A nonlinear thermodynamical formalism J er ome BUZZI (CNRS Orsay) - - PowerPoint PPT Presentation

A nonlinear thermodynamical formalism

J´ erˆ

• me BUZZI (CNRS Orsay)

Joint work with Renaud LEPLAIDEUR Seminar Resistˆ encia Dinˆ amica July 10, 2020

• J. Buzzi

A nonlinear thermodynamical formalism July 2020 1 / 11

Outline

1

Setup A little statistical mechanics Thermodynamical formalism

2

Results Variational principle Equidistribution Equilibrium states

3

Ingredients of proofs Variational principle and Equidistribution Reduction of the nonlinear to the linear equilibrium states

4

Conclusion

• J. Buzzi

A nonlinear thermodynamical formalism July 2020 2 / 11

Setup A little statistical mechanics

A little statistical mechanics: Ising model

One-dimensional spin system: X = {+1, −1}Z, n ≥ 1 with σ : (xn)n∈Z → (xn+1)n∈Z Each spin interacts with its immediate neighbors: Energy of x ∈ XN = {x ∈ X : σNx = x}: EN(x) :=

• 0<p<N

φ ◦ σp(x) with φ(x) = −x0(x−1 + x+1) A Gibbs ensemble is µN ∈ P(XN) such that µN(x) = 1 ZN e−βEN(x) where ZN :=

• z∈XN

e−βEN (z) (β = (temperature)−1) Theorem (No phase transition for 1-D short-range) The Gibbs ensembles converge as n → ∞ to the unique µ0 ∈ P(σ) which maximizes the free energy: P(µ) := h(µ) − βµ(φ) µ0 is the equilibrium state. Moreover, P(µ0) = 1

β limn→∞ − 1 n log Zn

Thermodynamical formalism for Subshifts of Finite Type (Sinai, Ruelle, Bowen) See: Ruelle, Thermodynamical formalism, Cambridge Mathematical Library, 1978, 2004

• J. Buzzi

A nonlinear thermodynamical formalism July 2020 3 / 11

Setup A little statistical mechanics

A little statistical mechanics: Ising model

One-dimensional spin system: X = {+1, −1}Z, n ≥ 1 with σ : (xn)n∈Z → (xn+1)n∈Z Each spin interacts with its immediate neighbors: Energy of x ∈ XN = {x ∈ X : σNx = x}: EN(x) :=

• 0<p<N

φ ◦ σp(x) with φ(x) = −x0(x−1 + x+1) A Gibbs ensemble is µN ∈ P(XN) such that µN(x) = 1 ZN e−βEN(x) where ZN :=

• z∈XN

e−βEN (z) (β = (temperature)−1) Theorem (No phase transition for 1-D short-range) The Gibbs ensembles converge as n → ∞ to the unique µ0 ∈ P(σ) which maximizes the free energy: P(µ) := h(µ) − βµ(φ) µ0 is the equilibrium state. Moreover, P(µ0) = 1

β limn→∞ − 1 n log Zn

Thermodynamical formalism for Subshifts of Finite Type (Sinai, Ruelle, Bowen) See: Ruelle, Thermodynamical formalism, Cambridge Mathematical Library, 1978, 2004

• J. Buzzi

A nonlinear thermodynamical formalism July 2020 3 / 11

Setup A little statistical mechanics

A little statistical mechanics: Ising model

One-dimensional spin system: X = {+1, −1}Z, n ≥ 1 with σ : (xn)n∈Z → (xn+1)n∈Z Each spin interacts with its immediate neighbors: Energy of x ∈ XN = {x ∈ X : σNx = x}: EN(x) :=

• 0<p<N

φ ◦ σp(x) with φ(x) = −x0(x−1 + x+1) A Gibbs ensemble is µN ∈ P(XN) such that µN(x) = 1 ZN e−βEN(x) where ZN :=

• z∈XN

e−βEN (z) (β = (temperature)−1) Theorem (No phase transition for 1-D short-range) The Gibbs ensembles converge as n → ∞ to the unique µ0 ∈ P(σ) which maximizes the free energy: P(µ) := h(µ) − βµ(φ) µ0 is the equilibrium state. Moreover, P(µ0) = 1

β limn→∞ − 1 n log Zn

Thermodynamical formalism for Subshifts of Finite Type (Sinai, Ruelle, Bowen) See: Ruelle, Thermodynamical formalism, Cambridge Mathematical Library, 1978, 2004

• J. Buzzi

A nonlinear thermodynamical formalism July 2020 3 / 11

Setup A little statistical mechanics

A little statistical mechanics: Curie-Weiss mean-field model

Each spin interacts with the average spin: EN(x) :=

• 0≤p<N

−xp · 1 N

• 0≤q<N

xq = N   1 N

• 0≤p<N

φ ◦ σp(x)  

2

where φ(x) := x0. A Gibbs ensemble is µN ∈ P(XN) such that, for some y ∈ X, µn(x) = 1 ζN e−βEN (x) where ζN :=

• z∈XN

e−βEN (z) Theorem (Phase transition in 1-D mean-field) For β large, the Gibbs ensembles converge as n → ∞ to the average of two equilibrium states µ+, µ− ∈ P(σ) which are the two maximizers of the nonlinear free energy: Π(µ) := h(µ) − βµ(φ)2 Moreover, Π(µ0) = limn→∞ 1

n log ζn

See: R. Ellis, Entropy, large deviation and statistical mechanics, Springer-Verlag, 1985 Leplaideur-Watbled (2019): quadratic thermodynamical formalism for SFTs Is there a nonlinear thermodynamical formalism for dynamical systems?

