RENORMALIZATION, FREEZING PHASE TRANSITIONS AND FIBONACCI QUASICRYSTALS.

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1 RENORMALIZATION, FREEZING PHASE TRANSITIONS AND FIBONACCI QUASICRYSTALS. HENK BRUIN AND RENAUD LEPLAIDEUR Abstract. We examine the renormalization operator determined by the Fibonacci substitution within the full shift on two symbols Σ := {0, } N. We exhibit a fixed point and determine its stable leaf under iteration of the operator acting on potentials V : Σ R, which is completely determined by the germ near the attractor of the substitution. Then we study the thermodynamic formalism for potentials in this stable leaf, and prove they have a freezing phase transition at finite temperature, with ground state supported on the attracting quasi-crystal associated to the Fibonacci substitution. Resumé. Nous étudions les relations entre renormalisation, substitutions et transitions de phase: nous montrons que la substitution de Fibonacci dans le shift plein à deux symboles Σ := {0, } N génère un opérateur de renormalisation sur les potentiels V : Σ R. Nous montrons que cet opérateur possède un point fixe, uniquement déterminé par son germe proche de l attracteur associé à la substitution de Fibonacci. Nous déterminons aussi la feuille stable de ce point fixe. Dans un second temps, nous montrons que tous les potentiels dans cette feuille stable présentent une transition de phase congelante. En particulier, cela donne un nouvel exemple d obtention d un état fondamental porté par un quasi-cristal avant le zéro absolu.. Introduction.. Background. The present paper studies phase transitions from an ergodic theory and dynamical systems point of view. It investigates renormalization, substitutions and phase transition initiated in [2] and continued in [5]. Phase transitions are an important topic in statistical mechanics and also in probability theory see e.g. [0,, 22, 26]. The viewpoint presented here is different for several reasons. One of them is that, here, the geometry of the lattice is not relevant, whereas in statistical mechanics, the geometry of the lattice is the most important part. Date: Version of May 9, Mathematics Subject Classification. 37A35, 37A60, 37D20, 37D35, 47N0. Key words and phrases. thermodynamic formalism, equilibrium states, phase transition, substitution, Fibonacci numbers. and we only consider a one-dimensional lattice.

2 2 HENK BRUIN AND RENAUD LEPLAIDEUR During the 970 s, motivated by problems in statistical mechanics, Bowen, Ruelle and Sinai see [3, 23, 25] introduced thermodynamic formalism into ergodic theory. Given a dynamical system X, T and a potential ϕ : X R, the pressure function is given by { Pβ := sup h µ T + β } ϕ dµ, where the supremum is taken over the invariant probability measures µ, h µ T is the Kolmogorov entropy and β is a real parameter. Any measure realizing the supremum is then called an equilibrium state for βϕ. For a uniformly hyperbolic dynamical system X, T and a Hölder continuous potential ϕ, the pressure function β Pβ is analytic see e.g. [3, 23, 4] and there is a unique equilibrium state µ βϕ for every β. This equilibrium also satisfies a Gibbs property; in the dynamical systems language this condition expresses how the measure of n-cylinders scale: There is K > 0 such that K µ βϕ Z n expβ n i=0 ϕ T i x npβ K for all n, all n-cylinders Z n and x Z n. Since the late 970s, people in dynamical systems focused on extending the notions and results of thermodynamics to non-uniformly hyperbolic dynamical systems. This started with the work of Hofbauer [2, 2] proving non-analyticity of pressure for a non-hölder potential ϕ on the shift-space {0, } N, σ. This example is closely related to the Manneville-Pomeau map, and an associated renormalization procedure, presented in [2], was the starting point of the project this paper is part of. Ledrappier [6] showed that any finite number of equilibrium can co-exist in similar examples cf. also [4, 9] for the co-existence of multiple equilibirum states in other settings. Weakening the Gibbs property may be necessary as well. For instance, Yuri [29], in the setting of maps with neutral fixed points, used a version of weak Gibbs in which the K in is replaced by K n with lim n log K n n = 0. See [6, Section 3.] for similar results to smooth interval maps with critical points. More recently, the original motivation came back into focus, and the question of phase transitions is now a very active theme in ergodic theory. Nevertheless, due to equivalences or interdependences in the classical settings between unique existence of Gibbs measures, unique existence of equilibrium states and regularity of the pressure function, and also due to historical or inspiration models e.g. the Ising model in probability or Erhenfest vs. Gibbs classification in statistical mechanics, the notion of phase transition may vary in the literature. In this paper, we adopt a largely accepted definition now in dynamical systems: a phase transition is characterized by a lack of analyticity of the pressure function Although analyticity is usually considered as a very rigid property and thus quite rare, it turns out that proving non-analyticity for the pressure function is not so

3 RENORMALIZATION AND THERMODYNAMICS 3 easy. Currently, this has become an important challenge in smooth ergodic theory to produce and study phase transitions, see e.g. [8, 8, 5] and also [3, Sec. 6] for the possible shapes of the pressure function. We also refer to [24] for results on the regularity of the pressure in the non-compact setting and [29, 20] for uniqueness of the equilibrium state, again in the non-compact case. To observe phase transitions, one has to weaken hyperbolicity of the system or of regularity of the potential; it is the latter one that we continue to investigate here. Our dynamical system is the full shift, which is uniformly hyperbolic. The first main question we want to investigate is thus which potentials ϕ will produce phase transitions. More precisely, we are looking for a machinery to produce potentials with phase transitions. The main purpose of [2] was to investigate possible relation between renormalization and phase transitions. In the shift space {0, } N, σ, a renormalization is a function H for which there is an integer k 2 such that 2 σ k H = H σ. The link with potentials was made in [2] by introducing a renormalization operator R acting on potentials and related to a solution H for 2. It is easy to check that constant length k substitutions are solutions to 2. In [5], we studied the Thue- Morse substitution, which has constant length 2. Here we investigate the Fibonacci substitution, which is not of constant length. Several reasons led us to study the Fibonacci case: Together with the Thue-Morse substitution, the Fibonacci substitution is the most famous substitution and it has been well-studied. In particular, the dynamical properties of their respective attracting sets are well-known and this will be used extensively in this paper. As a result, we were able to describe the relevant fixed point of renormalization exactly. Information of the left and right-special words in these attractors is a key ingredient to prove existence of a phase transition; it is a crucial issue in the relations between substitutions and phase transitions. The type of phase transition we establish is a freezing phase transition. This means that beyond the phase transition i.e., for large β, the pressure function is affine and equal to its asymptote, and the equilibrium state i.e., ground state is the unique shift-invariant measure supported on an aperiodic subshift space, sometimes called quasi-crystal. One open question in statistical mechanics see [27] is whether freezing phase transitions can happen and whether quasi-crystal ground state can be reach at positive temperature. An affirmative answer was given for the Thue-Morse quasi-crystal in [5]; here we show that this also holds for the Fibonacci quasi-crystal. We think that Fibonacci shift opens the door to study more general cases. One natural question is whether any quasi-crystal can be reached as a ground state at positive temperature. In this context we emphasize that the Fibonacci substitution space is also Sturmian shift, that is, it encodes the irrational rotation with angle

