Subsystem entropies of shifts of finite type and sofic shifts on countable amenable groups

In this work we study the entropies of subsystems of shifts of finite type (SFTs) and sofic shifts on countable amenable groups. We prove that for any countable amenable group $G$, if $X$ is a $G$-SFT with positive topological entropy $h(X)>0$, then the entropies of the SFT subsystems of $X$ are dense in the interval $[0, h(X)]$. In fact, we prove a"relative"version of the same result: if $X$ is a $G$-SFT and $Y \subset X$ is a subshift such that $h(Y)<h(X)$, then the entropies of the SFTs $Z$ for which $Y \subset Z \subset X$ are dense in $[h(Y), h(X)]$. We also establish analogous results for sofic $G$-shifts.


Introduction
Let G be a countable group and let A be a finite alphabet of symbols. In symbolic dynamics, the central objects of study are the subsystems of the so-called full shift, the dynamical system (A G , σ), where σ denotes the action of G on A G by translations (Definition 2.7). Shifts of finite type (Definition 2.16) and sofic shifts (Definition 2.17) are the most widely studied and well understood examples of symbolic dynamical systems. In each of these cases, the system of interest is completely specified by a finite amount of information. This allows for combinatorial, finitary arguments to be applied to the analysis of the dynamics of such systems.
Entropy is one of the most fundamental invariants of a topological dynamical system. Many fundamental results from classical entropy theory (i.e., in the case where G = Z) only generalize if G is an amenable group (Definition 2.2). Amenability allows one to "approximate" the group by a sequence of finite subsets in a way that is useful for studying dynamics. See Definition 2.22 for the definition of the entropy of a symbolic dynamical system on an amenable group.
In general, one would like to understand the structure of the collection of subsystems of a given subshift. In this paper we study the entropies of the SFT subsystems of a given SFT, as well as the entropies of the sofic subsystems of sofic shifts. There are many existing results in the literature in the case where G = Z. For example, the Krieger Embedding Theorem [12] characterizes the irreducible SFT subsystems of a given irreducible Z-SFT. Additionally, Lind [13] has provided an algebraic characterization of the real numbers that are realized as the entropy of a Z-SFT.
However, the situation is very different in cases where G = Z. Even in the case where G = Z d for d > 1, the classes of SFTs and sofic shifts behave quite differently. For example, Boyle, Pavlov, and Schraudner [5] have shown by example that the subsystems of Z d sofic shifts can be badly behaved for d > 1 (in contrast with the case where d = 1). Moreover, Hochman and Meyerovitch [9] have characterized the real numbers that are realized as entropy of a Z d -SFT (with d > 1), but in contrast to the result of Lind mentioned above, the characterization is in algorithmic terms and unavoidably involves concepts from computability and recursion theory. Nonetheless, Desai [6] has shown that a Z d -SFT with positive entropy has a wealth of SFT subsystems (sharpening an earlier result of Quas and Trow [16]). In recent years, several results of the G = Z and G = Z d cases have seen extensions to larger classes of groups, especially amenable groups. To name a few: Barbieri [2] has classified the real numbers that are realized as the entropy of a G-SFT for many types of amenable G (extending the result of Hochman and Meyerovitch mentioned above); Frisch and Tamuz [8] have investigated the (topologically) generic properties of G-subshifts for arbitrary amenable G; Barbieri and Sablik [3] have shown how an arbitrary effective G-subshift, where G is finitely generated, may be simulated by a G -SFT, where G is the semidirect product G = Z 2 G; and Huczek and Kopacz [10] have (very recently) obtained a partial generalization of Boyle's lower entropy factor Theorem [4] to countable amenable groups with the comparison property. In this vein, we prove the following generalization of Theorem 1.1 to arbitrary countable amenable groups. Choosing G = Z d and Y = ∅ in the above theorem recovers the result of Desai (Theorem 1.1 above). Note that a shift space X ⊂ A G has at most countably many SFT subsystems, and therefore the set of entropies of SFT subsystems is at most countable. In this sense, Theorem 4.2 is "the most one could hope for." Remark 1.2. After a preprint of this work was made public, the authors of [8] made us aware that a short alternate proof of Theorem 4.2 can be derived from their main results. Specifically, they prove there that for any countable amenable group G and any real c ≥ 0, the set of G-subshifts with entropy c is dense (in fact residual) within the space of G-subshifts with entropy at least c with respect to the Hausdorff topology. This result immediately implies that for any G-SFT X, there exist G-subshifts contained in X that achieve all possible entropies in [0, h(X)]; then, some simple approximations with G-SFTs (in the sense of our Theorem 2.27) can be used to obtain a proof of Theorem 4.2.
For sofic shifts, we obtain the following result. From this result, we can quickly derive the fact (Corollary 5.3) that if X is a sofic G-shift, then each real number in [0, h(X)] can be realized as the entropy of some (not necessarily sofic) subsystem of X. (Recall that the alternate proof of Theorem 4.2 described in Remark 1.2 above relies on a version of this result requiring X to be an SFT.) The tool for proving Theorem 5.2 (from Theorem 4.2) is provided by the following theorem, which may be of independent interest. We note that this result generalizes another theorem of Desai [6,Proposition 4.3], which addressed the case G = Z d .
Theorem 5.1. Let G be a countable amenable group and let W be a sofic G-shift. For every ε > 0, there exists an SFTX and a one-block codeφ :X → W such that the maximal entropy gap ofφ satisfies H(φ) < ε.
The maximal entropy gap H(φ) is defined in §2 (Definition 2.28). In particular, this result implies that if Y is sofic and ε > 0, then there is an SFT X that factors onto Y and satisfies h(X) < h(Y ) + ε.
Our proofs of Theorems 4.2, 5.1, and 5.2 take the same general approach as the arguments given by Desai for the G = Z d case. However, the extension to the general amenable setting requires substantial new techniques. Indeed, our proofs are made possible by the existence of exact tilings (Definition 3.1) of the group G that possess nice dynamical properties. Such exact tilings are trivial to find for Z d (by tiling the group using large hypercubes), but for arbitrary amenable groups were only recently constructed by Downarowicz, Huczek, and Zhang [7]; their construction is the main technical tool employed in this paper.
As mentioned in Remark 1.2 above, Theorem 4.2 can be alternately derived from results in [8]. We present a self-contained proof here for two reasons. Firstly, we would like to present a direct adaptation of the techniques from [6], since it demonstrates the power of the improved tiling results of [7]. Secondly, this presentation provides a unified approach to all of our proofs, since our proofs in the sofic setting (where we are not aware of alternative proofs) also rely on tiling-based constructions that are similar to those in our proof of Theorem 4.2.