• J. Buzzi

A nonlinear thermodynamical formalism July 2020 4 / 11

Setup A little statistical mechanics

A little statistical mechanics: Curie-Weiss mean-field model

Each spin interacts with the average spin: EN(x) :=

• 0≤p<N

−xp · 1 N

• 0≤q<N

xq = N   1 N

• 0≤p<N

φ ◦ σp(x)  

2

where φ(x) := x0. A Gibbs ensemble is µN ∈ P(XN) such that, for some y ∈ X, µn(x) = 1 ζN e−βEN (x) where ζN :=

• z∈XN

e−βEN (z) Theorem (Phase transition in 1-D mean-field) For β large, the Gibbs ensembles converge as n → ∞ to the average of two equilibrium states µ+, µ− ∈ P(σ) which are the two maximizers of the nonlinear free energy: Π(µ) := h(µ) − βµ(φ)2 Moreover, Π(µ0) = limn→∞ 1

n log ζn

See: R. Ellis, Entropy, large deviation and statistical mechanics, Springer-Verlag, 1985 Leplaideur-Watbled (2019): quadratic thermodynamical formalism for SFTs Is there a nonlinear thermodynamical formalism for dynamical systems?

• J. Buzzi

A nonlinear thermodynamical formalism July 2020 4 / 11

Setup A little statistical mechanics

A little statistical mechanics: Curie-Weiss mean-field model

Each spin interacts with the average spin: EN(x) :=

• 0≤p<N

−xp · 1 N

• 0≤q<N

xq = N   1 N

• 0≤p<N

φ ◦ σp(x)  

2

where φ(x) := x0. A Gibbs ensemble is µN ∈ P(XN) such that, for some y ∈ X, µn(x) = 1 ζN e−βEN (x) where ζN :=

• z∈XN

e−βEN (z) Theorem (Phase transition in 1-D mean-field) For β large, the Gibbs ensembles converge as n → ∞ to the average of two equilibrium states µ+, µ− ∈ P(σ) which are the two maximizers of the nonlinear free energy: Π(µ) := h(µ) − βµ(φ)2 Moreover, Π(µ0) = limn→∞ 1

n log ζn

See: R. Ellis, Entropy, large deviation and statistical mechanics, Springer-Verlag, 1985 Leplaideur-Watbled (2019): quadratic thermodynamical formalism for SFTs Is there a nonlinear thermodynamical formalism for dynamical systems?

• J. Buzzi

A nonlinear thermodynamical formalism July 2020 4 / 11

Setup Thermodynamical formalism

Nonlinear Thermodynamical formalism

Weighted dynamics (X, T, φ) T : X → X continuous on compact metric space with potential φ : X → R continuous and nonlinearity F : R → R continuous For probability measures Kolmogorov-Sinai entropy: hT : P(X) → R ∪ {−∞} linear pressure: Pφ(T, µ) = hT(µ) + µ(φ) nonlinear pressure: Πφ,F(T, µ) = hT(µ) + F(µ(φ)) Equilibrium states µ ∈ P(T) such that Pφ(T, µ) = supν∈P(T) Pφ(T, ν) (linear) µ ∈ P(T) such that Πφ,F(T, µ) = supν∈P(T) Πφ,F(T, ν) (nonlinear) Counting BT(x, ǫ, n) := {y ∈ X : ∀0 ≤ k < n d(f ky, f kx) < ǫ}; (ǫ, n)-cover of X:

x∈C BT(x, ǫ, n) = X

weight of order n: wφ(C, n) :=

x∈C exp :=Snφ(x)

• 0≤k<n φ(σkx)

nonlinear weight of order n: ωφ(C, n) :=

x∈C nF( 1 n exp 0≤k<n φ(σkx))

Topological pressure Ptop

φ (T) := limǫ→0 lim supn→∞ 1 n log minC (ǫ, n)-cover wφ(C, n)

Πtop

φ,F(T) := limǫ→0 lim supn→∞ 1 n log minC (ǫ, n)-cover ωφ,F(C, n)

• J. Buzzi

A nonlinear thermodynamical formalism July 2020 5 / 11

Setup Thermodynamical formalism

Nonlinear Thermodynamical formalism

Weighted dynamics (X, T, φ) T : X → X continuous on compact metric space with potential φ : X → R continuous and nonlinearity F : R → R continuous For probability measures Kolmogorov-Sinai entropy: hT : P(X) → R ∪ {−∞} linear pressure: Pφ(T, µ) = hT(µ) + µ(φ) nonlinear pressure: Πφ,F(T, µ) = hT(µ) + F(µ(φ)) Equilibrium states µ ∈ P(T) such that Pφ(T, µ) = supν∈P(T) Pφ(T, ν) (linear) µ ∈ P(T) such that Πφ,F(T, µ) = supν∈P(T) Πφ,F(T, ν) (nonlinear) Counting BT(x, ǫ, n) := {y ∈ X : ∀0 ≤ k < n d(f ky, f kx) < ǫ}; (ǫ, n)-cover of X:

x∈C BT(x, ǫ, n) = X

weight of order n: wφ(C, n) :=

x∈C exp :=Snφ(x)

• 0≤k<n φ(σkx)

nonlinear weight of order n: ωφ(C, n) :=

x∈C nF( 1 n exp 0≤k<n φ(σkx))

Topological pressure Ptop

φ (T) := limǫ→0 lim supn→∞ 1 n log minC (ǫ, n)-cover wφ(C, n)

Πtop

φ,F(T) := limǫ→0 lim supn→∞ 1 n log minC (ǫ, n)-cover ωφ,F(C, n)

• J. Buzzi

A nonlinear thermodynamical formalism July 2020 5 / 11

Setup Thermodynamical formalism

Nonlinear Thermodynamical formalism

Weighted dynamics (X, T, φ) T : X → X continuous on compact metric space with potential φ : X → R continuous and nonlinearity F : R → R continuous For probability measures Kolmogorov-Sinai entropy: hT : P(X) → R ∪ {−∞} linear pressure: Pφ(T, µ) = hT(µ) + µ(φ) nonlinear pressure: Πφ,F(T, µ) = hT(µ) + F(µ(φ)) Equilibrium states µ ∈ P(T) such that Pφ(T, µ) = supν∈P(T) Pφ(T, ν) (linear) µ ∈ P(T) such that Πφ,F(T, µ) = supν∈P(T) Πφ,F(T, ν) (nonlinear) Counting BT(x, ǫ, n) := {y ∈ X : ∀0 ≤ k < n d(f ky, f kx) < ǫ}; (ǫ, n)-cover of X:

x∈C BT(x, ǫ, n) = X

weight of order n: wφ(C, n) :=

x∈C exp :=Snφ(x)