4 4 HENK BRUIN AND RENAUD LEPLAIDEUR the golden mean := We expect that the machinery developed here can be extended to the Sturmian shift associated to general irrational rotation numbers although those with bounded entries in the continued fraction expansion will be the easiest, possibly to rotations on higher-dimensional tori, and also to more general substitutions..2. Results. Let Σ = {0, } N be the full shift space; points in Σ are sequences x := x n n 0 or equivalently infinite words x 0 x.... Throughout, let x j = x j denote the opposite symbol. The dynamics is the left-shift σ : x = x 0 x x 2... x x Given a word w = w 0... w n of length w = n, the corresponding cylinder or n-cylinder is the set of infinite words starting as w 0... w n. We use the notation C n x = [x 0... x n ] for the n-cylinder containing x = x 0 x... If w = w 0... w n is a word with length n and w = w 0... a word of any length, the concatenation ww is the word w 0... w n w 0... The Fibonacci substitution on Σ is defined by: { 0 0 H : 0 and extended to words by the concatenation rule Hww = HwHw. convenient for us to count the Fibonacci numbers starting with index 2: 3 F 2 =, F = 0, F 0 =, F =, F 2 = 2, F n+2 = F n+ + F n. We have 4 F a n := H n a = { F n+ if a = 0, F n if a =. The Fibonacci substitution has a unique fixed point ρ = It is We define the orbit closure K = n σ n ρ; it forms a subshift of Σ, σ associated to ρ, supporting a unique shift-invariant probability measure µ K. More properties on K are given in Section 2. We define the renormalization operator acting on potentials V : Σ R by { V σ Hx + V Hx if x [0], RV x = V Hx if x []. We are interested in finding fixed points for R and, where possible, studying their stable leaves, i.e., potentials converging to the fixed point under iterations of R.

5 RENORMALIZATION AND THERMODYNAMICS 5 Contrary to the Thue-Morse substitution, the Fibonacci substitution is not of constant length. This is the source of several complications, in particular for the correct expression for R n. For α > 0, let X α be the set of functions V : Σ R such that V x n α if dx, K = 2 n. More precisely, X α is the set of functions V such that: V is continuous and non-negative. 2 There exist two continuous functions g, h : Σ R, satisfying h K 0 and g > 0, such that V x = gx n α + hx n α when dx, K = 2 n. We call g the α-density, or just the density of V X α. Continuity and the assumption h K 0 imply that hx/n α = on α. Our first theorem achieves the existence of a fixed point for R and shows that the germ of V close to K, i.e., its α-density, allows us to determine the stable leaf of that fixed point. Given a finite word w, let κ a w denote the number of symbols a {0, } in w. If x Σ\K, we denote by κ a x the number of symbols a in the finite word x 0... x n where dx, K = 2 n. Theorem. If V X α, with α-density function g, then lim k Rk V x = for all x Σ \ K if α < ; 0 for all x Σ if α > ; g dµk Ṽ x for all x Σ if α =, where Ṽ X is a fixed point for R, given by log 5 Ṽ x = log κ 0 x+ κ x+ κ 0 x+ κ x+ κ0 x+ κ x+ 2 κ 0 x+ κ x+ 2 if x [0]; if x []. This precise expression of Ṽ corresponds to a α-density gx = 2 /2 if x [0] K and gx = /2 if x [] K, and Ṽ xdµ K =. A freezing phase transition is characterized by the fact that the pressure is of the form Pβ = aβ + b for β β c and that the equilibrium state is fixed for β β c. The word freezing comes from the fact that in statistical mechanics β is the inverse of the temperature so the temperature goes to 0 as β + and that a ground-state is reached at positive

6 6 HENK BRUIN AND RENAUD LEPLAIDEUR temperature /β c, see [26, Chap. 2]. In mathematics, one usually talks about maximizing measure see e.g. [7, Prop. 29 ] Theorem 2. Any potential ϕ := V with V X admits a freezing phase transition at finite β: there exists β c > 0 such that for 0 β < β c the map Pβ is analytic, there exists a unique equilibrium state for βϕ and this measure has full support; 2 for β β c, the pressure Pβ = 0 and µ K is an equilibrium state for βϕ; it is unique for β > β c. The equilibrium states µ βϕ for β < β c satisfy a local Gibbs property. Namely, remains true but only for cylinders of the form Z r n xx J, where J is a cylinder with empty intersection with K and r n is a n th -return time into J. In that case, the constant K also depends on the cylinder J. This holds because these equilibrium states have conditional measures with respect to J satisfying a Gibbs property for the induced map see Section 4 and [7]. The Gibbs property is less practical for the equilibrium state µ K for β β c because the pressure Pβ = 0 and the potential ϕ is identically zero on the support of µ K. Therefore a weak Gibbs property holds trivially with K n = 2n for every n-cylinder intersecting K = suppµ K. The proof of the existence of a phase transition depends on the estimate of a complicated series; its convergence for large β with limit value strictly less than ensures that a phase transition has taken place at some β c. Regularity of the pressure function or number of equilibrium states for β = β c depends on the convergence of another complicated series related to the first one. Unfortunately, it requires a far more precise understanding of these series to obtain a reliable estimate on the value of β c and whether the second series converges or diverges at β = β c. The phenomenon that the same class of potentials are both fixed by the renormalization operator R and exhibit phase transitions remains intriguing. We are not aware of a proof that the one implies the other. Rather, we believe that they both follow from a deeper structure, basically that the power α in the definition of V α expresses the dimension of the lattice. This is one more incentive to study the structure of renormalization and phase transitions for substitutive systems in higher dimensional lattices..3. Outline of the paper. In Section 2 we recall and prove various properties of the Fibonacci subshift and its special words. We establish the form of H n and R n V for arbitrary n. We define a notion of accident and link it to special words in the Fibonacci shift. In Section 3, after clarifying the role of accidents on the computation of R n V, we prove Theorem. Section 4 deals with the thermodynamic formalism. Following the strategy of [7] we specify and estimate the required quite involved quantities that are the core of the proof of Theorem 2.

