Phase transition in one-dimensional subshifts

In this note we give simple examples of a one-dimensional mixing subshift with positive topological entropy which have two distinct measures of maximal entropy. We also give examples of subshifts which have two mutually singular equilibrium states for H\"{o}lder continuous functions. We also indicate how the construction can be extended to yield examples with any number of measures of maximal entropy of equilibrium states.


Introduction
It is a well known fact that a wide class of uniformly hyperbolic transformation that satisfy some irreducibility condition have unique measures of maximal entropy. Irreducible subshifts have unique measures of maximal entropy if they are for instance of finite type or sofic (factors of shifts of finite type). More general hyperbolic systems which satisfy specification have unique measures of maximal entropy [1]. In either case the systems have a 'uniform' mixing property. Hofbauer had shown in [5] that even subshifts of finite type don't necessarily have unique equilibrium states. However the potentials in the provided examples lack of course regularity and one of the equilibrium states is supported on a single fixed point. On infinite alphabets there are examples of Markov shifts that have many Gibbs states although all are non-invariant [4] with only one exception. For a large class of Hölder continuous functions we get here multiple equilibrium states which are supported on embedded subshifts.
Here we produce a simple example which shows that even in the case of an irreducible subshift with positive entropy one cannot generally expect to have a unique measure of maximal entropy. Also Krieger [6] gave an example with a non-unique measure of maximal entropyalthough not as simple and straightforward as this one. For two-dimensional subshifts of finite type, Burton and Steiff [2,3] have given examples whose measures of maximal entropy form simplices that consist of more than single points.
In section 2 we give an example of a subshift which has two measures of maximal entropy. We then extend it to examples which can have any number of measures of maximal entropy including one with infinitely many. In the third section we construct an example that has two equilibrium states for Hölder continuous potentials which we then extend to examples with any number of equilibrium states and also one with infinitely many.

Measures of maximal entropy
We will first describe an example with two distinct ergodic measures of maximal entropy and then show how to construct an example with any number of distinct ergodic measures including one with infinitely many.
Let ν ≥ 2 be some positive integer and let us construct a subshift over the symbolset A = {0, 1, 2, . . . , 2ν}. We shall refer to the symbols {1, 2, . . . , ν} as the green symbols and denote them by G. Similarly we call {ν + 1, . . . , 2ν} the yellow symbols and write Y for them. Hence A = {0} ∪ G ∪ Y. Let us denote by Σ G = G Z the (green) full ν-shift over the alphabet G. Similarly we write Σ Y for the (yellow) full ν-shift Y Z .
Let τ > 0. The subshift Σ ⊂ A Z will consist of the union of the two monochromatic shift spaces Σ G and Σ Y , to which we add bicoloured sequences in the following way: A sequence in Σ in which a word α of one colour (green or yellow) is followed by a word β of the other colour (yellow or green) will have a string of zeroes γ separating the two words α and β. The length |γ| of that string of zeros is then at least, τ (a + b), where a = |α|, b = |β| are the lengths of the coloured words to the 'left' and 'right' of γ. If in particular the word α or β is infinitely long then it can only be followed by an infinite string of zeros and cannot 'transition' to a string of another colour.
If ω = ω m ω m+1 . . . ω m+n−1 is an allowed word of some length n in Σ, then we denote by U(ω) the cylinder set {x ∈ Σ : x i = ω i , m ≤ i < m + n}. The shift transformation σ on Σ is defined in the usual way by σ(x) i = s i+1 , i ∈ Z, for x ∈ Σ. Notice that the transition rules are shift invariant. Thus Σ is indeed a subshift which we endow with the usual topology generated by the cylinder sets U(ω), where ω runs over all finite words in Σ. If τ = 0 then Σ is a sofic system in which every string of zeroes connects to coloured symbols of different colours, and whose topological entropy is 2 log ν (see lemma 2). Lemma 1. The shift transformation σ on Σ is topologically mixing for every τ > 0.
Proof. We have to show that for any two finite words ω and η there exists a number N so that U(ω) ∩ σ n (U(η)) is non-empty for all n ≥ N. This is evidently true if ω and η are both monochromatic of the same colour. If the last symbol of η and the first symbol of ω are of different colour then consider the word π = η0 λ ω. Clearly, π is an admissable word for every λ > τ (|η| + |ω|), and the cylinder set U(π) ⊂ U(ω) ∩ σ n (U(η)) is non-empty for all n ≥ N, where N can then be chosen to be equal to (1 + τ )(|η| + |ω|).
If the last symbol of η and the first symbol of ω are of the same colour, then let us consider the word π = η0 κ ε0 λ ω, where ε is a symbol (or some other short word) of the other colour.
Proof. Let us first get a lower bound on the topological entropy h of Σ. For that purpose let us observe that since Σ contains two full ν-shifts, Σ Y and Σ G , the topological entropy of Σ must be at least log ν.
Let us now estimate the number of the remaining words of length n from above. We have two cases to consider: (i) Monochromatic words that might also contain zeroes. According to our rules such words begin or end with strings of zeroes. Thus we obtain purely yellow and green words of lengths k = 0, . . . , n which on at least one side are framed by strings of zeroes. Their number turns out to be 2 n k=0 (n − k)ν k ≤ 2nν n n k=0 ν −k which has exponential growth rate log ν. (ii) To estimate the number of words of length n that genuinely contain symbols of both colours, let us observe that since any such word has at least one transition from yellow to green or vice versa, it therefore must also contain at least n τ τ +1 zeroes, that is at most . The coloured symbols come in monochromatic blocks of alternating colour. Denote by P k,ℓ the number of possibilities in which one can arrange ℓ symbols in k blocks (separated by the appropriate number of zeroes), where k = 1, 2, . . . , ℓ. One finds that which is the number of possibilities of picking the first element of every block but the very first one. The number of n-words in Σ which contain ℓ coloured symbols arranged in k ≤ ℓ monochromatic blocks of alternating colour is 2P k,ℓ ν ℓ . Distributing ℓ coloured symbols out of n ′ in k blocks can be done in P k,ℓ many ways, where 1 ≤ k ≤ ℓ ≤ n ′ . This leaves m = n − (τ + 1)ℓ zeroes to be distributed on k + 1 intervals, namely the k − 1 gaps between the blocks of coloured symbols plus the two ends of the entire word. There are Q k,m = m + k k many possibilities. By Stirling's formula we have We can thus estimate the total number of bi-coloured strings of length n by Proof. The topological entropy of Σ is at least log ν as this is the topological entropy of the embedded subshifts Σ j . To get an upper bound on the topological entropy of Σ we estimate the number Q(n) of words of length n that are not 'monochromatic', i.e. don't belong to a single Σ j . We proceed as before and obtain: where m = n − [(τ + 1)ℓ], ℓ counts the number of 'coloured' symbols (i.e. symbols in A \ {0}), k − 1 is the number of changes of colour that is where there is a transition from a word in some Σ j to Σ j±1 (the factor 2 k−1 accounts for number of possible switches in colour). As before we thus get (with n ′ = [n/(τ + 1)]) Consequently we get that and thus lim n→∞ 1 n log Q(n) ≤ max log 2, log 5ν 1 + τ .