The paper is organized as follows. In §2 we discuss basic notions and elementary theorems of symbolic dynamics, set in terms appropriate for countable amenable groups. In §3 we define and explore the concept of tilings and exact tilings of amenable groups, appealing to Downarowicz, Huczek, and Zhang for the existence of certain desirable tilings. In §4 we prove our main results for G-SFTs, and in §5 we prove our main results for sofic G-shifts. Finally, in §6 we provide a example of a Z 2 sofic shift whose only SFT subsystem is a fixed point.

Basics of symbolic dynamics
2.1. Amenable groups. We begin with a brief overview of amenable groups. Definition 2.1 (Group theory notations). Let G be a group and let K, F ⊂ G be subsets. We employ the following notations.
i. The group identity is denoted by the symbol e ∈ G, ii. KF = {kf : k ∈ K and f ∈ F }, iii. K −1 = {k −1 : k ∈ K}, iv. Kg = {kg : k ∈ K} for each g ∈ G, v. K F expresses that K and F are disjoint, and is their (disjoint) union, vi. K F = (K \ F ) (F \ K) is the symmetric difference of K and F , and vii. |K| is the cardinality of the (finite) set K. Definition 2.2 (Følner condition for amenability). Let G be a countable group. A Følner sequence is a sequence (F n ) n of finite subsets F n ⊂ G which exhausts G (in the sense that for each g ∈ G, we have g ∈ F n for all sufficiently large n) and for which it holds that lim n→∞ |KF n F n | |F n | = 0 for every finite subset K ⊂ G. If such a sequence exists, then G is said to be an amenable group.
Throughout this paper, G denotes a fixed countably infinite amenable group and (F n ) n is a fixed Følner sequence for G.
If e ∈ K and F is (K, ε)-invariant, then F is also (K , ε )-invariant for any ε > ε and any K ⊂ K such that e ∈ K . If F is (K, ε)-invariant, then so is the translate F g for each fixed g ∈ G. Invariance is the primary way by which we say a large finite subset F ⊂ G is a "good finite approximation" of G, according to the finitary quantifiers K and ε. The amenability of G provides a wealth of nearly invariant sets, which enables such approximation for the purpose of studying the dynamics of G-actions.
Next we develop concepts related to the geometry of finite subsets of G.
Definition 2.4 (Boundary and interior). Let K, F ⊂ G be finite subsets. The K-boundary of F is the set and the K-interior of F is the set If F is sufficiently invariant with respect to K, then the K-boundary of F is a small subset of F (proportionally), by the following lemma.
Lemma 2.5. Suppose K, F ⊂ G are nonempty finite subsets and e ∈ K. Then Proof. If e ∈ K, then KF F = KF \ F . If g ∈ KF \ F , then g = kf for some k ∈ K and f ∈ ∂ K F , by Definition 2.4. Therefore For the second inequality, note that f ∈ ∂ K F implies ∃k ∈ K such that kf ∈ F , therefore g = kf ∈ KF \ F is a point such that Given finite subsets K, F ⊂ G, in this paper we focus on the KK −1 -boundary and KK −1 -interior of F (rather than the K-boundary and K-interior), and we make use of the following lemma.

Shift spaces.
Here we present necessary definitions from symbolic dynamics. See Lind and Marcus [14] for an introductory treatment of these concepts.
Definition 2.7 (Shifts and subshifts). Let A be a finite set of symbols equipped with the discrete topology. A function x : G → A is called an A-labelling of G. By convention, we write x g for the symbol x(g) ∈ A which is placed by x at g ∈ G.
The set of all A-labellings of G is denoted A G , which we equip with the product topology. For each g ∈ G, let σ g : A G → A G denote the map given by The subshift generated by x is the topological closure of O(x) as a subset of A G , and is denoted O(x) ⊂ A G .
Definition 2.8 (Codes and factors). Let A X , A W be finite alphabets and let X ⊂ A G X and W ⊂ A G W be subshifts. A map φ : X → W is shift-commuting if φ • σ g = σ g • φ for each g ∈ G; the map φ is said to be a sliding block code if it is continuous and shift-commuting; and φ is said to be a factor map if it is a surjective sliding block code. If a factor map exists from X to W , then W is said to be a factor of X and X is said to factor onto W . If a sliding block code φ : X → W is invertible and bi-continuous, then φ is said to be a topological conjugacy, in which case X and W are said to be topologically conjugate. Definition 2.9 (Products of shifts). If A and Σ are finite alphabets, then A × Σ is also a finite alphabet (of ordered pairs). If X ⊂ A G and T ⊂ Σ G are subshifts, then we view the dynamical direct product X × T as a subshift of (A × Σ) G , defined by (x, t) ∈ X × T if and only if x ∈ X and t ∈ T . The shift space X × T factors onto both X and T via the projection maps π X and π T , given by π X (x, t) = x and π T (x, t) = t for each (x, t) ∈ X × T .
Remark 2.10. Definition 2.9 above introduces an abuse of notation, as technically we have (x, t) ∈ A G × Σ G = (A × Σ) G . However, if equipped with the G-action ς given by ς g (x, t) = (σ g x, σ g t), then A G × Σ G becomes a dynamical system that is topologically conjugate to (A × Σ) G .

2.3.
Patterns. In this section we describe patterns and their related combinatorics. Definition 2.11 (Patterns). Let A be a finite alphabet and let F ⊂ G be a finite set. A function p : F → A is called a pattern, said to be of shape F . The set of all patterns of shape F is denoted A F . The set of all patterns of any finite shape is denoted A * = F A F , where the union is taken over all finite subsets F ⊂ G.
Remark 2.12. Given a point x ∈ A G and a finite subset F ⊂ G, we take x(F ) to mean the restriction of x to F , which is itself a pattern of shape F . Usually this is denoted x| F ∈ A F , but we raise F from the subscript for readability. Definition 2.13 (One-block code). Let A X and A W be finite alphabets and let X ⊂ A G X and W ⊂ A G W be subshifts. A factor map φ : X → W is said to be a one-block code if there exists a function Φ : A X → A W with the property that Definition 2.14 (Occurrence). Let A be a finite alphabet and let F ⊂ G be a finite set. A pattern p ∈ A F is said to occur in a point x ∈ A G if there exists an element g ∈ G such that (σ g x)(F ) = p. If X ⊂ A G is a subshift, then the collection of all patterns of shape F occurring in any point of X is denoted by Definition 2.15 (Forbidden patterns). Let A be a finite alphabet, let F ⊂ G be a finite set and let X ⊂ A G be a subshift. A pattern p ∈ A F is said to be allowed in X if p ∈ P(F, X) (if p occurs in at least one point of X).
