Entropy for zero-temperature limits of Gibbs-equilibrium states for countable-alphabet subshifts of finite type

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1 Entropy for zero-temperature limits of Gibbs-equilibrium states for countable-alphabet subshifts of finite type I. D. Morris August 22, 2006 Abstract Let Σ A be a finitely primitive subshift of finite type over a countable alphabet. For suitable potentials f : Σ A R we can associate an invariant Gibbs equilibrium state µ tf to the potential tf for each t 1. In this note, we show that the entropy hµ tf converges in the limit t to the maximum entropy of those invariant measures which maximize R f dµ. We further show that every weak-* accumulation point of the family of measures µ tf has entropy equal to this value. This answers a pair of questions posed by O. Jenkinson, R. D. Mauldin and M. Urbański. Key words: Gibbs state; equilibrium state; ground state; maximizing measure; countable alphabet subshift of finite type; entropy. 1 Introduction Let σ : Σ A Σ A be a subshift of finite type on a countable alphabet I N, and let f : Σ A R be uniformly continuous. Under suitable conditions to be made precise below it may be shown that f admits a unique σ-invariant Gibbs state µ f. In this note we study the behaviour of families of invariant Gibbs states µ tf t 1 in the limit t. Our result provides a complete characterisation of the limiting behaviour of the entropy hµ tf, and extends previous results known in the case where the alphabet I is finite. In particular, we answer two questions posed by O. Jenkinson, R. D. Mauldin and M. Urbański in a previous investigation of this topic [6]. School of Mathematics, University of Manchester, Sackville Street, Manchester M60 1QD, U.K. 1

2 In the thermodynamic interpretation, our parameter t corresponds to an inverse temperature of a system, and the measure µ tf corresponds to the equilibrium of the system at temperature 1/t. The limit t is therefore a zero temperature limit, and accumulation points of the measures µ tf constitute ground states. The aim of the present note is to show that the entropy µ tf is continuous in the zero temperature limit, in the sense that hµ = lim t hµ tf whenever µ is an accumulation point at infinity of the measures µ tf. The history of results of this type in the finite-alphabet case is somewhat convoluted; results analogous to those in the present paper and in [6] were proved independently on a number of occasions, e.g. [1, 2, 3, 5]. The history of these and related results is discussed briefly in [4, 5]. 2 Definitions and statement of theorem In this section we review some necessary background from the thermodynamic formalism of countable-alphabet subshifts of finite type. Our reference for this section is the work of R.D. Mauldin and M. Urbański [7, 2] but see also [8, 11, 12]. We then give a formal statement of our results. Let I N be a countably infinite set, and let A: I I {0, 1} be a square matrix. We define the shift space Σ A associated to A to be the set { } Σ A := x = x n n 1 : x n I and Ax n, x n+1 = 1 for all n 1, and define the shift map σ : Σ A Σ A by ] σ [x n n 1 = x n+1 n 1. When i 1,..., i m I m we define the corresponding cylinder set by [i 1,..., i m ] := {x = x n n 1 Σ A : x j = i j for all 1 j m}. We then define a topology on Σ A by declaring every cylinder set to be open. Given f : Σ A R and n > 0, define var n f := fx fy. i 1,...,i n I n x,y [i 1,...,i n] We say that f has summable variations if the quantity f var = n=1 var n f is finite. Note that if f var < then f is continuous, but need not be bounded. For n > 0 and x Σ A, we use the abbreviation S n fx to denote the ergodic sum n 1 j=0 fσj x. We denote the set of σ-invariant Borel probability measures on Σ A by M σ. We equip M σ with the following weak-* topology: we say that 2

3 µ n n 1 converges weakly-* to µ if and only if f dµ n f dµ for every bounded continuous f : Σ A R. In our case, an equivalent condition is that µ n B µb for all cylinder sets B. Given f : Σ A R, we define the maximum ergodic average of f to be the quantity { βf = f dν : ν M σ }. When f dµ = βf for µ M σ, we say that µ is a maximising measure for f. We denote the set of all maximising measures for f by M max f. The study of maximising measures is a topic of substantial recent research interest: for further information on this topic we direct the reader to the survey article of O. Jenkinson [4]. For the purposes of this article, we will define the pressure of a continuous function f : Σ A R to be the quantity P f = { hν + f dν : ν M σ and } f dν, 1 where hν denotes the entropy of the measure ν see e.g. [13]. Some alternative formulations of the pressure may be found in [7, 2]. We say that a measure µ M σ is an equilibrium state for f if P f = hµ+ f dµ, and say that µ is an invariant Gibbs measure for f if µ M σ and there is a real constant Q > 1 such that Q 1 µ [x 1,..., x n ] esnfx np f Q for every n > 0 and x Σ A. When I is finite and σ is topologically mixing, it is well known that every Hölder continuous f possesses a unique invariant Gibbs measure, and that this measure is also the unique equilibrium state for f [9, 10]. When I is infinite, it may be the case that certain Hölder continuous functions fail to have invariant Gibbs states. To ensure the existence of invariant Gibbs states we impose the following additional condition on f. Definition 2.1. We call a continuous function f : Σ A R summable if it satisfies the condition exp fx <. x [i] i I We write S A for the set of all summable functions f : Σ A R which satisfy the additional condition f var <. 3

