Factors of Gibbs measures on subshifts What is a Gibbs measure? - - PowerPoint PPT Presentation

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Factors of Gibbs measures on subshifts

Sophie MacDonald

UBC Mathematics

West Coast Dynamics Seminar, May 2020

1 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Acknowledgments

Thanks to Lior and Jayadev for inviting me to speak. I am grateful to be supervised by Lior, Brian Marcus, and Omer Angel. All work joint with Lu´ ısa Borsato, PhD student at Universidade de S˜ ao Paulo, visiting at UBC for the year, supported by grants 2018/21067-0 and 2019/08349-9, S˜ ao Paulo Research Foundation (FAPESP). We are very grateful to Brian for his generous support and supervision, and to Tom Meyerovitch for his generous advice throughout this work.

2 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Goals for this presentation

In this talk, I hope to communicate to you:

• Roughly two definitions of a Gibbs measure on a subshift

and why they are equivalent

• A property defining a class of factor maps that preserve

Gibbsianness, and some elements of the proof

• A Lanford-Ruelle theorem for irreducible sofic shifts on Z

On Thursday, we can go into more detail, as interest dictates

3 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Subshifts on groups

• Finite (discrete) alphabet A, countable group G
• Product topology on full shift AG (compact metrizable)

4 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Subshifts on groups

• Finite (discrete) alphabet A, countable group G
• Product topology on full shift AG (compact metrizable)
• Shift action of G on AG via (x · g)h = xgh

4 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Subshifts on groups

• Finite (discrete) alphabet A, countable group G
• Product topology on full shift AG (compact metrizable)
• Shift action of G on AG via (x · g)h = xgh
• When G = Z, (σnx)0 = xn

4 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Subshifts on groups

• Finite (discrete) alphabet A, countable group G
• Product topology on full shift AG (compact metrizable)
• Shift action of G on AG via (x · g)h = xgh
• When G = Z, (σnx)0 = xn
• A subshift is a closed, shift-invariant set X ⊆ AG

4 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Subshifts on groups

• Finite (discrete) alphabet A, countable group G
• Product topology on full shift AG (compact metrizable)
• Shift action of G on AG via (x · g)h = xgh
• When G = Z, (σnx)0 = xn
• A subshift is a closed, shift-invariant set X ⊆ AG
• Shift of finite type (SFT): subshift obtained by

forbidding finitely many finite patterns from AG

4 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Subshifts on groups

• Finite (discrete) alphabet A, countable group G
• Product topology on full shift AG (compact metrizable)
• Shift action of G on AG via (x · g)h = xgh
• When G = Z, (σnx)0 = xn
• A subshift is a closed, shift-invariant set X ⊆ AG
• Shift of finite type (SFT): subshift obtained by

forbidding finitely many finite patterns from AG

• Sliding block code: π : X → Y with π(x · g) = π(x) · g

4 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Subshifts on groups

• Finite (discrete) alphabet A, countable group G
• Product topology on full shift AG (compact metrizable)
• Shift action of G on AG via (x · g)h = xgh
• When G = Z, (σnx)0 = xn
• A subshift is a closed, shift-invariant set X ⊆ AG
• Shift of finite type (SFT): subshift obtained by

forbidding finitely many finite patterns from AG

• Sliding block code: π : X → Y with π(x · g) = π(x) · g
• Mostly care about π surjective (hence notation π),

called a factor map

4 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Subshifts on groups

• Finite (discrete) alphabet A, countable group G
• Product topology on full shift AG (compact metrizable)
• Shift action of G on AG via (x · g)h = xgh
• When G = Z, (σnx)0 = xn
• A subshift is a closed, shift-invariant set X ⊆ AG
• Shift of finite type (SFT): subshift obtained by

forbidding finitely many finite patterns from AG

• Sliding block code: π : X → Y with π(x · g) = π(x) · g
• Mostly care about π surjective (hence notation π),

called a factor map

• Sofic shift: factor of an SFT

4 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Subshifts on groups

• Finite (discrete) alphabet A, countable group G
• Product topology on full shift AG (compact metrizable)
• Shift action of G on AG via (x · g)h = xgh
• When G = Z, (σnx)0 = xn
• A subshift is a closed, shift-invariant set X ⊆ AG
• Shift of finite type (SFT): subshift obtained by

forbidding finitely many finite patterns from AG

• Sliding block code: π : X → Y with π(x · g) = π(x) · g
• Mostly care about π surjective (hence notation π),

called a factor map

• Sofic shift: factor of an SFT
• All measures G-invariant Borel probability measures