7 RENORMALIZATION AND THERMODYNAMICS 7.4. Acknowledgement. The authors want to thank the anonymous referee for the valuable suggestions and references that helped to improve our paper. 2. Properties of H, K and R 2.. The set K as Sturmian subshift. In addition to being a substitution subshift, K, σ is the Sturmian subshift associated to the golden mean rotation, T : x x + mod. The golden mean is = and it satisfies 2 = +. Fixing an orientation on the circle S, let ab denote the arc of points between a and b in the circle in that orientation. Define the itinerary ex = e 0 e... of a point x S as { 0 if T i e i = x 0 if Tx i 0. Then it turns out that e2 = ρ, the fixed point of the substitution. 0 2 X X 0 X Figure. Coding for Fibonacci Sturmian subshift. There is an almost i.e., up to a countable set one-to-one correspondence between points in K and codes of orbits of S, T, expressed by the commutative diagram S T S π π K σ K and π is a bijection, except at points T n S, n 0. Since Lebesgue measure is the unique T -invariant probability measure, µ K := Leb π is the unique invariant probability measure of K, σ.

8 8 HENK BRUIN AND RENAUD LEPLAIDEUR We will use the same terminology for both K and S. For instance, a cylinder C n x for x S is an interval, with the convention that C n x = π C n πx, and we may confuse a point x S and its image πx K. Definition 2.. Let A K denote the set of finite words that appear in ρ. A word ω := ω 0... ω n A K is said to be left-special if 0w and w both appear in A K. It is right-special if w0 and w both appear in A K. A left and right-special word is called bi-special. A special word is either left-special or right-special. Since ρ has n+ subwords of length n a characterization of Sturmian words, there is exactly one left-special and one right-special word of length n. They are of the form ρ 0... ρ n and ρ n... ρ 0 respectively, which can be seen from the forward itinerary ex for x S close to and backward itinerary ex for x S close to 0. Sometimes the left and right-special word merge into a single bi-special word ω, but only one of the two words 0ω0, and ω appears in A K, see [, Section ]. Proposition 2.2. Bi-special words in A K are of the form ρ 0... ρ Fm 3 and for each m 3, ρ 0... ρ Fm 3 is bi-special. We prove this proposition at the end of Section Results for H n. We recall that κ a w is the number of symbol a in the finite word w. Lemma 2.3. For any finite word w, the following recursive relations hold: κ 0 H n w = F n κ 0 w + F n κ w; κ H n w = F n κ 0 w + F n 2 κ w; H n w = F n+ κ 0 w + F n κ w = H n w + H n 2 w, where H 0 w = w, H w = Hw. Since we have defined F 2 = and F = 0, see 3, these formulas hold for n = 0 and n = as well. Proof. Since H n 0 contains F n+ zeroes and F n ones, while H n 0 contains F n zeroes and F n 2 ones, the first two lines follow from concatenation. The third line is the sum of the first two, and naturally the recursive relation follows from the same recursive relation for Fibonacci numbers. Since K, σ, µ K is uniquely ergodic, and isomorphic to S, T, Leb, we immediately get that { κ a H n w 0 = if a = 0, 6 lim = n + H n w 0 = if a =.

9 RENORMALIZATION AND THERMODYNAMICS 9 Lemma 2.4. Assume that x and y have a maximal common prefix w. Then H n x and H n y coincide for T n w + F n+2 2 digits, where T n w is defined by 7 T 0 w = w, T w = Hw, T n+2 w = T n+ w + T n w. Proof. For x = w0... and y = w..., we find w w 0 H Hw 0 Hw 0 0 H H2 w 0 0 H 2 w H H3 w H 3 w H where we used that Ha starts with 0 for both a = 0 and a =. For T n w := H n w, the recursive formula 7 follows as in Lemma 2.3. Iterating the words 0 and 0 by H, we find: H H H Thus H n 0 = H n 0 = F n+2 and the common prefix of H n 0 and H n 0 has length F n+2 2. Therefore, if x = w0 and y = w0, then H n x and H n y coincide for T n w + F n+2 2 digits. A similar argument works for x = w00 and y = w0. Corollary 2.5. For x K and n N, H n x and ρ coincide for at least F n+3 2 digits if x [0] and for at least F n+2 2 digits if x []. Proof. If x [0], then, by Lemma 2.4, H n x coincides with H n ρ = ρ for at least T n 0+F n+2 2 digits. But T n 0 = H n 0 = F n+, so T n 0+F n+2 2 = F n+3 2. If x [], then Hx [0] and the previous argument gives that H n x coincides with H n ρ = ρ for at least F n+2 2 digits. Proof of Proposition 2.2. We iterate the blocks 0 0, 0 0 and 0 under H: H H H, so the common central block here is bi-special, and it is the same as the common block v of H n 0 and H n 0 of length F n+2 2 in the proof of Lemma 2.4. Thus we have found the bi-special word of length F n+2 2, and every prefix and suffix of v is left and right-special respectively. The fact that these are the only bi-special words can be derived from the Rauzy graph for this Sturmian shift, see e.g. [, Sec. ]. In their notation, there is a bi-special word of length k if the two special nodes

