Nonlinear thermodynamical formalism

We define a nonlinear thermodynamical formalism which translates into dynamical system theory the statistical mechanics of generalized mean-field models, extending investigation of the quadratic case by Leplaideur and Watbled. Under suitable conditions, we prove a variational principle for the nonlinear pressure and we characterize the nonlinear equilibrium measures and relate them to specific classical equilibrium measures. In this non-linear thermodynamical formalism, which can, e.g., model mean-field approximation of large systems, several kind of phase transitions appear, some of which cannot happen in the linear case. We use our correspondence between non-linear and linear equilibrium measures to further the understanding of phase transitions, { both in previously known cases (Curie-Weiss and Potts models) and in} new examples (metastable phase transition). Finally, we apply some of the ideas introduced to the classical thermodynamical formalism, proving that freezing phase transitions can occur over \emph{any} zero-entropy invariant compact subset of the phase space.


Introduction
In the 1970s, Sinai, Ruelle, Bowen, and others (see, e.g., [28,26,4]) developed a thermodynamical approach to dynamical systems inspired by the statistical mechanics of lattice systems.In a recent work [16], the third named author and Watbled applied this program to the Curie-Weiss mean-field theory: they introduced a new thermodynamical formalism over the full shift where the energy functional is quadratic.They obtained precise results using the specific structure of this setting.
Our goal in this paper is to understand the generality of their results.It turns out that we can define the nonlinear pressure of a measure as the sum of its entropy and its "energy", defined as any weak-star continuous function of the measure.We are in particular interested in the case when the energy is a smooth function of the integrals of one or several potentials, in which case we call it an energy with potential(s).Assuming only that the classical thermodynamical formalism is well-behaved, we can analyze this nonlinear thermodynamics using suitable convex analysis.
We first prove a variational principle: the supremum of the nonlinear pressure of the measures is given by a combinatorial formula involving the classical separated sets for the Bowen-Dinaburg dynamical metric (Theorem A), then defining equilibrium measure as those measures achieving the previous supremum.It is easy to show that equilibrium measures exist and, in the expansive case, we relate them to Gibbs ensembles (Theorem B).In the case of energies with potentials we show that equilibrium measures are classical equilibrium measures for some specific linear combination of these potentials (Theorem C).When the nonlinearity is a real-anaytic function of the integral of a single potential, we obtain finiteness of the set of equilibrium measures (Theorem D).As is well-known from physics and examples including the Curie-Weiss theory, phase transitions can occur in this setting, e.g., there may be several equilibrium measures that may depend non-analytically on parameters giving rise to freezing (Theorem E and Section 5.4) or metastable phase transitions.(Theorem F in Section 5.2).
1.1.Classical thermodynamical formalism.We recall the classical definitions (see, e.g., [31]).We will sometimes call these notions linear to distinguish them from the ones we introduce in this paper.
Let (, ) be a continuous system, i.e., a continuous self-map  :  →  of a compact metric space together with a continuous function  ∈ (, R).The function  is called the potential.We denote by P the set of Borel probability measures on , endowed with the weak star topology, by P( ) the subset of invariant measures and by P erg ( ) the subset of ergodic and invariant measures.
An equilibrium measure for (, ) is then an invariant probability measure  such that  (, , ) =  top (, ), i.e., a measure that achieves the above supremum.
The (linear) pressure function is the function  ↦ →  top (, ) where  is a real parameter, called the inverse of temperature.
1.2.Nonlinear formalism.We propose the following generalization.It will prove convenient to write () for ∫︀  d.We consider again a continuous map  acting on a compact metric space .
An energy is a function ℰ : P → R which is continuous in the weak star topology; note that we will need the energy of non-invariant measures.We say that ℰ is an energy with potential  (a continuous function defined on ) if it can be written for some continuous function  :  → R defined on an interval containing all values taken by .More generally, an energy with potentials takes the form (1.4) ℰ() =  (︀ ( 1 ), . . ., (  ) )︀ where  1 , . . .,   are continuous functions defined on  and  :  → R is a continuous function on some set  ⊂ R  .For ℰ to be well-defined on the whole of P, the set  must contain the convex hull of the set of values taken by ⃗  = ( 1 , . . .,   ) :  → R  .We add the adjective "  " ( ≥ 1), "smooth" or "analytic" to ℰ whenever the domain  of  is open and  is   ( = ∞ meaning smooth,  =  meaning analytic) on  .An energy is said to be convex when for all Borel probability measure  on P (hence,  is a measure of measures): For example, if ℰ is an energy with potentials, it is convex whenever  is.
Not assuming potentials, we first need to replace Birkhoff sums.Given  ∈  and  ∈ N, we define an empirical measure Observe that for any potential , ∆   () = 1    () is the averaged Birkhoff sum.We thus define the nonlinear weight of order  of a finite set  and the nonlinear partition function as where the supremum is taken over all (, )-separated sets .
Again, an (, )-separated set  is said to be adapted if it realizes the maximum in   and we define an nonlinear (, )-Gibbs ensemble (note that the continuity of  ensures that the maximum in  is realized for all (, )).
The nonlinear topological pressure, to be thought of as an analogue of topological entropy weighted by energy, is In Theorem A we will show that under suitable hypotheses, replacing the supremum limit by an infimum limit: gives the same quantity as Π ℰ top ( ).Meanwhile the nonlinear pressure is defined for all invariant probability measures  by Π ℰ (, ) = ℎ(, ) + ℰ().
This condition is satisfied by uniformly hyperbolic diffeomorphisms that have a single basic set in their spectral decomposition as any invariant probability measure can be approximated by an ergodic one both in the weak star topology and in entropy.It is also satisfied for arbitrary continuous systems (, ℰ) with convex energy, since, in this case, for any  ∈ P( ), using the ergodic decomposition  = ∫︀    ().
Recall that, in the invertible case,  is said to be an expansive homeomorphism when there exist a number  0 > 0 (called an expansivity constant for  ) such that (see, e.g., [13] Definition 3.2.11;note that here we use a ≤ sign, making the expansivity constants possibly slightly smaller).This notion is generalized to nonnecessarily invertible maps under the name of positive expansivity by considering only the positive orbits: and the results we state below for expansive homeomorphisms could be extended to positively expansive map with the same proofs.
Our first result establishes a variational principle generalizing eq. ( 1.3) to all energies.
Theorem A (Variational principle).Let  :  →  be a continuous map of a compact space and let ℰ : P → R be an energy.Assume that (, ℰ) has an abundance of ergodic measures, Then the nonlinear topological pressure satisfies: If, additionally,  is an expansive homeomorphism with some constant  0 > 0, then When the conclusion sup P( ) Π ℰ (, •) = Π ℰ top ( ) of the above theorem holds, we define a nonlinear equilibrium measure as any measure  ∈ P( ) realizing this supremum: As in the classical setting, existence of an equilibrium measure is easily obtained when entropy is upper semicontinuous, and in the expansive case equilibrium measures prescribe the asymptotic behavior of Gibbs ensembles.
Theorem B. Let  :  →  be a continuous map of a compact space and let ℰ : P → R be an energy.Assume that (, ℰ) has an abundance of ergodic measures.
If  ↦ → ℎ(, ) is upper semicontinuous1 , then there exists at least one nonlinear equilibrium measure.
If additionally  is an expansive homeomorphism for some constant  0 > 0, then any accumulation point  of any sequence (  ) ∈N of nonlinear Gibbs ( 0 , )-ensembles belongs to the closure of the convex span of all nonlinear equilibrium measures.
The last statement means that there exists a probability measure  on P (a measure of measures), concentrated on the set EM of equilibrium measures, such that (see, e.g., [24], Proposition 1.2.)The accumulation points can indeed fail to be equilibrium measures, e.g., in the Curie-Weiss model when there are two asymmetric equilibrium measures and one chooses symmetric Gibbs ensembles, see [16].
Next we study the uniqueness and nature of the nonlinear equilibrium measures in the case of an energy with potentials as in eq.(1.4).Our main point here is that we can use classical convex analysis to reduce the nonlinear thermodynamical formalism to the linear one.
More precisely, we will use the classical Legendre duality between entropy and pressure; using the vector of integral of potentials (( 1 ), . . ., (  )) as intermediate coordinates, this will reduce to finite-dimensional Legendre duality.This duality holds for the class of   Legendre systems (, ⃗ ) (where  ∈ N * ∪ {∞, } and   means analytic), see Definitions 4.7, 4.9.When additionally each linear combination of the (  ) admits a unique linear equilibrium measure, one says that (, ⃗ ) is   Legendre with unique linear equilibrium measures.Let us note that classical examples fulfill these requirements: if  is a topologically transitive Anosov diffeomorphism or expanding map, and ( 1 , . . .,   ) is a family of Hölder-continuous potentials whose linear combinations are not cohomologuous to a constant, i.e., for all  1 , . . .,   ∈ R  : Observe that as a consequence, even though nonlinear equilibrium measures may fail to be unique, under the hypotheses of Theorem C they are ergodic as soon as linear equilibrium measures are (and more, see Corollary 1.11).Addendum 1.9.In the above setting, the set Y can be computed from the linear pressure function defined over R  by P( 1 , . . .,   ) =  top (, ∑︀      ).More precisely Y = (∇P) −1 (V ) where ∇P is the gradient of P and The function h can also be computed from P, as −h is the Legendre dual of  .
Remarks 1.10.Given (, ⃗ ) a smooth Legendre system, any compact subset of R  can be realized as the set Y above by choosing a suitable  ∞ smooth nonlinearity  (Corollary 4.22).
Our proof will apply to a more general notion of equilibrium measures, see eq. (4.3).
Theorem D. If (, ) is a   Legendre system with unique linear equilibrium measures and  is   with a single potential ( = 1), then there are only finitely many nonlinear equilibrium measures.
Note that we do not simply claim that EM is finite-dimensional, but that it is finite, even though it can contain several equilibrium measures.In fact, this failure of uniqueness can occur even for a topologically transitive subshift of finite type with a Hölder-continuous potential (see e.g.[16] and Section 5 below).However uniqueness holds for generic non-linearities for any  ≥ 1 (Proposition 4.20).
The above characterization shows that for many systems with expanding or hyperbolic properties, such as mixing subshifts of finite type, the nonlinear equilibrium measures share the good ergodic properties of the classical equilibrium measures.Let us recall some of them.
where the two last properties are understood to hold with respect to Hölder-continuous observables.
These results are folklore in the sense that some of them are immediate consequences of the founding results of Sinai, Ruelle, and Bowen, while others were first considered in more general settings.The following are convenient references: ergodicity, mixing, and exponential decay of correlation follow from Ruelle's Perron-Frobenius theorem (see, e.g., [1, chapter 1]), the almost sure invariance principle, which implies many limit theorems was proved in [19] in much greater generality.

