ON THE ABSENCE OF ZERO-TEMPERATURE LIMIT OF

EQUILIBRIUM STATES FOR FINITE-RANGE INTERACTIONS

ON THE LATTICE Z2

JEAN-RENE CHAZOTTES AND MAO SHINODA

Abstract. We construct finite-range interactions on SZ2

, where S isa finite set, for which the associated equilibrium states (i.e., the shift-invariant Gibbs states) fail to converge as temperature goes to zero.More precisely, if we pick any one-parameter family pµβqβą0 in whichµβ is an equilibrium state at inverse temperature β for this interaction,then limβÑ8 µβ does not exist. This settles a question posed by the firstauthor and Hochman who obtained such a non-convergence behaviorwhen d ě 3, d being the dimension of the lattice.

Keywords. lattice systems, equilibrium states, Gibbs measures, ther-modynamic formalism, multidimensional subshift of finite type, Wangtilings, effective subshifts, Turing machine.

Contents

1. Introduction and main result 21.1. Setting 21.2. Known results and main theorem 21.3. More comments 31.4. On the role of symbolic dynamics 41.5. Organization of the paper 42. Equilibrium states, subshifts and tilings 52.1. Equilibrium states 52.2. Subshifts, 2D subshifts of finite type and Wang tilings 63. Imbedding a 1D effective subshift into a 2D subshift of finite type 73.1. Self-simulating structure 93.2. Simulation to check forbidden strings 124. Construction of a some 1D effective subshift 135. Estimates on admissible patterns 156. Proof of the main theorem 18Appendix A. On partial sums of binomial coefficients 23Appendix B. A measure with entropy log |G

rY˘,`k`1|{λ`k`1

24

References 24

Date: October 21, 2020.We thank Mike Hochman for having drawn our attention to the work of Durand et al.

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2 J. R. CHAZOTTES AND M. SHINODA

1. Introduction and main result

1.1. Setting. Leaving precise definitions till the next section, we work inthe context of Gibbs measures for shift-invariant absolutely summable in-

teractions on a space of configurations of the form SZd , where S is a finiteset. As explained below, we will in fact deal with equilibrium states, that is,shift-invariant Gibbs measures. The problem we consider is the following.Given an interaction Φ and an inverse temperature β ą 0, there is a sim-plex of equilibrium states ESpβq associated with βΦ (which might not be asingleton for large values of β, as for instance in the Ising model). We askthe following:

What is the behavior of ESpβq when β Ñ `8?

When there is a single equilibrium state µβ for each β, this question issimply: Does the limit of pµβqβą0 exist? If it does, what is the limitingmeasure? (The natural topology in this problem is the weak topology, seebelow.) This question is connected with ground states. We need not explainwhat they are because they will play no explicit role in the present paper.Let us only say that a ground state for an interaction Φ is a probability

measure supported on a certain closed subset of SZd , possibly uncountable,which is invariant under the shift action (that is, a subshift), and determinedby the “maximizing configurations” of the Hamiltonian of Φ. We refer to[GT15] for details. (Notice that in this paper we use the convention signfollowed in dynamical systems and also in [Rue04], namely we “prefer” tomaximize instead of minimizing.)

1.2. Known results and main theorem. If the answer to the above ques-tion is known in a number of particular examples, notably in relation withphase transitions, see, e.g., [DS85,vEFS93,Geo11], the general study of thisproblem is pretty recent, and it was started by people working in ergodictheory and dynamical systems. They considered ‘potentials’ on SN (or SZ)for which there is a single equilibrium state, which is also a Gibbs measure,for each inverse temperature [Bow08]. 1 In a nutshell, the situation is thefollowing. For locally constant ‘potentials’, which correspond to finite-rangeinteractions, convergence always takes place, and it is possible to describethe limit measures [Bre03,CGU11,Lep05]. For Lipschitz ‘potentials’, whichcorrespond for instance to exponentially decaying pair interactions (as afunction of the distance between sites), there is a rather surprising negativeresult. Recall that, in this class, equilibrium states and Gibbs measures co-incide, and for a given potential and for each β there is exactly one Gibbsmeasure. It was first proved in [CH10] that there do exist potentials (orinteractions) in this class such that the limit of pµβqβą0 does not exist whenβ Ñ `8. Another construction was given in [CRL15].

1A caution on the terminology is in order. In statistical physics, an interaction or a

potential is a family of real-valued functions on SZd

indexed by the finite subsets of Zd.For equilibrium states, a function deriving from the potential appears naturally and canbe interpreted as the ‘mean energy per site’. In dynamical systems, people consider d “ 1(the shift representing time evolution), and they only consider this mean energy per sitethat they call a potential.

3

What happens when d ě 2? In sharp contrast with the case d “ 1, it wasproved in [CH10] that, when d ě 3, then one can construct a finite-rangeinteraction (with S “ t0, 1u) such that, for any family pµβqβą0 in whichµβ is an equilibrium state for this interaction at inverse temperature β, thelimit limβÑ8 µβ does not exist. (We will comment below on this rathersubtle statement.) The case d “ 2 is left as an open problem in [CH10] (forreasons that will be explained later on), and in this paper, we solve it. Moreprecisely, our main theorem is the following.

Theorem 1.1 (Main theorem).

There exists a finite set S and a finite-range interaction on SZ2, such that

for any one-parameter family pµβqβą0 in which µβ is an equilibrium state(i.e., a shift-invariant Gibbs measure) at inverse temperature β, the limitlimβÑ8 µβ does not exist.

Several comments are in order. First, if there were a unique equilibriumstate/Gibbs measure for each β, then there would be a unique choice forµβ, and the previous result could be formulated more transparently: thereexist finite range interactions such that the limit limβÑ8 µβ does not exist.But in our example we didn’t look if uniqueness holds at low temperature.Second, by compactness (in the weak topology) of the space of probabilitymeasures, if we take any sequence pβ`q`ě1 of inverse temperatures such thatβ` Ñ `8, there exists a subsequence p`iq such that the sequence pµβ`i qiě1,

in which µβ`i is an equilibrium state, has a limit. Our result, as well as

the one in [CH10] mentioned above, is about continuous-parameter families.Third, and last, there is nothing new in the fact that one can choose somedivergent family of equilibrium states. Consider for instance the nearest-neighbor Ising model in which one can choose a family which alternates,when β is large enough, between the ` and ´ phases. However, it is alwayspossible to choose families which converge to either δ´ or δ`. In our example,it is not possible to choose any family which converges to a ground state.Let us also mention that in [CRL15] such a non-convergence result (for anyd ě 2) was obtained, but for non-locally constant Lipschitz ‘potentials’.

1.3. More comments. The fact that the Gibbs measures of an interactioncan behave in a ‘chaotic’ way when temperature goes to zero seems to havebeen first proved in [vER07] for a class of examples of nearest-neighbor,bounded-spin models, in any dimension. In that example, S is the unitcircle. The paper [CH10] was the first to exhibit this kind of behavior formodels with a finite number of ‘spin’ values at each site. In the abovementioned paper [CRL15], a stronger property is studied namely ‘sensitivedependence’. Roughly, it means that the non-convergence can indeed occuralong any prescribed sequence of temperatures going to zero, by making anarbitrarily small perturbation of the original interaction. We believe thatour example exhibits this property but we did not try to prove it. Finally,let us mention that we only deal with equilibrium states, that is, shift-invariant Gibbs measures. It is well known that there can exist non-shiftinvariant Gibbs measures at low temperature, e.g., in the three-dimensionalIsing model where the so-called ‘Dobrushin states’ appear [Dob73, DS85].The situation is unclear in that case.