10 0 HENK BRUIN AND RENAUD LEPLAIDEUR in the Rauzy graph coincide: D k = G k. The lengths of the two buckles of nonspecial nodes between D k = G k are two consecutive Fibonacci numbers minus one, as follows from the continued fraction expansion Therefore, the complexity satisfies = k + = pk = #{nodes of Rauzy graph of order k} = F n + F n +, so indeed only the numbers k = F n+ 2 can be the lengths of bi-special words Iterations of the renormalization operator. The renormalization operator for potentials can be rewritten as recall the definition of F a n, from 4 9 RV [a] = F a V σ j H [a]. This leads to an expression for R n V. The main result here is Lemma 2.7, which shows that 0 R n V x = where n = F n V σ j H n x, { n + if x [0], n if x []. The substitution H solves a renormalization equation of the form 2. If x = 0x..., then Hx = 0Hx... and σ 2 Hx = H σx. If x = x... then σ Hx = H σx. The renormalization equation is thus more complicated than for the constant length case. We need an expression for iterates of H and σ. Lemma 2.6. Given k 0 and a = 0,, let w = w w 2... w F a k for every 0 i < F a k. H σ i H k [a] = σ Hw...w i H k+ [a] = H k a. Then Proof. For k = 0 this is true by default and for k =, this is precisely what is done in the paragraph before the lemma. Let us continue by induction, assuming that the statement is true for k. Then σ i removes the first i symbols of w = H k a, which otherwise, under H, would be extended to a word of length Hw... w i. We need this number of shifts to remove Hw... w i from H[w] = H k+ [a].

11 RENORMALIZATION AND THERMODYNAMICS Lemma 2.7. For every k 0 and a = 0,, we have R k V [a] = S F a k V H k [a], where S n V = n i=0 V σi denotes the n-th ergodic sum. Proof. For k = 0 this is true by default. For k =, this follows by the definition of the renormalization operator R. Let us continue by induction, assuming that the statement is true for k. Write w = H k a and t i = Hw i = F wi. Then R k+ V [a] = RV S F a k V H k [a] induction assumption Fk a ti = V σ j H σ i H k [a] by formula 9 as required. = = i=0 F k a ti i=0 F a k+ l=0 V σ j+ Hw...w i H V σ l H k+ [a], H k [a] by Lemma Special words are sources of accidents. Overlaps of ρ with itself are strongly related to bi-special words. They are of prime importance to determine the fixed points of R and their stable leaves, see e.g. formula 2 below. Dynamically, they correspond to what we call accident in the time-evolution of the distance between the orbit and K. For most x close to K, dσx, K = 2dx, K, but the variation of dσ j x, K is not always monotone with respect to j. When it decreases, it generates an accident: Definition 2.8. Let x Σ and dx, K = 2 n. If dσx, K 2 n, we say that we have an accident at σx. If there is an accident at σ j x, then we shall simply say we have an accident at j. The next lemma allows us to detect accidents. Lemma 2.9. Let x = x 0 x... coincide with some y K for d digits. Assume that the first accident occurs at b. Then x b... x d is a bi-special word in A K. Moreover, the word x 0... x d is not right-special. Proof. By definition of accident, there exists y and y in K such that dx, K = dx, y and dσ b x, K = dσ b x, y. Figure 2 shows that the word x b... x d is bi-special because its two extensions y and y in K have different suffix and prefix for this word.

12 2 HENK BRUIN AND RENAUD LEPLAIDEUR y d y x b Figure 2. Accident and bi-special words It remains to prove that x 0... x d is not right-special. If it was, then x 0... x d x d = y 0... y d y d would a K-admissible word, thus dx, K 2 d+ 2 d. 3. Proof of Theorem 3.. Control of the accidents under iterations of R. We compute R n V and show that accidents do not crucially perturb the Birkhoff sum involved. This will follow from Corollaries 3.2 and 2.5. Note that Lemma 2.4 shows that H is one-to-one. The next proposition explains the relation between the attractor K and its image by H. Proposition 3.. The subshift K is contained in HK σ HK. More precisely, if [0] K HK and [] K σ HK. Proof. First note that Lemma 2.4 shows that H is one-to-one and recall that the word is forbidden in K. Hence, each digit in x = x 0 x x 2... K is followed and preceded by a 0 unless the is in first position. Assuming x 0 = 0, we can uniquely split x into blocks of the form 0 and 0. In this splitting, we replace each single 0 by and each pair 0 by 0. This produces a new word, say y, and Hy = x by construction. Denote this operation by H. It can be used on finite words too, provided that the last digit is. If x 0 =, we repeat the above construction with 0x, and x = σ Hy. It remains to prove that y K. For every x K, there is a sequence k n such that σ kn ρ x. Assume again that x 0 = 0. Then we can find a sequence l n k n / such that H σ ln ρ = σ kn ρ. Therefore y := lim n σ ln ρ K satisfies Hy = x. Finally, for x 0 =, we repeat the argument with 0x. Corollary 3.2. If dx, K = dx, y with y K, then dh n x, K = dh n x, H n y for n 0.

13 RENORMALIZATION AND THERMODYNAMICS 3 Proof. Write x = wa and y = wa where a is an unknown digit and a its opposite. Note that H n x starts with 0 for any n. Assume that there is some z K such that dhx, z < dhx, Hy. Case : x = w0... and y = w... Necessarily, y = w0. Therefore Hx = Hw0... and Hy = Hw By assumption, z coincides with Hx longer than Hy, which shows that z starts as z = Hw0... Consequently, H z = w0... and this contradicts that dx, K = dx, y. Case 2: x = wb... and y = w0.... Then Hx = Hw00... since Hb starts with 0 regardless what b is and Hy = Hw0... Again z coincides with Hx longer than Hy and thus z starts as Hw00. The 0 before last position is necessarily a single zero for the H -procedure and thus H z coincide with x for longer than y. This is a contradiction. Consequently, for both cases we have shown dhx, K = dhx, Hy. result follows by induction. The By 0, R n V is given by a Birkhoff sum of F n terms where n = n + or n as in. To compute R n V x, we need an estimate for dσ j H n x, K, for 0 j F n. The key point is that no accident can occur for these j. This follows from the next lemma. Lemma 3.3. The sequence H n x has no accident in the first F n entries. Proof. We give the proof for x [], so n = n. The proof for x [0] is analogous. By Corollary 2.5, H n x coincides for at least F n+2 2 digits with ρ. If an accident happens in the first F n digits, say at entry 0 j < F n, then by Lemma 2.9, a bi-special word starts at j, which by Proposition 2.2 is a suffix of ρ of length F m 2 for some m. Since we have an accident, j + F m 2 F n+2, so m > n +. Hence ρ 0... ρ Fn+2 can be written as BBB where B is the suffix of ρ of length j and B is a suffix of ρ of length B /. Clearly B starts with 0. We can split it uniquely into blocks 0 and 0, and B fits an integer number of such blocks, because if the final block would overlap with the second appearance of B, then B would start with, which it does not. Therefore we can perform an inverse substitution H, for each block B and also for B because we can apply H on ρ 0... ρ Fn+2. This gives H BBB = CCC which has the same characteristics. Repeating this inverse iteration, we find that ρ starts with 00, or with 00, a contradiction. Let Nx, n be the integer such that 2 Nx,n = dh n x, K. By the previous lemma dσ j H n xk = 2 Nx,n j for every j < F n. For the largest value j = F N, we have dσ j H n x, K = 2 Tn+F n+2 2 F n. Therefore, if g is the α-density function