• 0≤k<n φ(σkx)

nonlinear weight of order n: ωφ(C, n) :=

x∈C nF( 1 n exp 0≤k<n φ(σkx))

Topological pressure Ptop

φ (T) := limǫ→0 lim supn→∞ 1 n log minC (ǫ, n)-cover wφ(C, n)

Πtop

φ,F(T) := limǫ→0 lim supn→∞ 1 n log minC (ǫ, n)-cover ωφ,F(C, n)

• J. Buzzi

A nonlinear thermodynamical formalism July 2020 5 / 11

Setup Thermodynamical formalism

Nonlinear Thermodynamical formalism

Weighted dynamics (X, T, φ) T : X → X continuous on compact metric space with potential φ : X → R continuous and nonlinearity F : R → R continuous For probability measures Kolmogorov-Sinai entropy: hT : P(X) → R ∪ {−∞} linear pressure: Pφ(T, µ) = hT(µ) + µ(φ) nonlinear pressure: Πφ,F(T, µ) = hT(µ) + F(µ(φ)) Equilibrium states µ ∈ P(T) such that Pφ(T, µ) = supν∈P(T) Pφ(T, ν) (linear) µ ∈ P(T) such that Πφ,F(T, µ) = supν∈P(T) Πφ,F(T, ν) (nonlinear) Counting BT(x, ǫ, n) := {y ∈ X : ∀0 ≤ k < n d(f ky, f kx) < ǫ}; (ǫ, n)-cover of X:

x∈C BT(x, ǫ, n) = X

weight of order n: wφ(C, n) :=

x∈C exp :=Snφ(x)

• 0≤k<n φ(σkx)

nonlinear weight of order n: ωφ(C, n) :=

x∈C nF( 1 n exp 0≤k<n φ(σkx))

Topological pressure Ptop

φ (T) := limǫ→0 lim supn→∞ 1 n log minC (ǫ, n)-cover wφ(C, n)

Πtop

φ,F(T) := limǫ→0 lim supn→∞ 1 n log minC (ǫ, n)-cover ωφ,F(C, n)

• J. Buzzi

A nonlinear thermodynamical formalism July 2020 5 / 11

Setup Thermodynamical formalism

Nonlinear Thermodynamical formalism

Weighted dynamics (X, T, φ) T : X → X continuous on compact metric space with potential φ : X → R continuous and nonlinearity F : R → R continuous For probability measures Kolmogorov-Sinai entropy: hT : P(X) → R ∪ {−∞} linear pressure: Pφ(T, µ) = hT(µ) + µ(φ) nonlinear pressure: Πφ,F(T, µ) = hT(µ) + F(µ(φ)) Equilibrium states µ ∈ P(T) such that Pφ(T, µ) = supν∈P(T) Pφ(T, ν) (linear) µ ∈ P(T) such that Πφ,F(T, µ) = supν∈P(T) Πφ,F(T, ν) (nonlinear) Counting BT(x, ǫ, n) := {y ∈ X : ∀0 ≤ k < n d(f ky, f kx) < ǫ}; (ǫ, n)-cover of X:

x∈C BT(x, ǫ, n) = X

weight of order n: wφ(C, n) :=

x∈C exp :=Snφ(x)

• 0≤k<n φ(σkx)

nonlinear weight of order n: ωφ(C, n) :=

x∈C nF( 1 n exp 0≤k<n φ(σkx))

Topological pressure Ptop

φ (T) := limǫ→0 lim supn→∞ 1 n log minC (ǫ, n)-cover wφ(C, n)

Πtop

φ,F(T) := limǫ→0 lim supn→∞ 1 n log minC (ǫ, n)-cover ωφ,F(C, n)

• J. Buzzi

A nonlinear thermodynamical formalism July 2020 5 / 11

Results Variational principle

Nonlinear Variational principle

Similarly as in the linear case due to Walters, Theorem (B-Leplaideur) For any (X, T, φ, F) continuous system whose ergodic measures are dense in the following sense: (*) ∀µ ∈ P(T)∀ǫ > 0 ∃ν ∈ Perg(T) |hT(ν) − hT(µ)| + |ν(φ) − µ(φ)| < ǫ the variational principle holds: Πtop

φ,F(T) : = lim ǫ→0 lim sup n→∞

1 n log min

C (ǫ, n)-cover ωφ,F(C, n)

= lim

ǫ→0 lim inf n→∞

1 n log min

C (ǫ, n)-cover ωφ,F(C, n)

= sup

µ∈P(T)

ΠT,φ(µ) Example In contrast to the linear case, the variational principle may fail without (*). If X = {−1, +1}, T(x) = φ(x) = x, F(x) = −x2 then Πtop

φ,F(T) = −1 < Πφ,F( 1 2(δ−1 + δ+1)) = 0

• J. Buzzi

A nonlinear thermodynamical formalism July 2020 6 / 11

Results Variational principle

Nonlinear Variational principle

Similarly as in the linear case due to Walters, Theorem (B-Leplaideur) For any (X, T, φ, F) continuous system whose ergodic measures are dense in the following sense: (*) ∀µ ∈ P(T)∀ǫ > 0 ∃ν ∈ Perg(T) |hT(ν) − hT(µ)| + |ν(φ) − µ(φ)| < ǫ the variational principle holds: Πtop

φ,F(T) : = lim ǫ→0 lim sup n→∞

1 n log min

C (ǫ, n)-cover ωφ,F(C, n)

= lim

ǫ→0 lim inf n→∞

1 n log min

C (ǫ, n)-cover ωφ,F(C, n)

= sup

µ∈P(T)

ΠT,φ(µ) Example In contrast to the linear case, the variational principle may fail without (*). If X = {−1, +1}, T(x) = φ(x) = x, F(x) = −x2 then Πtop