4 In the case where I is infinite, the existence of invariant Gibbs states requires a further condition on A. We say that the matrix A is finitely primitive if there exist an integer N and a finite set J I such that for each a, b I we can find i 1,..., i n J N such that the cylinder set [a, i 1,..., i n, b] Σ A is nonempty. R. D. Mauldin and M. Urbański proved the following theorem see [7, 8]: Theorem Mauldin-Urbański. If g S A and A is finitely primitive, then P g is finite and g has a unique invariant Gibbs measure µ g. This measure satisfies for every x Σ A and n > 0. e 4 g var µ g [x 1,..., x n ] e Sngx np g e 4 g var 2 In fact, the above theorem was proved under the stronger assumption that var n g = Oθ n for some 0 < θ < 1, but as remarked in [6] the proof generalises without difficulty. The requirement of finite primitivity was later shown by O. Sarig to be necessary as well as sufficient [12]. It should be noted that while every g S A has finite pressure and possesses an invariant Gibbs state, in some cases this invariant Gibbs state may have infinite entropy, and hence is not an equilibrium state in the sense defined above. However, Mauldin and Urbański were able to show the following: Proposition 1. Let g S A and pose that the additional condition gx exp gx < 3 x [i] x [i] i holds. Then the invariant measure µ g satisfies hµ g < and g dµ g <, and µ g is an equilibrium state for g in the sense that hµ g + g dµ g = P g. We will see in the following section that if f S A, then for each t > 1 the function tf satisfies condition 3 above automatically. Hence the map t µ tf which will be studied in this article is well-defined, and for every t > 1 the measure µ tf has finite entropy and is both an equilibrium state and a Gibbs state. O. Jenkinson, R.D. Mauldin and M. Urbański showed in [6] that the family of measures µ tf t 1 has at least one accumulation point in M σ, and showed that every accumulation point must be a maximising measure for f. They then asked whether the entropy of these accumulation points could be characterised either as the remum of the entropies of the maximising measures, or as the limit of the entropies hµ tf. In this note we prove the following: 4

5 Theorem 1. Let σ : Σ A Σ A be a finitely primitive subshift of finite type, and let f S A. For each t > 1, let µ tf be the unique Gibbs equilibrium state associated to the potential tf, and let µ be any accumulation point at infinity of µ tf t 1. Then, hµ = lim P tf tβf = lim hµ tf = hν. t t ν M maxf This theorem constitutes a positive answer to Questions 1 and 2 in [6]. 3 Proof of Theorem 1 It is convenient to break up the proof of Theorem 1 into several separate statements. In particular, the theorem is obtained by combining Lemmata 3.2 and 3.5 below. We include some observations from [6]. Throughout this section we assume that the matrix A is finitely primitive and that f S A. Lemma 3.1. For each t > 1 we have tfx exp i x [i] tfx x [i] so that the conclusions of Proposition 1 apply to µ tf. <, Proof. Let t > 1. Since f S A it is clear that there exists C t > 0 such that tfx e t 1 x [i] fx C t. i Hence, i x [i] tfx exp x [i] tfx x [i] C t exp i fx x [i] <. Lemma 3.2. The maps t hµ tf and t P tf tβf are decreasing and bounded below on the interval 1,, and satisfy lim hµ tf = lim P tf tβf hν. t t ν M maxf Proof. Recall from [7, 2] that the map t P tf is analytic, and that its first and second derivatives admit the characterisations P tf = f dµ tf βf 5

6 and [ P 1 tf = lim S n f n n f dµ tf ] 2 dµ tf 0. Note also that P tf tβf = P tf tβf as a consequence of 1. We have P tf tβf = P tf βf = f dµ tf βf 0 for every t > 1, and so P tf tβf is decreasing. To see that t hµ tf is decreasing we note that for every t > 1, so that hµ tf = P tf tp tf h µ tf = tp tf 0. Clearly hµ tf 0 for all t > 1, and P tf tβf 0 by the variational principle 1. Since both maps are decreasing, we conclude that lim t hµ tf and lim t P tf tβf both exist. It follows that the limit lim tp tf tβf = lim P tf tβf hµ tf t t exists also. Since P tf tβf is convergent and monotone, the integral t=2 P tf βf dt must be finite; it follows that lim t tp tf tβf cannot be nonzero, and so lim hµ tf = lim P tf tβf t t as required. To complete the proof, we observe that P tf tβf ν Mmaxf hν for all t > 1 as a consequence of the variational principle 1. Lemma 3.3. Let f S A. Then, lim t t 1 P tf = βf. Proof. This follows from Lemma 3.2 and the relation P tf = P tf tβf + tβf. Lemma 3.4. Let µ be an accumulation point at infinity of µ tf t>1 in the weak-* topology. Then µ is a probability measure and is maximising for f. Proof. See [6, Theorem 1]. Lemma 3.5. Let µ = lim k µ tk f be an accumulation point at infinity of µ tf t>1 in the weak-* topology. Then, hµ = lim hµ tf = hν. t ν M maxf 6