1.4.
Examples.We will give a few examples to which the above theorems apply, mostly inspired by physics.These examples involves an additional real parameter, the inverse temperature  > 0: the energy function is then ℰ() = ℰ 1 () where ℰ 1 is a reference energy and  tunes the balance between entropy and that energy, in agreement with thermodynamics. 2This leads to the natural question of how the existence, the number, or the equilibrium measures themselves depend on this parameter , leading to the physical notion of phase transitions.
1.4.5.Wassertein distance to the maximal entropy measure.We can go beyond the case with potentials: let us give a simple but intriguing example.Consider the map  :  ↦ → 2 mod 1 on the circle, with reference energy ℰ 1 () = W  (, ) where  denotes the Lebesgue measure,  ∈ [1, +∞) and W  is the Wasserstein distance of exponent .
Theorems A and B ensure that the nonlinear topological pressure is achieved by at least one invariant measure.The main question, which we leave open, is then to describe the non-empty set of equilibrium measures for ℰ 1 , in particular determine uniqueness.
1.5.More Phase Transitions.A phase transition can be defined from any of a number of different phenomena that often occur simultaneously: loss of the analyticity of the pressure with respect to physical parameters, multiple equilibrium measures, or failure of the central limit theorem for example.
Sarig [27] has studied such equivalences in the setting of Markov shifts.In contrast, we see here (Section 5.1) that non-analyticity of pressure and multiplicity of equilibrium measures can occur though the central limit theorem continues to hold (Corollary 1.11).Such distinctions have been observed before in [15] and [29].The key point of view in the definition of Legendre regular systems and the proof of Theorems C and D is to consider a certain convex set, the entropypotential diagram (defined in Section 4, see figures 1, 2), which describes the pairs (ℎ(, ); (⃗ )) that can be achieved when  runs over P( ).Phase transitions then occur when the nonlinearity "becomes more convex" than the diagram.
In Section 5.4, we shall illustrate more broadly the benefits of this diagram by considering freezing phase transitions, by which we mean that for all  >  0 for some  0 > 0, the set of equilibrium measures is non-empty and independent of ; its elements are called "ground states" as they must maximize the energy.
Theorem E. Let  :  →  be a continuous dynamical system of finite, positive topological entropy, and assume that  ↦ → ℎ(, ) is upper semi-continuous.
The first item is not directly related to the non-linear thermodynamical formalism, but its analysis is a simple application of the tools developed here (more precisely, we rely on the entropy-potential diagram introduced in Section 4 which is central to our non-linear study).1.6.Questions.We close this introduction with a few more open questions.
(See Remark 2.3.)• Can one find a subshift of finite type, Hölder-continuous potentials and a real-analytic nonlinearity3 such that there exist infinitely many nonlinear equilibrium measures?What if we additionally impose the quadratic nonlinearity, i.e.,  () = 1 2 ‖‖ 2 ?• Can one find a "natural" energy (necessarily not an energy with potentials) on some subshift of finite type such that the non-linear equilibrium measure is unique but not ergodic?• For the doubling map and the Wasserstein energy   (•, ) from Section 1.4.5, is there a finite  > 0 at which  0 is an equilibrium measure?Is  still an equilibrium measure for some  > 0? What happens just after  ceases to be an equilibrium?