14 4 HENK BRUIN AND RENAUD LEPLAIDEUR for V, then we obtain 2 R n V x = F n g σ j H n Fn x Nx, n j + o α g σ j H n x. Nx, n j α 3.2. Proof of Theorem Ṽ is a fixed point. Take Ṽ as in 5. We show that R fixes Ṽ. Assume x / K is such that κ 0 x = n and κ x = m see the definition of κ a above the statement of Theorem. Then, by Lemmas 2.3 and 2.4 and the fact that Hx starts with 0, we get κ 0 Hx = n + m + κ 0 σ Hx = n + m, κ Hx = n κ σ Hx = n. If x starts with 0, then Hx starts with 0 and RṼ x = Ṽ Hx + Ṽ σhx n + m + + = log n + n + m + + n + n + m + + = log n + n + m + n + + log n + m + + = log n + n + m + n + n + = log + m + + n + + m + n + m + 2 = log = log n + m + n + m + n log n + m + n + 2 n + m + n + n + m + n + since 2 = + n + m + n + m + = Ṽ x.

15 RENORMALIZATION AND THERMODYNAMICS 5 If x starts with, then Hx starts with 0 and n + m + + RṼ x = Ṽ Hx = log n + n + m + + n + n + m + + n + 2 = log n + m + + n + 2 n + + m = log n + + m n + m + 3 = log 2 n + m + 2 n + m + 2 n + m + 2 = log = log = n + m + n + m + 2 Ṽ x Toeplitz summation. Next, we compute R n V for arbitrary V X. For V X α with α, we can perform the same computation, and obtain an extra factor Fn α which will push the limit to 0 or according to whether α > or α <. We will show that F n g σ j H n x actually converges, which immediately yields Nx,n j that o converges to 0. We can thus ignore the little o terms. Fn g σ j H n x Nx,n j Lemmas 2.3 and 2.4 and Corollary 3.2 give Nx, n := log 2 dh n x, K = T n + F n+2 2 We thus have to compute the limit of F n g σ j H n x F n+ κ 0 x + F n κ x + F n+2 j + 2 for T n := F n+ κ 0 x + F n κ x. as n +, where n = n + if x [0] and n = n if x []. Moreover g is a non-negative continuous function, hence uniformly continuous. For y K closest to x, the point σ F n H n x coincides with σ F n H n y for at least F n 2 digits. There exists a sequence ε n 0 such that g σ j H n x g σ j H n y εn, for every j F n. Finally, Binet s formula F n+ F n = 5 n shows that F n+ κ 0 x + F n κ x + F n+2 j + 2 = F n+ κ 0 x + κ x + j F n+ + ε n = F n κ 0 x + κ x + 2 j F n + ε n,

16 6 HENK BRUIN AND RENAUD LEPLAIDEUR where ε n and ε n tend to 0 as n +. Combining ε, ε and ε in a single o, we can rewrite the above equalities to 3 F n g σ j H n x F n+ κ 0 x + F n κ x + F n+2 j + 2 = + o F n F n g σ j H n y, X n where X n = κ 0 x + κ x + if x [0] and X n = κ 0 x + κ x + 2 if x [] Convergence of the weighted sum in 3. The reader can verify that we are here considering a Toeplitz summation method, with a regular matrix see [5, Definition 7.5] and [28], up to a renormalization factor, which is the limit of F n F n X n j. F n This expression is a Riemann sum, and converges to Ṽ x as n. Using notations from [28], coefficients of the Toeplitz matrix are a n,k := Ṽ x F n X n k for k F n and a n,k = 0 otherwise, and they satisfy the Müller criterion see [9] and [28, p. equality.4] + lim a n,k a n,k+ log k = 0. n + k= Therefore 4 lim n + F n F n g σ j z X n j F n = Ṽ x g dµ K µ K -a.e. Nevertheless, our expression in 3 is different, because the point z = H n y depends on n. A priori, this may generate fluctuations in the convergence, but we prove here that this is not the case. The main argument is that K, σ is uniquely ergodic. This implies that the convergence in 4 is uniform in z. Indeed, if it is not uniform, we can find ε > 0 and a sequence of z n such that for every n, Fn g σ j z n F n Ṽ x g dµ X n j K > ε for F n every n. Then any accumulation point µ of the family of measures µ n := F n F n X n j δ σ j z n F n is σ-invariant because F n +, supported on K, and g dµ g dµ K. This contradicts the unique ergodicity for K, σ. j F n