Example with infinitely many distinct ergodic measures of maximal entropy.
Let ν ≥ 2. We now let L go to infinity and obtain a subshift Σ over the alphabet A = N 0 = {0, 1, 2, . . . }. There are infinitly many embedded full ν-shift for j = 1, 2, . . . . As before we only transition from a word α in Σ j a word β in Σ j±1 by keeping them separted by a string of zeros of length ≥ τ (|α| + |β|). Again the resulting shift space Σ is topologically mixing but it is not a compact space. However we can consider its one-point-compactificationΣ = Σ ∪ {∞} and a suitable metric which in particular implies that neighbourhoods of ∞ contain the embedded subshifts Σ j for large enough j. The topological entropy can then be defined by (ε, n)-separated sets E ε,n (see e.g. [7]) and is given by h top (Σ) = lim ε→0 + lim n→∞ log |Eε,n| n . Example 3. If ν ≥ 2 and τ ≥ log 5 log ν then there are infinitely many distinct ergodic measures of maximal entropy on Σ.
Proof. Evidently h top (Σ) ≥ log ν. Moreover if we denote by R L (n) the number of words of length n in Σ whose first coordinate lie in A L = {0, 1, . . . , Lν} then we get that h top (Σ) = lim L→∞ lim n→∞ log R L (n) n . We have that

Equilibrium states
Let us construct now subshifts with non unique equilibrium states for Hölder continuous potentials. We say a function f : Σ → R is Hölder continuous on the shift space Σ if there exists a ϑ ∈ (0, 1) and a constant C so that for every n > 0 one has where the supremum is over all pairs of sequences x = · · · x −1 x 0 x 1 x 2 . . . , y = · · · y −1 y 0 x 1 y 2 · · · ∈ Σ for which x i = y i ∀|i| < n and either x −n = y −n or x n = y n .
3.1. Example with two distinct equilibrium states. Let τ > 0 and ν ≥ 3 and integer. Then we put as above A = {0} ∪ G ∪ Y = {0, 1, . . . , 2ν} and define as above the subshift Σ ⊂ A Z which has the two embedded full ν-shifts Σ * , * = G, Y and where the transitions are strings of zeros of lengths ≥ τ (|α| + |β|) with α, β being the adjacent monochromatic words of different colours. Let f be a real valued Hölder continuous function on the full two-shift Σ G so that P G (f ) > sup f + log 2, where P G (f ) is the pressure of f on the full shift Σ G (see [7]). For ν ≥ 3 this is possible since P G (f ) − sup f is bounded above by log ν and equal to log ν for constant f and the pressure function is continuous.
Let us define a Hölder continuous function g on Σ in two steps. We first extend the function to Σ Y by identification: that is g( y) = f ( x) for x ∈ Σ G , y ∈ Σ Y for which y i = x i + ν∀i. Now we extend g to bi-coloured words in the following way: If x 0 = 0 then we put g( x) = sup f . If x 0 = 0 and x ∈ Σ is a sequence which contains symbols of both colours then let x −n · · · x n be the longest monochromatic string of either green or yellow symbols (that is x −n · · · x n is a word in either Σ G or Σ Y and either x −n−1 = 0 or x n+1 = 0). We then pick a monochromatic point y ∈ U(x −n · · · x n ) ⊂ Σ * ( * = G, Y) and define g( x) = f ( y). In this way g is a Hölder continuous function on Σ. Let us now prove that the pressure of g (on Σ) is in fact equal to P G (f ) (which equals P Y (g) for a suitable choice of τ .