Variational principle
In this section we prove Theorem A. We first introduce some convenient notations.We fix a compact metric space , a map  :  →  and an energy ℰ.In order to be as general as possible, we do not assume  to be continuous for now, but only Borel-measurable.Note that   being compact, every subset is totally bounded; this ensures the finiteness of (, )-separated sets even when  is not assumed to be continuous.We often omit , ℰ from the notation, i.e., Π top = Π ℰ top ( ), Π() = Π ℰ (, ) etc. Recall the definitions of the empirical measures of a point  ∈ , of the nonlinear weight of a subset  ⊂ , and of the partition function: We define for use in this section the following notation: 2.1.Preliminaries.We will use the Wasserstein distance on the set P of probability measures on .Proofs of the statements we need can be found in many places, e.g., [30].
The distance between  1 ,  2 ∈ P can be defined as The "Kantorovich duality" states that this definition is equivalent to where  is the distance on  and Γ( 1 ,  2 ) is the set of 'transport plans", i.e., Borel probability measures on  ×  with marginals  1 and  2 .Moreover in these definitions both the supremum and the infimum are reached; a transport plan realizing the Wasserstein distance is said to be optimal.The compactness of  implies that the Wassertein distance induces the weak-star topology on P, and that Wasserstein distance can be bounded above by total variation distance: We will also use the following reformulation of Birkhoff's ergodic theorem.
We will first prove Inequality 1 is proved in Proposition 2.4.Inequality 3 is proved in Proposition 2.9.Inequality 2 immediatlely follows from the definitions of Π top and Π top .
The missing inequality sup is proved in Proposition 2.5 assuming an abundance of ergodic measures.
Remark 2.3.If (, ℰ) is continuous but without abundance of ergodicity, the following example shows that inequality

2.2.1.
Bounding below the nonlinear topological pressure.We prove Inequality 1 , then Inequality 4 assuming an abundance of ergodic measures.Note that continuity of  is not needed at that stage.
By Lemma 2.1, there is a set  ⊂  with () ≥ 3 4 and   ∈ N such that for all  ∈  and all  ≥   we have W(∆   , ) ≤ .By the Brin-Katok entropy formula [6], there exist  ⊂  with () ≥ 3  4 and   ∈ N such that for all  ∈  and all  ≥   we have Consider any  ≥ max(  ,   ) and any 0 <  ≤ .Let  be any (, )separated set of  that is maximal with respect to inclusion; in particular,  is an (, )-cover, hence a (, )-cover.Let  ′ be a minimal subset of  that is an (, )-cover of  ∩ .
Observe that we only used lower-semicontinuity for ℰ here; but its uppersemicontinuity ensures it reaches its supremum, a desirable feature.This motivates the continuity requirement in the definition of an energy.Proposition 2.5 (Inequality 4 ).If  is Borel-measurable and (, ℰ) has an abundance of ergodic measures, then sup Proof.Let  be any invariant probability measure.Since (, ℰ) has an abundance of ergodic measures, there is a sequence of measures holds for every  in P( ).

2.2.2.
Bounding from above the nonlinear topological pressure: Inequality 3 .To end the proof of equality (1.8), it remains to construct measures almost realizing the nonlinear topological pressure.We divide the proof into several lemmas that we shall reuse in Section 3. We follow the strategy of Misiurewicz' proof of the linear variational principle, from which we extract the following result.We recall that   () stands for the entropy for the measure  ∈ P( ) of the partition .
Lemma 2.6 (Misiurewicz [20]).Fix  > 0 and let (  ) ∈N be a sequence of (,   )separated sets where   → ∞.Assume that for each ,   is a probability measure concentrated on   and that converges in the weak star topology to some measure  ∞ .Fix any finite partition  of  into subsets of diameter less than  and with negligible boundaries with respect to  ∞ (such an  always exists).Then for all  ∈ N, The proof is not reproduced here, let us simply mention that it consists in partitioning in  different ways the integer interval 0,   − 1 into subintervals of length  plus a small remainder at the start and end.Note that the hypothesis that   is (,   )-separated is intended to make the computation of    (   ) a formality: each element of    contains at most one element of   .
To address the nonlinearity, we now divide the space of measures into parts where the energy is almost constant, and then split (, )-separated sets according to this partition.Lemma 2.7.Let  > 0,  ∈ (0, 1) and (  ) ∈N be a sequence of (,   )-separated subsets of  where   → ∞.There exist  =  () ∈ N, real numbers (  ) 1≤≤ , a sequence of partitions D  = ( , ) 1≤≤ of   and  ⊂ 1,  such that, up to extracting a subsequence (still denoted by (  )  ), for all : Let  = { 1 , . . .,   } be a -covering of (P, W) and set   := ℰ(  ).For each  ∈ P we can define () = min{ | W(,   ) ≤ }.We then set   = { ∈ P | () = }; the (  ) form a partition of P, and for all  ∈   we have The sequence given by could be preferred to (μ  )  for the proof of 3 , and can be treated in pretty much the same way.However, we will need (μ  )  in Section 3 to describe the accumulation points of Gibbs ensembles.
Proof.Let first μ∞ be an accumulation point of (μ  ); up to extracting a further subsequence, we assume μ∞ = lim  μ .
To check that μ∞ ∈ P( ), first observe that W(∆    ,  * ∆    ) ≤ diam    by the total variation bound (i.e., using a coupling that leaves the common part ∑︀ 1≤<      in place and moves the remaining mass 1   from  to    ) and conclude using an averaged coupling as in the proof of Lemma 2.7 above that W(μ  ,  * μ ) → 0. Up to this point, no use was made of the continuity assumption on  .But we want to pass to the limit in the arguments of W, and the continuity ensures that  * μ →  * μ∞ .Then we get W(μ ∞ ,  * μ∞ ) = 0, and thus μ∞ ∈ P( ).Note also that ℰ(μ  ) ≤   +  for all , so that the same holds for μ∞ .
Consider a partition  of  whose element have diameter at most  and whose boundaries have zero measure with respect to μ∞ .Setting log|| for all  ∈ N and all  such that   ≥ 2.It follows that for all  and all  large enough (then taking successive limits as  → ∞ and  → ∞): Proposition 2.9.If  is continuous, then we have sup Proof.Let  > 0, and choose  > 0 small enough to ensure For each  ∈ N, let   be an (, )-separated subset of  realizing (, ).Let (  )  be a sequence of integers such that   → ∞ and 1   log (,   ) → Π top ().We apply Lemma 2.7, fix any  ∈ , define μ as in Lemma 2.8 and let μ∞ be any of its accumulation points.We then have Letting  go to zero ends the proof.
Assuming  is continuous and abundance of ergodic measures, we have shown that: Since, obviously, Π top ≤ Π top , the above inequalities must be equalities.This proves eq.(2.2) under the assumptions of Theorem A.

2.3.
Proof of Theorem A: the expansive case.We assume that  is a homeomorphism admitting the expansivity constant  0 > 0. To begin with, we let 0 <  ≤  0 and show that (2.10) Let us prove that Π top () ≤ Π top ( 0 ) by extracting an ( 0 , )-separated set from an (, )-separated one and comparing their weights.
We first fix  > 0 arbitrarily small.By the uniform continuity of ℰ on P, there is 0 We need the following version of the Theorem of uniform expansivity.
Claim 2.12.There exists  ≥ 1 such that for all  ≥ 2 , for any  ∈ , Proof of the Claim.If this does not hold, pick for every  : Pick  0 and  ≥  0 .Note the following inclusions: Then, consider any accumulation point  for   :=    (  ).This yields This is in contraction with the fact that  0 is an expansivity constant.
We now fix some finite (/2,  )-cover   of  and some large enough integer  ≥ 1 (exactly how large will be specified later on; in particular we assume equation (2.13) holds).

