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FLATNESS IS A CRITERION FOR SELECTION OF MAXIMIZING MEASURES RENAUD LEPLAIDEUR 2 1 0 2 Abstract. ForafullshiftwithNp+1symbolsandforanon-positivepotential,locally n proportionaltothedistancetooneofN disjointfullshiftswithpsymbols,weprovethat a J the equilibrium state converges as the temperature goes to 0. Themainresultisthatthelimitisaconvexcombinationofthetwoergodicmeasures 6 withmaximalentropyamongmaximizingmeasuresandwhosesupportsarethetwoshifts 2 where the potential is the flattest. ] In particular, this is a hint to solve the open problem of selection, and this indicates S thatflatnessisprobablya/thecriterionforselectionasitwasconjecturedbyA.O.Lopes. D As a by product we get convergence of the eigenfunction at the log-scale to a unique . calibrated subaction. h t a m 1. Introduction [ 1 1.1. Background. In this paper we deal with the problem of ergodic optimization and, v more precisely with the study of grounds states. Ergodic optimization is a relatively new 2 5 branch of ergodic theory and is very active since the 2000’s. For a given dynamical system 4 (X,T), the goal is to study existence and properties of the invariant probabilities which 5 maximize a given potential φ : X R. We refer the reader to [15] for a survey about . 1 ergodic optimization. → 0 2 Ground states are particular maximizing measures which can be reached by freezing the 1 : system as limit of equilibrium states. Namely, for β > 0, which in statistical mechanics v represents the inverse of the temperature, we consider the/an equilibrium state associated i X to βφ, that is a T-invariant probability whose free energy r (cid:90) a h (T)+β φdν, ν is maximal (where h is the Kolmogorov entropy of the measure ν). Then, considering an ν equilibrium state µ , it is easy to check (see [12]) that any accumulation point for µ as β β β goes1 to + is a maximizing measure for φ. ∞ The first main question is to know if µ converges. It is known (see [8]) that for an β uniformly hyperbolic dynamical systems, generically in the 0-topology, φ has a unique C maximizing measure. Therefore, convergence of µ obviously holds in that case. β Date: Version of January 27, 2012. Part of this research was supported by DynEurBraz IRSES FP7 230844. 1β is the inverse of the temperature, thus β → +∞ means Temp → 0, which means we freeze the system. 1 2 RENAUDLEPLAIDEUR Nevertheless, generic results do not concern all the possibilities, and it is very easy to build examples, which at least for the mathematical point of view are meaningful, and for which the set of ergodic maximizing measures is as wild as wanted. For these situations, the question of convergence is of course fully relevant. Cases of convergence or non-convergence are known (see [9, 16, 11, 10]), but the general theory is far away of being solved. In particular, no criterion which guaranties convergence (except the uniqueness of the maximizing measure or the locally constant case) is known, say e.g. for the Lipschitz continuous case. The second main question, and that is the one we want to focus on here, is the problem of the selection. Assuming that φ has several ergodic maximizing measures and that µ β converges, what is the limit. In other words, is there a way to predict the limit from the potential, or equivalently, what makes the equilibrium state select one locus instead of another one ? Inspired by a similar study for the Lagrangian Mechanics ([2]), it was conjectured by A.O. Lopes that flatness of the potential would be a criterion for selection and that the equilibrium state always selects the locus where the potential is the flattest. In [3] it is actually proved that the conjecture is not entirely correct. Authors consider in the full 3-shift a negative potential except on the two fixed points 0∞ and 1∞, where it vanishes but is sharper in 1∞ than in 0∞. Then, they prove that the equilibrium state actually converges, but not necessarily to the Dirac measure at 0∞. The first part of the conjecture is however not (yet) invalided and the question to know if flatness is a criterion for selection is still relevant. Here, we make a step in the direction of proving that flatness is indeed a criterion for selection. Precisestatementsaregiveninthenextsubsection. Weconsiderinthefullshift with Np+1 symbols a potential, negative everywhere except on N Bernoulli subshifts2 with p symbols. Figure 1 illustrates the dynamics. Σ1 Σ2 u Σ Σ N j Figure 1. Our system 2Note that these Bernoulli shifts are empty interior compact sets. SELECTION OF MAXIMIZING MEASURE 3 Then, flatness is ordered on these N Bernoulli subshifts : the potential is flatter on the first one than on the second one, then, it is flatter on the second one than on the third one, and then so on (see below for complete settings). Any of these Bernoulli shifts has a unique measure of maximal entropy, and the set of ground states is contained in the convex hull of these N-measures. We show here that the equilibrium state converges and selects a convex combination of the two ergodic measures with supports in the two flattest Bernoulli subshifts. We emphasize that this result is absolutely not in contradiction with [3]. Indeed, in [3] it is proved that the equilibrium state converges to a convex combination of the Dirac measures at 0∞ and 1∞, which are obviously the two flattest loci ! 1.2. Settings. 1.2.1. The set Σ. We consider the full-shift Σ with Np+1 symbols, with N and p two positive integers. We also consider Np+1 positive real numbers, 0 < α < α (cid:54) ... (cid:54) α , 1 2 p 0 < α < α (cid:54) ... (cid:54) α , ...,0 < α < α (cid:54) ... (cid:54) α and α. We p+1 p+2 2p (N−1)p+1 (N−1)p+2 Np assume α < α < α (cid:54) α . 1 p+1 2p+1 (N−1)p+1 N N N We set Σ := 1,...,p , Σ := p+1,...,2p , ...,Σ := (N 1)p+1,...,Np . 1 2 N { } { } { − } For simplicity the last letter Np+1 is denoted by u. The letter (j 1)p+i will be denoted − by u . ij The set of letters defining Σ is := u ,1 (cid:54) i (cid:54) p . Hence, Σ = N, and a word j Aj { ij } j Aj admissible for Σ is a word (finite or infinite) in letters u . The length of a word is the j ij number of digit (or letter) it contains. The length of the word w is denoted by w . | | Ifw = w w ...w andw(cid:48) = w(cid:48),w(cid:48),...w(cid:48) aretwofinitewords,wedefinetheconcatenated 0 1 n 0 1 n(cid:48) word ww(cid:48) = w w ...w w(cid:48),w(cid:48),...w(cid:48) . This is easily extended to the case w(cid:48) = + . If 0 1 n 0 1 n(cid:48) | | ∞ m is a finite-length admissible word for Σ , [m ] denotes the set of points starting with j ∗ the same m letters than m and whose next letter is not in . j | | A The distance in Σ is defined (as usually) by d(x,y) = θmin{j, xi(cid:54)=yi}, whereθ isafixedrealnumberin(0,1). Thisdistanceissometimesgraphicallyrepresented as in Figure 2. n 1 − y (cid:0) (cid:0) x =y (cid:64) 0 0 (cid:64) x x =y n−1 n−1 Figure 2. The sequence x and y coincide for digits 0 up to n 1 and then split. − 4 RENAUDLEPLAIDEUR Weemphasizehere, thatcontrarilyto[3]wehavenotchosenθ = 1 inviewtogetthemost 2 general result as possible. Indeed, in [3] it was not clear if some results where independent ornotofθ’svalue. Moreover,thisalsomeansthatweareconsideringallHöldercontinuous functions, and not only the Lipschitz ones, because in Σ, a Hölder continuous function can be considered as a Lipschitz continuous, up to a change of θ’s value. 