4 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Finite thermodynamics

Take a finite set {1, . . . , N} (e.g. patterns on Λ ⋐ G) with “energy function” u ∈ RN and probability vector p

5 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Finite thermodynamics

Take a finite set {1, . . . , N} (e.g. patterns on Λ ⋐ G) with “energy function” u ∈ RN and probability vector p The free energy (volume derivative is called pressure) −

N

• i=1

pi log pi

• entropy H(p)

−

N

• i=1

piui

5 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Finite thermodynamics

Take a finite set {1, . . . , N} (e.g. patterns on Λ ⋐ G) with “energy function” u ∈ RN and probability vector p The free energy (volume derivative is called pressure) −

N

• i=1

pi log pi

• entropy H(p)

−

N

• i=1

piui is uniquely maximized by the Gibbs distribution, pi = Z −1 exp(−ui)

5 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Finite thermodynamics

Take a finite set {1, . . . , N} (e.g. patterns on Λ ⋐ G) with “energy function” u ∈ RN and probability vector p The free energy (volume derivative is called pressure) −

N

• i=1

pi log pi

• entropy H(p)

−

N

• i=1

piui is uniquely maximized by the Gibbs distribution, pi = Z −1 exp(−ui) What about infinite volume?

5 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Interactions

Define on every finite set Λ ⋐ G an interaction ΦΛ : X → R where ΦΛ(x) depends only on xΛ.

6 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Interactions

Define on every finite set Λ ⋐ G an interaction ΦΛ : X → R where ΦΛ(x) depends only on xΛ. We assume translation-invariance, ΦgΛ(x) = ΦΛ(x · g).

6 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Interactions

Define on every finite set Λ ⋐ G an interaction ΦΛ : X → R where ΦΛ(x) depends only on xΛ. We assume translation-invariance, ΦgΛ(x) = ΦΛ(x · g).

• Example: Ising interaction on Zd

6 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Interactions

Define on every finite set Λ ⋐ G an interaction ΦΛ : X → R where ΦΛ(x) depends only on xΛ. We assume translation-invariance, ΦgΛ(x) = ΦΛ(x · g).

• Example: Ising interaction on Zd

Then the Hamiltonian series gives the energy of xΛ HΦ

Λ (x) =

• ∆⋐G

∆∩Λ=∅

Φ∆(x)

6 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Interactions

Define on every finite set Λ ⋐ G an interaction ΦΛ : X → R where ΦΛ(x) depends only on xΛ. We assume translation-invariance, ΦgΛ(x) = ΦΛ(x · g).

• Example: Ising interaction on Zd

Then the Hamiltonian series gives the energy of xΛ HΦ

Λ (x) =

• ∆⋐G

∆∩Λ=∅

Φ∆(x) This converges when Φ is absolutely summable Φ =

• Λ⋐G

e∈Λ

ΦΛ∞ < ∞

6 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Potentials

Define the energy at e directly via a potential f ∈ C(X).

7 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Potentials

Define the energy at e directly via a potential f ∈ C(X). Regularity: if G has polynomial growth |Bn| ∼ nd, define vk(f ) = sup{|f (x) − f (x′)| | xBk = x′

Bk}

f SVd(X) =

∞

• k=0

kd−1vk−1(f )

7 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Potentials

Define the energy at e directly via a potential f ∈ C(X). Regularity: if G has polynomial growth |Bn| ∼ nd, define vk(f ) = sup{|f (x) − f (x′)| | xBk = x′

Bk}

f SVd(X) =

∞

• k=0

kd−1vk−1(f ) We called this the shell norm, vs. the volume norm

∞

• k=0

kdvk−1(f )

7 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Potentials ⇐ ⇒ interactions

Interactions are more convenient for Gibbs measures; potentials are more convenient for equilibrium measures.

8 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Potentials ⇐ ⇒ interactions

Interactions are more convenient for Gibbs measures; potentials are more convenient for equilibrium measures. An interaction Φ induces a potential AΦ ∈ SVd(X): AΦ(x) = −

• Λ⋐G, e∈Λ

aΛΦΛ(x) where aΛ ≥ 0 are weights with

g∈G ag−1Λ = 1.

8 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Potentials ⇐ ⇒ interactions

Interactions are more convenient for Gibbs measures; potentials are more convenient for equilibrium measures. An interaction Φ induces a potential AΦ ∈ SVd(X): AΦ(x) = −

• Λ⋐G, e∈Λ

aΛΦΛ(x) where aΛ ≥ 0 are weights with

g∈G ag−1Λ = 1.

This works if Φ < ∞ and diam(Λ)d/|Λ| is bounded above for ΦΛ ≡ 0 (thanks to Nishant Chandgotia).