φ,F(T) = −1 < Πφ,F( 1 2(δ−1 + δ+1)) = 0

• J. Buzzi

A nonlinear thermodynamical formalism July 2020 6 / 11

Results Equidistribution

Nonlinear Equidistribution

(X, T, φ, F) is a continuous systems whose ergodic measures are dense Guided by statistical mechanics, we define a (ǫ, n)-Gibbs ensemble as 1 ωφ,F(C, n)

• x∈C

enF( 1

n Snφ(x)) δx + δTx + · · · + δT n−1x

n ∈ P(X) for any (ǫ, n)-cover C of X with minimum weight. Theorem If (X, T) is ǫ-expansive, i.e., ∀x, y ∈ X (∀p ∈ Z d(T px, T py) < ǫ) = ⇒ x = y Then (ǫ, n)-Gibbs ensembles accumulate only on averages of equilibrium states as n → ∞ Example For the Curie-Weiss model one can choose Gibbs-ensembles which are symmetric under the flip (xn)n∈Z → (−xn)n∈Z. But none of the two equilibrium states satisfies this symmetry for large β.

• J. Buzzi

A nonlinear thermodynamical formalism July 2020 7 / 11

Results Equidistribution

Nonlinear Equidistribution

(X, T, φ, F) is a continuous systems whose ergodic measures are dense Guided by statistical mechanics, we define a (ǫ, n)-Gibbs ensemble as 1 ωφ,F(C, n)

• x∈C

enF( 1

n Snφ(x)) δx + δTx + · · · + δT n−1x

n ∈ P(X) for any (ǫ, n)-cover C of X with minimum weight. Theorem If (X, T) is ǫ-expansive, i.e., ∀x, y ∈ X (∀p ∈ Z d(T px, T py) < ǫ) = ⇒ x = y Then (ǫ, n)-Gibbs ensembles accumulate only on averages of equilibrium states as n → ∞ Example For the Curie-Weiss model one can choose Gibbs-ensembles which are symmetric under the flip (xn)n∈Z → (−xn)n∈Z. But none of the two equilibrium states satisfies this symmetry for large β.

• J. Buzzi

A nonlinear thermodynamical formalism July 2020 7 / 11

Results Equidistribution

Nonlinear Equidistribution

(X, T, φ, F) is a continuous systems whose ergodic measures are dense Guided by statistical mechanics, we define a (ǫ, n)-Gibbs ensemble as 1 ωφ,F(C, n)

• x∈C

enF( 1

n Snφ(x)) δx + δTx + · · · + δT n−1x

n ∈ P(X) for any (ǫ, n)-cover C of X with minimum weight. Theorem If (X, T) is ǫ-expansive, i.e., ∀x, y ∈ X (∀p ∈ Z d(T px, T py) < ǫ) = ⇒ x = y Then (ǫ, n)-Gibbs ensembles accumulate only on averages of equilibrium states as n → ∞ Example For the Curie-Weiss model one can choose Gibbs-ensembles which are symmetric under the flip (xn)n∈Z → (−xn)n∈Z. But none of the two equilibrium states satisfies this symmetry for large β.

• J. Buzzi

A nonlinear thermodynamical formalism July 2020 7 / 11

Results Equilibrium states

Nonlinear Equilibrium States

(X, T, φ) strongly regular if: – hT : P(T) → R upper semicontinuous – for each β ∈ R, ∃!νβ s.t. Pβφ(T, νβ) = Ptop

βφ(T)

– p : β ∈ R → Ptop

βφ(T) is strictly convex and real-analytic over R

Note: According to results of Sinai, Ruelle and Bowen, the above holds for transitive uniformly hyperbolic dynamics Theorem (B-Leplaideur) Let (X, T, φ) be strongly regular and let F : R → R be a nonlinearity. If F is real analytic, then there are finitely many nonlinear equilibrium states. Moreover, each equilibrium state coincides with the linear equilibrium states for some multiple of the potential φ. Remark The linear uniqueness may be lost, but we keep finiteness Nonlinear equilibrium states are ergodic and more generally inherit the good properties of the linear equilibrium states.

• J. Buzzi

A nonlinear thermodynamical formalism July 2020 8 / 11

Results Equilibrium states

Nonlinear Equilibrium States

(X, T, φ) strongly regular if: – hT : P(T) → R upper semicontinuous – for each β ∈ R, ∃!νβ s.t. Pβφ(T, νβ) = Ptop

βφ(T)

– p : β ∈ R → Ptop

βφ(T) is strictly convex and real-analytic over R

Note: According to results of Sinai, Ruelle and Bowen, the above holds for transitive uniformly hyperbolic dynamics Theorem (B-Leplaideur) Let (X, T, φ) be strongly regular and let F : R → R be a nonlinearity. If F is real analytic, then there are finitely many nonlinear equilibrium states. Moreover, each equilibrium state coincides with the linear equilibrium states for some multiple of the potential φ. Remark The linear uniqueness may be lost, but we keep finiteness Nonlinear equilibrium states are ergodic and more generally inherit the good properties of the linear equilibrium states.

• J. Buzzi

A nonlinear thermodynamical formalism July 2020 8 / 11

Results Equilibrium states

Nonlinear Equilibrium States

(X, T, φ) strongly regular if: – hT : P(T) → R upper semicontinuous – for each β ∈ R, ∃!νβ s.t. Pβφ(T, νβ) = Ptop

βφ(T)

– p : β ∈ R → Ptop

βφ(T) is strictly convex and real-analytic over R

Note: According to results of Sinai, Ruelle and Bowen, the above holds for transitive uniformly hyperbolic dynamics Theorem (B-Leplaideur) Let (X, T, φ) be strongly regular and let F : R → R be a nonlinearity. If F is real analytic, then there are finitely many nonlinear equilibrium states. Moreover, each equilibrium state coincides with the linear equilibrium states for some multiple of the potential φ. Remark The linear uniqueness may be lost, but we keep finiteness Nonlinear equilibrium states are ergodic and more generally inherit the good properties of the linear equilibrium states.