1.2.2. The potential, the Transfer operator and the Gibbs measures. The potential A is defined by (cid:26) α d(x,Σ ), if x [u ] A(x) = − j i ∈ ij A(x) = α, if x [u]. − ∈ The potential is negative on Σ but on each Σ where it is constant to 0. j The transfer operator, also called the Ruelle-Perron-Frobenius operator, is defined by k p (cid:88)(cid:88) (ϕ)(x) := eA(uijx)ϕ(u x)+eA(ux)ϕ(ux). β ij L j=1 i=1 It acts on continuous functions ϕ. We refer the reader to Bowen’s book [7] for detailed theoryoftransferoperator,GibbsmeasuresandequilibriumstatesforLipschitzpotentials. The eigenfunction is H and the eigenmeasure is ν . They satisfy: β β (H ) = eP(β)H , ∗(ν ) = eP(β)ν . β β β β β β L L The eigenmeasure and the eigenfunction are uniquely determined if we required the as- (cid:90) sumption that ν is a probability measure and H dν = 1. β β β The Gibbs state µ is defined by dµ := H dν . The measure µ is also the equilibrium β β β β β state for the potential β.A : it satisfies (cid:26) (cid:90) (cid:27) (cid:90) max h (σ)+β Adµ = h (σ)+β Adµ µ σ−inv µ µβ β and this maximum is (β) and is called the pressure of βA. eP(β) is also the spectral P radius of . It is a single dominating eigenvalue. β L 1.3. Results. In each Σ we get a measure of maximal entropy µ . As each Σ is a j top,j j subshift of finite type, µ is again of the form top,j dµ = H dν , top,j top,j top,j where H and ν are respectively the eigenfunction and the eigen-probability asso- top,j top,j ciated to the transfer operator in Σ for the potential constant to 0. j Note that, as β goes to + , µ has only N possible ergodic accumulation points, which β ∞ are the measures of maximal entropy in each Σ , µ . j top,j Our results are Theorem 1 The eigenmeasure ν converges to the eigenmeasure ν for the weak* β top,1 topology as β goes to + . ∞ SELECTION OF MAXIMIZING MEASURE 5 Theorem 2 The Gibbs measure µ converges to a convex combination of µ and µ β top,1 top,2 for the weak* topology as β goes to + . This combination depends in which zone Z 1 ∞ ∪ Z Z Z (see Figure 3) the parameters are: 2 3 4 ∪ ∪ 1 (1) For parameters in Z , µ converges to (µ +p2µ ). 1 β 1+p2 top,1 top,2 (2) For parameters in Z , µ converges to µ . 2 β top,1 1 (3) For parameters in Z Z µ converges to (µ +ρ2µ ) for some ρ > 0 3∪ 4 β 1+ρ2 top,1 i top,2 i i locally constant in Z Z , Z Z and Z Z (see Equalities (27) page 29, (29) 3 4 4 3 3 4 \ \ ∩ page 30 and (30) page 30). Z α 4 Z 1 θ α p + 1 Z = 2 α α α 1 p+1 α 1 2θ 1 Z − 3 Figure 3. Ratio between µ and µ top,1 top,2 Zone Z corresponds to 0 < α (cid:54) α θ and α = θ αp+1−α1. Zone Z corresponds to 3 p+1 1−θ 2 4 α α = 1 and α (cid:62) α θ. We emphasize that Z exists if and only if θ > 1. p+1 2θ 1 p+1 4 2 − As a by product of Theorem 1 and Theorem 2 we get the exact convergence for the eigenfunction to a unique subaction (see Section 2 for definition): Corollary 3 The calibrated subactions are all equal up to an additive constant. Moreover, the eigenfunction H converges at the log-scale to a single calibrated subaction : β 1 V := lim log(H ). β β→+∞ β The question of convergence and uniqueness of a subaction seems to be important for the theory of ergodic optimization. It appeared very recently in [4]. We point out that 6 RENAUDLEPLAIDEUR convergenceoftheeigenfunctiontoasubactionisrelatedtothestudyofaLargeDeviation Principlefortheconvergenceofµ (seee.g.