Measures of maximal entropy of bounded density shifts

We ﬁnd suﬃcient conditions for bounded density shifts to have a unique measure of maximal entropy. We also prove that every measure of maximal entropy of a bounded density shift is fully supported. As a consequence of this we obtain that bounded density shifts are surjunctive.


Introduction
The concept of entropy is of particular interest when trying to define formally how a system behaves at equilibrium.Given a dynamical system, we say that an invariant measure is a uniform equilibrium state if it achieves the maximal possible entropy.It has been of interest to physicist and mathematicians to determine whether a system has a unique equilibrium state or not.When this happens some mathematicians say the system is intrinsically ergodic and some physicist say the system does not have phase transition.
In this paper we are interested in trying to determine if bounded density shifts are intrinsically ergodic.Bounded density shifts were introduced by Stanley in [22].These subshifts are defined somewhat similarly to the classical β-shifts in that they both are hereditary ( [14]), meaning that membership in the shift is preserved under coordinatewise reduction of letters.Whereas β-shifts are 'bounded from above' by a specific sequence coming from a βexpansion, bounded density shifts are restricted by length-dependent bounds on the sums of letters in subwords.
2 Definitions and preliminary results

Subshifts
We devote this section to collect some basic definitions in symbolic dynamics.For a broader introduction to subshifts, languages and their properties, see [15].Let A be a finite set of symbols.We say that w is a word if there exists n ∈ N such that w ∈ A n and we denote the length of w by |w|.Let ε denote the empty word, i.e. the word with no symbols.
A word u is a subword of w if u = w k w k+1 . . .w l for some 1 ≤ k ≤ l ≤ |w|.For words w (1) , . . ., w (n) , we use w (1) . . .w (n) to represent their concatenation.We say that a word u is a prefix of w if u = w 1 . . .w k for some 1 ≤ k ≤ |w| and a suffix if u = w k . . .w |w| for some 1 ≤ k ≤ |w|, denote by Suf(w) and Pre(w) the sets of nonempty suffixes and prefixes respectively for w.
We endow A Z with the product topology.When describing a point x ∈ A Z as a sequence, we use a dot to indicate the central position as follows, x = . . .x −1 .x0 x 1 . .., where x i to represent the ith coordinate of x.We represent intervals of integers with [i, j], and x [i,j] = x i x i+1 ...x j .
The shift map σ : For any subshift X, let We define L(X) = ∞ i=0 L n (X) as the language of the subshift X.Given a word w and k ∈ Z, we define its cylinder set as [w] k = {x ∈ X : x [k,k+|w|−1] = w}.The cylinder sets form a basis of the topology of A Z .

Specification properties
A subshift X is specified if there exists M ∈ N such that for all u, w ∈ L(X), there is a v ∈ L M (X) such that uvw ∈ L(X).Following [9], we also define specification for subsets of the language.
Let X be a subshift, G ⊂ L(X) and t ∈ N 0 .We say that G has specification (with gap size t) if for all m ∈ N and w (1) , . . ., w (m) ∈ G, there exists v (1) , . . ., v (m−1) ∈ L t (X) such that Moreover, if the cylinder [w] 0 contains a periodic point of period exactly |w| + t, then we say that G has periodic specification.

Measures of maximal entropy
For any subshift X, we denote by M (X) the set of Borel probability measures on X. Equipped with the weak* topology M (X) is a compact topological space.
For any µ ∈ M (X) and any finite measurable partition ξ of X, the entropy of ξ (with respect to µ), denoted by H µ (ξ), is defined by where terms with µ(A) = 0 are omitted.
Given a subshift X we denote the σ-invariant Borel probability measures with M (X, σ).For µ ∈ M (X, σ), the entropy of µ (for the shift map σ) is defined by where ξ (n) represents the partition of X into cylinder sets from the first n letters, i.e.
We note for future reference that ξ (n) = n−1 i=0 σ −i ξ (1) , where ξ (1) is the partition based on x 0 and ∨ is the join of partitions.We will later need to make use of the following basic facts about entropy; for proofs and general introduction to entropy theory, see [24].
Theorem 2.3 ([24], p. 184).For any subshift X, finite measurable partition ξ of X, measures µ i ∈ M (X), and . By the well-known Variational Principle, the supremum of h µ (X) over all µ ∈ M (X, σ) is the topological entropy h top (X) of X.For any subshift X, we have that For general topological dynamical systems, the supremum above may not be achieved.However, every subshift has at least one measure of maximal entropy, that is ν ∈ M (X, σ) achieving the supremum above, meaning that h ν (X) = h top (X) (e.g.see [24,Remark (2), pg 192]).
We say a subshift is intrinsically ergodic if there is only one (probability) measure of maximal entropy.
Every specified subshift is intrinsically ergodic [1].This result was generalized in several works, including [9] and [18].Before stating the result we need some extra definitions.
Given a collection of words D ⊆ L(X) and n ≥ 1, we define D n = D ∩ L n (X).We denote the growth rate of D by Note that h(L(X)) = h top (X).Following [9], we say that L(X) admits a decomposition C p GC s for C p , G, C s ⊂ L(X) if every w ∈ L(X) can be written as uvw for some u ∈ C p , v ∈ G, w ∈ C s .For such a decomposition, we define the collection of words G(M ) for each M ∈ N by Recall that Per(n) denotes the set of points with period at most n under σ.
Theorem 2.4.(Climenhaga and Thompson [9]) .Let X be a subshift whose language L(X) admits a decomposition L(X) = C p GC s , and suppose that the following conditions are satisfied: 1. G has specification.