8 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Potentials ⇐ ⇒ interactions

Interactions are more convenient for Gibbs measures; potentials are more convenient for equilibrium measures. An interaction Φ induces a potential AΦ ∈ SVd(X): AΦ(x) = −

• Λ⋐G, e∈Λ

aΛΦΛ(x) where aΛ ≥ 0 are weights with

g∈G ag−1Λ = 1.

This works if Φ < ∞ and diam(Λ)d/|Λ| is bounded above for ΦΛ ≡ 0 (thanks to Nishant Chandgotia). A potential f with finite volume norm induces an interaction Φf (with f = AΦf ) by a telescoping construction due to Muir, building on Ruelle.

8 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

The Gibbs relation

Let (ΛN)∞

N=1 be a sequence of finite sets exhausting G,

and define relations TX,N ⊂ X 2 by (x, x′) ∈ TX,N ⇐ ⇒ xΛc

N = x′

Λc

N 9 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

The Gibbs relation

Let (ΛN)∞

N=1 be a sequence of finite sets exhausting G,

and define relations TX,N ⊂ X 2 by (x, x′) ∈ TX,N ⇐ ⇒ xΛc

N = x′

Λc

N

Let TX = ∞

N=1 TX,N (tail/asymptotic/Gibbs relation)

9 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

The Gibbs relation

Let (ΛN)∞

N=1 be a sequence of finite sets exhausting G,

and define relations TX,N ⊂ X 2 by (x, x′) ∈ TX,N ⇐ ⇒ xΛc

N = x′

Λc

N

Let TX = ∞

N=1 TX,N (tail/asymptotic/Gibbs relation)

Equivalently, for all x ∈ X, (x, x′) ∈ TX ⇐ ⇒ lim

g→∞ d(x · g, x′ · g) = 0

9 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Cocycles

A cocycle is a measurable function φ : TX → R with φ(x, x′′) = φ(x, x′) + φ(x′, x′′)

10 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Cocycles

A cocycle is a measurable function φ : TX → R with φ(x, x′′) = φ(x, x′) + φ(x′, x′′) An interaction Φ induces a cocycle via φΦ(x, x′) =

• Λ⋐G

[ΦΛ(x) − ΦΛ(x′)]

10 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Cocycles

A cocycle is a measurable function φ : TX → R with φ(x, x′′) = φ(x, x′) + φ(x′, x′′) An interaction Φ induces a cocycle via φΦ(x, x′) =

• Λ⋐G

[ΦΛ(x) − ΦΛ(x′)] A potential f induces a cocycle via φf (x, x′) =

• g∈G

[f (x′ · g) − f (x · g)]

10 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Cocycles

A cocycle is a measurable function φ : TX → R with φ(x, x′′) = φ(x, x′) + φ(x′, x′′) An interaction Φ induces a cocycle via φΦ(x, x′) =

• Λ⋐G

[ΦΛ(x) − ΦΛ(x′)] A potential f induces a cocycle via φf (x, x′) =

• g∈G

[f (x′ · g) − f (x · g)] If Φ < ∞ and diam(Λ)d/|Λ| ≤ C then these agree, φΦ = φAΦ

10 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

The DLR equations

Definition

For a measure µ, a cocycle φ, a finite Λ ⋐ G, and a Borel A ⊆ X, the Dobrushin-Lanford-Ruelle equation reads µ(A | FΛc)(x) =

• η∈AΛ

 

ζ∈AΛ

exp(φ(ηxΛc, ζxΛc))1X(ζxΛc)  

−1

1A(ηxΛc)

11 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

The DLR equations

Definition

For a measure µ, a cocycle φ, a finite Λ ⋐ G, and a Borel A ⊆ X, the Dobrushin-Lanford-Ruelle equation reads µ(A | FΛc)(x) =

• η∈AΛ

 

ζ∈AΛ

exp(φ(ηxΛc, ζxΛc))1X(ζxΛc)  

−1

1A(ηxΛc) Examples (with Φ ≡ 0)

11 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

The DLR equations

Definition

For a measure µ, a cocycle φ, a finite Λ ⋐ G, and a Borel A ⊆ X, the Dobrushin-Lanford-Ruelle equation reads µ(A | FΛc)(x) =

• η∈AΛ

 

ζ∈AΛ

exp(φ(ηxΛc, ζxΛc))1X(ζxΛc)  

−1

1A(ηxΛc) Examples (with Φ ≡ 0)

• yes: Parry measure on irreducible edge shift

(uniform on paths of length n between two states)

