are dynamically generating, then hΣ(GX, ρ1) = hΣ(GX, ρ2). So hΣ(GX) := hΣ(GX, ρ1). Moreover, if G is amenable, then this coincides with classical topological entropy.

Lewis Bowen (Texas A&M) Sofic entropy theory 17 / 29

Measure-entropy for sofic groups

Suppose GX preserves a probability measure µ on X. For a finite set L ⊂ C(X), let Orb(W, L, δ, σi) :=

• φ ∈ Orb(W, δ, σi) :
• f dµ − 1

di

• q∈di

f(φ(q))

• < δ ∀f ∈ L
• .

Lewis Bowen (Texas A&M) Sofic entropy theory 18 / 29

Measure-entropy for sofic groups

hΣ,µ(GX, ρ) := sup

ǫ>0

inf

W⊂G

inf

L⊂C(X) inf δ>0 lim sup i→∞

d−1

i

log Nǫ(Orb(W, L, δ, σi), ρ∞).

Lewis Bowen (Texas A&M) Sofic entropy theory 19 / 29

Measure-entropy for sofic groups

hΣ,µ(GX, ρ) := sup

ǫ>0

inf

W⊂G

inf

L⊂C(X) inf δ>0 lim sup i→∞

d−1

i

log Nǫ(Orb(W, L, δ, σi), ρ∞).

Theorem (Kerr-Li, 2011)

If ρ1, ρ2 are dynamically generating, then hΣ,µ(GX, ρ1) = hΣ,µ(GX, ρ2). So hΣ,µ(GX) := hΣ,µ(GX, ρ1). Moreover, if G is amenable, then this coincides with classical measure-entropy. Also, hΣ,κG(GK G) = H(K, κ).

Lewis Bowen (Texas A&M) Sofic entropy theory 19 / 29

The variational principle

Theorem (Kerr-Li, 2011)

Let GX be an action by homeomorphisms on a compact metrizable

• space. For any sofic approximation Σ,

hΣ(GX) = sup

µ

hΣ,µ(GX).

Lewis Bowen (Texas A&M) Sofic entropy theory 20 / 29

The variational principle

Theorem (Kerr-Li, 2011)

Let GX be an action by homeomorphisms on a compact metrizable

• space. For any sofic approximation Σ,

hΣ(GX) = sup

µ

hΣ,µ(GX).

Conjecture (Gottschalk’s surjunctivity conjecture)

For any countable discrete group G, any finite set K and any continuous G-equivariant map φ : K G → K G, if φ is injective then must also be surjective. This is known to hold for sofic groups: Gromov (1999), Weiss (2000), Kerr-Li (2011).

Lewis Bowen (Texas A&M) Sofic entropy theory 20 / 29

The annealed sofic entropy of free group actions

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

Lewis Bowen (Texas A&M) Sofic entropy theory 21 / 29

The annealed sofic entropy of free group actions

For each n ≥ 1, let σn : F = s1, . . . , sr → Sym(n) be chosen uniformly at random. Suppose FX is an action by homeomorphisms on a compact metrizable space X with pseudo-metric ρ. Define h∗

µ(GX, ρ)

:= sup

ǫ>0

inf

W⊂G

inf

L⊂C(X) inf δ>0 lim sup n→∞

n−1 log E[Nǫ(Orb(W, L, δ, σn), ρ∞)].

Lewis Bowen (Texas A&M) Sofic entropy theory 21 / 29

Free Groups: a special case

Given a finite partition P of X define Fµ(P) := −(2r − 1)Hµ(P) +

r

• i=1

Hµ(P ∨ siP); fµ(FX, P) := inf

W⊂F Fµ g∈W

gP

• .

Lewis Bowen (Texas A&M) Sofic entropy theory 22 / 29

Free Groups: a special case

Given a finite partition P of X define Fµ(P) := −(2r − 1)Hµ(P) +

r

• i=1

Hµ(P ∨ siP); fµ(FX, P) := inf

W⊂F Fµ g∈W

gP

• .