17 RENORMALIZATION AND THERMODYNAMICS 7 Therefore, the convergence in 4 is uniform in z and this shows that F n g σ j H n y F n X n Ṽ x g dµ K. This finishes the proof of Theorem. j F n 4. Proof of Theorem The case log n+ n+. We first consider the potential ϕx = log n n dx, K = 2 n, leaving general potentials in X for later. when 4... Strategy, local equilibria. Fix some cylinder J such that the associated word, say ω J, does not appear in ρ as e.g.. We follow the induction method presented in [7]. Let τ be the first return time into J possibly τx = +, and consider the family of transfer operators L Z,β : ψ L Z,β ψ x L Z,β ψx := + n= y J τy=n σ n y=x e β Snϕy nz ψy, which acts on the set of continuous functions ψ : J R. Following [7, Proposition ], for each β there exists Z c β such that L Z,β is well defined for every Z > Z c β. By [7, Theorem ], Z c β 0 because the pressure of the dotted system which in the terminology of [7] is the system restricted to the trajectories that avoid J is larger or equal to the pressure of K which is zero. We shall prove Proposition 4.. There exists β 0 such that L 0,β J x < for every β > β 0 and x J. We claim that if Proposition 4. holds, then [7, Theorem 4] proves that Pβ = 0 for every β > β 0, and µ K is the unique equilibrium state for βϕ. To summarize [7] and adapt it to our context, the pressure function 2 satisfies Z c β Pβ, Pβ 0 and Pβ = 0 if logl 0,β J < 0 see Figure 3. As long as Pβ > 0, there is a unique equilibrium state and it has full support. In particular this shows that the construction does not depend on the choice of J. If Proposition 4. holds, then Pβ = 0, for β > β c and no equilibrium state gives positive weight to J and also to any cylinder which does not intersect K. Therefore, µ K is the unique equilibrium state. 2 We will see that L 0,β J is a constant function on J.

18 8 HENK BRUIN AND RENAUD LEPLAIDEUR z z =logλ 0,β z = Pβ Z c β β Figure 3. The pressure between Z c β and log λ 0,β := L 0,β J Proof of Proposition 4.-Step. We reduce the problem to the computation of a series depending on β. Note that ϕx only depends on the distance from x to K. This shows that if x, x J and y, y J are such that with ω {0, } n, τy = τy = n, then y = ωx, y = ωx, S n ϕy = S n ϕy. In other words, L Z,β J is a constant function, and then equal to the spectral radius λ Z,β of L Z,β. Consequently, to compute λ Z,β, it suffices to compute the sum of all e β Snϕω nz, where ω is a word of length n + ω J, starting and finishing with ω J. Such a word ω can also be seen as a path of length n starting from J and returning for the first time to J at time n. We split such a path in several sub-paths. We fix an integer N and say that the path is free at time k if ω k... ω n ω J is at distance larger than 2 N to K. Otherwise, we say that we have an excursion. The path is thus split into intervals of free times and excursions. We assume that N is chosen so large that 0 is a free time. This also shows that for every k n, dσ k ωω J, K is determined by ω k... ω n. If k is a free time, then ϕσ k ωω J A N := log + N. Denote by k0 the maximal integer such that k is a free time for every k k 0. Then S k0 +ϕ k 0 + A N and there at most 2 k0+ such prefixes of length k 0 +. Now, assume that every j for k 0 + j k 0 + k is an excursion time, and assume that k is the maximal integer with this property. To the contribution S k0 +ϕωω J we must add the contribution S k ϕσ k 0+ ωω J of the excursion. Then we have a new interval of free times, and so on. We can compute L 0,β J by grouping together paths with the same number of free periods and excursions. If we denote by C E the total contribution of all paths with exactly one excursion and only starting at the

19 RENORMALIZATION AND THERMODYNAMICS 9 beginning of the excursion, then we have 5 λ 0,β = L 0,β J + k= + k 0 =0 e k 0+βA N +log 2 k+ C k E. The sum in k accounts for k + free intervals with k excursions between them. The sum in k 0 accounts for the possible length k 0 + for an interval of free times. These events are potentially not independent but the sum in 5 includes all paths, possible or not, and therefore yields an upper bound. The integer N is fixed, and we can take β so large that βa N < log 2. This shows that the sum in k 0 in 5 converges and is as close to 0 as we want if β is taken sufficiently large. To prove Proposition 4., it is thus sufficient to prove that C E can be made as small as we want if β increases Proof of Proposition 4.-Step 2. We split excursions according to their number of accidents, see Definition 2.8. Let x be a point at a beginning of an excursion. Let B 0 := 0 = b 0, B := b > b 0, B 2 := b + b 2 > b, B 3 := b + b 2 + b 3,..., B M := b + b b M be the times of accidents in the excursion. There is y 0 K such that x shadows y 0 at the beginning of the excursion, say for d 0 iterates. Let y i K, i =,..., M, be the points that x starts to shadow at the i-th accident, for d i iterates. y 0 x y d y2 y 3 d 0 d 2 b b 2 b 3 Figure 4. Accidents during an excursion. By Lemma 2.9, x bi+... x di is bi-special and by Proposition 2.2, d i b i+ = F ni+ 2 for some n i+. Remark. We emphasize that the first d i entries of y i form neither a right-special word due to Lemma 2.9 nor a left-special word, because otherwise there would be an accident earlier.

20 20 HENK BRUIN AND RENAUD LEPLAIDEUR If there are M + accidents counting the first as 0, the ergodic sums for ϕ are S bi+ ϕσ B i x = = b i+ k=0 b i+ k=0 = log ϕ σ B i+k x log d i + k d i k d i + b i+ = log +, d i + b i+ d i + b i+ for 0 i M, while the ergodic sum of the tail of the excursion is 6 S dm ϕσ B M x = d M k=0 ϕ σ B M +k x = log d M + N +. We set e i := e β S b i ϕσ B i x for i =... M and e M+ := e β S d M ϕσ B M x. Computing C E, we can order excursions according to their number of accidents M + and then according to the contribution of each accident. Let E i stand for the total contribution of all possible e i s between accidents i and i. Then 7 C E = + M= Proof of Proposition 4.-Step 3. Let us now find an upper bound for E i. By definition, E i is the sum over the possible d i and b i of e i. Recall that d i b i = F ni 2, so b i and F ni determine d i. The key idea is that F ni and F ni+ determine the possible values of b i. This implies that E i can be written as an expression over the F ni and F ni+. For 2 i M each e i depends on F ni and b i. Let us show that for 2 i M, b i depends on n i and n i. Indeed, the sequence y i K coincides with ρ for F ni 2 initial symbols, and from entry b i+ has another d i b i+ = F ni+ 2 symbols in common with the head of ρ, but differs from x Bi +d i at entry d i, see Figure 4. Thus we need to find all the values of d i > F ni 2 such that ρ 0... ρ di ends the bi-special word ρ 0... ρ Fni+ 3 but is itself not bi-special. The possible starting positions of this appearance of ρ 0... ρ Fni+ 3 are the required numbers b i+. Lemma 4.2. Let us denote by b i+ j, j, the j-th value that b i+ can assume. Then M+ 8 b i+ j maxf ni F ni+, F ni + jf ni+ 2. i= E i.