4 J. R. CHAZOTTES AND M. SHINODA

1.4. On the role of symbolic dynamics. In [CH10], [CRL15] and thepresent work, a central role is played by symbolic dynamics, in particularthe construction of subshifts with certain properties. Informally, a subshift

is a subset of configurations in SZd defined by a (finite or infinite) set of‘patterns’ which cannot appear anywhere in these configurations. Whend “ 1, we say that we have a 1D subshift. A prominent class of 1D subshiftsis that of subshifts of finite type for which there are finitely many forbiddenpatterns. They play a central role to ‘encode’ certain differential dynamicalsystems such as Axiom A diffeomorphisms [Bow08]. There are many 1Dsubshifts that are not of finite type which were introduced for various pur-poses (for instance the Thue-Morse subshift defined by substitution rules);see for instance [Kur03].

There is a striking and dramatic difference between 1D and 2D subshiftsof finite type. For instance, it is formally undecidable whether a 2D subshiftof finite type is empty or not. This undecidability problem is closely relatedto the existence of nonempty shifts of finite type without periodic points, or,equivalently, the existence of Wang tile sets (their definition is given below)such that one can tile the plane but never in a periodic fashion [LS02].

1D subshifts of finite type are closely related to the zero temperature limitof (one-dimensional) Gibbs measures of finite-range potentials: the limitingmeasure, which always exists, is necessarily supported on a subshift of finitetype. The above mentioned examples of non-convergence for non-finite-range potentials [CH10, CRL15] rely on the construction of some subshiftswhich are necessarily not of finite type. Roughly speaking, the idea is to cookup two subshifts of SZ, each carrying only one shift-invariant probabilitymeasure (among other properties), and a (non-finite-range) potential suchthat the corresponding one-parameter family of Gibbs measures pµβqβą0

accumulates at the same time on the two measures as β Ñ `8.In dimension higher than one, we previously said that this non-convergence

phenomenon can arise for finite-range potentials. The underlying phenom-enon which we exploit is that one can imbed (in a way precised below) any(effective) 1D subshift into a higher-dimensional subshift of finite type. In[Hoc09], there is a construction which allows to imbed a 1D effective sub-shift into a 3D subshift of finite type, which is the one used in [CH10]. Inthis paper we use another construction from [DRS12] based on ‘hierarchicalself-simulating tilings’. It permits to imbed any (effective) 1D subshift intoa 2D subshift of finite type. This is a rather cumbersome construction (thatwe will partly describe it below), although the underlying ideas are simple.(Let us mention that a different embedding construction is given in [AS13].)Moreover, we use the construction of certain 1D subshifts given in [CRL15].It is somewhat more flexible than the one used in [CH10]. Once we havea 2D subshift of finite type built up from a certain 1D subshift, we canthen define a finite-range potential which ‘penalizes’ the forbidden patterns(which are finitely many).

1.5. Organization of the paper. In Section 2 we set the necessary def-initions and notations for equilibrium states, subshifts and Wang tilings.In Section 3 we state the embedding theorem of Durand et al. [DRS12]and establish a key proposition which results from their construction. In

5

particular, we explain some of the ideas of the proof of the embedding the-orem. Next, we construct in Section 4 a certain 1D effective subshift thatserves as a ‘base’ for a 2D subshift of finite type, and we define an associ-ated finite-range interaction. Section 5 contains some estimates involvingthe admissible patterns of the 2D subshift of finite type. Finally, we provein Section 6 our main result (Theorem 1.1), namely that that for everyone-parameter family pµβqβą0 in which µβ is an equilibrium state at inversetemperature β for the above interaction, limβÑ8 µβ does not exist.

2. Equilibrium states, subshifts and tilings

The configuration space is SZd , where S is a finite set and d ě 1 is aninteger. Regarding equilibrium states, we are interested in d “ 2, but wewill also consider the case d “ 1 to construct some subshifts needed in theproof of the main result. On SZd , we have the shift operator σ defined by

σipxqj “ xi`j

where x “ pxkqkPZd P SZd , i, j P Zd. In the language of symbolic dynamics

[LS02], pSZd , σq is the d-dimensional full shift over S. As usual, SZd is giventhe product topology, which is generated by the cylinder sets, and thus itis a compact metrizable space. We denote by B the Borel σ-algebra whichcoincides with the σ-algebra generated by cylinder sets.

2.1. Equilibrium states. We only recall a few definitions and facts, mainlyto set notations. We refer to [Geo11,Rue04] for details, as well as to [Kel98]for a viewpoint from ergodic theory for Zd-actions. The basic ingredientis a shift-invariant summable interaction Φ “ pΦΛqΛŤZd . (Λ Ť Zd means

that Λ is a nonempty finite subset of Zd.) More precisely, for each Λ Ť Zd,ΦΛ : SZd Ñ R is BΛ-measurable,2 ΦΛpxq “ ΦΛpxq whenever x and x coincideon Λ, ΦΛ`i “ ΦΛ ˝σ

i for all i P Zd, andř

ΛQ 0 }ΦΛ}8 ă 8. We say that Φ isof finite range if there exists R ą 0 such that ΦΛ ” 0 whenever diampΛq ą R.

Given Φ, define the function φ : SZd Ñ R by

(1) φpxq “ÿ

ΛQ 0ΛŤZd

ΦΛpxq

|Λ|.

By definition, the equilibrium states of βΦ are the shift-invariant probabilitymeasures which maximize the quantity

ż

βφdν ` hpνq

over all shift-invariant probability measures ν on SZd . Here hpνq is theentropy of ν (also called the mean entropy per site in statistical physics),and the supremum (which is attained) is called the (topological) pressureand is denoted by P pβφq “ P pσ, βφq. For the class of interactions we con-sider, shift-invariant Gibbs measures coincide with equilibrium states (see[Geo11, Chapter 15] or [Rue04, Theorem 4.2]). We use the terminologyand convention of dynamical systems and thermodynamic formalism where

2BΛ is the σ-algebra generated by the coordinate maps ω ÞÑ ωx when x is restrictedto Λ.

6 J. R. CHAZOTTES AND M. SHINODA

one “prefers” to maximize, whereas in statistical physics one “prefers” tominimize. Note that if Φ is of finite range then φ is locally constant, whichmeans that the values of φpxq are determined by finitely many coordinatesof x.

2.2. Subshifts, 2D subshifts of finite type and Wang tilings. Werefer to [LS02] for more details.

We now turn to symbolic dynamics. For Λ Ť Zd and x P SZd , xΛ isthe restriction of the configuration x to Λ. An element ω P SΛ is called aΛ-pattern, or simply a pattern, and Λ is its support. But only the “shape”of Λ matters, not its “location” in Zd. More precisely, define the equivalencerelation „ on the set of finite subsets of Zd by setting Λ „ Λ1 if and only ifthere exists i P Zd such that Λ1 “ Λ ` i. Hence, a support of a pattern isan equivalence class for „. Let us also denote by „ the equivalence relationsaying that two patterns ω P SΛ and ω1 P SΛ1 are congruent if there existsj P Zd such that Λ1 “ Λ ` j and ω1i`j “ ωi for all i P Λ. In the sequel, forthe sake of simplicity, we will several times consider a “localized” Λ, and byω P SΛ we will mean any pattern ω1 „ ω.