• J. Buzzi

A nonlinear thermodynamical formalism July 2020 8 / 11

Results Equilibrium states

Nonlinear Equilibrium States

(X, T, φ) strongly regular if: – hT : P(T) → R upper semicontinuous – for each β ∈ R, ∃!νβ s.t. Pβφ(T, νβ) = Ptop

βφ(T)

– p : β ∈ R → Ptop

βφ(T) is strictly convex and real-analytic over R

Note: According to results of Sinai, Ruelle and Bowen, the above holds for transitive uniformly hyperbolic dynamics Theorem (B-Leplaideur) Let (X, T, φ) be strongly regular and let F : R → R be a nonlinearity. If F is real analytic, then there are finitely many nonlinear equilibrium states. Moreover, each equilibrium state coincides with the linear equilibrium states for some multiple of the potential φ. Remark The linear uniqueness may be lost, but we keep finiteness Nonlinear equilibrium states are ergodic and more generally inherit the good properties of the linear equilibrium states.

• J. Buzzi

A nonlinear thermodynamical formalism July 2020 8 / 11

Ingredients of proofs Variational principle and Equidistribution

Proof of the variational principle and the equidistribution

Recall T, φ, F continuous with dense ergodic measures (1) supµ∈Perg(T) Π(T, µ) ≤ Πtop(T) Direct estimate from ergodic and Shannon-McMillan-Breiman theorems (2) supµ∈P(T) Π(T, µ) ≤ Πtop(T) Extends 1) by continuity and density (or convexity): trivial but needs an extra assumption (3) Πtop(T) ≤ supµ∈P(T) Π(T, µ) Decompose an (ǫ, n)-cover Cn according to the φ-average ≈ α Cn,α From the definition of the weight: |Cn,α| ≥ exp n

• Π

top(T) − F(α)

• Apply Misiurewicz argument to Cn,α

µα ∈ P(T) with µ(φ) ≈ α and h(T, µ) ≈ lim supn

1 n log |Cn,α|.

Remark The proof of (2) is trivial but requires the density of ergodicity or convexity of F Question Is Πtop(T) = supµ∈Perg(T) Π(T, µ)?

• J. Buzzi

A nonlinear thermodynamical formalism July 2020 9 / 11

Ingredients of proofs Variational principle and Equidistribution

Proof of the variational principle and the equidistribution

Recall T, φ, F continuous with dense ergodic measures (1) supµ∈Perg(T) Π(T, µ) ≤ Πtop(T) Direct estimate from ergodic and Shannon-McMillan-Breiman theorems (2) supµ∈P(T) Π(T, µ) ≤ Πtop(T) Extends 1) by continuity and density (or convexity): trivial but needs an extra assumption (3) Πtop(T) ≤ supµ∈P(T) Π(T, µ) Decompose an (ǫ, n)-cover Cn according to the φ-average ≈ α Cn,α From the definition of the weight: |Cn,α| ≥ exp n

• Π

top(T) − F(α)

• Apply Misiurewicz argument to Cn,α

µα ∈ P(T) with µ(φ) ≈ α and h(T, µ) ≈ lim supn

1 n log |Cn,α|.

Remark The proof of (2) is trivial but requires the density of ergodicity or convexity of F Question Is Πtop(T) = supµ∈Perg(T) Π(T, µ)?

• J. Buzzi

A nonlinear thermodynamical formalism July 2020 9 / 11

Ingredients of proofs Variational principle and Equidistribution

Proof of the variational principle and the equidistribution

Recall T, φ, F continuous with dense ergodic measures (1) supµ∈Perg(T) Π(T, µ) ≤ Πtop(T) Direct estimate from ergodic and Shannon-McMillan-Breiman theorems (2) supµ∈P(T) Π(T, µ) ≤ Πtop(T) Extends 1) by continuity and density (or convexity): trivial but needs an extra assumption (3) Πtop(T) ≤ supµ∈P(T) Π(T, µ) Decompose an (ǫ, n)-cover Cn according to the φ-average ≈ α Cn,α From the definition of the weight: |Cn,α| ≥ exp n

• Π

top(T) − F(α)

• Apply Misiurewicz argument to Cn,α

µα ∈ P(T) with µ(φ) ≈ α and h(T, µ) ≈ lim supn

1 n log |Cn,α|.

Remark The proof of (2) is trivial but requires the density of ergodicity or convexity of F Question Is Πtop(T) = supµ∈Perg(T) Π(T, µ)?

• J. Buzzi

A nonlinear thermodynamical formalism July 2020 9 / 11

Ingredients of proofs Variational principle and Equidistribution

Proof of the variational principle and the equidistribution

Recall T, φ, F continuous with dense ergodic measures (1) supµ∈Perg(T) Π(T, µ) ≤ Πtop(T) Direct estimate from ergodic and Shannon-McMillan-Breiman theorems (2) supµ∈P(T) Π(T, µ) ≤ Πtop(T) Extends 1) by continuity and density (or convexity): trivial but needs an extra assumption (3) Πtop(T) ≤ supµ∈P(T) Π(T, µ) Decompose an (ǫ, n)-cover Cn according to the φ-average ≈ α Cn,α From the definition of the weight: |Cn,α| ≥ exp n

• Π

top(T) − F(α)

• Apply Misiurewicz argument to Cn,α

µα ∈ P(T) with µ(φ) ≈ α and h(T, µ) ≈ lim supn

1 n log |Cn,α|.

Remark The proof of (2) is trivial but requires the density of ergodicity or convexity of F Question Is Πtop(T) = supµ∈Perg(T) Π(T, µ)?