11 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

The DLR equations

Definition

For a measure µ, a cocycle φ, a finite Λ ⋐ G, and a Borel A ⊆ X, the Dobrushin-Lanford-Ruelle equation reads µ(A | FΛc)(x) =

• η∈AΛ

 

ζ∈AΛ

exp(φ(ηxΛc, ζxΛc))1X(ζxΛc)  

−1

1A(ηxΛc) Examples (with Φ ≡ 0)

• yes: Parry measure on irreducible edge shift

(uniform on paths of length n between two states)

• no: point mass on sunny-side-up shift (the measure

doesn’t know about the yolk)

11 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Conformal measures

A holonomy of TX is a Borel isomorphism ψ : A → B between Borel sets A, B ⊆ X with (x, ψ(x)) ∈ TX

12 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Conformal measures

A holonomy of TX is a Borel isomorphism ψ : A → B between Borel sets A, B ⊆ X with (x, ψ(x)) ∈ TX A measure µ is conformal with respect to a cocycle φ if for any holonomy ψ : A → B and µ-a.e. x ∈ A, d(µ ◦ ψ) dµ (x) = exp(φ(x, ψ(x)))

12 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Conformal measures

A holonomy of TX is a Borel isomorphism ψ : A → B between Borel sets A, B ⊆ X with (x, ψ(x)) ∈ TX A measure µ is conformal with respect to a cocycle φ if for any holonomy ψ : A → B and µ-a.e. x ∈ A, d(µ ◦ ψ) dµ (x) = exp(φ(x, ψ(x))) Requires nonsingularity: µ(A) = 0 = ⇒ µ(TX(A)) = 0

12 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Equivalence of the definitions

• Dobrushin (1969) and Lanford-Ruelle (1969) introduced

the DLR equations

• Capocaccia (1976) introduced conformal measures

13 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Equivalence of the definitions

• Dobrushin (1969) and Lanford-Ruelle (1969) introduced

the DLR equations

• Capocaccia (1976) introduced conformal measures
• Keller (1998): conformal ⇐

⇒ satisfies the DLR equations (for f ∈ SVd(X), for a full shift X on Zd)

13 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Equivalence of the definitions

• Dobrushin (1969) and Lanford-Ruelle (1969) introduced

the DLR equations

• Capocaccia (1976) introduced conformal measures
• Keller (1998): conformal ⇐

⇒ satisfies the DLR equations (for f ∈ SVd(X), for a full shift X on Zd)

• Kimura (2015): any subshift on Zd, SVd potential:
• conformal =

⇒ satisfies the DLR equations

13 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Equivalence of the definitions

• Dobrushin (1969) and Lanford-Ruelle (1969) introduced

the DLR equations

• Capocaccia (1976) introduced conformal measures
• Keller (1998): conformal ⇐

⇒ satisfies the DLR equations (for f ∈ SVd(X), for a full shift X on Zd)

• Kimura (2015): any subshift on Zd, SVd potential:
• conformal =

⇒ satisfies the DLR equations

• satisfies the DLR equations =

⇒ “topologically Gibbs” (defined by Meyerovitch (2013))

13 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Equivalence of the definitions

• Dobrushin (1969) and Lanford-Ruelle (1969) introduced

the DLR equations

• Capocaccia (1976) introduced conformal measures
• Keller (1998): conformal ⇐

⇒ satisfies the DLR equations (for f ∈ SVd(X), for a full shift X on Zd)

• Kimura (2015): any subshift on Zd, SVd potential:
• conformal =

⇒ satisfies the DLR equations

• satisfies the DLR equations =

⇒ “topologically Gibbs” (defined by Meyerovitch (2013))

• M.-Borsato (2020): DLR equations =

⇒ conformal (any countable group, any subshift, any cocycle)

13 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Equivalence of the definitions

• Dobrushin (1969) and Lanford-Ruelle (1969) introduced

the DLR equations

• Capocaccia (1976) introduced conformal measures
• Keller (1998): conformal ⇐

⇒ satisfies the DLR equations (for f ∈ SVd(X), for a full shift X on Zd)

• Kimura (2015): any subshift on Zd, SVd potential:
• conformal =

⇒ satisfies the DLR equations

• satisfies the DLR equations =

⇒ “topologically Gibbs” (defined by Meyerovitch (2013))

• M.-Borsato (2020): DLR equations =

⇒ conformal (any countable group, any subshift, any cocycle)

13 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Equivalence of the definitions

• Dobrushin (1969) and Lanford-Ruelle (1969) introduced

the DLR equations

• Capocaccia (1976) introduced conformal measures
• Keller (1998): conformal ⇐

⇒ satisfies the DLR equations (for f ∈ SVd(X), for a full shift X on Zd)