Given a (finite or infinite) collection of patterns F ⊂ A * , a new subshift X ⊂ X may be constructed by expressly forbidding the patterns in F from occurring in points of X. We denote this by X = R(X, F) = {x ∈ X : ∀p ∈ F, p does not occur in x}.
For a single pattern p, we abbreviate R(X, {p}) as X \ p. The shift X is said to be specified by the collection F if X = R(A G , F).

2.4.
Shifts of finite type. In this section, we define shifts of finite type and sofic shifts over G. We also discuss many related elementary facts.
For an SFT, it is always possible to take F in the form F = A K \ P(K, X) for some large finite subset K ⊂ G. In this case, we say X is specified by (patterns of shape) K.
If X ⊂ A G is an SFT specified by a finite subset K ⊂ G, then it holds that x ∈ X ⇐⇒ ∀g ∈ G (σ g x)(K) ∈ P(K, X) for each x ∈ A G . If K specifies X, then so does K for any (finite) subset K ⊃ K. If X and T are SFTs, then so is the dynamical direct product X × T . The following elementary facts are needed; we abbreviate the proofs as they are similar to the well-known the proofs in the case where G = Z (see [14]). Proposition 2.18. Let X be an SFT, let W be a sofic shift, and let φ : X → W be a factor map. Then there exists an SFTX and a topological conjugacyφ :X → X such that the composition φ •φ :X → W is a one-block code.
Proof. Because φ is continuous and shift-commuting, there exists a large finite subset K ⊂ G such that for each x, x ∈ X and each g ∈ G, it holds that Suppose that e ∈ K and that P(K, X) specifies X as an SFT. LetÃ = P(K, X) be a new finite alphabet, and letX ⊂Ã G be the set of all pointsx ∈Ã G such that ∃x ∈ X, ∀g ∈ G,x g = (σ g x)(K).
ThenX is an SFT specified by patterns of shape K −1 K. The mapφ :X → X desired for the theorem is given bỹ Proposition 2.19. For any subshift X ⊂ A G , there is a descending family of SFTs (X n ) n such that X = n X n .
Proof. Let (p n ) n enumerate {p ∈ A * : p does not occur in X}, and for each n let Then (X n ) n witnesses the result.
Proposition 2.20. Let X ⊂ A G be a subshift and let X 0 ⊂ A G be an SFT such that X ⊂ X 0 . If (X n ) n is any descending family of subshifts such that X = n X n , then X n ⊂ X 0 for all sufficiently large n.
Proof. Take K ⊂ G to specify X 0 as an SFT. Note P(K, X n ) n is a descending family of finite sets, and it is therefore eventually constant. In particular, we have P(K, X n ) = P(K, X) ⊂ P(K, X 0 ) for all sufficiently large n.
When G = Z d , SFTs are often reduced via conjugacy to so-called 1-step SFTs, in which the allowed patterns are specified by a d-hypercube of side-length 1. Such SFTs are often desired because they allow for a kind of "surgery" of patterns. If two patterns occur in two different labellings from a 1-step SFT, and yet they agree on their 1-boundaries, then the first may be excised and replaced by the second. This yields a new labelling which also belongs to the 1-step SFT. Although there is no obvious notion of 1-step SFTs when G = Z d , we do have the following result which allows for this sort of excision and replacement of patterns.
Lemma 2.21. Let X ⊂ A G be an SFT specified by K ⊂ G, let F ⊂ G be a finite subset, and let x, y ∈ X be two points such that x and y agree on ∂ KK −1 F . Then the point z, defined by z g = y g if g ∈ F and z g = x g if g / ∈ F , also belongs to X.
In the first case, we have (σ g z)(K) = (σ g y)(K) which is an allowed pattern in X. In the second case, we have Kg ⊂ (F c ) (∂ KK −1 F ). Since x and y agree on ∂ KK −1 F , we have (σ g z)(K) = (σ g x)(K) which is again an allowed pattern in X. In either case, (σ g z)(K) is allowed in X for every g, hence z ∈ X.

2.5.
Entropy. Let X ⊂ A G be a nonempty subshift. Recall that for a given large finite set F ⊂ G, the number of patterns of shape F that occur in any point of X is |P(F, X)|, which is at most |A| |F | . As this grows exponentially (with respect to |F |), we are interested in the exponential growth rate of |P(F, X)| as F becomes very large and approaches the whole group G. For nonempty finite sets F ⊂ G, we let Definition 2.22 (Entropy). Let X be a nonempty subshift. The (topological) entropy of X is the nonnegative real number h(X) given by the limit where (F n ) n is again the Følner sequence of G. For the empty subshift, we adopt the convention that h(∅) = 0.
It is well-known that the limit above exists, does not depend on the choice of Følner sequence for G, and is an invariant of topological conjugacy (see [11]).
For any subshift X ⊂ A G and any finite subset F ⊂ G it holds that h(F, X) ≤ log |A|, and consequently h(X) ≤ log |A|. More generally, if X and X are subshifts such that X ⊂ X , then h(F, X) ≤ h(F, X ) for every finite subset F ⊂ G and consequently h(X) ≤ h(X ). If X and X are subshifts over A, then so is X ∪ X and h(X ∪ X ) = max h(X), h(X ) .
The following proposition is a classical fact; a proof is given in [11].
Proposition 2.23. Let G be a countable amenable group. If a G-shift W is a factor of a G-shift X, then h(W ) ≤ h(X).
Frequently in this paper we refer to "measuring" or approximating the entropy of a subshift via a large set F . We give a precise definition as follows.
Infinitely many such sets exist for any δ, as provided by the Følner sequence and the definition of h(X). We introduce this notion so that we may layer invariance conditions and entropy-approximating conditions as needed.
Proposition 2.25. For finitely many choices of i, let K i ⊂ G be any finite subsets, and let ε i > 0 be any positive constants. For finitely many choices of j, let X j ⊂ A G j be any subshifts over any finite alphabets, and let δ j > 0 be any positive constants. There exists a finite subset F ⊂ G which is (K i , ε i )-invariant for every i, and which δ j -approximates the entropy of X j for every j.