21 RENORMALIZATION AND THERMODYNAMICS 2 This will allow us to find an upper bound for E i for i M later in this section. Proof. We abbreviate the bi-special words L k = ρ 0... ρ Fk 3 for k 4. For the smallest value d i F ni 2 so that ρ 0... ρ di ends in but is not identical to a block L ni+, this block starts at entry: F ni F ni+ if n i+ < n i and n i n i+ is even, b i+ 0 = F ni F ni+ if n i+ < n i and n i n i+ is odd, F ni if n i+ n i. However, if n i+ < n i then d i = F ni 2, and if n i+ n i then d i = F ni+ + 2 and thus ρ 0... ρ di is right-special, contradicting Lemma 2.9. Therefore we need to wait for the next appearance of L ni+. For the Rauzy graph of the Fibonacci shift, the bi-special word L k is the single node connecting loops of length F k and F k 2, see [, Section ]. Therefore the gap between two appearances of L k is always F k 2 or F k. This gives b i+ j + b i+ j + F ni+ 2 for all j 0 and 8 follows. For i =, we introduce the quantity n 0, coinciding with the overlap of the end of the previous fictitious word, say y. The point is that y 0 is the beginning of the excursion, thus the first accident. Then F n0 N and F n > N which yields n 0 < n. Formula 8 can now be applied. Therefore b = F n0 2 + j F n 2 with j 0. The estimate E M+ = d e β log Fn M +d N+ = d follows from by 6, with d M = F nm + d and d. β FnM + d N + β FnM N + β N + Recall that within excursions, all F nj N + for all j, where N can be chosen as large as we want. We also remind Binet s formula F n = 5 n+ / n+, which allows to replace F n by n+ 5, and we to treat the quantities as negligible compared to n i for large n i, as it is the case. Therefore, assuming that β >, there exists a constant C such that E i = j β log e + maxfn i Fn i+,f n i +jf n i+ 2 Fn i+ C j + max n i n i+, n i n i+ + j/ 2 β for 2 i M C2 + max n i n i+, n i n i+ β. β

22 22 HENK BRUIN AND RENAUD LEPLAIDEUR Let P = log N 5 log = max{n N : F n N}. Then 7 yields 9 C E + M=0 2 M N + β β E + max n i n i+, n i n i+ β P n M β. n,...n M >P n 0 P Proof of Proposition 4.-Step 4. We show that C E 0 as β +. Proposition 4.3. There exists A = Aβ 0, with lim β + A = 0 such that C E 2P N + β + n= nβ + M=0 A M M i=0 n i i!. Before proving this proposition, we show that it finishes the proof of Proposition 4.. The series has only positive terms. Clearly, + M=0 AM M n i i=0 i! A en, so the main sum converges if β > e. Thus Proposition 4.3 implies that C E 0 as β +. Therefore, inequality 5 shows that if β +, then λ 0,β 0 too, and hence Proposition 4. is proved. The rest of this subsection is devoted to the proof of Proposition 4.3. The following lemma follows easily by induction: Lemma 4.4. Let η and y be positive real numbers. Then for every n, y x n e ηx y dx = n n! j! y j η n+ j. For some positive integer n and real numbers ξ, ζ > 0, let D n = d n,i,j n+,n i=,j= be the matrix with n + rows and n columns defined by j! i! ζj i+ if i j, ξ d n,i,j := if i = j +, j 0 if i > j +,

23 or in other words: D n = RENORMALIZATION AND THERMODYNAMICS 23 0!ζ!ζ 2 2!ζ 3... j!ζ j... n!ζ n ξ ζ... n 2!ζ n ξ ζ 2 ξ j! i! ζj i ζ ζ 2 j ξ 0 0 ζ n We call w non-negative and write w 0 if all a entries of w are non-negative. This defines a partial ordering on vectors by w w w w 0. Lemma 4.5. Assume 0 < ζ < and set K :=. Then, for every n, ζ D n K n 0! K n! K n 2!. K n n! ξ K n 0! K n! K n 2!. K n n!. Proof. This is just a computation. For the first row we get For row i > we get n j!ζ j K n. j! ζ Kn. ζ Kn. j= K n n i i 2! + j! Kn ζj i+ i! j! = Kn + ζ + ζ 2... Kn i! i!. j=i Proposition 4.6. Set ζ := and K =. For every M 2, consider β log ζ integers n,... n M, with n M > P. Then, n,...,n M >P M i= + max n i n i+, n i n i+ M β K M. ξ n i=0. n M P i i!

24 24 HENK BRUIN AND RENAUD LEPLAIDEUR Proof. Note that M + max n i n i+, n i n i+ β n,...,n M >P i= =... + max n n 2, n n 2 β n M = n 2 = n = + max n 2 n 3, n 2 n 3 β... + max n M n M, n M n M β. This means that we can proceed by induction. Now n =P + + max n n 2, n n 2 β n2 P n 2 P + = n 2 P + + x n 2 β dx + n 2 n 2 n 2 e β x n 2 log dx e x n 2 ζ dx, x n 2 β dx because ζ =. This shows that the result holds for M = 2. β log Assuming that the sum for M = p is of the form p a jn p P j, we compute the sum for M = p +. p a j n p=p + j j n p P j + max np n p+, n p n p+ β a j np+ P x P j a j n p+ P j+ + j + + x n p+ β dx + j n p+ x P j e a j x n p+ ζ n p+ dx. x P j x n p+ β dx

25 RENORMALIZATION AND THERMODYNAMICS 25 Set w w = w i w i, for vectors w = w,..., w p+ and w = w,..., w p+. Lemma 4.4 yields p a j n p=p + j D p n p P j + max np n p+, n p n p+ β a j j + n p+ P j+ + a 0 a n p+. a p. n p p+. Lemma 4.5 concludes the proof of the induction. j i=0 j! i! ζj i+ n p+ P i Proof of Proposition 4.3. We have just proved that + max n i n i+, n i n i+ β P n M β n,...n M >P K M + n M =P + M n M P j n M P β. j! It remains to sum over n 0. Note that in that case, there are only P terms of the form + because n + n 0 n 2 + j β 0 P < n for each possible n 0, + β = + n 0 n + j 2 for β 2. Finally, inequality 9 yields with A := β K = C E 2P N + β + + n 0 n 2 + β j= + 2 β + n 0 n 2 β + β, 2 + M=0 A M + n= M n β β + n 0 n 2 + j 2 n j j!,. This tends to 0 as β +. β log 4.2. End of the proof of Theorem 2.