h(C
3. For every M ∈ N, there exists τ such that given v ∈ G(M ), there exists words u, w with |u| ≤ τ, |w| ≤ τ for which uvw ∈ G.
Then X is intrinsically ergodic.Furthermore, if G has periodic specification, then converges to the measure of maximal entropy in the weak* topology.
Remark.Using results from [17], Climenhaga explained in a blog post [7] that condition 3 is actually not required to prove uniqueness of the measure of maximal entropy.However, this condition is not difficult to check for bounded density shifts with positive entropy (Lemma 3.4) and so we verify it regardless.

Bounded density shifts
Bounded density shifts were introduced in [22] (see also [2,Chapter 3.4]). Let for all m ≥ 0, and The bounded density shift associated to a canonical function, f , is defined as follows: Note that X f is a subshift on the alphabet A = {0, 1, ..., ⌊f (1)⌋}.
Actually, bounded density shifts can be defined for any function f : N 0 → [0, ∞), but it was shown in [22] that every bounded density shift can be defined by some canonical f .Definition.Let X f be a bounded density shift, the limit is called the limiting gradient and is denoted by α f .
The existence of the limit is given by Fekete's lemma and the definition of canonical function; furthermore, the limit is an infimum, and so f (n) ≥ α f n for all n.
There exist bounded density shifts with α f = 0 but they are fairly trivial systems where the upper density of non-zero coordinates is always 0. A bounded density shift has positive topological entropy if and only if α f > 0 (see [14,Theorem 12]) if and only if it is coded (determined by a labeled irreducible graph with possibly countably many vertices) ([22, Theorem 3.1]).
As we mentioned in the previous section, the specification property guarantees intrinsic ergodicity.For bounded density shifts, X f is specified with specification constant M if and only if 0 M is intrinsically synchronizing ([22, Theorem 5.1]).Bounded density shifts with positive topological entropy without specification can easily be constructed ( [22]).
As we mentioned in the previous section, the specification property guarantees intrinsic ergodicity.For bounded density shifts, X f is specified with specification constant M if and only if 0 M is intrinsically synchronizing [22, Theorem 5.1].There exist bounded density shifts with positive topological entropy without specification ( [22]).
A subshift X with alphabet A = {0, 1, ..., n} is hereditary if every time there is x ∈ X and y ∈ A Z with y i ≤ x i ∀i ∈ Z, then y ∈ X.It is not difficult to check that bounded density shifts are hereditary.

Intrinsic ergodicity
In this section we fix a binary bounded density shift X f .We define where ǫ denotes the empty word.
Lemma 3.1.The language L(X f ) admits a decomposition BGB.
Proof.Let z ∈ L (X f ).Define u to be the prefix of z in B of maximal length (which may be the empty word ǫ), and denote its length by M ≥ 0. Let z ′ be the maximal proper subword of z that does not overlap with u, i.e.
. Similarly, define w to be the suffix of z ′ in B of maximal length (which may be the empty word ǫ), and denote its length by N ≥ 0. We write y = z [M +1,|z|−N ] , and assume for a contradiction that y / ∈ G. Then by definition, there exists a word v ∈ Pre(y) ∪ Suf(y) with If v ∈ Pre(y), then uv would be a prefix of z in B longer than u, contradicting minimality of u.Similarly, if v ∈ Suf(y), then vw would be a suffix of z ′ in B longer than w, contradicting minimality of w.Therefore, we have a contradiction and y ∈ G, and so z = uyw ∈ BGB.
Lemma 3.2.The set G has specification.
Proof.We will show that G has periodic specification with gap size t = 0. Let m ∈ N, w (1) , . . ., w (m) ∈ G, v (a) ∈ Suf(w (m) ), v (b) ∈ Pre(w (1) ) and This implies that any periodic point made from concatenations of words from G is in X f .We conclude that G has periodic specification.
In the second part of the following proposition we use techniques from Misiurewicz's proof of the variational principle [16] to build measures with entropy higher or equal than that of a sub-language.These applications of the tools from [16] have already been noted in [4, Proposition 5.1] and [17,Lemma 6.8].