Proof. Choose F = F n for sufficiently large n.
The following theorem is an elementary generalization of a classical statement (see [14] for a proof in the case where G = Z). We omit the proof here for brevity. Proposition 2.26. Let (X n ) n be a descending family of subshifts, and let X = n X n . Then It is desirable to work with SFTs as much as possible while preserving (or, in our case, approximating) relevant dynamical quantities. We shall make frequent use of the next theorem, which we justify with several of the above results.
Proof. By Proposition 2.19, there is a descending family of SFTs (X n ) n such that X = n X n . By Proposition 2.20, we have X n ⊂ X 0 for all sufficiently large n. By Proposition 2.26, we have h(X) ≤ h(X n ) < h(X) + ε for all sufficiently large n. Choose Z = X n for n large enough to meet both conditions.
If φ : X → W is a factor map of subshifts, then we have already seen that h(W ) ≤ h(X). The "entropy drop" or entropy gap between X and W is the quantity h(X) − h(W ). A subsystem X ⊂ X induces a corresponding subsystem φ(X ) = W ⊂ W , and later in this paper we will want a uniform bound for the entropy gap between every X and W pair. We make this idea precise in the following definition.
Definition 2.28. Suppose φ : X → W is a factor map. The maximal entropy gap of φ is the quantity where the supremum is taken over all subshifts X ⊂ X. In particular, it holds that Recall that if X and T are subshifts, then the dynamical direct product X × T factors onto both X and T via the projection map(s) π X (x, t) = x and π T (x, t) = t.
Proposition 2.29. Let X and T be shift spaces. The maximal entropy gap of the projection map π X : where the supremum is taken over all subshifts Z ⊂ X × T .
A quick corollary is that when h(T ) = 0, we have h(Z) = h(π X (Z)) for any subsystem Z ⊂ X × T .

Tilings of amenable groups
3.1. Definition and encoding. In this section we consider the notion of tilings of G. The existence of tilings of G with certain properties is essential in our constructions in subsequent sections.
Definition 3.1 (Quasi-tilings and exact tilings). A quasi-tiling of G is a pair (S, C), where S is a finite collection of finite subsets of G (called the shapes of the tiling) and C is a function that assigns each shape S ∈ S to a subset C(S) ⊂ G, called the set of centers or center-set attributed to S. We require that e is in S for each S ∈ S. The following properties are also required.
i. For distinct shapes S, S ∈ S, the subsets C(S) and C(S ) are disjoint.
ii. The shapes in S are "translate-unique", in the sense that We may refer to both the pair (S, C) and the collection as "the quasi-tiling." Each subset τ = Sc ∈ T is called a tile. For a quasi-tiling T , we denote the union of all the tiles by T . A quasi-tiling T may not necessarily cover G in the sense that T = G; nor is it necessary for any two distinct tiles τ , τ ∈ T to be disjoint. However, if both of these conditions are met (that is, if T is a partition of G), then T is called an exact tiling of G.
Ornstein and Weiss [15] previously constructed quasi-tilings of G with good dynamical properties, and this construction has become a fundamental tool for analyzing the dynamics of G-actions. Downarowicz, Huczek, and Zhang [7] sharpened this construction, showing that a countable amenable group exhibits many exact tilings with good dynamical properties, as we describe below (see Theorem 3.6).
A quasi-tiling T of G may be encoded in symbolic form, allowing for dynamical properties to be attributed to and studied for quasi-tilings. The encoding method presented here differs from the one presented in [7], as we will only require exact tilings in this paper. See Remark 3.4 below for further discussion of the relation between our encoding and the encoding given in [7]. which we view as a finite alphabet. If T is an exact tiling of G over S, then it corresponds to a unique point t ∈ Σ G as follows. For each g ∈ G, there is a unique tile Sc ∈ T containing g; let s = gc −1 ∈ S and set t g = (S, s).
In the above definition, note that s is the "relative position" of g in the translate Sc of S. In other words, t labels each element g of G with both the type of shape of the tile containing g and the relative position of g within that tile. In particular, g ∈ C(S) ⇐⇒ t g = (S, e).
Note that the correspondence T → t ∈ Σ G , when regarded as a map on the set of all exact tilings of G over S, is injective. However, the correspondence is not surjective in general. Let Σ E ⊂ Σ G be the set of all encodings of exact tilings of G over S. It may be the case that no exact tiling of G over S exists, in which case Σ E = ∅. In general, we have the following useful theorem.
Proof. Let Σ 1 be the set of all points t ∈ Σ G that satisfy the following local rule: where c = s −1 0 g. It is easy to see that Σ 1 is an SFT, and from Definition 3 For the reverse inclusion, let t ∈ Σ 1 be an arbitrary point satisfying the local rule (R1) everywhere. For each S ∈ S, let C(S) = {g ∈ G : t g = (S, e)}. Then T = T (S, C) is a quasi-tiling. To complete the proof, it suffices to show that T is exact and encoded by t, since that would give t ∈ Σ E and then Σ E (S) = Σ 1 .
Let g ∈ G, suppose t g = (S, s), and let c = s −1 g. By rule (R1) and the fact that e ∈ S, we have t c = t ec = (S, e) and therefore c ∈ C(S). Hence, g = sc ∈ Sc ∈ T . This demonstrates that T = G. Next, suppose Sc, S c ∈ T are not disjoint and let g ∈ Sc ∩ S c . Then g = sc = s c for some s ∈ S and s ∈ S . From c ∈ C(S) we have t c = (S, e), and by the rule (R1) we have t g = t sc = (S, s).
By identical proof we have t g = (S , s ), from which it follows that S = S and s = s . The latter implies that and hence Sc and S c are the same tile. This demonstrates that T is a partition of G, and therefore T is an exact tiling of G over S. Finally, we note that it is straightforward to check that T is encoded by t, which completes the proof.
Remark 3.4. Before we move on, we note here that the encoding method presented above (Definition 3.2) differs from the one presented in [7]. The encoding method in that work gives symbolic encodings for all quasi-tilings, which is not necessary for our present purposes. Indeed, the encoding in [7] uses the alphabet Λ = S ∪ {0}, and a point λ ∈ Λ G encodes a quasi-tiling (S, C) when λ g = S ⇐⇒ g ∈ C(S) and λ g = 0 otherwise. This is a prudent encoding method for the study of general quasitilings, as any quasi-tiling may be encoded in this manner. Our encoding method works only for exact tilings, but is well-suited to our purposes. In fact, if one is only interested in exact tilings, then the two encodings are equivalent. Indeed, if Λ E ⊂ Λ G is the collection of all encodings of exact tilings of G over S, then there is a topological conjugacy φ : Σ E → Λ E given by φ(t) g = S ⇐⇒ t g = (S, e) and φ(t) g = 0 otherwise.