[5]and[17]). Nevertheless,inthesetwopapers, β Lopes et al. always assume the uniqueness of the maximizing measure, which yields the uniqueness of the calibrated subaction (up to a constant). Here we prove convergence to a unique subaction without the assumption of the uniqueness of the maximizing measure. It it thus allowed to hope that we could get a more direct proof of a Large Deviation Principle, without using the very indirect machinery of dual shift (see [5]). 1.4. Further improvements: discussion on hypothesis. The present work is part of a work in progress. The situation described here is far away from the most general case and our goal is to prove the next conjecture. In[14], Garibaldiet al. introducethesetofnon-wanderingpointswithrespecttoaHölder continuouspotentialA, Ω(A). Thissetcontainstheunionofthesupportsofalloptimizing measures (here we consider maximizing measures, there they consider minimizing measures). The set Ω(A) is invariant and compact. Under the assumption that it can be decomposed in finitely many irreducible pieces, it is shown that calibrated subactions are constant on these irreducible pieces and their global value is given by these local values and the Peierls barrier (see Section 2 below). We believe that, under the same hypothesis, it is possible to determine which irreducible component have measure at temperature zero: Conjecture. For A : Σ R Hölder continuous, if Ω(A) has finitely many irreducible → components, Ω(A) ...Ω(A) , then, µ (Ω(A) ) goes to 0 if Ω(A) is not one of the two 1 N β i i flattest loci for A. We emphasize that this conjecture does not mean that there is convergence “into” the components. It may be (as in [10]) that an irreducible components has several maximizing measures and that there is no selection between these measures. Thisconjectureisforthemomentfarofbeingproved,inparticularbecauseseveralnotions are not yet completely clear. In particular the notion of flatness has to be specified. Moreover, the components are not necessarily subshifts of finite type, which is an obstacle to study their (for instance) measures of maximal entropy. The work presented here, is for a specific form of potential for which flatness is easily defined. The dynamics into the irreducible components and also the global dynamics are easy. We believe that the main issue here is to identify flatness as a criterion for selection. The next step would be to release assumptions on the dynamics; in particular we would like that theses components are not full shift and that the global dynamics is not a full shift. It is also highly probable that the conjecture should be adapted after we have solve this case. Distortion into the dynamics could perhaps favor other components. The last step would be to get the result for general (or as general as possible) potential. SELECTION OF MAXIMIZING MEASURE 7 Nevertheless, and even if the present work is presented as a work in progress and an intermediate step before a more general statement, we want to moderate the specificity of the potential we consider here. For a uniformly hyperbolic system (X,T) and for any Hölder function φ, there exists two Hölder continuous ψ and ψ such that φ = 1 2 (cid:26)(cid:90) (cid:27) ψ + m(φ) + ψ T ψ , where m(φ) = max φdµ and ψ is non-negative and 1 2 2 1 ◦ − vanishes only on the Aubry set. This means that, up to consider a cohomologous function, the assumption on the sign of A is free. Now, if we would consider a very regular potential (say at least 1) on a geometrical dynamical systems, the fact that ψ vanishes on the 1 C AubrysetmeansthatclosetothatAubryset,ψ (x)isproportionaltothedistancebetween 1 x and the Aubry set (with coefficient related to the derivative of ψ ). Consequently, the 1 potential A we consider here is a kind of discrete version for the symbolic case of a 1 C potential on a Manifold. 