26 26 HENK BRUIN AND RENAUD LEPLAIDEUR End of the case log n+. Proposition 4. shows that there exists some minimal β 0 such that λ 0,β < for every β > β 0. This also shows that Pβ = 0 for n β > β 0. Since Pβ is a continuous and convex function, it is constant for β > β 0. As P0 = log 2, there exists a minimal β c > 0 such that Pβ > 0 for every 0 β < β c. Clearly, β c β 0. We claim that for β < β c, there exists a unique equilibrium state and that it has full support. Indeed, there exists at least one equilibrium state, say µ β, and at least one cylinder, say J, has positive µ β -measure. Therefore, we can induce on this cylinder, and the form of potential see [7, Theorem 4] shows that there exists a unique local equilibrium state. It is a local Gibbs measure and therefore µ β is uniquely determined on each cylinder, and unique and with full support due to the mixing property. We claim that the pressure function Pβ is analytic on [0, β c ]. Indeed, each cylinder J has positive µ β -measure and the associated Z c β is the pressure of the dotted system that is: restricted to the trajectories that avoid J. This set of trajectories has a pressure strictly smaller than Pβ because otherwise, several equilibrium states would coexist. Therefore Pβ is determined by the implicit equation λ Pβ,β = and Pβ > Z c β for β [0, β c ]. The Implicit Function Theorem shows that Pβ is analytic. For β β c, the pressure Pβ = 0 and for cylinders J as above, we have Z c β 0. This shows that Z c β = 0 for every β β c. Due to the form of the potential, λ 0,β is continuous and decreasing in β. Now, the next result finishes the proof of Theorem 2 in the case that V x = log n+ n. Lemma 4.7. The parameters β c and β 0 coincide. Proof. As mentioned above β c β 0. Assume by contradiction β c < β 0. We claim that λ 0,βc >. Indeed, β λ 0,β is strictly decreasing, and λ 0,βc would yield that λ 0,β < for every β > β c. This would imply β c β 0 because β 0 is minimal with this property, and we have assumed β c < β 0. Now, for fixed β, Z λ Z,β is continuous and strictly decreasing and goes to 0 as Z +. Therefore, since λ 0,βc >, there exists Z > 0 such that λ Z,βc =. The local equilibrium state for this Z generates some σ-invariant probability measure 3 with pressure for β c ϕ equal to Z, thus positive, and this contradicts Pβ c = 0. This proves that β c = β 0 3 Since Z c β c = Pβ c = 0 < Z, the expectation of the return time is comparable to which converges. L Z,βc J Z,

27 RENORMALIZATION AND THERMODYNAMICS The general case V X. For V X, there exists κ > 0 such that V κϕ. This shows that the pressure function is constant equal to zero for β β 0 /κ. Again, the pressure is convex, thus non-increasing and continuous. We can define β c such that Pβ > 0 for 0 β β c and Pβ = 0 for β β c. The rest of the argument is relatively similar to the previous discussion. We deduce that for β < β c, there exists a unique equilibrium state, it has full support and Pβ is analytic on this interval. For β β c, it is not clear that λ 0,β decreases in β. However, we do not really need this argument, because if λ 0,β >, then the decrease of Z λ Z,β which follows from convexity argument and lim Z + λ Z,β = 0, is sufficient to produce a contradiction. References [] P. Arnoux and G. Rauzy. Représentation géométrique de suites de complexité 2n +. Bull. Soc. Math. France, 92:99 25, 99. [2] R. Baraviera, A. Leplaideur and A. O. Lopes. The potential point of view for renormalization. Stoch. Dyn., 24:250005, 34, 202. [3] R. Bowen. Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Lecture Notes in Mathematics, Vol Springer-Verlag, Berlin, nd ed by JR Chazottes. [4] M. Bramson and S. Kalikow. Nonuniqueness in g-functions. Israel J. Math., 84-2:53 60, 993. [5] H. Bruin and R. Leplaideur. Renormalization, thermodynamic formalism and quasi-crystals in subshifts. Comm. Math. Phys., 32: , 203. [6] H. Bruin and S. Vaienti. Return time statistics for unimodal maps. Fund. Math., 76:77 94, [7] A. O. Contreras, G. Lopes and Ph. Thieullen. Lyapunov minimizing measures for expanding maps of the circle. Ergodic Theory Dynam. Systems, 25: , 200. [8] D. Coronel and J. Rivera-Letelier. Low-temperature phase transitions in the quadratic family. Adv. Math., 248: , 203. [9] M. I. Cortez and J. Rivera-Letelier. Invariant measures of minimal post-critical sets of logistic maps. Israel J. Math., 76:57 93, 200. [0] H.-O. Georgii. Gibbs measures and phase transitions, volume 9 of de Gruyter Studies in Mathematics. Walter de Gruyter & Co., Berlin, second edition, 20. [] G. Grimmett. Probability on graphs, volume of Institute of Mathematical Statistics Textbooks. Cambridge University Press, Cambridge, 200. Random processes on graphs and lattices. [2] F. Hofbauer. Examples for the nonuniqueness of the equilibrium state. Trans. Amer. Math. Soc., , 977. [3] G. Iommi and M. Todd. Transience in dynamical systems. Ergodic Theory and Dynam. Systems, 33: , 203. [4] G. Keller. Equilibrium states in ergodic theory, volume 42 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge, 998. [5] L. Kuipers and H. Niederreiter. Uniform distribution of sequences. Wiley-Interscience [John Wiley & Sons], New York, 974. Pure and Applied Mathematics. [6] F. Ledrappier. Un exemple de transition de phase. Monatsh. Math., 832:47 53, 977.

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