For n ě 1 let

Λn “ t´n` 1, . . . , 0, . . . , n´ 1ud

which is the discrete d-dimensional cube with volume λn “ |Λn| “ p2n´1qd.When d “ 1, patterns of the form ω0 ¨ ¨ ¨ωn´1, where ωi P S and n ě 0, arecalled n-strings or simply strings. Given a Λ-pattern ω, let

rωs “

x P SZd : xΛ “ ω(

denote the corresponding cylinder set. Given a finite set of patterns P , wewrite rP s “

Ť

p PP r ps.

A (nonempty) subset X of SZd is a subshift if it is closed and σ-invariant.

Equivalently, X Ď SZd is a subshift if there exists a set F of patterns suchthat X “ XF where

XF “

!

x P SZd : no pattern from F appears in x)

.

Thus F is the set of “forbidden” patterns. Note that XF may be emptyand that different forbidden sets may generate the same subshift. If F is

empty, XF “ SZd . A subshift X is a of finite type if there exists a finite setF such that X “ XF . We will use the abbreviation SFT for “subshift offinite type”. A subshift X is effective if there exists a recursively enumerableset F such that X “ XF , that is, if we can have a Turing machine which,given no input, lists out the elements of F . Let us remark that the class ofeffective subshifts is countable, so we apparently rule out “most” subshifts,but all known examples are in this class, provided that they are definedusing computable parameters, which is not a restriction in practice.

Given a subshift X and an integer n ě 1, define the set of (locally)admissible Λn-patterns by

PX,n “

ω P SΛn : no forbidden pattern of X appears in ω(

.

7

Finally, the set of probability measures on SZd is given the weak topology. Asequence of probability measures pµkqkě1 converges to a probability measureµ if, for any cylinder set B, µkpBq Ñ µpBq, as k Ñ `8.

Let us briefly explain how a two-dimensional SFT can be seen as a Wangtiling, and vice versa. Working with tilings is better adapted for some con-structions we use later on. We consider tiles which are unit squares withcolored sides. The colors are taken from a finite set C . For visualizationpurposes, one can actually use colors, but it can be more convenient to usesymbols or integers. Hence, the word “color” means any element from afinite set of symbols C . Hence a tile is a quadruple of colors (left, right, topand bottom ones), i.e., an element of C 4. A tile set is a subset τ Ă C 4. AWang tiling with tiles from τ is a mapping x : Z2 Ñ τ which respects thecolor matching condition: abutting edges of adjacent tiles must have thesame color. We shall simply say that it is a τ -tiling. See Fig. 1 for an exam-ple (where the colors are not only put on edges to ease visualization). We

can naturally identify each such tiling with a point x “ pxi,jqpi,jqPZ2 P τZ2,

interpreting τ as an alphabet. The set W Ă τZ2

of all τ -tilings is obviouslya subshift of finite type (called the Wang shift of τ). Conversely, a SFT canbe regarded as a Wang shift. In this paper, a tiling will mean a Wang tiling.

Figure 1. An example of tile set τ and a region of Z2 legallytiled using this tile set.

3. Imbedding a 1D effective subshift into a 2D subshift offinite type

We are going to outline how to imbed a one-dimensional effective subshiftinto a two-dimensional SFT, as explained in detail in [DRS12]. We definethe “vertical extension” of a subshift X Ă AZ as

pX “

x “ pxi,jqpi,jqPZ2 P AZ2: @j, pxi,jqiPZ P X, @i, j, xi,j “ xi,j`1

(

.

The following theorem and proposition are key-results in our construction.

Theorem 3.1. [DRS12, Theorem 10]Let X be a one-dimensional effective subshift over a finite alphabet set A.

8 J. R. CHAZOTTES AND M. SHINODA

Then there exist finite alphabets C and B Ă A ˆ C, and a two-dimensional

SFT Y Ă BZ2such that πpY q “ pX where π is the projection from B to A.3

In plain words, every sequence from a 1D effective subshift can be obtainedas a projection of a configuration of some 2D subshift of finite type in anextended alphabet.

Configurations in Y can be seen as configurations from pX marked withsome extra symbols taken from C. These symbols form a new “layer” which

is superimposed on top of pX. For reasons explained below, the layer where

configurations of pX appear is called the input layer, whereas the layer wherepatterns over C appear is called the computation layer. Then π erases thesuperimposed layer of data corresponding to C.

The following proposition will play an important role in some computa-tions later on.

Proposition 3.2.For each n ě 1 there exists cn ď p2n´ 1q2 such that

|π´1pω X PY,n| “ cn

for all ω P PX,n, where pω P PpX,n

.

This proposition says that the number of admissible patterns in the com-putation layer does not depend on the input layer. To prove it, we need someelements of the proof of Theorem 3.1 found in [DRS12]. The basic idea ofthe proof is to run a Turing machine MX which checks the forbidden stringsof X. The transition rules of MX are converted into tiling constraints, sincethey are described locally. Then a ‘space-time diagram’ of MX is (almost)a tiling based on the corresponding tile set. We can consider the horizontaldimension as ‘space’, given by the symbols on the tape of the Turing ma-chine, whereas the vertical dimension is ‘time’ which is given by successivecomputations of the Turing machine.

There are two difficulties to check forbidden strings of X. First, we haveto check arbitrary long input strings, since the length of forbidden stringsmay not be bounded. Second, we need to start the Turing machine at everysite, since we should check every string starting at every site. In orderto overcome the first difficulty, we will consider a tile set which admits ahierarchical structure, the self-simulating structure defined in Subsection3.1. The way to solve the second difficulty is explained in Subsection 3.2.

Colloquially, the idea to organize the computations uses fixed-point self-similar tilings. The idea of a self-similar fixed-point tile set can be sketchedas follows. It is well known that tilings can be used to simulate computa-tions, in the sense that for any Turing machine one can construct a tile setsimulating it: use rows of tiles to simulate the tape in the machine, withsuccessive rows corresponding to consecutive states of the machine. In turn,these computations can be used to guarantee the desired behavior of biggerblocks, called macro-tiles. So, for a desired behavior of macro-tiles, we canconstruct a tile set which guarantees this behavior. If these tiling rules co-incide with the rules for macro-tiles, we get self-similarity as a consequence.

3We define in the obvious way the projection acting on patterns or configurations byapplying π to every symbol.

9

The way to achieve this is to use an idea very close to the classical Kleenefixed-point theorem in computability theory (and which was for instanceused to construct self-reproducing automata by von Neumann).

3.1. Self-simulating structure. In order to get the self-simulating struc-ture, we consider tilings with “macro tiles”.

Definition 3.3 (Macro tiles). Consider a tile set τ and an integer N ě 1.

A pattern ω over τ t0,1,...,N´1u2 is called a τ -macro tile with zoom factor N ,if all tiles in ω satisfy the color matching property. Denote by τ pNq the setof all τ -macro tiles with zoom factor N . Every side of a macro-tile consistsof a sequence of N colors and we call it a macro-color.