Existence of an equilibrium measure and convergence of the Gibbs ensembles
In this section we prove Theorem B. Its existence claim is a simple consequence of the variational principle we just established as Theorem A.
Lemma 3.1.Assume that  is continuous with  ↦ → ℎ(, ) upper semicontinuous, and that (, ℰ) has an abundance of ergodic measures.Then the set EM of nonlinear equilibrium measures is non-empty.
The second part of Theorem B is proven along the same lines than Proposition 2.9.Proposition 3.2.Assume that  is an expansive homeomorphism with  0 > 0 an expansivity constant, and that (, ℰ) has an abundance of ergodic measures.Let  be an accumulation point of ( 0 , )-Gibbs ensembles as  → ∞.Then  can be approximated in the weak-star topology by linear combinations of nonlinear equilibrium measures.
Proof.Note that  being an expansive homeomorphism, entropy is upper semicontinuous.By the second half of Lemma 3.1, we are reduced to approximate  by convex combination of measures that almost achieve the nonlinear topological pressure.
By definition  is the limit of measures of the form Theorem B is established.

Convexity and nonlinear equilibrium measures
In this section, independently of Sections 2.2 and 3, we prove an extended version of Theorem C, i.e., we study the nonlinear formalism for an energy with potentials.Specifically, we consider a continuous map  :  →  with finite entropy ℎ top ( ) < ∞ together with an energy defined as for all  ∈ P( ) where, for some positive integer , •  1 , . . .,   :  → R are continuous functions called the potentials; The rest of this section is divided as follows.First, we introduce a "fully nonlinear formalism" which is the natural setting of our technique and describe the entropy-potential diagram which is a useful visualization.Second we recall the relevant background concerning Legendre duality and we set up appropriate definitions to use this duality and we provide examples of dynamical system satisfying them.Thirdly we weave all this together and apply Legendre duality in the dynamical context to reach the main goal of this section, Theorem 4.15 (which contains Theorem C).Finally we deduce some uniqueness results (Corollary 4.19, Propositions 4.20 and 4.21).
4.1.Fully nonlinear pressure.Our approach applies to the following more general setting: Definition 4.2.Given a continuous system  with potentials ⃗ , a fully nonlinear pressure is a function with   (⃗ ) = .The first point is immediate given the assumption that  0  > 0. The second and third point will follow from some convex analysis; the second point more precisely follows from the assumption that the gradient of entropy diverges at the boundary in the definition of   Legendre systems (Definitions 4.9 and 4.7) , see the proof of Theorem 4.15.Remark 4.5.To find the largest value of h +  is to find the largest  such that there exists  ∈ (⃗ ) at which h() = − () + , i.e., to find the highest vertical translate of the graph of − that touches the entropy-potential diagram.This makes apparent that the nonlinear equilibrium measures will correspond to linear equilibrium measures associated to one or several linear combinations of potentials, whose coefficients are given by the equations of the tangent hyperplanes at the touching points, see e.g., figure 3.