Proposition 3.3. There exists
Proof.For each n ∈ N and w ∈ L n (X f ) ∩ B, consider the set: ) be the atomic measure concentrated uniformly on the points of K n , i.e.
Let µ n ∈ M (X f ) be defined by Since M (X f ) is compact (in the weak* topology), we can choose a subsequence such that and µ n j → µ ∈ M (X f ).By the definition of µ n , it is routine to check that µ ∈ M (X f , σ), i.e. µ is σ-invariant.
So, for any such t, we can rewrite {0, 1, . . ., n − 1} as follows Observe that Thus, the cardinality of S t is at most 2q.
Using (11) we get Combining ( 10), ( 12) and Theorem 2.1 we obtain log For the inequality l∈St H νn j (σ −l ξ) ≤ 2q log(l) we apply Theorem 2.2.We note that for each 0 ≤ t ≤ q − 1, we have Summing the first term in the last line of (13) over t from 0 to q − 1, and using that the numbers {t + rq : 0 ≤ t ≤ q − 1, 0 ≤ r ≤ a(t) − 1} are all distinct and are all no greater than n − q, yields Using ( 13) and ( 15) we get Now, we divide by n j and apply Theorem 2.3 (with p i = 1 n j ), to obtain We will also use that which is obtained using the definition of weak* convergence.Then, combining ( 16) and ( 17) yields Lemma 3.4.For every M ∈ N, there exists τ such that given v ∈ G(M ), there exist words u, w with |u| ≤ τ , |w| ≤ τ for which uvw ∈ G.
Proof.Let M ∈ N and v ∈ G(M ).This implies that there exist Note that N 2 corresponds to the section where u ′ and w ′ appear and N 3 where v ′ appears.Also, we can assume that |z| ≥ τ (otherwise we are considering that z ∈ Pre(0 τ )), then Here, the first inequality holds since v ′ ∈ G, the second equality holds because |N 1 ∩ [1, |z|]| = τ (using |z| ≥ τ ), and the second inequality holds since The proof for z ∈ Suf(0 τ u ′ v ′ w ′ 0 τ ) is similar.
Theorem 3.5.Let X f be a bounded density shift.If every measure of maximal entropy µ has the property that ⌊f (1)⌋ i iµ([i] 0 ) < α f , then X f is intrinsically ergodic, and converges to the measure of maximal entropy in the weak* topology.
Proof.If α f = 0, then since all sequences have frequency 0 of non-0 symbols, the unique invariant measure is the delta measure of ∞ 0 ∞ .If α f > 0 we will obtain the result using Theorem 2.4.First note that B = C p = C s .Using Lemma 3.1 we obtain L(X) = C p GC s .Now we will check the numbered hypotheses of Theorem 2.4.
1. Lemma 3.2 gives us that G has specification.
2. Let µ ′ be the measure constructed in Lemma 3.3.By hypothesis it cannot be a measure of maximal entropy.Thus, 3. We obtain this property using Lemma 3.4.
The main application of the previous result that we have is the following.
Corollary 3.6.Let X f be a bounded density shift.If for every measure of maximal entropy µ.This implies that X f is intrinsically ergodic, and converges to the measure of maximal entropy in the weak* topology.
Proof.Using [12, Corollary 4.6] and the fact that bounded density shifts are hereditary we have that for any measure of maximal entropy Since µ is a probability measure this implies that µ([i] 0 ) ≤ 1/(i + 1).Thus, ⌊f .
We obtain the result using Theorem 3.5.
Remark.In particular, every binary bounded density shift with α f > 1/2 is intrinsically ergodic.
Furthermore, we suspect that the hypothesis of Theorem 3.5 may always be satisfied, at least for binary subshifts, leading to the following questions.
Question 3.7.Let X be a hereditary binary subshift with positive topological entropy.Is it true that for any measure of maximal entropy µ we have that A reason to suspect Question 3.7 is true is that if X is hereditary and µ([1] 0 ) achieves its (positive) supremum, then it should be possible to increase the entropy of µ by allowing a small proportion of randomly chosen 1 symbols to change to 0s.Some circumstantial evidence is given by the class of B-free shifts, for which it is known that maximal entropy is achieved by such a procedure (cf.Theorem 2.1.8 of [13]).We also ask the corresponding question for bounded density shifts on larger alphabets.Question 3.8.Is it true that for every bounded density shift we have that for every measure of maximal entropy?
One more natural question is whether we can prove stronger properties on the unique measure of maximal entropy via arguments such as those in [4] and [19].Question 3.9.Let X f be an intrinsically ergodic bounded density shift.Does the measure of maximal entropy have the K-property?Is it Bernoulli?
We don't know how to approach this question with current techniques.All arguments we're aware of which prove Bernoulli require connection to countable-state Markov shifts, which do not seem clear for bounded density shifts.And the usual argument to prove K-property (without Bernoulli) is to show that the product of (X f , σ) with itself has a unique measure of maximal entropy, but in general Climenhaga-Thompson decompositions are not preserved under products, and we do not see any reason that bounded density structure improves the situation.We note that purely being hereditary does not necessarily imply either property, as in [13] it was shown that for B-free shifts, the unique measure of maximal entropy factors onto the so-called Mirsky measure, which is of zero entropy; this precludes the K-property.