It is easy to see that a τ -tiling can be seen as a τ pNq-tiling. In particular,we pay attention to the situation when a τ -tiling can be split uniquely intomacro-tiles which acts like tiles from another tile set ρ.

Definition 3.4 (Simulation). Let ρ and τ be tile sets and N ě 1. Thetile set ρ is simulated by a tile set τ with zoom factor N if there exists aninjective map r : ρÑ τ pNq such that

‚ t, s P ρ satisfy the color matching property if and only if rptq, rpsqsatisfy the color matching property;

‚ for every τ -tiling there exists a unique vertical and horizontal NˆNsplit such that every pattern in the N ˆN square is the image of anelement in ρ by r.

Example 3.5 (Coordinate tile). [DRS12] Consider a tile set ρ whose ele-ment is colored by only one color, namely “0”, and consider ρ “ t0u4. Wedefine a tile set which simulates ρ. Let N ě 2 and C “ pZ{NZq2. Define atile set τ by

τ “

t P C 4 : tb “ t` “ pi, jq, tr “ pi` 1, jq, tt “ pi, j ` 1q pi, jq P C(

.

Define a map r : ρÑ τ by rpp0, 0, 0, 0qq “ the τ -macro tile with zoom factorN whose macro colors are p0, 0qp1, 0q ¨ ¨ ¨ pN ´ 1, 0q for the bottom and top,p0, 0qp0, 1q ¨ ¨ ¨ p0, N ´ 1q for the left and right; see Fig. 2. We call the tilerpp0, 0, 0, 0qq the coordinate tile with size N .

We will consider a sequence tτkukě0 of tile sets with the following prop-erties: the level k` 1 tile set τk`1 is simulated by the level k tile set τk withzoom factor Nk`1; for every k the tile set τk describes simulation of MX ; thezoom factors Nk increase and macro tiles of τk can treat long input stringsas k increases. See Fig. 4.

We start with the construction of a tile set which simulates a `-bits coloredtile set, ρ Ă pt0, 1u`q4. Consider a Turing machine Mρ which checks whethera given four `-bits input represents a tile in ρ. Superimposing other tileson coordinate tiles (Example 3.5), we define a tile set which simulates ρ.The superimposed tiles make another “layer” on the coordinate tiles. Letc “ pcb, cl, ct, crq be a four `-bits input. Consider a macro tile with size Nconsisting of coordinate tiles with size N . Tiles whose color contain 0 arecalled boundary tiles. (In Fig. 3 the grey and green zones are the boundarytiles.) On the middle ` tiles of the bottom (respectively left, top and right)side of the boundary, we distribute a `-bits color cb (resp. cl, ct and cr). For

10 J. R. CHAZOTTES AND M. SHINODA

(i, j)

(i, j) (i + 1, j)

(i, j + 1)

(0, 0)

(0, 0)

(1, 0) (N � 1, 0)

(0, 1)

(0, N � 1) (0, N � 1)

(N � 1, 0)

Figure 2. A tile in τ (left) and a τ macro tile with zoomfactor N (right).

the rest of the boundary tiles we distribute 0. Since we use the coordinatetiles, each tile “knows” its coordinate and we can distribute colors like this.

In the middle square of the macro tile (the red and yellow zones in Fig. 3)we put tiles which describes a universal Turing machine with a program ofMρ. Since the rule of a universal Turing machine is given by local rules, theycan be embedded into tiles. Conveying the `-bits colors on the boundary tothe middle square by wires, we let the universal Turing machine to know theinput. Then a space-time diagram with the input c “ pcb, cl, ct, crq appears inthe square. Since each tile “knows” its coordinate, this structure is arrangedeasily. The size N of macro tiles is chosen to be large enough to contain thesestructures and to finish the simulation on the computation zone. If c P ρ,the simulation doesn’t halt and we have a macro tile. If c R ρ, the simulationhalts and there is no macro tile with this structure. Since a universal Turingmachine is deterministic, there is a one-to-one correspondence between aninput and the pattern of the simulation. Hence the tile set, say η, whichmakes this macro tile simulates ρ.

Note that the number of tiles in η does not depend on Mρ since we usea universal Turing machine. Moreover, tiles for boundaries, wires and thespace-time diagram do not depend on Mρ and only those for the program do.By using this fact, we modify the program on a macro tile of η and get self-simulating structure. We replace the program of Mρ with the following threeprograms: the program to make the boundaries, wires and computationstructures; the program of MX ; and the program which rewrites itself. Thenη simulates a tile set whose macro tiles have the same structure as in Figure3 and carries the same program as in the macro tiles of η.

Using this construction, we can make a sequence of tile sets which simulatetiles in the next level and carry the same program. The first level tile set

11

Simulation Zone

Program Zone

Wires

Wires

-bits colourk

-bits colourk

-bits colourk

-bits colourk

(1, 0)(0, 0)

(0, 0)

(N � 1, 0)

(0, N � 1)

Figure 3. A macro tile simulating a tile colored by k-bits.

τ0 simulates the second one τ1 with zoom factor N0, then τ1 simulates thethird one τ2 with zoom factor N1, and so on and so forth. See Fig. 4. Sinceτ0 simulates τ1 and τ1 simulates τ2, the patterns of τ0 in N0 ˆ N0 squaresmust represent tiles in τ2. If we zoom out, we can see τ2-tiling in a τ0-tilingand so on.

�0 �1 �2

N0

N1

N2

Figure 4. Sequence of macro tiles which simulate the pre-vious level tile sets.

12 J. R. CHAZOTTES AND M. SHINODA

3.2. Simulation to check forbidden strings. According to the previousdiscussion we can construct a sequence tτku of tile sets which simulate thetiles in the next level. Moreover, macro tiles in any level simulate the sameprogram which includes MX . We superimpose tiles which carry alphabetA on tiles in τ0. The new layer is called the input layer and the otherone is called the computation layer. Then MX in the macro tiles of τ0 canaccess the input layer and it checks whether the input is forbidden or not.However it is not clear how the programs in the macro tiles of higher leveltile sets know the input. We distribute the infinite strings in the inputlayer in the following way. We call by a level k macro tile a macro tileconsists of tiles in τk. Consider a τ0-tiling and zoom out to see τk-tiling. Let`k “ N0N1 ¨ ¨ ¨Nk´1. Then a level k macro tile is represented by Nk tilesconsisting of `k ˆ `k tiles in τ0. Each `k ˆ `k tile represents a level k ´ 1macro tile. We distribute `k bits in the input string to Nk level k´ 1 macrotiles in the following way: The ith bit from the left is distributed to the ithlevel k ´ 1 macro tile from the bottom. See Fig. 5. Not only one bit butalso string can be distributed. Moreover we can change the length of thedistributed strings, while it should be very short with regard to the size ofthe macro tile to which it is distributed. Hence the main program in eachmacro tile simulates the distributed substring of the input. Note that theway of distributing does not depend on inputs. By the above discussion, wecan obtain the set of tilings whose input layers are the vertical extension ofX.

Input string

Nk Lktiles with size

Figure 5. Sequence of macro tiles which simulate the pre-vious level tile sets.

We now prove Proposition 3.2.