Theorem

If P is generating then fµ(P) = h∗

µ(GX).

Lewis Bowen (Texas A&M) Sofic entropy theory 22 / 29

A Markov chain example

Lewis Bowen (Texas A&M) Sofic entropy theory 23 / 29

A Markov chain example

Lewis Bowen (Texas A&M) Sofic entropy theory 23 / 29

The Cayley graph

Lewis Bowen (Texas A&M) Sofic entropy theory 24 / 29

The Ising model

Lewis Bowen (Texas A&M) Sofic entropy theory 25 / 29

Example

h∗(F{magenta, brown}F) = −2ǫ log(ǫ) − 2(1 − ǫ) log(1 − ǫ) − log(2).

Lewis Bowen (Texas A&M) Sofic entropy theory 26 / 29

Formulae

Let α : F → Aut(X, µ), β : F → Aut(Y, ν) be actions.

Lewis Bowen (Texas A&M) Sofic entropy theory 27 / 29

Formulae

Let α : F → Aut(X, µ), β : F → Aut(Y, ν) be actions. ergodic decomposition: f(tα + (1 − t)β) = tf(α) + (1 − t)f(β) + (1 − r)H(t, 1 − t) (Seward).

Lewis Bowen (Texas A&M) Sofic entropy theory 27 / 29

Formulae

Let α : F → Aut(X, µ), β : F → Aut(Y, ν) be actions. ergodic decomposition: f(tα + (1 − t)β) = tf(α) + (1 − t)f(β) + (1 − r)H(t, 1 − t) (Seward). subgroup: f(α|G) = [F : G]f(α) (Seward).

Lewis Bowen (Texas A&M) Sofic entropy theory 27 / 29

Formulae

Let α : F → Aut(X, µ), β : F → Aut(Y, ν) be actions. ergodic decomposition: f(tα + (1 − t)β) = tf(α) + (1 − t)f(β) + (1 − r)H(t, 1 − t) (Seward). subgroup: f(α|G) = [F : G]f(α) (Seward). direct product: f(α × β) = f(α) + f(β).

Lewis Bowen (Texas A&M) Sofic entropy theory 27 / 29

Formulae

Let α : F → Aut(X, µ), β : F → Aut(Y, ν) be actions. ergodic decomposition: f(tα + (1 − t)β) = tf(α) + (1 − t)f(β) + (1 − r)H(t, 1 − t) (Seward). subgroup: f(α|G) = [F : G]f(α) (Seward). direct product: f(α × β) = f(α) + f(β). n-1 factor maps: if π : (X, µ) → (Y, ν) is an n-to-1 factor map then f(β) = f(α) + (r − 1) log(n).

Lewis Bowen (Texas A&M) Sofic entropy theory 27 / 29

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) Sofic entropy theory 28 / 29

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) Sofic entropy theory 28 / 29

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, 2011)

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) Sofic entropy theory 28 / 29

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, 2011)

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) Sofic entropy theory 28 / 29

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, 2011)

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) Sofic entropy theory 28 / 29

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, 2011)

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) Sofic entropy theory 28 / 29

Further Results & Open Questions

There is a relative entropy theory for extensions of sofic groupoids. Is there an Ornstein theory for free groups ? e.g., does every system of positive entropy factor onto a Bernoulli shift? Sofic entropy of algebraic dynamical systems (L.B., Kerr-Li, L.B.-Li). Sofic topological pressure (Nhan-Phu Chung) Local sofic entropy theory (Zhang). Sofic dimension (Dykema-Kerr-Pichot) (analogous to Voiculescu’s free entropy dimension). Sofic mean dimension (H. Li). lp-version of von Neumann dimension for Banach space representations of sofic groups (Ben Hayes).

Lewis Bowen (Texas A&M) Sofic entropy theory 29 / 29