Entropy minimality and surjunctivity
We will now prove a property called entropy minimality for all bounded density shifts for α f > 0 using results from [12].We first need some definitions.
A subshift X is entropy minimal if every subshift strictly contained in X has lower topological entropy.Equivalently, X is entropy minimal if every measure of maximal entropy on X is fully supported.
Let X be a subshift and v ∈ L(X).The extender set of v is defined by Theorem 4.1 (García-Ramos and Pavlov [12]).Let X be a subshift with h top (X) > 0, µ a measure of maximal entropy and v, w ∈ L(X).
Proof.Let X f be a bounded density shift, µ ∈ M (X f , σ) a measure of maximal entropy and w ∈ L(X f ).Since the topological entropy of X f is positive then 1 ∈ L(X f ), and µ([1] 0 ) > 0 (otherwise µ([0] 0 ) = 1 and the entropy cannot be positive).By Poincaré's recurrence theorem, there exists v ′ ∈ L(X f ) for which µ([v ′ ] 0 ) > 0 and We can then define v which is coordinatewise less than or equal to w with This implies that x ′ [n,m] ∈ L(X f ).Thus, x ′ ∈ X f , and so y ∈ E X f (0 |v| w0 |v| ).Since y was arbitrary, A subshift is synchronized if there exists v ∈ L(X) such that v is an intrinsically synchronizing word.
Every entropy minimal synchronized subshift is intrinsically ergodic [23,12] and every synchronized subshift is coded [11].Hence, we obtain the following corollary.Another application of entropy minimality is surjunctivity.Given a subshift X, we say φ : X → X is a shift-endomorphism if it's continuous and it commutes with the shift.If a shift-endomorphism is bijective we say it is a shift-automorphism.
A subshift X is said to be surjunctive if every injective shift-endomorphism of X is a shift-automorphism.Every full shift is surjunctive ([10, Chapter 3].The following result is known (e.g.see [5]) but it is not explicitly stated.We write the proof since the argument is simple.Lemma 4.4.Every entropy minimal subshift is surjunctive.
Proof.Let X be a subshift and φ : X → X an injective shift-endomorphism.This implies that φ(X) is a subshift which is topologically conjugate to X. Since topological entropy is conjugacy-invariant, φ(X) has the same topological entropy as X.If X is entropy minimal then φ(X) = X.
Using this and Theorem 4.2 we obtain the following.

Universality
A dynamical system is said to be universal if every system with smaller entropy can be embedded in the original system (this can be studied either in the topological or measure-theoretic category).For instance, measuretheoretic universality of the full shift follows from Krieger's generator theorem.Results about both types (topological and measure-theoretical) of universality have been proved for systems with specification in [20,3,6], and we can prove a topological universality result for bounded density subshifts as well.We first need some basic definitions about topological dynamical systems.
A topological dynamical system is a pair (X, T ) where X is a compact metrizable space and T : X → X is a continuous function.Let (X, T ) and (X ′ , T ′ ) be two topological dynamical systems.We say X and X ′ are conjugated if there exists a homeomorphism f : X → X ′ such that For any TDS (X, T ) one can assign a topological entropy h top (X, T ).When the system is a subshift the notion coincides with the definition in Section 2.3.For the definition see [24,Chapter 7].
Let α ∈ R + .We define X α as the bounded density shift obtained with the function f (n) = ⌊nα⌋.Using [22,Theorem 1.3] we have that X α has specification.
Given a bounded density shift X f , one can check that X α f ⊂ X f .Let x ∈ X α f , then for every i ∈ Z and for every p ∈ N we have Therefore x ∈ X f and X α f ⊂ X f .Corollary 5.2.Let X f be a bounded density shift.We have that X f is h top (X α f )-universal.