Proof of Proposition 3.2. Fix n ě 1 and ω P PX,n. Take p P π´1pω X PY,n

and ω1 P PX,n. Consider a macro tile in the computation layer of p whose

13

size is the maximum size included in the n ˆ n square. The macro tilesimulates a very short substring of ω. Replace the input layer of p bypω1. Then the macro tile simulate the substring of ω1 in the same positionand with the same length as the simulated substring of ω. Since ω1 is anadmissible string of X, the simulation does not halt. Hence we have a newmacro tile. This change of the macro tile influences macro tiles in higherlevels to change the colors on the boundary and their simulations are alsorenewed. Since there is a one-to-one correspondence between inputs and thepatterns of simulation, a map from the patterns in the computation layer ofπ´1

pω X PY,n to π´1pω1 X PY,n is injective. Since ω is arbitrary, this proves

that the number of patterns in π´1pω X PY,n does not depend on the input

ω. The number of possible patterns in the computation layer depends onthe positions of the macro tile with the maximum size included in a squareof size p2n´ 1q ˆ p2n´ 1q. Hence cn is at most p2n´ 1q2.

b!<latexit sha1_base64="DdfsZw3L51Y6fDHjuvxcSBPi0R0=">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</latexit>

the input layer

the computation layer

Figure 6. An element p P π´1pω X Pn,Y .

�

4. Construction of a some 1D effective subshift

We construct a particular 1D subshift which we will imbed into a two-dimensional SFT by using Theorem 3.1. Then we define a finite-range in-teraction which somewhat penalizes the admissible patterns of this SFT.

We consider the alphabet Σ “ t´1, 0,`1u. Let Σ` “ t0,`1u and Σ´ “t0,´1u. We construct a sequence tXkukě1 of SFTs by giving the sequencetFku of forbidden sets.

We will consider an increasing sequence t`ku of lengths of forbidden stringsand a decreasing sequence trku of frequencies of 0 in forbidden strings. Wewill choose these sequences to satisfy limkÑ8 `k “ 8 and limkÑ8 rk “ 0.

14 J. R. CHAZOTTES AND M. SHINODA

We will give the precise conditions on t`ku and trku in Section 5. Herewe explain the basic idea to make a sequence of SFTs by controlling thefrequency of 0 in forbidden sets. For each n set Σn “ Σt0,1,...,n´1u. Forω P Σn denote by f0 the frequency with which 0 appears in ω, i.e.,

f0pωq “1

n

ˇ

ˇ

i P t0, 1, . . . , n´ 1u : ωi “ 0(ˇ

ˇ.

Let

F1 “ Σ2`1´1 zΣ2`1´1` Y

ω P Σ2`1´1` : f0pωq ě r1

(

.

The first set forbids the strings including ´1, while the second set forbidsthe strings with many zeros. Similarly define F2 by

F2 “ Σ2`2´1 zΣ2`2´1´ Y

ω P Σ2`2´1´ : f0pωq ě r2

(

.

For m ě 2 define F2m´1 by

F2m´1 “ Σ2`2m´1´1 zΣ2`2m´1´1` Y

ω P Σ2`2m´1´1` : f0pωq ě r2m´1

(

Y

#

ω P Σ2`2m´1´1` : there exists a subsequence η of ω s.t. η P

m´1ď

i“1

F2i´1

+

and F2m by

F2m “ Σ2`2m´1 zΣ2`2m´1´ Y

ω P Σ2`2m´1´ : f0pωq ě r2m

(

Y

#

ω P Σ2`2m´1´ : there exists a subsequence η of ω s.t. η P

m´1ď

i“1

F2i

+

.

Let Xk “ XFk . By construction we have

X1 Ą X3 Ą ¨ ¨ ¨ Ą X2m´1 Ą ¨ ¨ ¨

X2 Ą X4 Ą ¨ ¨ ¨ Ą X2m Ą ¨ ¨ ¨ .

Let X` “Ş

mě1X2m´1, X´ “Ş

mě1X2m and X “ X` Y X´. Notethat X˘ ‰ H since ˘18 P X˘. By construction we have PX`,`2m´1 “

Σ2l2m´1´1` zF2m´1 and PX´,`2m “ Σ2l2m´1

` zF2m for m ě 1. We will chooset`ku and trku in such a way that htoppX1q ą htoppX2q ą htoppX3q ą ¨ ¨ ¨ ą

htoppX2m´1q ą htoppX2mq ą htoppX2m`1q ą ¨ ¨ ¨ and to keep this kind ofcondition after imbedding in two dimensions.

Applying Theorem 3.1 to the subshift X, we know that there exist analphabet B Ă Σˆ C and a two-dimensional SFT Y over B.

We need to blow up the number of patterns in Y . We replace Σ in B

by rΣ “ trω “ pω, sq P Σ ˆ t0, 0, ˚u : s “ ˚ if ω ‰ 0u. Let rB be thecorresponding alphabet. We have two possibilities when ω “ 0, which blowsup the number of patterns corresponding to the numbers of 0. Denote byrY the corresponding SFT. Let rY˘ “ rπ´1X˘ X rY where rπ is the projection

from rB to Σ.Let F be the (finite) set of forbidden patterns for rY . Elements in F

are “cross-shaped”. Define ΛF “ tp0, 0q, p˘1, 0q, p0,˘1qu, and a finite-rangeinteraction Φ by

ΦΛpxq “

#

´|Λ| if Λ “ ΛF and xΛF P F

0 otherwise.

15

X+ X�

X1

X2

X3

X4

X5

X6

X2m�1

X2m

Figure 7. Sketch of the sequences of nested subshifts offinite type.

Then, the corresponding locally constant potential φ : rB Z2Ñ R is (recall

(1))

φpxq “

#

´1 if xΛF P F

0 otherwise.

That φ is locally constant means that for every p P F , φpxq “ φpx1q whenever

x, x1 P rps. Observe that φ ” 0 on rY , since configurations in rY have nopattern from F .

5. Estimates on admissible patterns

We start with the conditions we have to impose on t`ku and trku. Foreach k ě 1 we require that

(S1) p2`k ´ 1q rk ě 1,

which ensure that there exists at least one admissible string with 0 of anysize. For t P r0, 1s, let

(2) Hptq “ ´t log t´ p1´ tq logp1´ tq

with the usual convention 0 log 0 “ 0. Let `1 “ 22 and r1 “ 2´1. Define`k`1 and rk`1 inductively by the following conditions:

(S2) 2´1rk log 2 ě 10Hprk`1q

(S3) p4`k ´ 2q´1 ě 10`

rk`1 ` 2p2`k`1 ´ 1q´2 logp2`k`1 ´ 1q˘

(S4) `k`1 ě 24`k .

Note that (S3) implies

p4`k ´ 2q´1 ě 10`

rk`1 ` p2`k`1 ´ 1q´2 log c`k`1

˘

16 J. R. CHAZOTTES AND M. SHINODA

since cn ď p2n ´ 1q2. Since we can rewrite conditions (S1), (S2), (S3) and(S4) by using recursive functions, the subshift X we defined in previoussection is effective.

The input layer of an admissible pattern of rY can contain forbidden stringssince macro tiles on the computation layer only checks very short substringsof the input. Such input patterns are said to be locally admissible. Forlater use we define the globally admissible set. For n ě 1 define the globally

admissible set GrY ,n

of rY with size n by

GrY ,n“ π´1 P

pX,nX P

rY , n.