Next we turn our attention to the dynamical properties of tilings, as derived from their encodings. Definition 3.5 (Dynamical tiling system). Let S be a finite collection of finite shapes, let T be an exact tiling of G over S, and let T be encoded by the point t ∈ Σ E (S). The dynamical tiling system generated by T is the subshift generated by This allows for the dynamical properties (e.g., entropy) of Σ T as a subshift of Σ G to be ascribed to T . The tiling entropy of T is h(T ) = h(Σ T ), the entropy of Σ T as a subshift of Σ G .
The tiling entropy of T is a measure of the "complexity" of tile patterns that occur in large regions of G. In particular, when T has entropy zero, the number of ways to cover a large region F ⊂ G by tiles in T grows subexponentially (with respect to |F |).
The following theorem is quickly deduced from the main result of Downarowicz, Huczek, and Zhang [7], which we state in this form for convenience. It is this result that allows us to utilize exact tilings of G in this paper. i. Each shape S ∈ S is (K, ε)-invariant.
The point t 0 encodes an exact tiling T 0 of G over S with tiling entropy h(T 0 ) = 0.

3.2.
Approximating sets with tiles. Entropy and other dynamical properties of G-shifts are well measured by sets with strong invariance properties (the Følner sequence F n provides a wealth of such sets). However, we would instead like to utilize an (appropriately selected) exact tiling T for this purpose. In this section, we build good tile approximations of sets: finite collections of tiles T * ⊂ T attributed to large, suitably invariant subsets F ⊂ G that are good in the sense that the symmetric difference F T * is small (as a proportion of |F |).
Lemma 3.8. Let S be a finite collection of shapes from G, and let U = S. Let ε > 0, and choose δ > 0 such that δ|U ||U U −1 | < ε. Let F ⊂ G be a finite subset that is (U U −1 , δ)-invariant. For any exact tiling T of G over S, the following statements hold: Proof. First, we observe that each tile τ ∈ T is contained in a translate U g for some g ∈ G; indeed, we have τ = Sc for some S ∈ S and c ∈ C(S) ⊂ G, then S ⊂ U implies τ ⊂ U c. This fact also gives that |τ | ≤ |U | for every tile τ ∈ T . We claim that every tile τ ∈ T × (F ) \ T • (F ) intersects ∂ U U −1 F . To establish the claim, we first note that for each such tile τ it holds that τ ∩ F = ∅ and τ ⊂ F . So let f ∈ τ ∩ F , and note that f ∈ τ ⊂ U g for some g ∈ G. From τ ⊂ F we also have U g ⊂ F . By Lemma 2.6 we have f ∈ U g ⊂ int U U −1 F c , and hence By the claim in the previous paragraph, there is a map γ : T × \ T • → ∂ U U −1 F with the property that γ(τ ) ∈ τ for each τ . Observe that γ is injective, as distinct tiles are disjoint, and therefore |T × \ T • | ≤ |∂ U U −1 F |. We also have that by the invariance hypothesis on F and Lemma 2.5. Then This establishes statement (i.). The remaining two statements are easy to check One more notion is necessary to develop before moving on from tilings: the frame of a given subset with respect to a given tiling.
where the union ranges over all τ ∈ T • (F ). See Figure 1 for an illustration.

Results for SFTs
Having discussed everything about tilings relevant for our purposes, we are now ready to begin discussing our main results. In this section we present our results for SFTs, and in the following section we turn our attention to sofic shifts. Theorem 4.1. Let G be a countable amenable group, and let X be a G-SFT such that h(X) > 0. Then Before we begin the proof, let us give a short outline of the main ideas. The broad strokes of this proof come from Desai [6], whose argument in the case where G = Z d we are able to extend to the case where G is an arbitrary countable amenable group. This is possible by utilizing the exact tilings of G constructed by Downarowicz, Huczek and Zhang [7].
Given an arbitrary ε > 0, we produce a family of SFT subshifts of X whose entropies are 2ε-dense in [0, h(X)]. We accomplish this by first selecting an exact, zero entropy tiling T 0 of G with suitably large, invariant tiles. Then we build subshifts with strongly controlled entropies inside the product system Z 0 = X ×Σ 0 , where Σ 0 is the dynamical tiling system generated by T 0 .
The T -interior of F is shaded.
(c) The inner frame of F (with respect to T , K) is shaded. The K-boundary of each tile inside F is taken. To construct these subshifts from Z 0 , we control which patterns in the X layer can appear in the "interior" of the tiles in the Σ 0 layer. We are able to finely comb away entropy from Z 0 by forbidding these patterns one at a time. This process generates a descending family of subsystems for which the entropy drop between consecutive subshifts is less than ε. After enough such patterns have been forbidden, the overall entropy is less than ε. This collection of subshifts therefore has entropies that are ε-dense in [0, h(Z 0 )]. Then we project the subshifts into X and utilize Theorem 2.27 to produce SFTs subsystems of X with entropies that are 2ε-dense in [0, h(X)].
Proof. Let X ⊂ A G be an SFT such that h(X) > 0, let K ⊂ G be a large finite subset such that P(K, X) specifies X as an SFT, and let ε be any constant such that 0 < ε < h(X). Choose δ > 0 such that 2δ + δ log 2 + 2δ log |A| < ε.
By Theorem 3.6, there exists a finite collection S of finite subsets of G with the following properties.
iii. There is a point t 0 ∈ Σ E (S) such that h(O(t 0 )) = 0. Consequently, t 0 encodes an exact tiling T 0 of G over S with tiling entropy zero.
For the remainder of this proof, these are all fixed. We shall abbreviate ∂F = ∂ KK −1 F for any finite subset F ⊂ G. For a pattern p on F , we take ∂p to mean p(∂F ) and call this the border of p (with respect to KK −1 ).