• Kimura (2015): any subshift on Zd, SVd potential:
• conformal =

⇒ satisfies the DLR equations

• satisfies the DLR equations =

⇒ “topologically Gibbs” (defined by Meyerovitch (2013))

• M.-Borsato (2020): DLR equations =

⇒ conformal (any countable group, any subshift, any cocycle) Going forward, we’ll use the term Gibbs measure

13 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Equilibrium measures

Let G = Zd, X ⊆ AG, f ∈ SVd(X), µ a measure on X

14 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Equilibrium measures

Let G = Zd, X ⊆ AG, f ∈ SVd(X), µ a measure on X The pressure of f is PX(f ) = sup

µ

• h(µ) +
• f dµ
• (This is really a theorem, rather than a definition, but

we won’t need the definition)

14 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Equilibrium measures

Let G = Zd, X ⊆ AG, f ∈ SVd(X), µ a measure on X The pressure of f is PX(f ) = sup

µ

• h(µ) +
• f dµ
• (This is really a theorem, rather than a definition, but

we won’t need the definition) A measure that attains the supremum is an equilibrium measure for f

14 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Equilibrium measures

Let G = Zd, X ⊆ AG, f ∈ SVd(X), µ a measure on X The pressure of f is PX(f ) = sup

µ

• h(µ) +
• f dµ
• (This is really a theorem, rather than a definition, but

we won’t need the definition) A measure that attains the supremum is an equilibrium measure for f Problem: find sufficient topological conditions on X such that Gibbs ⇐ ⇒ equilibrium

14 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Irreducibility and mixing

A subshift X ⊆ AG is irreducible if any two patterns η, ζ ∈ B(X) appear at different positions in some x ∈ X

15 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Irreducibility and mixing

A subshift X ⊆ AG is irreducible if any two patterns η, ζ ∈ B(X) appear at different positions in some x ∈ X

• This is a kind of topological transitivity

15 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Irreducibility and mixing

A subshift X ⊆ AG is irreducible if any two patterns η, ζ ∈ B(X) appear at different positions in some x ∈ X

• This is a kind of topological transitivity

A subshift X ⊆ AG is strongly irreducible if there is a finite ∆ ⋐ G such that if ∆Λ ∩ Λ′ = ∅ then for any η ∈ BΛ(X), ζ ∈ BΛ′(X), [η]Λ ∩ [ζ]′

Λ = ∅

15 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Irreducibility and mixing

A subshift X ⊆ AG is irreducible if any two patterns η, ζ ∈ B(X) appear at different positions in some x ∈ X

• This is a kind of topological transitivity

A subshift X ⊆ AG is strongly irreducible if there is a finite ∆ ⋐ G such that if ∆Λ ∩ Λ′ = ∅ then for any η ∈ BΛ(X), ζ ∈ BΛ′(X), [η]Λ ∩ [ζ]′

Λ = ∅

• Over Z, strongly irreducible ⇐

⇒ mixing (irreducible and aperiodic)

15 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Irreducibility and mixing

A subshift X ⊆ AG is irreducible if any two patterns η, ζ ∈ B(X) appear at different positions in some x ∈ X

• This is a kind of topological transitivity

A subshift X ⊆ AG is strongly irreducible if there is a finite ∆ ⋐ G such that if ∆Λ ∩ Λ′ = ∅ then for any η ∈ BΛ(X), ζ ∈ BΛ′(X), [η]Λ ∩ [ζ]′

Λ = ∅

• Over Z, strongly irreducible ⇐

⇒ mixing (irreducible and aperiodic)

• Strong irreducibility implies condition (D): any

x, x′ ∈ X can be glued along a narrow border

15 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Irreducibility and mixing

A subshift X ⊆ AG is irreducible if any two patterns η, ζ ∈ B(X) appear at different positions in some x ∈ X

• This is a kind of topological transitivity

A subshift X ⊆ AG is strongly irreducible if there is a finite ∆ ⋐ G such that if ∆Λ ∩ Λ′ = ∅ then for any η ∈ BΛ(X), ζ ∈ BΛ′(X), [η]Λ ∩ [ζ]′

Λ = ∅

• Over Z, strongly irreducible ⇐

⇒ mixing (irreducible and aperiodic)

• Strong irreducibility implies condition (D): any

x, x′ ∈ X can be glued along a narrow border

Theorem (Dobrushin, 1969; formulation due to Ruelle)

If X ⊆ AZd satisfies condition (D) and Φ < ∞, then any Gibbs measure on X for Φ is an equilibrium measure for AΦ.