• J. Buzzi

A nonlinear thermodynamical formalism July 2020 9 / 11

Ingredients of proofs Variational principle and Equidistribution

Proof of the variational principle and the equidistribution

Recall T, φ, F continuous with dense ergodic measures (1) supµ∈Perg(T) Π(T, µ) ≤ Πtop(T) Direct estimate from ergodic and Shannon-McMillan-Breiman theorems (2) supµ∈P(T) Π(T, µ) ≤ Πtop(T) Extends 1) by continuity and density (or convexity): trivial but needs an extra assumption (3) Πtop(T) ≤ supµ∈P(T) Π(T, µ) Decompose an (ǫ, n)-cover Cn according to the φ-average ≈ α Cn,α From the definition of the weight: |Cn,α| ≥ exp n

• Π

top(T) − F(α)

• Apply Misiurewicz argument to Cn,α

µα ∈ P(T) with µ(φ) ≈ α and h(T, µ) ≈ lim supn

1 n log |Cn,α|.

Remark The proof of (2) is trivial but requires the density of ergodicity or convexity of F Question Is Πtop(T) = supµ∈Perg(T) Π(T, µ)?

• J. Buzzi

A nonlinear thermodynamical formalism July 2020 9 / 11

Ingredients of proofs Reduction of the nonlinear to the linear equilibrium states

Characterization of the equilibrium states

(X, T, φ, F) strongly regular: (a) hT : P(X) → R ∪ {−∞} is upper semicontinuous (b) pT,φ : β → Ptop

T (βφ) is real-analytic and strictly convex

(c) for each β ∈ R, there is a unique linear equilibrium state νβ for βφ Idea: condition on z = µ(φ) ∈ Z := {µ(φ) : µ ∈ P(T)} = [A, B] M(z) := {µ ∈ P(T) : µ(φ) = z}, a non-empty compact set for z ∈ Z

1

p′

T,φ : R → (A, B) is a real-analytic increasing diffeo such that p′ T,φ(β) = νβ(φ)

recall (b) −hT convex and lsc hence −hT = (−hT )∗∗ = −P∗

T so:

hT (µ) = infψ∈C(X)(Ptop

T (ψ) − µ(ψ))

hence {subdifferentials of PT : C(X) → R at βφ} = {equilibrium states for βφ} thus p′

T,φ(β) = νβ(φ) as claimed and (b) concludes.

Let β : (A, B) → R be the real-analytic increasing diffeo defined as β = (p′

T,φ)−1

2

νβ(z) = unique maximizer for: entropy, linear pressure, nonlinear pressure in M(z)

3

h(z) := sup{h(µ) : µ ∈ M(z)} = hT(νβ(z)) is continuous and concave over [A, B]

4

h : (A, B) → R is real-analytic with h′(A+) = +∞ and h′(B−) = −∞

5

Z∗ := {z ∈ Z : h + F is maximum at z} is a finite subset of (A, B)

6

{nonlinear equilibrium states} = {νβ(z) : z ∈ Z∗}

• J. Buzzi

A nonlinear thermodynamical formalism July 2020 10 / 11

Ingredients of proofs Reduction of the nonlinear to the linear equilibrium states

Characterization of the equilibrium states

(X, T, φ, F) strongly regular: (a) hT : P(X) → R ∪ {−∞} is upper semicontinuous (b) pT,φ : β → Ptop

T (βφ) is real-analytic and strictly convex

(c) for each β ∈ R, there is a unique linear equilibrium state νβ for βφ Idea: condition on z = µ(φ) ∈ Z := {µ(φ) : µ ∈ P(T)} = [A, B] M(z) := {µ ∈ P(T) : µ(φ) = z}, a non-empty compact set for z ∈ Z

1

p′

T,φ : R → (A, B) is a real-analytic increasing diffeo such that p′ T,φ(β) = νβ(φ)

Let β : (A, B) → R be the real-analytic increasing diffeo defined as β = (p′

T,φ)−1

2

νβ(z) = unique maximizer for: entropy, linear pressure, nonlinear pressure in M(z)

3

h(z) := sup{h(µ) : µ ∈ M(z)} = hT(νβ(z)) is continuous and concave over [A, B]

4

h : (A, B) → R is real-analytic with h′(A+) = +∞ and h′(B−) = −∞

5

Z∗ := {z ∈ Z : h + F is maximum at z} is a finite subset of (A, B)

6

{nonlinear equilibrium states} = {νβ(z) : z ∈ Z∗}

• J. Buzzi

A nonlinear thermodynamical formalism July 2020 10 / 11

Ingredients of proofs Reduction of the nonlinear to the linear equilibrium states

Characterization of the equilibrium states

(X, T, φ, F) strongly regular: (a) hT : P(X) → R ∪ {−∞} is upper semicontinuous (b) pT,φ : β → Ptop

T (βφ) is real-analytic and strictly convex

(c) for each β ∈ R, there is a unique linear equilibrium state νβ for βφ Idea: condition on z = µ(φ) ∈ Z := {µ(φ) : µ ∈ P(T)} = [A, B] M(z) := {µ ∈ P(T) : µ(φ) = z}, a non-empty compact set for z ∈ Z

1

p′

T,φ : R → (A, B) is a real-analytic increasing diffeo such that p′ T,φ(β) = νβ(φ)

Let β : (A, B) → R be the real-analytic increasing diffeo defined as β = (p′

T,φ)−1

2

νβ(z) = unique maximizer for: entropy, linear pressure, nonlinear pressure in M(z)

3

h(z) := sup{h(µ) : µ ∈ M(z)} = hT(νβ(z)) is continuous and concave over [A, B]

4

h : (A, B) → R is real-analytic with h′(A+) = +∞ and h′(B−) = −∞

5

Z∗ := {z ∈ Z : h + F is maximum at z} is a finite subset of (A, B)

6

{nonlinear equilibrium states} = {νβ(z) : z ∈ Z∗}

• J. Buzzi

A nonlinear thermodynamical formalism July 2020 10 / 11

Ingredients of proofs Reduction of the nonlinear to the linear equilibrium states

Characterization of the equilibrium states

(X, T, φ, F) strongly regular: (a) hT : P(X) → R ∪ {−∞} is upper semicontinuous (b) pT,φ : β → Ptop

T (βφ) is real-analytic and strictly convex

(c) for each β ∈ R, there is a unique linear equilibrium state νβ for βφ Idea: condition on z = µ(φ) ∈ Z := {µ(φ) : µ ∈ P(T)} = [A, B] M(z) := {µ ∈ P(T) : µ(φ) = z}, a non-empty compact set for z ∈ Z