Since we restrict the admissible set of rY to the elements whose input layerconsists of admissible strings of X, an element of G

rY ,ncan be extended to

bigger and bigger squares.Define mk “ p2`k ´ 1qp2`k´1 ´ 1q´1. Then an admissible string of X´

with length 2`k ´ 1 consists of mk admissible strings of X´ with length2`k´1´ 1. However Pmk

X´, `k´1‰ PX´, `k , since we may find forbidden strings

in the concatenated part. Let k be even. Let B`,k´1 be the set of admissible

strings with more than one 0, i.e., B`, k´1 “ PX`, `k´1zt`1`k´1u. Let C`, k

be the set of strings obtained by concatenating alternatively strings fromB`, k´1 and the string `12`k´1´1:

C`, k “!

ω1 ¨ ¨ ¨ωmk P PmkX`, `k´1

: ωm P B`,k´1 if m is odd

and ωm “ p`1q2`k´1´1 if m is even)

.

Since the string p`1q2`k´1´1 appears between strings from B`, k´1, no forbid-den string appears in the concatenated parts. Hence we have C`, k Ă PX`, `k .We define B´, k and C´, k for odd k in the same way.

We use the following lemma to show Proposition 5.2.

Lemma 5.1. We have

(3) |PX´, `k |10 ď |C`, k| if k is odd

and

(4) |PX`, `k |10 ď |C´, k| if k is even .

Proof. Assume k is odd. By (S2) we have the following.

|PX´,`k |10 ď

¨

˝

p2`k´1q rkÿ

i“0

ˆ

2`k ´ 1

i

˙

˛

‚

10

ď

´

ep2`k´1qHprkq¯10

“ e10p2`k´1qHprkq

ď e2´1p2`k´1q rk´1 .

17

Now

|C`,k| “ |B`, k´1|2´1mk “

¨

˝

ÿ

iďp2`k´1´1q rk´1

ˆ

2`k´1 ´ 1

i

˙

´ 1

˛

‚

2´1mk

ě

¨

˝

ÿ

iďp2`k´1´1q rk´1

ˆ

p2`k´1 ´ 1q rk´1

i

˙

˛

‚

2´1mk

“ ep2`k´1´1q rk´12´1mk “ e2´1p2`k´1q rk´1 .

Hence we get (3). The proof of (4) is very similar and thus left to thereader. �

For each n ě 1 set the globally admissible set GrY˘,n

of rY˘ with size n by

GrY˘,n

“ π´1 PpX˘,n

X PrY , n

.

Proposition 5.2. We have

|GrY´, `k

|10 ď |GrY`, `k

| if k is odd

and|G

rY`, `k|10 ď |G

rY´, `k| if k is even.

Proof. By Proposition 3.2 we have

|GrY˘,`k

| “ÿ

p PY˘, `k

enumber of 0’s in πppq

“ c`kÿ

pω P PxX˘,`k

enumber of 0’s in pω

“ c`kÿ

ω PPX˘, `k

ef0pωqp2`k´1q2 .

Assume k is odd. By (4) we have

|GrY´,`k

|10 “

¨

˝c`kÿ

ω PPX ,`k

ef0pωqp2`k´1q2

˛

‚

10

ď |PX´,`k |10 e10 rkp2`k´1q2`10 log c`k

ď |C , k| e10prkp2`k´1q`p2`k´1q´1 log c`k qp2`k´1q .

Note that every ω in C ,k satisfies f0pωqp2`k´1q ě 2´1mk. By (S3) we have

f0pωqp2`k ´ 1q2 ě2`k ´ 1

2p2`k´1 ´ 1qp2`k ´ 1q

ě 10prkp2`k ´ 1q ` p2`k ´ 1q´1 log c`kqp2`k ´ 1q

for ω P C ,k. Hence we have

|GrY´,`k

|10 ďÿ

ω PC ,k

ef0pωqp2`k´1q2 ď c`kÿ

ω PPX ,`k

ef0pωqp2`k´1q2 “ |GrY ,`k

|.

(5)

18 J. R. CHAZOTTES AND M. SHINODA

Therefore, for odd k, the proof is finished. We can do very similar calcula-tions for even k. �

6. Proof of the main theorem

We now prove Theorem 1.1. For each β ą 0, we pick an arbitrary equi-librium state for βφ (there can be several) and we consider the resultingone-parameter family pµβqβą0. Let Mk be the minimum size of the macro-tiles whose simulation checks all substrings with length less than `k. For eachk ě 1 we denote by θpkq the smallest index which satisfies 2`θpkq´1 ě 2Mk.By this choice at least one macro tile with size Mk should appear in the boxwith size `θpkq.

Our main theorem follows from the following proposition and the rest ofthis paper is dedicated to prove it.

Proposition 6.1. Take an arbitrary δ P p0, 1s. Then for all k large enoughwe have

µβk

´

rGrY´, `k`1

s

¯

ě 1´ δ if k is even,

µβk

´

rGrY`, `k`1

s

¯

ě 1´ δ if k is odd.

Invoking the ergodic decomposition [Geo11, Chapter 14], we can restrictourselves to ergodic equilibrium states in the proof of Proposition 6.1.

SinceŞ

kě1rGrY´,`ks “ rY´,

Ş

kě1rGrY`,`ks “ rY` and rY´ X rY` “ H, the

proposition shows that the one-parameter family pµβqβą0 does not converge.To prepare the proof of the above proposition, we need the following three

lemmas.

Lemma 6.2. Let C “ logˇ

ˇ rBˇ

ˇ. Thenş

φ dµβ ě ´Cβ´1 for any β ą 0.

Proof. There exists at least one shift-invariant measure µ whose support is

contained in rY . Since φ ” 0 on rY , this implies thatş

φ dµ “ 0. Hence, sincehpµq ě 0, we get

P pβφq ě hpµq ` β

ż

φ dµ ě 0.

Since µβ is an equilibrium state for βφ, we have

hpµβq ` β

ż

φ dµβ “ P pβφq ě 0.

Thereforeż

φ dµβ ě ´β´1hpµβq ě ´β

´1 logˇ

ˇ rBˇ

ˇ.

�

Given n ě 1 and a function ψ : rB Z2Ñ R, let Snψ “

ř

iPΛnψ ˝ σi.

Let C 1 “ 2C and for n, k ě 1 define En,k as the set of configurations

y P rB Z2satisfying the following conditions:

Sn``θpk`1q´1φpyq

λn``θpk`1q´1ě

ż

φ dµβk ´ Cβ´1`k

(6)

19

and

1

λnSnχrG

rY´,`k`1spyq ď µβk

´

rGrY´,`k`1

s

¯

` C 12´2`k .(7)

Lemma 6.3. Take k ě 1 and ε ą 0. Let µβk be an ergodic equilibrium statefor βkφ. Then for all n large enough

µβkpEn,kq ą 1´ ε.(8)

Proof. Since µβk is ergodic, we have

limnÑ8

1

λnSnφpyq “

ż

φ dµβk

limnÑ8

1

λnSnχrG

rY´,`k`1spyq “ µβk

´

rGrY´,`k`1

s

¯

for µβk almost every point y, which completes the proof. �

We estimate the number of globally admissible patterns in the configura-tions of En,k.