15 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Topological Markov properties

• A subshift X ⊆ AG has the topological Markov property

(TMP) if xΛ1x′

Λc

2 ∈ X whenever x, x′ agree on Λ2 \ Λ1 for

Λ2 large enough depending on Λ1

16 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Topological Markov properties

• A subshift X ⊆ AG has the topological Markov property

(TMP) if xΛ1x′

Λc

2 ∈ X whenever x, x′ agree on Λ2 \ Λ1 for

Λ2 large enough depending on Λ1

• X has the strong TMP if we can take Λ2 = ∆Λ1 for a

fixed finite ∆ ⋐ G

16 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Topological Markov properties

• A subshift X ⊆ AG has the topological Markov property

(TMP) if xΛ1x′

Λc

2 ∈ X whenever x, x′ agree on Λ2 \ Λ1 for

Λ2 large enough depending on Λ1

• X has the strong TMP if we can take Λ2 = ∆Λ1 for a

fixed finite ∆ ⋐ G

• SFT =

⇒ strong TMP = ⇒ TMP, both strict

16 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Topological Markov properties

• A subshift X ⊆ AG has the topological Markov property

(TMP) if xΛ1x′

Λc

2 ∈ X whenever x, x′ agree on Λ2 \ Λ1 for

Λ2 large enough depending on Λ1

• X has the strong TMP if we can take Λ2 = ∆Λ1 for a

fixed finite ∆ ⋐ G

• SFT =

⇒ strong TMP = ⇒ TMP, both strict

• none of these properties preserved under factors (golden

mean SFT → even shift lacks the TMP)

16 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Topological Markov properties

• A subshift X ⊆ AG has the topological Markov property

(TMP) if xΛ1x′

Λc

2 ∈ X whenever x, x′ agree on Λ2 \ Λ1 for

Λ2 large enough depending on Λ1

• X has the strong TMP if we can take Λ2 = ∆Λ1 for a

fixed finite ∆ ⋐ G

• SFT =

⇒ strong TMP = ⇒ TMP, both strict

• none of these properties preserved under factors (golden

mean SFT → even shift lacks the TMP)

Theorem (Lanford-Ruelle, AZ; Bowen, Ruelle, Z-SFT)

For an SFT X ⊆ AZd and Φ < ∞, any equilibrium measure on X for Φ is a Gibbs measure for Φ.

16 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Topological Markov properties

• A subshift X ⊆ AG has the topological Markov property

(TMP) if xΛ1x′

Λc

2 ∈ X whenever x, x′ agree on Λ2 \ Λ1 for

Λ2 large enough depending on Λ1

• X has the strong TMP if we can take Λ2 = ∆Λ1 for a

fixed finite ∆ ⋐ G

• SFT =

⇒ strong TMP = ⇒ TMP, both strict

• none of these properties preserved under factors (golden

mean SFT → even shift lacks the TMP)

Theorem (Lanford-Ruelle, AZ; Bowen, Ruelle, Z-SFT)

For an SFT X ⊆ AZd and Φ < ∞, any equilibrium measure on X for Φ is a Gibbs measure for Φ.

Theorem (Meyerovitch, 2013)

For any subshift X ⊆ AZd and f ∈ SVd(X), any equilibrium measure for f is topologically Gibbs for f (⇐ ⇒ Gibbs when X has the TMP).

16 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Preservation of Gibbsianness

Chazottes-Ugalde, Kempton-Pollicott (both 2011): a symbol amalgamation map between full shifts over N preserves Gibbsianness (for regular potentials)

17 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Preservation of Gibbsianness

Chazottes-Ugalde, Kempton-Pollicott (both 2011): a symbol amalgamation map between full shifts over N preserves Gibbsianness (for regular potentials) Natural generalization of hidden Markov models

17 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Preservation of Gibbsianness

Chazottes-Ugalde, Kempton-Pollicott (both 2011): a symbol amalgamation map between full shifts over N preserves Gibbsianness (for regular potentials) Natural generalization of hidden Markov models Let π : X → Y be a continuous factor map, φ a cocycle

• n Y , and π∗φ(x, x′) = φ(π(x′), π(x′)).

17 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Preservation of Gibbsianness

Chazottes-Ugalde, Kempton-Pollicott (both 2011): a symbol amalgamation map between full shifts over N preserves Gibbsianness (for regular potentials) Natural generalization of hidden Markov models Let π : X → Y be a continuous factor map, φ a cocycle

• n Y , and π∗φ(x, x′) = φ(π(x′), π(x′)).

Question: for which X, Y , π, φ must π∗µ be Gibbs for φ whenever µ is Gibbs for π∗φ?