1

p′

T,φ : R → (A, B) is a real-analytic increasing diffeo such that p′ T,φ(β) = νβ(φ)

Let β : (A, B) → R be the real-analytic increasing diffeo defined as β = (p′

T,φ)−1

2

νβ(z) = unique maximizer for: entropy, linear pressure, nonlinear pressure in M(z)

3

h(z) := sup{h(µ) : µ ∈ M(z)} = hT(νβ(z)) is continuous and concave over [A, B]

concave follows from hT affine on P(T) bounded, concave, upper semicontinuous on [A, B] = ⇒ continuous

4

h : (A, B) → R is real-analytic with h′(A+) = +∞ and h′(B−) = −∞

5

Z∗ := {z ∈ Z : h + F is maximum at z} is a finite subset of (A, B)

6

{nonlinear equilibrium states} = {νβ(z) : z ∈ Z∗}

• J. Buzzi

A nonlinear thermodynamical formalism July 2020 10 / 11

Ingredients of proofs Reduction of the nonlinear to the linear equilibrium states

Characterization of the equilibrium states

(X, T, φ, F) strongly regular: (a) hT : P(X) → R ∪ {−∞} is upper semicontinuous (b) pT,φ : β → Ptop

T (βφ) is real-analytic and strictly convex

(c) for each β ∈ R, there is a unique linear equilibrium state νβ for βφ Idea: condition on z = µ(φ) ∈ Z := {µ(φ) : µ ∈ P(T)} = [A, B] M(z) := {µ ∈ P(T) : µ(φ) = z}, a non-empty compact set for z ∈ Z

1

p′

T,φ : R → (A, B) is a real-analytic increasing diffeo such that p′ T,φ(β) = νβ(φ)

Let β : (A, B) → R be the real-analytic increasing diffeo defined as β = (p′

T,φ)−1

2

νβ(z) = unique maximizer for: entropy, linear pressure, nonlinear pressure in M(z)

3

h(z) := sup{h(µ) : µ ∈ M(z)} = hT(νβ(z)) is continuous and concave over [A, B]

4

h : (A, B) → R is real-analytic with h′(A+) = +∞ and h′(B−) = −∞

pT,φ(β(z)) = hT (νβ(z)) + νβ(z)(β(z)ψ) = h(z) + β(z)z: h(z) = pT,φ(β(z)) − β(z)z h′(z) = β′(z)p′

T,φ(β(z)) − β′(z)z − β(z) = β′(z)νβ(z)(φ) − β′(z)z − β(z) = −β(z)

5

Z∗ := {z ∈ Z : h + F is maximum at z} is a finite subset of (A, B)

6

{nonlinear equilibrium states} = {νβ(z) : z ∈ Z∗}

• J. Buzzi

A nonlinear thermodynamical formalism July 2020 10 / 11

Ingredients of proofs Reduction of the nonlinear to the linear equilibrium states

Characterization of the equilibrium states

(X, T, φ, F) strongly regular: (a) hT : P(X) → R ∪ {−∞} is upper semicontinuous (b) pT,φ : β → Ptop

T (βφ) is real-analytic and strictly convex

(c) for each β ∈ R, there is a unique linear equilibrium state νβ for βφ Idea: condition on z = µ(φ) ∈ Z := {µ(φ) : µ ∈ P(T)} = [A, B] M(z) := {µ ∈ P(T) : µ(φ) = z}, a non-empty compact set for z ∈ Z

1

p′

T,φ : R → (A, B) is a real-analytic increasing diffeo such that p′ T,φ(β) = νβ(φ)

Let β : (A, B) → R be the real-analytic increasing diffeo defined as β = (p′

T,φ)−1

2

νβ(z) = unique maximizer for: entropy, linear pressure, nonlinear pressure in M(z)

3

h(z) := sup{h(µ) : µ ∈ M(z)} = hT(νβ(z)) is continuous and concave over [A, B]

4

h : (A, B) → R is real-analytic with h′(A+) = +∞ and h′(B−) = −∞

5

Z∗ := {z ∈ Z : h + F is maximum at z} is a finite subset of (A, B)

Z∗ ⊂ {z ∈ (A, B) : h′(z) + F ′(z) = 0} ∪ {A, B} can only accumulate on A or B h + F is increasing near A and decreasing near B

6

{nonlinear equilibrium states} = {νβ(z) : z ∈ Z∗}

• J. Buzzi

A nonlinear thermodynamical formalism July 2020 10 / 11

Ingredients of proofs Reduction of the nonlinear to the linear equilibrium states

Characterization of the equilibrium states

(X, T, φ, F) strongly regular: (a) hT : P(X) → R ∪ {−∞} is upper semicontinuous (b) pT,φ : β → Ptop

T (βφ) is real-analytic and strictly convex

(c) for each β ∈ R, there is a unique linear equilibrium state νβ for βφ Idea: condition on z = µ(φ) ∈ Z := {µ(φ) : µ ∈ P(T)} = [A, B] M(z) := {µ ∈ P(T) : µ(φ) = z}, a non-empty compact set for z ∈ Z

1

p′

T,φ : R → (A, B) is a real-analytic increasing diffeo such that p′ T,φ(β) = νβ(φ)

Let β : (A, B) → R be the real-analytic increasing diffeo defined as β = (p′

T,φ)−1

2

νβ(z) = unique maximizer for: entropy, linear pressure, nonlinear pressure in M(z)

3

h(z) := sup{h(µ) : µ ∈ M(z)} = hT(νβ(z)) is continuous and concave over [A, B]

4

h : (A, B) → R is real-analytic with h′(A+) = +∞ and h′(B−) = −∞

5

Z∗ := {z ∈ Z : h + F is maximum at z} is a finite subset of (A, B)

6

{nonlinear equilibrium states} = {νβ(z) : z ∈ Z∗}

• J. Buzzi

A nonlinear thermodynamical formalism July 2020 10 / 11

Conclusion

Remarks and Questions

generalizes the quadratic formalism of Leplaideur and Watbled to strongly regular systems (say transitive uniformly hyperbolic dynamics with C α potentials)

• ften ”everything” can be computed: the pressure function pT,φ(β) := Ptop(T, βφ)

can (say for SFTs with potential depending only on finitely many coordinates) Benoˆ ıt Kloeckner: for the variational principle and the convergence of Gibbs ensembles, F(µ(φ)) any continuous function P(X) → R with the same proofs pressure depending on several potentials F(µ(φ1), . . . , µ(φd)), e.g., Potts mean-field model (treated by Leplaideur-Watbled)

relation with ”dynamically mean field models” (G. Keller, F. Selley,...)?

dynamical applications??????