Let Σ 0 = O(t 0 ) ⊂ Σ E be the dynamical tiling system generated by the tiling T 0 , which has entropy zero. Of central importance to this proof is the product system X × Σ 0 , which factors onto X via the projection map π : X × Σ 0 → X given by π(x, t) = x for each (x, t) ∈ X × Σ 0 . Let us establish some terminology for certain patterns of interest which occur in this system. Given a shape S ∈ S, we shall refer to a pattern b = (b X , b T ) ∈ P(S, X × Σ 0 ) as a block (to distinguish from patterns of any general shape). If a block b ∈ (A × Σ) S satisfies b T s = (S, s) for every s ∈ S, then we shall say b is aligned. See Figure 2 for an illustration of the aligned property. For a subshift Z ⊂ X × Σ 0 , we denote the subcollection of aligned blocks of shape S that occur in Z by where the superscript a identifies the subcollection. Given a shape S ∈ S and an aligned block b of shape S, consider the border ∂b ∈ (A × Σ) ∂S . We are interested in the number of ways that the border ∂b may be extended to all of S -that is, the number of allowed (and in particular, aligned) interiors for S which agree with ∂b on the boundary ∂S. For a subshift Z ⊂ X × Σ 0 , we denote this collection by ints a (∂b, Z) = {b ∈ P a (S, Z) : ∂b = ∂b}.
We shall extend all the same terminology described above (blocks, aligned blocks, borders, interiors) to tiles τ = Sc ∈ T 0 , as there is a bijection between P(S, Z) and P(τ, Z) = P(Sc, Z). For a given tile τ ∈ T 0 , a block b ∈ (A × Σ) τ is aligned if b T sc = (S, s) for each sc ∈ Sc = τ . The subcollection of aligned blocks of shape τ occurring in a shift Z ⊂ X ×Σ 0 is denoted P a (τ, Z). Given a border ∂b ∈ (A×Σ) ∂τ , the collection of aligned blocks of shape τ occurring in Z agreeing with ∂b on ∂τ is also denoted ints a (∂b, Z) ⊂ P a (τ, Z).
For the theorem, we shall inductively construct a descending family of subshifts (Z n ) n of X × Σ 0 as follows. Begin with Z 0 = X × Σ 0 , then assume Z n has been constructed for n ≥ 0. If there exists a shape S n ∈ S and an aligned block β n ∈ P a (S n , Z n ) such that | ints a (∂β n , Z n )| > 1, then let Z n+1 = Z n \ β n . If no such block exists on any shape S ∈ S, then Z n is the final subshift in the chain and the chain is finite in length.
Let us first argue that in fact, the chain must be finite in length. For each n ≥ 0 we have Z n+1 ⊂ Z n , in which case P a (S, Z n+1 ) ⊂ P a (S, Z n ) for every shape S ∈ S. Moreover, for the distinguished shape S n (the shape of the forbidden block β n ), it holds that P a (S n , Z n+1 ) {β n } ⊂ P a (S n , Z n ). This implies that S∈S |P a (S, Z n )| strictly decreases with n. There is no infinite strictly decreasing sequence of positive integers, hence the descending chain must be finite in length. Let N ≥ 0 be the index of the terminal subshift, and note by construction that the shift Z N satisfies | ints a (∂b, Z N )| = 1 for every aligned block b ∈ P a (S, Z N ) on any shape S ∈ S.
Most of the rest of the proof aims to establish the following two statements: To begin, let F ⊂ G be a finite subset satisfying the following two conditions: (F1) F is (U U −1 , ϑ)-invariant, where U = S and ϑ is a positive constant such that ϑ|U ||U U −1 | < δ. Note this implies that F may be well approximated by tiles from any exact tiling of G over S, in the sense of Lemma 3.8. (F2) F is large enough to δ-approximate (Definition 2.24) the entropy of Σ 0 and Z n for every n ≤ N . This implies in particular that h(F, Σ 0 ) < δ. Such a set exists by Proposition 2.25. We fix F for the remainder of this proof. Now for each n ≤ N , we claim that | ints a (f (∂τ ), Z n )|, and (E1) where the indices t, f , and τ are as follows. The variable t ranges over P(F, Σ 0 ), and therefore t is the restriction to F of an encoding of an exact, zero entropy tiling T t of G over S. The variable f ranges over all (A×Σ)-labellings of the (T t , KK −1 )frame of F (Definition 3.9) that are allowed in Z 0 and for which f T agrees with t. Lastly, the variable τ ranges over the tiles in T • t (F ). To begin the argument towards the claims (E1) and (E2), let n ≤ N be arbitrary. To count patterns p ∈ P(F, Z n ), write p = (p X , p T ) and sum over all possible labellings in the tiling component. We have where the sum ranges over all t ∈ P(F, Σ 0 ). This is valid because Z n ⊂ Z 0 = X×Σ 0 , hence any z = (z X , z T ) ∈ Z n must have z T ∈ Σ 0 . Next, let t ∈ P(F, Σ 0 ) be fixed. The pattern t extends to/encodes an exact, zero entropy tiling T t of G over S, possibly distinct 1 from the original selected tiling T 0 .
Recall that F • (T t ) = T • t (F ) ⊂ F is the inner tile approximation of F by the tiling T t (Definition 3.7). Recall also that the (T t , KK −1 )-frame of F is the subset τ ∂τ where the union is taken over all τ ∈ T • t (F ) (Definition 3.9). Since K is fixed for this proof, we shall abbreviate the frame as fr t (F ). From Equation (1), we now split over all allowed labellings of fr t (F ). We have where the first sum is taken over all t ∈ P(F, Σ 0 ), and the second sum is taken over all f ∈ P(fr t (F ), Z 0 ) for which f T agrees with t.