17 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Preservation of Gibbsianness

Chazottes-Ugalde, Kempton-Pollicott (both 2011): a symbol amalgamation map between full shifts over N preserves Gibbsianness (for regular potentials) Natural generalization of hidden Markov models Let π : X → Y be a continuous factor map, φ a cocycle

• n Y , and π∗φ(x, x′) = φ(π(x′), π(x′)).

Question: for which X, Y , π, φ must π∗µ be Gibbs for φ whenever µ is Gibbs for π∗φ? Note that we need π∗TY = TX up to null sets (π essentially respects TX) for this to even make sense

17 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Preservation of Gibbsianness

Chazottes-Ugalde, Kempton-Pollicott (both 2011): a symbol amalgamation map between full shifts over N preserves Gibbsianness (for regular potentials) Natural generalization of hidden Markov models Let π : X → Y be a continuous factor map, φ a cocycle

• n Y , and π∗φ(x, x′) = φ(π(x′), π(x′)).

Question: for which X, Y , π, φ must π∗µ be Gibbs for φ whenever µ is Gibbs for π∗φ? Note that we need π∗TY = TX up to null sets (π essentially respects TX) for this to even make sense

Theorem (2020)

If X ⊂ AG is irreducible and has the TMP, and π essentially respects TX, then µ fully supported ergodic Gibbs for π∗φ = ⇒ π∗µ Gibbs for φ.

17 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Generalizing Lanford-Ruelle

Meyerovitch (2013) presents non-sofic examples with Lanford-Ruelle-like properties (equilibrium = ⇒ Gibbs)

• Skew products of Kalikow type (T-T −1)
• β-shifts
• the Dyck shift

18 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Generalizing Lanford-Ruelle

Meyerovitch (2013) presents non-sofic examples with Lanford-Ruelle-like properties (equilibrium = ⇒ Gibbs)

• Skew products of Kalikow type (T-T −1)
• β-shifts
• the Dyck shift

Problem: prove a Lanford-Ruelle theorem for a class of subshifts containing these examples

18 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Generalizing Lanford-Ruelle

Meyerovitch (2013) presents non-sofic examples with Lanford-Ruelle-like properties (equilibrium = ⇒ Gibbs)

• Skew products of Kalikow type (T-T −1)
• β-shifts
• the Dyck shift

Problem: prove a Lanford-Ruelle theorem for a class of subshifts containing these examples Natural first step: generalize beyond TMP; simplest class without TMP in general are the sofic shifts

18 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Generalizing Lanford-Ruelle

Meyerovitch (2013) presents non-sofic examples with Lanford-Ruelle-like properties (equilibrium = ⇒ Gibbs)

• Skew products of Kalikow type (T-T −1)
• β-shifts
• the Dyck shift

Problem: prove a Lanford-Ruelle theorem for a class of subshifts containing these examples Natural first step: generalize beyond TMP; simplest class without TMP in general are the sofic shifts

Theorem (2020)

For Y ⊆ AZ an irreducible sofic shift and f ∈ SVd(X), every equilibrium measure for f is Gibbs for f .

18 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Respecting the Gibbs relation

• If X ⊆ AG is irreducible and has the TMP, and

π : X → Y essentially respects TX, then π satisfies a weak almost invertibility property (doesn’t seem to imply that (X, µ) and (Y , π∗µ) are measurably conjugate)

19 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Respecting the Gibbs relation

• If X ⊆ AG is irreducible and has the TMP, and

π : X → Y essentially respects TX, then π satisfies a weak almost invertibility property (doesn’t seem to imply that (X, µ) and (Y , π∗µ) are measurably conjugate)

• If G is amenable, X ⊆ AG has the strong TMP, and

π : X → Y essentially respects TX, then h(X) = h(Y )

19 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Respecting the Gibbs relation

• If X ⊆ AG is irreducible and has the TMP, and

π : X → Y essentially respects TX, then π satisfies a weak almost invertibility property (doesn’t seem to imply that (X, µ) and (Y , π∗µ) are measurably conjugate)

• If G is amenable, X ⊆ AG has the strong TMP, and

π : X → Y essentially respects TX, then h(X) = h(Y )

• If X ⊆ AZ is an irreducible SFT then π : X → Y

essentially respects TX iff π has degree one

19 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Respecting the Gibbs relation

• If X ⊆ AG is irreducible and has the TMP, and

π : X → Y essentially respects TX, then π satisfies a weak almost invertibility property (doesn’t seem to imply that (X, µ) and (Y , π∗µ) are measurably conjugate)

• If G is amenable, X ⊆ AG has the strong TMP, and

π : X → Y essentially respects TX, then h(X) = h(Y )

• If X ⊆ AZ is an irreducible SFT then π : X → Y

essentially respects TX iff π has degree one

Theorem (2020)

Let X ⊆ AZ be a mixing SFT, π : X → Y a finite-to-one factor code, and f ∈ SV(Y ). If µ is a Gibbs measure for π∗f then π∗µ is a Gibbs measure for f .