Lemma 6.4. For every k there exists N ě 1 such that for every n ě N

1

λn#!

i P Λn : yi`Λ`k`1PG

rY ,`k`1

)

“1

λnSnχrG

rY ,`k`1spyq ą 1´ C 12´2`k(9)

for every y P En,k.

Proof. Take N ě 1 such that λ´1n λn``θpk`1q´1λ`θpk`1q

ă 2`k , for all n ě N .

Take n ě N and y P En,k. Since by definition rGrY ,`k`1

s Ą rPrY ,`θpk`1q

s, we

have

1´1

λnSnχrG

rY ,`k`1spyq “

1

λnSnχ

rBZ2zrG

rY ,`k`1spyq ď

1

λnSnχ

rBZ2zrP

rY ,`θpk`1qspyq.

SnχrBZ2zrP

rY ,`θpk`1qspyq is the number of positions in Λn for which a non-

admissible pattern with size `θpk`1q appears. Let i P Λn``θpk`1q´1 be aposition where a forbidden pattern appears, that is, yΛF`i P F . Then thepatterns in boxes with size `θpk`1q including i are non-admissible. The num-ber of such boxes is bounded by λ`θpk`1q

. Since the number of positions

in Λn``θpk`1q´1 where a forbidden pattern appears is ´Sn``θpk`1q´1φpyq, wehave

1

λnSnχ

rBZ2zrP

rY ,`θpk`1qspyq

ď ´λn``θpk`1q´1

λn

1

λn``θpk`1q´1Sn``θpk`1q´1φpyq ˆ λ`θpk`1q

.

See Fig. 8. By the choice of n, (6) and Lemma 6.2, we have

1´1

λnSnχrG

rY ,`k`1spyq ď 2`k

ˆ

´

ż

φ dµβ`k ` Cβ´1`k

˙

ď 2Cβ´1`k

2 `k “ C 1 2´2`k .

�

20 J. R. CHAZOTTES AND M. SHINODA

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A site which is a center ofa forbidden pattern

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A forbidden patternwhich overlaps withthe complement of ⇤n

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Figure 8. Estimate of the number of non-admissible patterns.

Proof of Proposition 6.1. By contradiction. We suppose that there existδ P p0, 1s and k large enough such that

µβk

´

rGrY´,`k`1

s

¯

ď 1´ δ if k is even,(10)

µβk

´

rGrY`,`k`1

s

¯

ď 1´ δ if k is odd.

We only deal with the case when k is even, since the other case is simi-lar. Without loss of generality we may assume k is large enough to satisfyC 12´2`k ď δ and Λ`k`1

Ą ΛF . By (7) and (10) we have

1

λnSnχrG

rY´,`k`1spyq ď 1´ δ ` C 12´2`k ď 1´

ˆ

1`1

100

˙

δ(11)

for every y P En,k.

Let h´k`1 “1

λ`k`1log

ˇ

ˇGrY´,`k`1

ˇ

ˇ. Using (5) in Proposition 5.2, we have

ˇ

ˇ

ˇG

rY´,`k`1

ˇ

ˇ

ˇě

ÿ

ωPC´,k`1

ef0pωqp2`k`1´1q2 ě eminωPC´,k`1

f0pωqp2`k`1´1q2.

By the definition of C´, k`1 we have

(12) h´k`1 “1

λ`k`1

logˇ

ˇ

ˇG

rY´,`k`1

ˇ

ˇ

ˇě min

ωPC´,k`1

f0pωq ětmk`1

2 u

`k`1ě Op`´1

k q.

Lemma 6.5. Take N ě 1 such that, for every n ě N and y P En,k, (9)holds. Then we have

1

λnlog |En,k| ď H

`

C 12´2`k˘

` p1´ δ1qh´k`1 ` oph´k`1q

where δ1 “ 91000 δ.

Proof. Fix n ě N . For y P En,k the number of positions in Λn for which

a pattern from rBΛ`k`1 zGrY , `k`1

appears is bounded above by λnC12´2`k by

(9):

λn

ˆ

1´1

λnSnχrG

rY ,`k`1spyq

˙

ď λnC12´2`k .

21

The number of possible places for such positions is bounded by

ÿ

răλnC12´2`k

ˆ

λnr

˙

ď eHpC12´2`k qλn λ2

n

(See Appendix A). SinceGrY, `k`1

is the disjoint union ofGrY´,`k`1

andGrY`,`k`1

,

(11) and (9) imply

SnχrGY 1`,`k`1

spyq “ SnχrGrY ,`k`1

spyq ´ SnχrGY 1´,`k`1

spyq

ě λn`

1´ C 12´2`k˘

´ λn

ˆ

1´´

1`1

100

¯

δ

˙

ěδ

100λn.

Considering overlapping parts, the possible choices of patterns from GrY`,`k`1

is bounded above by λnλ´1`k`1

and is bounded below by δ100

λnλ`k`1

.

Hence the number of ways to choose patterns from GrY ,`k`1

is bounded by

λnλ`k`1ÿ

r“ δ100

λnλ`k`1

ˆ

λnr

˙

ˇ

ˇ

ˇG

rY´,`k`1

ˇ

ˇ

ˇ

λnλ`k`1

´r ˇˇ

ˇG

rY`,`k`1

ˇ

ˇ

ˇ

r

ď

λnλ`k`1ÿ

r“ δ100

λnλ`k`1

ˆ

λnr

˙

ˇ

ˇ

ˇG

rY´,`k`1

ˇ

ˇ

ˇ

λnλ`k`1

´r ˇˇ

ˇG

rY´,`k`1

ˇ

ˇ

ˇ

r10

“

λnλ`k`1ÿ

r“ δ100

λnλ`k`1

ˆ

λnr

˙

ˇ

ˇ

ˇG

rY´,`k`1

ˇ

ˇ

ˇ

λnλ`k`1

´ 910r

ď

ˇ

ˇ

ˇG

rY´,`k`1

ˇ

ˇ

ˇ

λnλ`k`1

p1´ 910

δ100q

λnλ`k`1ÿ

r“ δ100

λnλ`k`1

ˆ

λnr

˙

ď

ˇ

ˇ

ˇG

rY´,`k`1

ˇ

ˇ

ˇ

λnλ`K

p1´δ1q δ

100

ˆ

λnλ`k`1

˙2

eλn

λ`k`1H`

δ100

˘

.

Hence

|En,k| ď eHpC12´2`k qλnλ2

n

ˇ

ˇGrY´,`k`1

ˇ

ˇ

λnλ`k`1

p1´δ1q δ

100

ˆ

λnλ`k`1

˙2

eλn

λ`k`1H`

δ100

˘

.

By taking logarithm and dividing out by λn we get

1

λnlog |En,k| ď H

`

C 12´2`k˘

`4

λnlog λn ` p1´ δ

1q1

λ`k`1

logˇ

ˇ

ˇG

rY´,`k`1

ˇ

ˇ

ˇ

`1

λnlog

δ

100` 2 log

1

λ`k`1

`1

λ`k`1

H

ˆ

δ

100

˙

.

22 J. R. CHAZOTTES AND M. SHINODA

Since δ P r0, 1s, we have λ´1n log δp100q´1 ă 0. Since log x ď x´ 1 for every

x ą 0, Hptq ď log 2 ď 1 and λn ě 9 for n ě 2, we have

2 log1

λ`k`1

`1

λ`k`1

H

ˆ

δ

100

˙

ď 2

ˆ

1

λ`k`1

´ 1

˙

`1

λ`k`1

“3

λ`k`1

´ 2 ď 0.