We have the pattern t ∈ P(F, Σ 0 ) fixed from before; next we fix a frame pattern f ∈ P(fr t (F ), Z 0 ) such that f T agrees with t. We wish to count the number of patterns p ∈ P(F, Z n ) such that p T = t and p(fr t (F )) = f . Let this collection be denoted by D n = D(t, f ; Z n ) ⊂ P(F, Z n ). Observe that each D n is finite and D n+1 ⊂ D n for each n < N . Consider the map γ : D 0 → τ P(τ, Z 0 ) given by γ(p) = (p(τ )) τ , which sends a pattern p ∈ D 0 to a vector of blocks indexed by T • t (F ). We claim the map γ is at most |A| δ|F | -to-1, and for each n ≤ N we have Together, these claims will provide a bound for |D n | from above and below, which combine with Equation (2) to yield the claims (E1) and (E2). First we argue that γ is at most |A| δ|F | -to-1. This is where we first invoke the invariance of F . Suppose (b τ ) τ ∈ τ P(τ, Z 0 ) is a fixed vector of blocks. If p ∈ D 0 is a pattern such that γ(p) = (b τ ) τ , then p T is determined by t and p(τ ) = b τ for each tile τ ∈ T • t (F ). Therefore, p is uniquely determined by p By property (F1) of the set F and by Lemma 3.8 we have |F \ F • t | < δ|F |, and thus the map γ is at most |A| δ|F | -to-1. Next, we shall prove the set equality (3). Let n ≤ N , and let p ∈ D n . For each tile τ ∈ T • t (F ), the block p(τ ) ∈ P(τ, Z n ) ⊂ (A × Σ) τ is aligned; this is because p T = t and t encodes the tiling T t itself. Moreover, p agrees with f on fr t (F ) by assumption that p ∈ D n = D(t, f ; Z n ), in which case p(∂τ ) = f (∂τ ) for each tile τ ∈ T • t (F ). This demonstrates that We shall prove the reverse inclusion by induction on n. For the n = 0 case, let (b τ ) τ be a vector of blocks such that b τ ∈ P a (τ, Z 0 ) and ∂b τ = f (∂τ ) for each τ ∈ T • t (F ). To construct a γ-preimage of (b τ ) τ in D 0 , begin with a point x ∈ X such that x(fr t (F )) = f X . Such a point exists because f occurs in some point of for each τ ∈ T • t (F ), because ∂τ ⊂ fr t (F ) for each τ . Moreover, for each τ it holds that the block b X τ occurs in a point of X, as each block b τ = (b X τ , b T τ ) occurs in a point of Z 0 = X × Σ 0 . Because X is an SFT specified by patterns of shape K, we may repeatedly apply Lemma 2.21 to excise the block x(τ ) and replace it with b X τ for every τ ∈ T • t . Every tile is disjoint, so the order in which the blocks are replaced does not matter. After at most finitely many steps, we obtain a new point . We have that p T = t and p(fr t (F )) = f by the selection of x . This implies that p ∈ D 0 . It also holds that p(τ ) = b τ for each τ ∈ T • t (F ), as t itself encodes the tiling T t from which the tiles τ ∈ T • t (F ) are drawn (and each block b τ is aligned, by assumption). We then finally have γ(p) = (b τ ) τ , which settles the case n = 0. Now suppose the set equality (3) holds for some fixed n < N , and let (b τ ) τ ∈ τ ints a (f (∂τ ), Z n+1 ). From the inclusion Z n+1 ⊂ Z n and the inductive hypothesis, it follows there is a pattern p ∈ D n such that γ(p) = (b τ ) τ . Suppose p = (x, t)(F ) for some (x, t) ∈ Z n (by induction, t is the point fixed from before). We need to modify p only slightly to find a γ-preimage of (b τ ) τ which occurs in Z n+1 (and hence belongs to D n+1 ).
Consider the block β n determined at the beginning of this proof, which is forbidden in the subshift Z n+1 . If β n occurs anywhere in the point (x, t), then (by the assumption that β n is aligned) it must occur on a tile 2 τ ∈ T t . It does not occur on any of the tiles from T • t (F ), because for each tile τ ∈ T • t (F ) we have (x, t)(τ ) = b τ which is allowed in Z n+1 by assumption.
Yet, β n may occur in (x, t) outside of F • t . By the construction of Z n+1 , we have | ints a (∂β n , Z n )| > 1, and therefore there is an aligned blockb which occurs in Z n such thatb = β n and ∂b = ∂β n . Apply Lemma 2.21 at most countably many times 3 to excise β X n wherever it may occur in x, replacing it withb X . This yields a new point x ∈ X.
Then (x , t) ∈ Z 0 also belongs to Z n+1 . It was already the case that none of the blocks β 0 , . . . , β n−1 could occur anywhere in (x, t) by the assumption (x, t) ∈ Z n , and now neither does β n occur anywhere in (x , t). The pattern p = (x , t)(F ) assumption that | ints a (∂β n , Z n )| > 1, we know there is an aligned blockb n = β n that occurs in Z n such that ∂b n = ∂β n .
Recall X is an SFT specified by patterns of shape K, andb X n is allowed in X. Again we may apply Lemma 2.21 at most countably many times, excising β X n wherever it may occur in x and replacing it withb X n . At the end we receive a new point x ∈ X, within which β X n does not occur. Then (x , t) is allowed in Z n+1 and (x , t)(S) = b, hence b ∈ P a (S, Z n+1 ).
The conclusion is that β n is the only aligned block lost from Z n to Z n+1 . For each b ∈ P a (S, Z n ), we have either ints a (∂b, Z n ) = ints a (∂b, Z n+1 ) or ints a (∂b, Z n ) = ints a (∂b, Z n+1 ) {β n }. If two positive integers differ by at most 1 then their ratio is at most 2, hence the inequality (4) follows.
Finally, we shall use the estimates (E1) and (E2) to argue for the ultimate claims (U1) and (U2) made before. For the first, consider a fixed n < N . It is clear that h(Z n+1 ) ≤ h(Z n ) by inclusion. For the second inequality in (U1), we have where the inequalities are justified by (E2), (4), the fact that |T • t (F )| < δ|F |, and (E1), respectively. Taking logs and dividing through by |F |, we obtain where we have used the property (F2) of F , the previous display, the property (F2) again, and our choice of δ. This inequality establishes (U1). For (U2), recall that the terminal shift Z N has the property that any aligned border ∂b ∈ P a (∂S, Z N ) on any shape S ∈ S has exactly 1 allowed aligned interior. Hence, we see that where the first inequality is justified by (E2) and the last inequality is justified by our bounds on the number of terms in the sums (established previously). Taking logs and dividing through by |F |, we finally have where we have used the property (F2) of the set F , the previous display, and our choice of δ. We have now established (U2). With (U1) and (U2) in hand, the rest of the proof is easy. By (U1) and (U2), we have that (Z n ) n≤N is a family of subshifts of For each n ≤ N , let X n = π(Z n ) ⊂ X, where π is the projection map π(x, t) = x. From Lemma 2.29, H(π) = h(Σ 0 ) = 0, hence h(X n ) = h(Z n ) for every n ≤ N .