1.5. Plan of the paper and acknowledgment. InSection2weprovethatthepressure behaveslikelogp+g(β)e−γβ forsomespecificγ andsomesub-exponentialfunctiong. The real number γ is obtained as an eigenvalue for the Max-Plus algebra (see Proposition 5). In Section 3, we define and study an auxiliary function F; This function gives the asymptotic both for the eigenmeasure and for the Gibbs measure. In Section 4 we prove Theorem 1 and in Section 5 we prove Theorem 2. As a by-product we give an asymptotic for the function g(β) (in the Pressure). In the last section, Section 6 we prove Corollary 3. Part of this work was done as I was visiting E. Garibaldi at Campinas (Brazil). I would like to thank him a lot here for the talks we get together and the attention he gave me to listen and correct some of my computations. 2. Peierls barrier and an expression for the pressure The main goal of this section is Proposition 5 where we prove that (β) converges expo- P nentially fast to logp. The exponential speed is obtained as an eigenvalue for a matrix in Max-Plus algebra. 2.1. The eigenfunction and the Peierls barrier. Lemma 1. The eigenfunction H is constant on the cylinder [u]. It is also constant on β cylinders of the form [m ], where m is an admissible word for some Σ . Furthermore, j ∗ if m(cid:48) is another admissible word for the same Σ with the same length than m, then H j β coincide on both cylinders [m ] and [m(cid:48) ]. ∗ ∗ Proof. The eigenfunction H is defined by β n−1 1 (cid:88) H := lim e−kP(β) (1I). β β n→∞ n L k=0 8 RENAUDLEPLAIDEUR For x and x(cid:48) in [m ] [m(cid:48) ], if wx is a preimage for x then wx(cid:48) is a preimage for x(cid:48) and ∗ ∪ ∗ A(wx) = A(wx(cid:48)). The same argument works on [u]. (cid:3) This Lemma allows us to set (cid:88)(cid:88) (1) τj := e−α(l−1)p+iθβHβ(uil )+e−αHβ(u). ∗ l(cid:54)=j i Thus, for x in [u ] we get i ij (cid:116) p (cid:88) eP(β)H (x) = eβA(uijx)H (u x)+τ . β β ij j i=1 τ Lemma 2. The function H is constant on each Σ : for any x in Σ , H (x) = j . β j j β eP(β) p − Proof. The function H is continuous on the compact Σ . It thus attains its minimum β j and its maximum. Let x and x be two points in Σ where H is respectively minimal j j j β and maximal. The transfer operator yields: (cid:32) p (cid:33) (cid:88) eP(β)H (x ) = H (u x ) +τ . β j β ij j j i=1 By definition, for each i, H (u x ) (cid:62) H (x ). This yields β ij j β j (2) (eP(β) p)H (x ) (cid:62) τ . β j j − Similarly we will get τ (cid:62) (eP(β) p)H (x ). As the potential is Lipschitz continuous, the j β j − pressure function (β) is analytic and decreasing (A is non-positive). Then (β) > logp. P P This shows H (x ) = H (x ). β j β j Let x be any point in Σ . Equality (H ) = eP(β)H yields j β β β L (cid:88) eP(β)H (x) = eβ.A(ix)H (ix)+τ . β β j i∈Aj For i , A(ix) = 0, and as H is constant on Σ we get j β j ∈ A (eP(β) p)H (x) = τ . β j − (cid:3) 1 The family of functions logH is uniformly bounded and equi-continuous; any {β β}β∈R+ 1 accumulation point V for logH as β goes to + (and for the 0-norm) is a calibrated β β ∞ C subaction, see [12]. SELECTION OF MAXIMIZING MEASURE 9 In the following, we consider a calibrated subaction V obtained as an accumulation point 1 for logH . Note that the convergence is uniform (on Σ) along the chosen subsequence β β for β. For simplicity we shall however write V = lim 1 logH . β→+∞ β β AdirectconsequenceofLemma2isthatthesubactionV isconstantoneachΣ . Actually, j it is proved in [13] that this holds for the more general case and, moreover, that any calibrated subaction satisfies (3) V(x) = max V(x )+h(x ,x) , j j j { } where h( , ) is the Peierls barrier and x is any point in Σ . It is