For all n large enough we have

4

λnlog λn ď 2´2`k .

Hence we have

1

λnlog |En,k| ď H

`

C 12´2`k˘

` p1´ δ1qh´k`1 ` 2´2`k

for all n large enough. �

Now we can finish the proof. We have a probability measure ν´k`1 which

is invariant ergodic under σΛ`k`1and whose entropy is h´k`1 “ log |G

rY ,`k`1|

and whose support is included in rGrY ,`k`1

s. (See Appendix B for details.)

Since no forbidden pattern appears in Λ`k`1for an element in rG

rY ,`k`1s, we

haveż

S`k`1φ dν´k`1 “

ż

rGrY´,`k`1

s

φ dν´k`1 ě ´#pΛ`k`1zΛ`k`1´1q “ ´8p`k`1 ´ 1q.

By the variational principle we have

P pβkφq “ P pσ, βkφq “ λ´1`k`1

P pσΛ`k`1, βkS`k`1

φq

ěhpν´k`1q

λ`k`1

`βkλ`k`1

ż

S`k`1φ dν´k`1.

The ‘block’ shift σΛ`kis defined in Appendix B. The second equality is a

general fact. (We refer [Wal82, Theorem 9.8] for a proof in the case ofa continuous transformation of a compact metric space and a continuousfunction φ. The proof in the present context is obtained by combining theproof of Theorem 9.8 in [Wal82] and Section 4.4 in [Kel98].) Since µβk is anequilibrium state for βkφ, we have

h´k`1 “ λ´1`k`1

hpν´k`1q ď P pβkφq ´βkλ`k`1

ż

S`k`1φ dν´k`1

ď hpµβkq ` βk

ż

φ dµβk `βk

p2`k`1 ´ 1q2ˆ 8p`k`1 ´ 1q

ď hpµβkq `Op2´`kq.(13)

We get the last inequality because φ ď 0 and (S4). The L1 version ofShannon-McMillan-Breiman theorem [Kel98] tells us that, for any η ą 0,there exists n0 such that for any n ě n0

ÿ

pP rBΛn

ˇ

ˇ

ˇ

ˇ

´1

λnlogµβkprpsq ´ hpµβkq

ˇ

ˇ

ˇ

ˇ

µβkprpsq ď η.

23

In particular

η ěÿ

pPrEn,ks

ˇ

ˇ

ˇ

ˇ

´1

λnlogµβkprpsq ´ hpµβkq

ˇ

ˇ

ˇ

ˇ

µβkprpsq

ě

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ÿ

pPrEn,ks

´

´1

λnlogµβkprpsq ´ hpµβkq

¯

µβkprpsq

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

from which it follows, by taking η “ 2´`k , that

hpµβkqµβkprEn,ksq ďÿ

pPrEn,ks

ˆ

´1

λnµβkprpsq logµβkprpsq

˙

` 2´`k .

Using Jensen inequality with the concave function t ÞÑ ´t log t and theweights 1{|rEn,ks| we get

hpµβkqµβkprEn,ksq ď1

λnlog |rEn,ks| ´

1

λnµβkprEn,ksq logµβkprEn,ksq ` 2´`k .

We now apply Lemma 6.3 with ε “ δ1

2 to get from (13) that, for n largeenough,

h´k`1

ˆ

1´δ1

2

˙

ď1

λnlog |rEn,ks|´

1

λn

ˆ

1´δ1

2

˙

log

ˆ

1´δ1

2

˙

`2´`k`Op2´`kq.

Then Lemma 6.5 implies

h´k`1

ˆ

1´δ1

2

˙

ď H`

C 12´2`k˘

` p1´ δ1qh´k`1 ` oph´k`1q `Op2

´`kq.

Dividing out by h´k`1, we have

(14) 1´δ1

2ď 1´ δ1 `

H`

C 12´2`k˘

h´k`1

`Op2´`kq

h´k`1

` op1q.

Using (12) and then letting k Ñ8 in (14), we get a contradiction. �

Appendix A. On partial sums of binomial coefficients

Let n ě 1 and α P s0, 1{2s. We have the following obvious lower bound:

αnÿ

r“0

ˆ

n

r

˙

ě

αnÿ

r“0

ˆ

tαnu

r

˙

“ 2tαnu.

(Notice that we make a slight abuse of notation by writing αn instead oftαnu.)Another bound we use is the following:

αnÿ

r“0

ˆ

n

r

˙

ď enHpαq

where Hpαq is defined in (2). For the reader’s convenience, we give a proofsince we could not find a handy reference. A straightforward calculationusing the fact that logα ´ logp1 ´ αq ď 0 for α ď 1{2 shows that for allr P r0, αns

r logα` pn´ rq logp1´ αq ě ´nHpαq .

24 J. R. CHAZOTTES AND M. SHINODA

Hence αrp1´ αqn´r ě e´nHpαq, therefore

1 “ pα` p1´ αqqn ěαnÿ

r“0

ˆ

n

r

˙

αrp1´ αqn´r ěαnÿ

r“0

ˆ

n

r

˙

e´nHpαq

which proves the desired bound.

Appendix B. A measure with entropy log |GrY˘,`k`1

|{λ`k`1

Let S be an alphabet. Take m ě 1 and consider the alphabet T “ SΛm .

Define σΛm : SZ2Ñ SZ2

by

pσiΛmpxqqj “ xiΛm`j “ xpj1`p2m`1qi1,j2`p2m`1qi2q

for i “ pi1, i2q P Z2 and j “ pj1, j2q P Λm. By iΛm we mean the dilatedsquare ti1p´m ` 1q, . . . , 0, . . . , i1pm ´ 1qu ˆ ti2p´m ` 1q, . . . , 0, . . . , i2pm ´1qu. The full shift over T is topologically conjugate with pA, σΛmq and the

conjugacies are the following: f : SZ2Ñ T Z2

defined by pfpxqqi “ pσiΛmxqΛm

and g : T Z2Ñ SZ2

defined by pgpyqqiΛm`j “ the alphabet at j in Λm for yi.Let P Ă T . Then we have

htoppZ, σΛmq “ log |P|.(15)

Let ν be an ergodic measure on the full shift over P whose entropy islog |P|. Let ν be the push forward of ν by g. Note that ν is σΛm-invariant.

In Section 6 we considered S “ rB, m “ `k and P “ GrY´,`k

and the

σΛm-invariant measure ν is ν´k`1.Since a generating pn, 1q-separated set for pX,σΛmq is pp2m ` 1qn, 1q-

separating set for pX,σq, we have

P pσΛm , Smφq “ λmP pσ, φq

for every continuous function φ on AZ2. See [Kel98, Section 4.4] for details.

Since the support of ν is PZ2and gpPZ2

q Ă rPs, the support of ν is includedin rPs.

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Centre de Physique Theorique, CNRS, Institut Polytechnique de Paris,France

Email address: jeanrene@cpht.polytechnique.fr

Department of Mathematics, Ochanomizu University, 2-1-1 Otsuka, Bunkyo-ku, Tokyo, 112-8610, Japan

Email address: shinoda.mao@ocha.ac.jp