4.3.
Legendre duality.To apply the well-rounded theory of Legendre duality, let us introduce its classical assumptions, following [25].
Recall that the Legendre transform  * of a convex function  : R  → R∪{−∞} is the convex function: If  is a concave function, we set i.e.,  # := (−) * , which is convex. 7e will use two classical duality results from [25].They ensure that the Legendre transform is an involution on suitable classes of semicontinuous or smooth convex functions.
Semicontinuous functions.A function is proper if it is finite at least at one point.
Theorem 4.6.The Legendre transform maps bijectively the class of upper semicontinuous, 8 proper concave functions to the class of lower semicontinuous proper convex functions.Moreover, this restriction of the Legendre transform is an involution up to sign: for all such  ,  = −( # ) * .
The above theorem implies that the Legendre transform is an involution over the class of lower semicontinuous proper convex functions  : ( * ) * = .Smooth functions.We consider the smoothness classes   for 1 ≤  ≤ , i.e., for any positive integer  as well as  = ∞ (infinitely differentiable) and  =  (real-analytic).The following abuses of notation will be convenient: for  = ∞ or ,  −1 just means   ; for  = 0, a   diffeomorphism is a homeomorphism.Definition 4.7.Let  : R  → R ∪ {−∞} be a function.Its (effective) domain is the set of points dom( ) in R  where it takes a finite value: dom( ) =  −1 (R).For 1 ≤  ≤ , the function  is said to be concave of   Legendre type when the following conditions are satisfied: (i) the function  is upper semicontinuous and concave; (ii) the interior int dom( ) is not empty and, on this set,  is strictly concave and   smooth; when  ≥ 2, we additionally ask that the Hessian of  is everywhere negative definite; (iii) for all sequences (  ) ∈N with   ∈ int(dom( )) which converge to a boundary point of dom( ), We say that a function  : Note that functions are convex of  1 Legendre type exactly when they are convex of Legendre type in the sense of Rockafellar [25,Chap. 26].Let us now extract the following result from the classical theory of Legendre duality.Theorem 4.8.For each 1 ≤  ≤ , the Legendre transform of any concave or convex function  of   Legendre type is a convex function  # or  * of   Legendre type.Moreover, the following holds for  concave: Proof.This statement follows from the results in [25,Chap. 26], except for the formula for  # () in (ii ).When  = 1, this is exactly Theorem 26.5 there applied to the convex function  = − .Indeed,  # =  * and ∇  =  ∘ ∇  with () = −.In particular, ∇  * = (∇ ) −1 , i.e., ∇  # = ( ∘ ∇  ) −1 = (∇  ) −1 ∘ , proving the first formula in claim (ii ).Now, ∇  is a  −1 map.From the same theorem, ∇  : dom( ) → dom( # ) is a homeomorphism.It is a  −1 -diffeomorphism, using, if  ≥ 2, that the Hessian of  is definite.The formula for ∇  # ensures that this gradient is also  −1 , thus  # is   .
To conclude, let  ∈ int(dom( # )).Note that  := (∇  ) −1 (−) ∈ int(dom( )) satisfies ∇  ( •  +  ()) = 0. Since  is strictly concave on int(dom( )) and concave everywhere,  must be the unique maximizer on dom( ), proving the second half of (ii ).4.4.Application to dynamical systems.Before exploiting Legendre duality further, let us discuss how the dynamical systems on which the linear Thermodynamical formalism is well-understood fit into our framework.We start with a convenient definition.Definition 4.9.For 1 ≤  ≤ , a continuous dynamical system with potentials (, ⃗ ) is   Legendre when: (i) the rotation set (⃗ ) has non-empty interior in R  , (ii) the topological entropy is finite: If moreover, for every  ∈ R  , there is exactly one linear equilibrium measure   for  and the potential  • ⃗ , then we say that (, ⃗ ) is   Legendre with unique linear equilibrium measures (  ) ∈R  .
The above classical theory of Legendre duality applied to such systems leads to the (finite-dimensional linear) pressure function: It is the Legendre transform of the concave finite-dimensional entropy function h: In particular, if (, ⃗ ) is   Legendre, then by applying Theorem 4.8 we obtain that the pressure is a   function.
In Definition 4.9, we took entropy as primary object, and then defined pressure by Legendre duality.However, it has been customary to discuss primarily the regularity of pressure -using Legendre duality, both points of view can be unified as follows.
Proposition 4.10.If (, ⃗ ) is a continuous system with potentials satisfying, for some 1 ≤  ≤ , • the rotation set (⃗ ) has nonempty interior in R  ; • the entropy function ℎ(, •) is upper semicontinuous and bounded over P( ); • the finite-dimensional pressure function P is finite over R  ,   smooth, strictly convex and, when  ≥ 2, with everywhere positive definite Hessian, then (, ⃗ ) is a   Legendre system.
It is now easy to check that many classical systems satisfy the thermodynamical formalism with   regularity.In many cases, the one point that needs checking is that the rotation set has non-empty interior (see Section 5.3 for an example where it does not).
Recall that a function  is cohomologous to a constant  if there is a continous function  such that  =  +  −  ∘  .Corollary 4.11.Let  be a mixing subshift of finite type or an Anosov diffeomorphism.Let ⃗  be a finite family of Hölder-continuous potentials  1 , . . .,   :  → R. Assume the following independence condition: for all  1 , . . .,   not all zero, ∑︀  =1     is not cohomologous to a constant.Then (, ⃗ ) is a   Legendre system with unique linear equilibrium measures.
Remark 4.12.Livsič theorem applies to such systems: a function is cohomologous to a constant if and only if on each periodic orbit, the average of the function is equal to that constant.The independence condition above is therefore equivalent to the existence of  + 1 periodic orbits with corresponding atomic measures  0 , . . .,   ∈ P( ) such that  0 (⃗ ), . . .,   (⃗ ) ∈ R  are affinely independent.
Proof of the corollary.Both subshifts of finite type and Anosov diffeomorphisms are Smale systems satisfying the regularity condition (SS3) in [26] in the sense of [26, 7.1, 7.11] and this will be enough for our purposes.
Since  has finite topological entropy and is expansive, the Kolmogorov-Sinai entropy function is upper semicontinuous and bounded over P( ).
If the rotation set, a convex set, had empty interior, it would be contained in some affine hyperplane, hence, there would be numbers  0 , . . .,   , not all zero, such that By Livsič theorem, this implies that ∑︀  =1     is cohomologuous to the constant  0 , contradicting the independence assumption.
Since  is a topologically mixing Smale system, its pressure function is realanalytic [26, 7.10].It has a semidefinite positive Hessian with kernel generated by the potentials cohomologous to constants.Hence the finite-dimensional pressure function P has definite positive Hessian in all of R  under the independence assumption above.In particular, P is strictly convex.
Thus, the assumptions of Proposition 4.10 are satisfied so that (, ⃗ ) is a   Legendre system.
Finally, for each  ∈ R  ,  • ⃗  is Hölder-continuous, hence there exists a unique linear equilibrium measure   .
The next statement follows immediately from [10, Corollary B, Theorems F & G], providing another family (intersecting the previous one) of dynamical systems to apply our framework to.We shall say that a Banach space X of functions  → R is a good Banach algebra of functions when: • X is stable by product and ‖ ‖ ≤ ‖ ‖‖‖ for all ,  ∈ X , • for every positive, bounded away from 0 function  ∈ X , log  is in X , • the norm of X dominates the uniform norm (in particular the elements of X are bounded), • the composition operator  ↦ →  ∘  is a continuous operator on X , • for every equilibrium measure  of a potential in X and every nonnegative  ∈ X , if ∫︀  d = 0 then  = 0, • every continuous function can be uniformly approximated by elements of X .(These assumptions are numerous, but many Banach spaces satisfy them, such as Hölder spaces or BV space on the interval, see [10] for some discussions of these hypotheses.)We refer to [10] for the notions of -to-1 map, simple dominant eigenvalue, and spectral gap appearing in the following statement.Theorem 4.13.Assume that  is -to-1 and  1 , . . .,   belong to some good Banach algebra of functions X and that for all  1 , . . .  not all zero, ∑︀  =1     is not cohomologous to a constant.If for all  ∈ R  the transfer operator defined by ℒ () = ∑︀  ′ ∈ −1 ()  •⃗ ( ′ )  ( ′ ) acts with a simple dominant eigenvalue and a spectral gap on X , then (, ⃗ ) is   Legendre with unique linear equilibrium measures.
4.5.Consequences of Legendre duality.Now that we have seen that Theorem 4.8 applies to plenty of dynamical systems, let us note some of the consequences.
Proposition 4.14.If (, ⃗ ) is a   Legendre system, then: (i) the finite-dimensional function h is continuous on the rotation set (⃗ ), (ii) ∇ h realizes a  −1 diffeomorphism from the interior of (⃗ ) onto R  with inverse  ↦ → ∇ P(−), (iii) the linear pressure function P has domain R  and is   , (iv) for all  ∈ R  , ∇ P() =  opt where  opt is the unique maximizer of h() + ⟨; ⟩ over int (⃗ ).If, additionally, (, ⃗ ) has unique equilibrium measures is the unique measure of maximum entropy in ℳ().
Proof.The function h is upper-semicontinuous, and since it is concave and finite it must be continuous on its domain, which coincides with the rotation set.
By assumption, h is a concave   Legendre function.Hence Theorem 4.8 ensures that the pressure P = h # is   .Since h is upper bounded as a continuous function with a compact domain, the domain of P() = sup ∈(⃗ ) h() + ⟨; ⟩ is the whole of R  .The same theorem tells us that ∇ h realizes a   diffeomorphism from the interior of (⃗ ) to R  , the interior of the domain of P, and that, for all  ∈ dom(P), ∇ P() = (∇ h) −1 (−).We further note that P() = ⟨ ;  opt ⟩+h( opt ) with  opt := (∇ h) −1 (−) = ∇  ().
We now assume that (, ⃗ ) has unique equilibrium measures Observe that   must maximize the entropy in ℳ() where  =   (⃗ ), hence ℎ(,   ) = h().By definition the linear pressure is Therefore, in Proposition 4.14, one must have:

This proves items (v ) and (vii ).
Note that {  (⃗ ) :  ∈ R  } = ∇ P(R  ), which is int((⃗ )), proving (vi ).z h(z) (1;y) 4.6.Set of nonlinear equilibrium measures.We now identify the fully nonlinear equilibrium measures, that is, the elements of EM (, , ⃗ ) (or just EM ) from Definition 4.2.We define the set of (, ⃗ )-equilibrium values to be For  ∈ (⃗ ), recall the notations ℳ() and h() from Definitions 4.4 and 4.9.We start with Theorem C, in a version generalized to fully nonlinear pressures (see Definition 4.2).We recall that  is defined on some open set  ⊂ R × R  and in the following    stands for /  ,  = 0, 1, . . ., .Theorem 4.15.Let (, ) be a   Legendre system for some 1 ≤  ≤  and let Π  be a fully nonlinear pressure defined by an admissible   function .
Then the set EM of (, ⃗ )-equilibrium measures is a nonempty and compact set of linear equilibrium measures.More precisely, (i) V = { ∈ int((⃗ )) : (h(); ) maximal } is a nonempty compact set on which Proof.We prove assertions (i ) and (ii ), the rest being immediate consequences.
Let us note that a measure  ∈ P( ) is a fully nonlinear equilibrium measure if and only if (ℎ(, ); (⃗ )) = sup () where () := (h(); ).Indeed, the first equality follows from the definitions and the second one follows from the fact that  0 ↦ → ( 0 ; ) is increasing for each  ∈ (⃗ ).Since  is continuous on the compact set (⃗ ), it follows that V is itself compact.
Proof of the claim.Consider a point  0 on the boundary of (⃗ ), and let us prove that it cannot maximize .Let ⃗  be any vector such that  0 + ⃗  ∈ int((⃗ )) and consider the function defined on [0, 1] by  () = h( 0 + ⃗ ).By concavity its derivative has a limit, finite or infinite, as  → 0. For all small enough  > 0, we have  ′ () = ⟨∇ h( 0 + ⃗ ), ⃗ ⟩.We know that |∇ h| → ∞ at the boundary, but it could a priori be that ∇ h becomes orthogonal to ⃗  as  → 0; we now prove that this cannot be the case.
At each small enough  > 0, the tangent space   to the upper boundary of  has (1, − ∇ h) as normal vector.As  → 0, |∇ h| → ∞ so that any accumulation point  0 of   is vertical, of the form R× where  is a hyperplane of R  (normal to an accumulation point of the direction of ∇ h( 0 + ⃗ )).Since  is contained in a half-space delimited by  0 ,  must be a supporting hyperplane of (⃗ ) at  0 .Since ⃗  has been chosen pointing to the interior of (⃗ ), the angle between ⃗  and  is bounded away from 0. It follows that for some constant  > 0 and all  > 0, ⟨∇ h( 0 + ⃗ ), ⃗ ⟩ ≥ |∇ h( 0 + ⃗ )||⃗ | → ∞.

It follows that
and eq.(4.16) follows and assertion (i ) is established.
Remark 4.17.The value max P( ) Π is a generalization of our previous definition of nonlinear pressure.Of course, one could decide to study the variational principle for full general  without any restriction.Nevertheless we point out that: (i) Assumption inf  0  > 0 is crucial: a change of sign would modify the nature of the problem, (ii) the case ( 0 ; ) =  0 +  () is of particular interest: in the classical variational principle, the term ℎ(, ) comes from the summation over (, )-covers in the Gibbs measures (see Formula (1.6)), and there is at the moment no candidate to replace this summation and define a topological pressure in the case of a general .
To state our next result, we recall that a subvariety of an open set  ⊂ R  is a subset defined by finitely many functions ℎ 1 , . . ., ℎ  ∈   ( ) as { ∈  : it is easy to see that any nontrivial subvariety has zero Lebesgue measure (see, e.g., [21] for a simple proof).
The previous theorem implies the following, which in particular contains Theorem D.
Corollary 4.18.Let (, ⃗ ) be a   Legendre system and  be a   admissible function defined on an open set  ⊂ R 1+ .Then the set V of (, ⃗ )-equilibrium values is a compact subset of an analytic sub-variety of R  .
In particular, it is a closed set with empty interior which is Lebesgue negligible.
Since a proper analytic sub-variety of a compact line segment is finite: Corollary 4.19.Let (, ) be a   Legendre system and  be a   admissible with  = 1, then the set EM of equilibrium measures is finite.
In full generality, we have a generic uniqueness: Proposition 4.20.Let (, ⃗ ) be a Legendre   regular system for some 2 ≤  ≤ .There is a unique nonlinear equilibrium measure in both of the following settings: (i) For  in some open and dense subset of { ∈   ( ) :  0  > 0} where  is a given admissible open subset of R × R  ; (ii) For ( 0 ; ) =  0 +  () with  in some open and dense subset of   ( ) where  is a given open neighborhood of (⃗ ) in R  .
Claim (ii ) above means that, for a generic nonlinearity  , there is a unique nonlinear equilibrium measure.It is not implied by the fully nonlinear case (i ) since the corresponding set of s has empty interior.It would be interesting to determine conditions on a fixed non-linearity  or  under which a generic ⃗  leads to a unique equilibrium measure In higher dimension  ≥ 2, we do not know any example with   regularity where finiteness does not hold.Beyond the real analytic case, even finiteness fails to hold for arbitrary nonlinearity: Proposition 4.21.Let (, ⃗ ) be a   Legendre system for some 2 ≤  ≤ ∞.For all compact  ⊂ int (⃗ ), there exists a   nonlinearity  such that the set of equilibrium values V equals .In particular the set of equilibrium measures can be infinite, even uncountable.
Before proving these two propositions, we recall some well-known facts about Morse functions.Given any open subset  ⊂ R  , a function  ∈   ( ) with 2 ≤  ≤  is Morse on  ⊂  if no critical point in  is degenerate and it is nonresonant if it takes distinct values at each of its critical points in  [22, Def.1.1.7 and 1.2.11].In particular, it has at most one maximizer on .Finally, the set of nonresonant Morse   functions on a compact set is open and dense (see the proofs in [22,Sect. 1.2]).This is to be understood with respect to the classical uniform topologies on   ( ) with finite , or the limit topology for  ∞ ( ), or the more complicated standard topology of   ( ) (see, e.g., [14, p. 53]).
Proof of Proposition 4.20.We prove Claim (i ).The proof of Claim (ii ) is entirely similar.Note that it is enough to prove the claim under the auxiliary assumptions  0  > 1/ and |∇ | <  for  > 0 arbitrary.
First, note that 0 = ∇  implies that |∇ ℎ| ≤  2 .Hence, it is enough to ensure that  is nonresonant Morse on the compact subset: Second, observe that  ↦ →  is continuous from   ( ) →   (int (⃗ )).Therefore the set  of  ∈   ( ) such that  is nonresonant and Morse on  is open.
Proof of Proposition 4.21.Let  : R  → [0, ∞) be a  ∞ function such that  = { ∈ R  |  () = 0} (such a function can be constructed as a convergent sum of functions that are each positive on one open balls, with the union of the balls equal to the complement of ).Let  be −1 outside int (⃗ ), coincide with − −h on a compact subset of int (⃗ ) containing  in its interior, and be lesser than −h in between; such a function exists since  does not approach the boundary of the rotation set.Then maximizing h() +  () is the same as minimizing  (), and is achieved precisely on .
Since Y = − ∇ ℎ(V ) where − ∇ ℎ : R  → int (⃗ ) is a diffeomorphism, Proposition 4.21 gives: Corollary 4.22.Let (, ⃗ ) be a Legendre   regular system for some 2 ≤  ≤ ∞.For all compact sets  ⊂ R  , there exists a   nonlinearity  such that the set Y from Theorem C equals .In particular the set of equilibrium measures can be infinite, even uncountable.

Examples of phase transitions
This section is devoted to the application of the framework developed above to a few families of systems whose energy depends on a real multiplicative parameter (i.e., an inverse temperature) and exhibiting various behaviors when this parameter is modified: changes in the number of equilibrium measures, piecewise analytic behavior with or without an affine piece.Most examples belong to the non-linear thermodynamical formalism, but even in the linear case we provide new insight thanks to the entropy-potential diagram , see Theorem 5.5.