19 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Preservation of Gibbsianness: proof ideas

• Lift finite-order holonomies from Y to X

20 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Preservation of Gibbsianness: proof ideas

• Lift finite-order holonomies from Y to X
• Building on Meester-Steif (2001): if π essentially

respects TX then it has no diamonds

20 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Preservation of Gibbsianness: proof ideas

• Lift finite-order holonomies from Y to X
• Building on Meester-Steif (2001): if π essentially

respects TX then it has no diamonds

• Hypotheses required to show that in almost every point,

every finite pattern appears infinitely often

20 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Preservation of Gibbsianness: proof ideas

• Lift finite-order holonomies from Y to X
• Building on Meester-Steif (2001): if π essentially

respects TX then it has no diamonds

• Hypotheses required to show that in almost every point,

every finite pattern appears infinitely often

• If X is strongly irreducible then every Gibbs

measure on X has full support

20 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Sofic Lanford-Ruelle: proof ideas

• Lift to the minimal right-resolving presentation, apply

Lanford-Ruelle, then push back down

21 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Sofic Lanford-Ruelle: proof ideas

• Lift to the minimal right-resolving presentation, apply

Lanford-Ruelle, then push back down

• Yoo (2018): on an irreducible sofic shift over Z, every
• eq. measure for f ∈ SVd has full support

21 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Sofic Lanford-Ruelle: proof ideas

• Lift to the minimal right-resolving presentation, apply

Lanford-Ruelle, then push back down

• Yoo (2018): on an irreducible sofic shift over Z, every
• eq. measure for f ∈ SVd has full support
• Yoo (2011): any fully supported (ergodic) measure on

an irreducible sofic shift lifts to a fully supported (ergodic) measure on any SFT cover

21 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Possible discussion topics

• Clarify statements

22 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Possible discussion topics

• Clarify statements
• More about the proofs

22 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Possible discussion topics

• Clarify statements
• More about the proofs
• Examples and pictures

22 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Possible discussion topics

• Clarify statements
• More about the proofs
• Examples and pictures
• Meyerovitch’s examples

22 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Possible discussion topics

• Clarify statements
• More about the proofs
• Examples and pictures
• Meyerovitch’s examples
• Further background on DLR theorems

22 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Possible discussion topics

• Clarify statements
• More about the proofs
• Examples and pictures
• Meyerovitch’s examples
• Further background on DLR theorems
• Anything else vaguely relevant, although the

probability of a sensible answer decays sharply with distance from the three theorems presented here

22 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Possible discussion topics

• Clarify statements
• More about the proofs
• Examples and pictures
• Meyerovitch’s examples
• Further background on DLR theorems
• Anything else vaguely relevant, although the

probability of a sensible answer decays sharply with distance from the three theorems presented here Feel free to reach out before Thursday afternoon! ⊲ sophmac at math dot ubc dot ca On Thursday we can discuss any questions or comments I have received, and see where the discussion goes. If it seems appropriate, I can take a poll, like Lior did last week, on prepared selections from the list above.

22 / 23

Factors of Gibbs measures on subshifts Sophie MacDonald Introduction What is a Gibbs measure?

Two-ish definitions Equivalence

DLR theorems Preservation of Gibbsianness

Introduction Results Proof ideas

Closing

Bibliography

⊲ Borsato, M (2020). arxiv.org/abs/2003.05532. ⊲ Kimura (2015). Master’s thesis, U. de S˜ ao Paulo. ⊲ Meester, Steif (2001). Pac. J. Math. 200 (2). ⊲ Meyerovitch (2013). Ergod. Th. & Dynam. Sys. 33. ⊲ Muir (2011). PhD dissertation, U. of North Texas. ⊲ Yoo (2011) Ergod. Th. & Dynam. Sys. 31. (2018) J. Mod. Dyn. 13. ⊲ In: Marcus, Petersen, Weismann (eds.) (2011). Chazottes, Ugalde. Preservation of Gibbsianness... Pollicott, Kempton. Factors of Gibbs measures... ⊲ For thermodynamic formalism: Ruelle (2004, 2nd ed). Thermodynamic formalism. Keller (1998). Equilibrium states in ergodic theory.

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