Lemma 3. If τ ≥ log 4ν
P G (f )−sup f − 1, then the pressure P (g) of g on Σ is equal to P G (f ).
Proof. Denote by W n the collection of Σ-words α = α 1 · · · α n of lengths n. The pressure of g is then given by the exponential growth rate of the partition function where x(α) is an arbitrary point in the cylinder set U(α) ⊂ Σ, and g n = g + gσ + gσ 2 + · · · + gσ n−1 is the n-th ergodic sum of g. If in the above sum we restrict to monochromatic words α of either colour, then we obtain the green or yellow partition function Z * n = α∈W * n e g n (x(α)) ( * = G, Y) whose growth rates are exactly P G (f ). We denoted by W * n the set of monochromatic words α = α 1 α 2 · · · α n in Σ * .
Let us note that the growth rate of n k=1 Z * k is also P G (f ). Hence, if we denote by W ′ n those words in W n that contain symbols of exactly one colour and otherwise at least one zero, then it follows that growth rate of Z n = α∈W ′ n e g n (x(α)) is also given by P G (f ). It thus remains to show that the exponential growth rate of α∈W ′′ n e g n (x(α)) is ≤ P G (f ), where W ′′ n denotes the genuinely bi-coloured words in W n (i.e. those words that contain green and yellow symbols). In Lemma 2 we estimated and since sup g = sup f , we obtain Then there are two mutually singular equilibrium states for g on Σ.
Proof. For ν ≥ 3 there are many potentials f : Σ G → R for which P G (f ) ≥ sup f + log 2. Then for the extension g : Σ → R there is an equilibrium state on each of the embedded full ν-shifts Σ G and Σ Y both of which, by Lemma 4, assume the pressure P (g) = P G (f ) in the variational principle. These two measures are both invariant and mutually singular. ✷
Let us define a Hölder continuous function g on Σ in two steps. We first extend the function from Σ 1 to Σ j for j = 2, . . . , L by identification: If x ∈ Σ 1 , y ∈ Σ j so that y i = x i +(j −1)ν∀i, then we put g( y) = f ( x). Then one extends g to bi-coloured words as before, namely we put g( x) = sup f if x 0 = 0 and if x 0 = 0 for a non-monochromatic x ∈ Σ then we put g( x) = f ( y) where y is a monochromatic point that realises the longest symmetric monochromatic word of x.
P G (f )−sup f − 1, then the pressure P (g) of g on Σ is equal to P G (f ). Proof. We have to find the exponential growth rate P (g) of the partition function Z n = α∈Wn e g n (x(α)) , where W n are the Σ-words of lengths n. Clearly P (g) ≥ P 1 (f ). In order to estimate the exponential growth rate of Z ′′ n = α∈W ′′ n e g n (x(α)) where W ′′ n are the genuinely multi-coloured words in W n we get as above that |W ′′ n | = Q(n) ≤ c 2 √ n2 n 1 + (5ν2 −1−τ ) n/(τ +1) , and therefore Z ′′ n = α∈W ′′ n e g n (x(α)) ≤ |W ′′ n |e n sup g ≤ c 2 e n sup f √ n2 n 1 + (5ν2 −1−τ )) n/(τ +1) which yields lim n→∞ log Z ′′ n n ≤ sup f + max log 2, log 5ν τ + 1 . Proof. For ν ≥ 3 there are many potentials f : Σ G → R for which P G (f ) ≥ sup f + log 2. Then for the extension g : Σ → R there is an equilibrium state on each of the embedded full ν-shifts Σ G and Σ Y both of which, by Lemma 4, assume the pressure P (g) = P G (f ) in the variational principle. These two measures are both invariant and mutually singular. ✷

3.3.
Example with infinitely many distinct equilibrium states. Let ν ≥ 3 and construct the subshift Σ ⊂ N Z 0 as was done above. A Hölder continuous function f on Σ 1 which satisfies P 1 (f ) ≥ sup f + log 2 is then as above extended to a Hölder function g on Σ so that it is a 'copy' on each of the embedded full shifts Σ j , j = 1, 2, . . . . Moreover we can extend g to the compact spaceΣ by defining g(∞) = sup f . If we denote by µ j the equilibrium state on Σ j for the potential g| Σ j then P j (g| Σ j ) = P 1 (f ) = P (g) for all j. We thus obtain: Example 6. If ν ≥ 3 and τ ≥ log 5 log ν and τ ≥ log 5ν P 1 (f )−sup f − 1 then there are infinitely many distinct equilibrium states µ j on Σ for g which are mutually singular.