5.1.
The Curie-Weiss Model -Symmetric case.The Curie-Weiss energy for a potential  is given by a quadratic nonlinearity, i.e., ℰ() = ℰ 1 () = 1  2 () 2 where  is a parameter called the inverse of temperature.For this specific case, we shall first use our general machinery above to recover an example treated in [16], then provide a second example exhibiting a "metastable" phase transition.
We consider here the left shift  on  := {, } N , endowed for example with the distance where  = (  ) ∈N ,  = (  ) ∈N , with the potential  :  → R defined by ).Among invariant measures in ℳ(), the one of maximal entropy is the Bernoulli measure with weights ( 1− 2 , 1+ 2 ), whose entropy is well-known: We thus are left with maximizing, given  ≥ 0, A simple computation shows that there are two cases (see Figure 3): (i ) For 0 ≤  ≤ 1, 0 is the unique critical point of P  and is indeed a maximum.Thus, V = {0}, there is a unique equilibrium state which is the Bernoulli measure of weights ( 1 2 , 1 2 ), and the nonlinear topological pressure is Π ℰ We have recovered the result of [16] that the nonlinear equilibrium measure is unique for 0 ≤  ≤ 1 but that there are two of them for  > 1, in line with the physical model.Note that any  2 Legendre system (, ) with an entropy-potential diagram that is symmetric with respect to the vertical axis will provide a similar example.Indeed the symmetry ensures that for all , 0 is a critical point; and as long as  < h ′′ (0), the graph of h being more concave at 0 than the graph of − , 0 will be a local maximum.It will then be a global maximum at least when  is close enough to 0. For  > h ′′ (0), 0 will be a local minimum and one will get (at least) two non-zero symmetric equilibrium values.)︁ )︂ so that h ′ is strictly decreasing, from +∞ when  → −2 to −∞ when  → 3; it has a single inflection point at  = 1 2 , is convex on (−2, 1  2 ] and concave on [ 1 2 , 3) (see its graph in Figure 4).
It follows that for  ≥ 0 small enough,   has only one critical point, which must be a maximum; in this regime, there is only one equilibrium state, with equilibrium value  < 0, and the pressure varies analytically.
Increasing , at some value  1 the line ℓ  touches the graph of h ′ on the right, and a second critical point appears.However, at this moment there is still only one equilibrium measure:   is unimodal, decreasing around the second critical point.Increasing  any further makes   bimodal, with three critical points: one local minimum located between two local maximums  1 () <  2 ().
At first,  1 () is the unique global maximum, but it ultimately gets surpassed by   ( 2 ()), precisely at the inverse temperature  0 when the vertical translate of the graph of −  2  2 touching the graph of h does so at two points.The choice of  has been made to ensure this happens, by giving the entropy-potential diagram a larger overhang to the right than to the left (see Figure 5): as  → ∞, the highest translate of the graph of − that touches the graph of h converges to the two vertical lines of equations ( = 3) and ( = −3).The latter of these vertical lines is far from the entropy-potential diagram since () = [−2, 3], and for large enough  the unique global maximum of   must be attained at  2 () → 3.
Again the pressure is analytic for  >  0 , but we have a phase transition at  0 : the pressure is  ↦ → max(  ( 1 ()),   ( 2 ())) and cannot be analytical at the point where the arguments of the max cross each other.Observe that the value  1 (<  0 ) does not correspond to a phase transition: pressure is analytic in the vicinity of  1 .
This example motivates the following definition.
Definition 5.2.A system (, ℰ 1 ) is said to exhibit a metastable phase transition at inverse temperature  0 > 0 when there are two curves of invariant probability measures (  ), (  ) defined on a neighborhood  of  0 with  ↦ → Π ℰ 1 (  ) and  ↦ → Π ℰ 1 (  ) both   , such that: (i) for all  ∈ , (  ), (  ) are local maximums of Π ℰ , (ii) for  <  0 ,   is an equilibrium measure of ℰ but   is not, and for  >  0 ,   is an equilibrium measure but   is not.
Observe that the pressure function  ↦ → Π ℰ 1 is not analytic at  0 , for otherwise Π ℰ 1 (  ) and Π ℰ 1 (  ) would have to coincide and both   and   would be equilibrium measures throughout .
The "metastable" terminology is suggested by the analogy with the physical phenomenon of the same name.A simple example of it is that of water remaining liquid below the freezing point in some circumstances.This is modeled by the liquid state (described by   ) admitting a continuation to  >  0 as a local maximum and the global maximal, the solid state (described by   ), being too far from   to allow the water to easily reorganize itself from one state to the other.
What we have proven can be summarized as follows.
Theorem F. There exists a locally constant potential  on a full shift  such that the Curie-Weiss energy ℰ 1 () = 1 2 () 2 exhibits a metastable phase transition.This gives another concrete example of multiple nonlinear equilibrium measures in a context where the linear thermodynamical formalism is long known to be flawless (analytic pressure, etc.) where, as above, [  ] is a cylinder, the set of words having the letter   in zeroth position.The framework developed above seems not to apply since the potentials are not linearly independent up to (coboundaries and) constants: ∑︀  1   ≡ 1, and the rotation set has empty interior.Let us take this as an opportunity to explain how this hypothesis is easily recovered: one simply extract a maximal independent subfamily of potentials, here ⃗  ∘ = (1  1 , . . . ,1  −1 ), and adjusts the nonlinearity to ensure  ∘ ((⃗  ∘ )) =  ((⃗ )) for all  ∈ P( ), here It is always possible to construct such an  ∘ , since by maximality each the potentials that are present in ⃗  can be expressed as linear combination of the potentials in ⃗  ∘ up to a coboundary and a constant, and a coboundary  −  ∘  can be neglected since ( −  ∘  ) = 0 for all invariant measures .Now (, ⃗  ∘ ) is   Legendre and we can apply Theorems B and C (recall that moreover (, ⃗  ∘ ) has unique linear equilibrium measures, hence each  ∈ V yields a unique nonlinear equilibrium measure), and these results translate to the original system (, ⃗ ) with the nonlinearity  : accumulation points of Gibbs ensembles are convex combinations of the nonlinear equilibrium measures, each of which coincides with a linear equilibrium measure for some linear combination of the (  ); however, due to the lack of independence, several different linear combinations lead to the same equilibrium state.
In the specific case of the mean-field Potts model one can work out the equilibrium measures by (nontrivial) direct computations.Given a vector  := ( 1 , . . .,   ) in the rotation set For  ≥ 0, the nonlinear pressure is We now summarize results from [9].For 0 ≤  <   : The value is  2 + log  and is achieved by a unique measure.
For  >   , Π ℰ 1 top is given by an implicit equation.It is realized by  equal to any permutation of ̃︀  defined by where  is the biggest solution for