Thank you! .

• J. Buzzi

A nonlinear thermodynamical formalism July 2020 11 / 11

Conclusion

Remarks and Questions

generalizes the quadratic formalism of Leplaideur and Watbled to strongly regular systems (say transitive uniformly hyperbolic dynamics with C α potentials)

• ften ”everything” can be computed: the pressure function pT,φ(β) := Ptop(T, βφ)

can (say for SFTs with potential depending only on finitely many coordinates) Benoˆ ıt Kloeckner: for the variational principle and the convergence of Gibbs ensembles, F(µ(φ)) any continuous function P(X) → R with the same proofs pressure depending on several potentials F(µ(φ1), . . . , µ(φd)), e.g., Potts mean-field model (treated by Leplaideur-Watbled)

relation with ”dynamically mean field models” (G. Keller, F. Selley,...)?

dynamical applications??????

Thank you! .

• J. Buzzi

A nonlinear thermodynamical formalism July 2020 11 / 11

Conclusion

Remarks and Questions

generalizes the quadratic formalism of Leplaideur and Watbled to strongly regular systems (say transitive uniformly hyperbolic dynamics with C α potentials)

• ften ”everything” can be computed: the pressure function pT,φ(β) := Ptop(T, βφ)

can (say for SFTs with potential depending only on finitely many coordinates) Benoˆ ıt Kloeckner: for the variational principle and the convergence of Gibbs ensembles, F(µ(φ)) any continuous function P(X) → R with the same proofs pressure depending on several potentials F(µ(φ1), . . . , µ(φd)), e.g., Potts mean-field model (treated by Leplaideur-Watbled)

relation with ”dynamically mean field models” (G. Keller, F. Selley,...)?

dynamical applications??????

Thank you! .

• J. Buzzi

A nonlinear thermodynamical formalism July 2020 11 / 11

Conclusion

Remarks and Questions

generalizes the quadratic formalism of Leplaideur and Watbled to strongly regular systems (say transitive uniformly hyperbolic dynamics with C α potentials)

• ften ”everything” can be computed: the pressure function pT,φ(β) := Ptop(T, βφ)

can (say for SFTs with potential depending only on finitely many coordinates) Benoˆ ıt Kloeckner: for the variational principle and the convergence of Gibbs ensembles, F(µ(φ)) any continuous function P(X) → R with the same proofs pressure depending on several potentials F(µ(φ1), . . . , µ(φd)), e.g., Potts mean-field model (treated by Leplaideur-Watbled)

relation with ”dynamically mean field models” (G. Keller, F. Selley,...)?

dynamical applications??????

Thank you! .

• J. Buzzi

A nonlinear thermodynamical formalism July 2020 11 / 11

Conclusion

Remarks and Questions

generalizes the quadratic formalism of Leplaideur and Watbled to strongly regular systems (say transitive uniformly hyperbolic dynamics with C α potentials)

• ften ”everything” can be computed: the pressure function pT,φ(β) := Ptop(T, βφ)

can (say for SFTs with potential depending only on finitely many coordinates) Benoˆ ıt Kloeckner: for the variational principle and the convergence of Gibbs ensembles, F(µ(φ)) any continuous function P(X) → R with the same proofs pressure depending on several potentials F(µ(φ1), . . . , µ(φd)), e.g., Potts mean-field model (treated by Leplaideur-Watbled)

relation with ”dynamically mean field models” (G. Keller, F. Selley,...)?

dynamical applications??????

Thank you! .

• J. Buzzi

A nonlinear thermodynamical formalism July 2020 11 / 11

Conclusion

Remarks and Questions

generalizes the quadratic formalism of Leplaideur and Watbled to strongly regular systems (say transitive uniformly hyperbolic dynamics with C α potentials)

• ften ”everything” can be computed: the pressure function pT,φ(β) := Ptop(T, βφ)

can (say for SFTs with potential depending only on finitely many coordinates) Benoˆ ıt Kloeckner: for the variational principle and the convergence of Gibbs ensembles, F(µ(φ)) any continuous function P(X) → R with the same proofs pressure depending on several potentials F(µ(φ1), . . . , µ(φd)), e.g., Potts mean-field model (treated by Leplaideur-Watbled)

relation with ”dynamically mean field models” (G. Keller, F. Selley,...)?

dynamical applications??????

Thank you! .

• J. Buzzi

A nonlinear thermodynamical formalism July 2020 11 / 11

Conclusion

Remarks and Questions

generalizes the quadratic formalism of Leplaideur and Watbled to strongly regular systems (say transitive uniformly hyperbolic dynamics with C α potentials)

• ften ”everything” can be computed: the pressure function pT,φ(β) := Ptop(T, βφ)

can (say for SFTs with potential depending only on finitely many coordinates) Benoˆ ıt Kloeckner: for the variational principle and the convergence of Gibbs ensembles, F(µ(φ)) any continuous function P(X) → R with the same proofs pressure depending on several potentials F(µ(φ1), . . . , µ(φd)), e.g., Potts mean-field model (treated by Leplaideur-Watbled)

relation with ”dynamically mean field models” (G. Keller, F. Selley,...)?

dynamical applications??????

Thank you! .

• J. Buzzi

A nonlinear thermodynamical formalism July 2020 11 / 11