7 Proof. By Lemmata 3.2 and 3.4 we have hµ and so it suffices to establish hν lim hµ tf, ν M t maxf lim hµ tf hµ. t For K I and n > 0, we use the symbol ˆKn to denote set of all n-tuples i 1,..., i n K n such that the cylinder [i 1,..., i n ] Σ A is nonempty. For each n > 0 we define A n = {[i 1,..., i n ] Σ A : i 1,..., i n În }. Clearly each A n is a countable measurable partition of Σ A. For x [0, 1] define φx = x log x when x 0, and φ0 = 0; note that φ is continuous. Given ν M σ and n > 0, we let H ν A n = φ ν [i 1,..., i n ]. i 1,...,i n I n Using standard facts from entropy theory, the entropy hν of an invariant 1 measure ν M σ is equal to hν = inf n 1 n H ν An. We claim that for each n > 0, lim H µ t k f A n = H µ A n. 4 k Since x [i] fx as i, we may choose a finite set J I such that 4 f var nβf n 1 f + fx < 0 5 i I\J x [i] and and 4 f var + nβf + 1 i 1,...,i n În \Ĵ n S n fx 6 i 1,...,i n În \Ĵ n x [i 1,...,i n] S n fx e x [i 1,...,in] Snfx 1. 7 x [i 1,...,i n] It is then easily seen that as t, te 4t f var ntβf+t e t 1 x [i] fx i I\J i I n 1 e t 1 x [i] fx 0, 7

8 as a consequence of 5 above. By Lemma 3.3, we may choose T > 2 such that t > T implies tnβf t np tf tnβf + t. 8 Let t > T and i 1,..., i n În \ J ˆn. Applying inequalities 2, 6 and 8, we obtain log µ tf [i 1,.., i n ] t 4t f var + np tf t 2t 4 f var + nβf x [i 1,...,i n] S n fx. x [i 1,...,i n] x [i 1,...,i n] S n fx S n fx Combining this with inequalities 2, 7 and 8, we deduce that for every t > T, φ µ tf [i 1,..., i n ] i 1,...,i n În \Ĵ n = log µ tf [i 1,..., i n ] µ tf [i 1,..., i n ] i 1,...,i n În \ J ˆn 2t S n fx µ tf [i 1,..., i n ] i 1,...,i n În \Ĵ n x [i 1,...,i n] so that 4t f var np tf 2te 2te 4t f var tnβf+t i 1,...,i n În \ J ˆ x [i 1,..,i n] n S n fx e t x [i 1,.,in] Snfx i 1,...,i n În \Ĵ n e t 1 x [i1,...,in] Snfx 2nt e n 1 4t f var tnβf+t e t 1 x [i] fx e t 1 x [i] fx i I\J lim φ µ tf [i 1,..., i n ] = 0. 9 t i 1,...,i n În \Ĵ n A similar, simpler argument using inequalities 2, 5 and 8 shows that when i 1,..., i n În \ J ˆn, one has µ [i 1,..., i n ] = 0, i I 8

9 from which it follows that Hµ A n = i 1,...,i n Ĵ n φ µ [i 1,..., i n ]. Combining this with 9 above, we deduce that to prove 4 we need only show that lim φµ tk f [i 1,..., i n ] = φµ [i 1,..., i n ]. k i 1,...,i n Ĵ n i 1,...,i n Ĵ n By weak-* convergence one has lim k µ tk f [i 1,..., i n ] = µ [i 1,..., i n ] for all cylinder sets; since φ is continuous and the sum is finite, the required convergence follows. The claim 4 is proved. To complete the proof of the Lemma, we pose for a contradiction that hµ < lim t hµ tf. Choose ε > 0 such that hµ lim t hµ tf 3ε, and choose N > 0 such that 1 N H µ A N h µ + ε. Using 4, we deduce that for all sufficiently large k > 0, hµ tk f 1 N H µ tk f A N 1 N H µ A N + ε hµ + 2ε lim t hµ tf ε. Taking the limit as k yields the required contradiction. Acknowledgments. The author gratefully acknowledges the hospitality and financial port of the Erwin Schrödinger Institute for Mathematical Physics, at which this research was conducted. The author would also like to thank O. Jenkinson for his careful reading of an earlier version of this article. References [1] Z. Coelho, Entropy and ergodicity of skew-products over subshifts of finite type and central limit asymptotics, PhD thesis, Warwick University [2] J.-P. Conze and Y. Guivarc h, Croissance des sommes ergodiques, unpublished manuscript c [3] G. Contreras, A. Lopes and P. Thieullen, Lyapunov minimizing measures for expanding maps of the circle, Ergod. Theory Dynam. Systems 5:

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