Each permutation of ̃︀
gives a distinct equilibrium measure.Thus we get exactly  equilibrium measures.
For  =   , the maximal value is simultaneously realized by ( 1  , . . ., 1  ) and by the  distinct permutations of z.Thus we get exactly +1 equilibrium measures.In this case, the convergence of Gibbs measures to a convex combination of these equilibrium measures was previously shown in [17].5.4.Freezing phase transitions.Let us explain how the entropy-potential diagram can be used to visualize "freezing phase transitions", i.e., situations where for some  0 , the set of equilibrium measures of the energy ℰ 1 is constant for  >  0 .These measures are called the ground states.The physical interpretation is that once the temperature goes below some positive value 1/ 0 , the system freezes in a macroscopic state corresponding to zero temperature, described by (one of) the ground states.In the linear thermodynamical formalism, the first freezing phase transition was exhibited by Hofbauer [11], motivated by giving examples with multiple equilibrium states (this is sometimes achieved at  =  0 ).Concretely, the typical examples are for the shift  on  = {, } N or  = {, } Z with potentials with  ∈ (0, 1], and the freezing equilibrium measure is  0 =  ... .It has more recently been shown by Bruin and Leplaideur [2,3] that one can produce in a similar way a freezing phase transition with more interesting ground states, supported on some uniquely ergodic, zero-entropy compact subsets of  such as given by the Thue-Morse or the Fibonacci substitutions.Let us interpret in the entropy-potential diagram  such a freezing phase transition, with potential  being maximized by some invariant measure  0 , say with  0 () = 0 for normalization.By definition, for  ≥  0 the pressure is affine and Figure 6.Freezing phase transitions in the linear thermodynamical formalism: for  >  0 , all support lines are concurrent, and  must exhibit an acute corner at its right end.Left: h is strictly concave, there might be a unique equilibrium measure throughout (case  = 1 in Hofbauer's example).Right: h has a flat part, at  0 there are (at least) two ergodic equilibrium measure, one at each end of the flat edge (case  < 1 in Hofbauer's example).achieved at  0 , meaning that all lines of slope < − 0 touching  do it at the same point (see Figure 6).
When these conditions are realized, the critical inverse temperature, i.e., the least possible value of  0 , is the least possible  in the entropy-potential inequality (iv).The intercept of the affine part of the graph of P is then the entropy of equilibrium measures after the freezing phase transition, and its slope is their energy () (here 0 is given by the chosen normalization of the rotation set).
Proof.The main here is the observation that (iv ) characterizes Freezing Phase Transitions, but for the sake of completeness we prove all the equivalences, through the cycle (i) =⇒ (iii) =⇒ (iv) =⇒ (ii) =⇒ (i).
Convex duality translates angular points to flat regions and vice-versa; that P is affine on an interval means that the entropy-potential diagram has an angular point with a supporting line of slope − for each  in the interval.Let us explain this, a simple case of what we left hidden behind the appeal to Legendre duality above.Using the notation h() = sup{ℎ(, ) : () = } for all  ∈ [, 0], h is concave thus continuous on (, 0), and has a continuous extension h on [, 0].We can the rewrite P() = max  h()+.Denoting by   an abscissa realizing P(), observe that for all  > 0, P( + ) ≥ h(  ) + ( + )  ≥ P() +   so that the right derivative of P is at least   .Similarly, P( − ) ≥ P() −   shows that the left derivative is at most   .Whenever P is differentiable, P ′ () =   .On an affine part, the derivative exists and is constant, therefore   is (locally) constant and h has an angular point.Moreover the abscissa of the angular point is the slope of the line extending the affine part of the graph of P, while the ordinate of that point is the intercept of that line.
Item (iii) thus implies that the entropy-potential diagram has an angular point with supporting lines of slope − for all  ≥  0 .Since slopes are arbitrarily high in magnitude, the abscissa of this angular point must be the supremum of the rotation set, i.e., 0. It must then have ordinate equal to the supremum of the realizable entropies for this energy, i.e., ℎ(,  0 ).In particular, the entropypotential diagram is constrained under a line of equation (ℎ(, ) = ℎ(,  0 ) −  0 ()), which is (iv).
Remark 5.4.If we consider several potentials  1 , . . .,   , the condition in Legendre regularity that |∇ h| goes to +∞ as one approaches the boundary is violated exactly when some linear combination of the (  ) exhibit a (linear) freezing phase transition.
The entropy-potential diagram makes it clear how to prove existence of freezing phase transition in both the linear and nonlinear settings.We divide Theorem E of the introduction in two parts.Theorem 5.5.Let  :  →  be a continuous map of finite, positive topological entropy such that  ↦ → ℎ(, ) is upper semi-continuous.Consider  0 ∈ P erg ( ) with zero entropy.Then there exists a continuous potential  :  → R such that the linear thermodynamical formalism of (, ) exhibits a freezing phase transition with ground state  0 .Moreover we can ensure that  0 is the unique ground state, and that at the critical inverse temperature  0 there are exactly two equilibrium states.
In particular, if  is a compact  -invariant set with zero topological entropy, then we can find a potential exhibiting a freezing phase transition supported on .This broadly extends [2,3] by proving existence of freezing phase transitions for all zero-entropy subshifts, instead of very specific ones; but it is not constructive, since the potential  is ultimately obtained through the Hahn-Banach theorem.
To have a second equilibrium state at the critical inverse temperature, it suffices to consider an arbitrary ergodic measure  1 of positive entropy: Jenkinson's theorem provides a continuous potential whose only ergodic equilibrium states (at  = 1) are  0 and  1 .This also fixes the critical inverse temperature at  0 = 1.Theorem 5.6.Let  :  →  be a continuous dynamical system of finite, positive topological entropy such that  ↦ → ℎ(, ) is upper semi-continuous.Let  :  → (−∞, 0] be a continuous potential such that  =  −1 (0) is  -invariant and has zero topological entropy.
Then there exists a continuous nonlinearity  1 : (−∞, 0] → (−∞, 0] with  (0) = 0 such that the energy ℰ 1 () =  1 (()) exhibits a "strong freezing phase transition" in the following sense.There is a  0 > 0 such that: • for each  <  0 the energy ℰ 1 has at least one equilibrium measure, and none of them are supported on , • at  =  0 there are several equilibrium measures, at least one supported on  and one not supported on , • for each  >  0 the equilibrium measures are exactly the -supported,  -invariant measures and the topological pressure function  ↦ → Π ℰ 1 top ( ) is affine.
Observe that here  1 will only be continuous at 0; we can extend it continuously to R, but we cannot make  1 differentiable in a neighborhood of 0.

4. 2 .z 1 z 2 Figure 1 .Definition 4 . 4 .
Figure 1.An entropy-potentials diagram in two dimensions (first coordinate represented by the vertical axis), in a case when the rotation set is not strictly convex.

Figure 2 .
Figure 2.An entropy-potential diagram  represented in the  = 1 case (first coordinate  0 represented by the vertical axis): P()is obtained by sliding a line along the normal vector (1; ) until it touches the hypograph of h, which happens above some  where ∇ h() = −.At this point ∇ P() = : changing the direction  makes the touching line "roll" along the upper side of ; this rolling combines the rotation of  and a normal translation given by scalar product with .Changed "variation in the amount of sliding" by a hopefully clearer explanation.
such that the nonlinear equilibrium measures are exactly the linear equilibrium measures with respect to each of the potentials ∑︀ Legendre with unique linear equilibrium measures.Theorem C. Assume that (, ⃗ ) is   Legendre, that  :  ⊂ R  → R is       where ( 1 , . . .,   ) ∈ Y .