thus important to j j · · compute what is the Peierls barrier here. Lemma3. Letx beanypointinΣ . ThePeierlsbarriersatisfiesh(x ,x) = α θ d(x,Σ ). j j j − (j−1)p+11−θ j For simplicity we shall set h (x) for h(x ,x). j j Proof. Let x be in Σ. Recall that h (x) is defined by j (cid:40)n−1 (cid:41) (cid:88) limsup A(σl(z)) : σn(z) = x, d(x ,z) < ε . j ε→0 n l=0 As we consider the limit as ε goes to 0, we can assume that ε < θ. Now, to compute h (x), we are looking for a preimage of x, which starts by some letter admissible for Σ j j (because ε < θ) and which maximizes the Birkhoff sum of the potential “until x”. As the potential is negative, this can be done if and only if one takes a preimage of x of the form mx, with m a Σ admissible word. Moreover, we always have to chose the letter u to j 1j get the maximal α possible. (j−1)p+i − In other word we claim that for every n (cid:62) 1 for every l (cid:62) 0 and for every word m of length n+l, S (A)(u ...u x) (cid:62) S (A)(mx). n 1j 1j n+l (cid:124) (cid:123)(cid:122) (cid:125) ntimes Let assume d(x,Σ ) = θa, i.e., the maximal admissible word for Σ of the form x x x ... j j 0 1 2 has length a. This yields +∞ (cid:88) θ θ h (x) = lim S (A)(u ...u x) = α θn+a = α θa = α d(x,Σ ). j n→∞ n (cid:124)1j (cid:123)(cid:122) 1(cid:125)j − (j−1)p+1 − (j−1)p+11 θ − (j−1)p+11 θ j n=1 − − ntimes (cid:3) Then, Lemma 3 and Equality (3) yield (cid:18) (cid:19) θ (4) V(x) = max V(x ) α d(x,Σ ) . j j − (j−1)p+11 θ j − Lemma 4. For every j, 1 (cid:16) (cid:17) β→lim+∞ β log e−α(j−1)p+1θβHβ(u1j∗)+...+e−αjpθβHβ(upj∗) = V(u1j∗)−α(j−1)p+1θ. 10 RENAUDLEPLAIDEUR |m| Proof. By Lemma 1, the eigenfunction H is constant on rings [m ] with m , hence β ∗ ∈ Aj V(u ) = ... = V(u ). Now, inequalities α < α (cid:54) α show 1j pj (j−1)p+1 (j−1p+)2 (j−1)p+i ∗ ∗ that e−α(j−1)p+1θβHβ(u1j ) is exponentially bigger than all the other terms as β goes to + . ∗ (cid:3) ∞ 2.2. Exponential speed of convergence of the pressure : Max-Plus formalism. Here we use the Max-Plus formalism. We refer the reader to [6] (in particular chapter 3) for basic notions on this theory. Some of the results we shall use here are not direct consequence of [6] (even if the proofs can easily be adapted) but can be found in [1]. Proposition 5. Let (cid:26) (cid:27) θ (α +α ) θ 1 p+1 γ = min min(α θ,α)+α , . p+1 1 1 θ 2 1 θ − − Then, there exists a positive sub-exponential function g such that (β) := logp+g(β)e−γβ. P Proof. We have seen (Lemma 2) that H is constant on the sets Σ . This shows that β j V := lim 1 logH isalsoconstantoftheΣ . Forsimplicitywesetu∞ := u u u .... β→+∞ β β j ij ij ij ij This is a point in Σ . Now we have j (cid:88)(cid:88) (5) (eP(β) p)Hβ(u∞ij) = e−α(l−1)p+iθβHβ(u1l ) +e−αHβ(u). − ∗ l(cid:54)=j i Note that the results we get concerning the subaction V are actually true for any calibrated subaction. We point out that we can first chose some subsequence of β such that 1 log( (β) logp) converges, and then take a new subsequence from the previous one to β P − 1 ensure that log(H ) also converges. β β We thus consider V := lim 1 logH and γ := lim 1 log( (β) logp). At β→+∞ β β − β→+∞ β P − that moment we do not claim that γ has the exact value set in the Proposition. It is only 1 1 an accumulation point for log( (β) logp). The convergence of log( (β) logp) β P − β P − will follow from the uniqueness of the value for γ. Then (5) and Lemma 4 yield for every j, (cid:18) (cid:19) (6) γ V(u∞) = max max(cid:0)V(u ) α θ(cid:1),V(u) α . ij il (l−1)p+1 − − l(cid:54)=j − − Consider the k rows and k+1 columns matrix   α θ α θ ... α θ α p+1 2p+1 (N−1)p+1 −∞ − − − −  α1θ α2p+1θ α  M1 :=  −.. −∞ − −.. .  . .  α θ ... α 1 − −∞ −