Then (X n ) n≤N is a descending family of subshifts of X such that (h(X n )) n≤N is ε-dense in [0, h(X)]. Though each X n may not be an SFT, we do know that X is an SFT. One may therefore apply Theorem 2.27 to construct a family of SFTs (Y n ) n≤N such that for each n ≤ N , we have X n ⊂ Y n ⊂ X and h( As ε was arbitrary, we conclude that the entropies of the SFT subsystems of X are dense in [0, h(X)].
The following "relative" version of Theorem 4.1 is stronger and easily obtained as a consequence of Theorem 4.1.
Corollary 4.2. Let G be a countable amenable group, let X be a G-SFT, and let Y ⊂ X be any subsystem such that h(Y ) < h(X). Then Proof. We prove the density directly. Suppose (a, b) ⊂ [h(Y ), h(X)] for positive reals a < b, and let ε < (b − a)/2. By Theorem 4.1, there exists an SFT Z 0 ⊂ X such that a < h(Z 0 ) < a + ε. Note that these inequalities give h(Y ) < h(Z 0 ). Consider the subshift Y ∪ Z 0 ⊂ X, which has entropy Because X is an SFT and Y ∪ Z 0 ⊂ X, by Theorem 2.27 there is an SFT Z such Since (a, b) was arbitrary, the proof is complete.

5.
1. An extension result for sofic shifts. In order to address the case of sofic shifts, we seek to leverage our results on SFTs. In particular, given a sofic shift W , we would like an SFT X such that W is a factor of X and such that the maximal entropy drop across the factor map is very small. The following theorem guarantees the existence of such SFTs.
Theorem 5.1. Let W ⊂ A G W be a sofic shift. For every ε > 0, there exists an SFTX and a one-block codeφ :X → W such that the maximal entropy gap ofφ satisfies H(φ) < ε.
Proof. Since W is sofic, there exists an SFT X ⊂ A G X and a factor map φ : X → W . Without loss of generality, we assume that i. φ is a one-block code, witnessed by the function Φ : A X → A W , and ii. A X and A W are disjoint. We abbreviate A XW = A X A W . Let ε > 0, and select δ > 0 such that Let K ⊂ G be a large finite subset that specifies X as an SFT. The set K is fixed for the remainder of this proof, and thus we shall denote ∂ KK −1 F by ∂F and int KK −1 F by int F for any finite set F ⊂ G. By Theorem 3.6, there exists a finite set of finite shapes S such that the following conditions are met. i. Each shape S ∈ S is (KK −1 , η)-invariant, where η > 0 is a constant such that η|KK −1 | < δ. By Lemma 2.5, this implies |∂S| < δ|S| for each S ∈ S. ii. KK −1 ⊂ S and |S| > δ −1 for each S ∈ S.
Recall by Proposition 3.3 that Σ E is an SFT. By Theorem 2.27, there is an SFT T such that O(t 0 ) ⊂ T ⊂ Σ E and h(T ) < h(O(t 0 )) + δ. Consequently, each point t ∈ T is an encoding of an exact tiling T t of G over S (possibly distinct from the original tiling T 0 ), with tiling system entropy Because X and T are SFTs, we have that X × T ⊂ (A X × Σ) G is also an SFT.
Let t ∈ T be arbitrary, and recall that T t is a partition of G. Thus, for each g ∈ G, there is a unique tile τ ∈ T t such that g ∈ τ . We define the notation T t (g) by setting T t (g) = τ . Next we define a map φ t : X → A G XW by the following rule: for each g ∈ G and x ∈ X, This map is well-defined, as τ = ∂τ int τ . The map φ t applies the one-block code φ to "most" of a point x, by relabelling the interiors of each tile τ ∈ T t .
We now define a sliding block code ϕ : X × T → (A XW × Σ) G by applying the map(s) φ t fiber-wise: for each point (x, t) ∈ X × T , let It is straightforward to check that ϕ is indeed a sliding block code (Definition 2.8). For the theorem, the desired shiftX is identified with the range of this map. Let See Figure 3 for an illustration of the construction. It remains to show that there is a one-block codeφ :X → W , that the shiftX is an SFT, and that H(φ) < ε.
First, let us show thatX factors onto W . The factor map is induced by the functionΦ : A XW → A W , which is an extension of Φ, defined by the following rule: It is clear thatφ is a one-block code. Let us now show thatφ(X) = W . Let x ∈ X and t ∈ T , in which case (φ t (x), t) ∈X is an arbitrary point. The effect of applying the map φ t to x is to apply the one-block code φ to "most" of x. The mapφ then "completes" the relabelling, via the extended functionΦ. In fact, we have thatφ(φ t (x), t) = φ(x) ∈ W , henceφ(X) ⊂ W . For the reverse inclusion, let w ∈ W . Since φ : X → W is onto, there exists a point x ∈ X such that φ(x) = w. Figure 3. A hypothetical point x ∈ X with a tiling t ∈ T overlayed; the partially-transformed point φ t (x) is pictured, which is labelled with symbols from both X and W ; finally, the whollytransformed image point φ(x) ∈ W is reached.
Let us now show thatX is an SFT. We repeat that the shiftX can be written in the following instructive form: In order to show thatX is an SFT, we will construct an SFTX 1 ⊂ (A XW × Σ) G and then prove thatX =X 1 . Recall that K ⊂ G specifies X as an SFT. Let K T ⊂ G be a finite subset such that P(K T , T ) specifies T . We defineX 1 to be the set of points (x, t) ∈ (A XW × Σ) G that satisfy the following local rules.
(R1) Any pattern of shape K T that occurs in t must belong to P(K T , T ), and any pattern of shape K that occurs inx and belongs to A K X must also belong to P(K, X) (recall P(K,X) ⊂ A K XW = (A X A W ) K in general). Note by Definition 2.14 that this condition is shift-invariant. (R2) For any shape S ∈ S and any c ∈ G, if t satisfies (σ c t) s = (S, s) for each s ∈ S, then ∃b ∈ P(S, X) such that (σ cx ) s = b s ∈ A X for all s ∈ ∂S and (σ cx ) s = Φ(b s ) ∈ A W for all s ∈ int S.
As these are local rules, they define an SFT; call itX 1 ⊂ (A XW × Σ) G . Moreover, it is easily checked that any point (φ t (x), t) ∈X satisfies these rules everywhere (by construction ofX), and so we haveX ⊂X 1 .
For the reverse inclusion, consider a point (x, t) ∈X 1 . From (R1) it follows that t ∈ T , as T is an SFT specified by K T . Therefore, t encodes an exact tiling T t of G over S with h(T t ) < δ. Let (τ n ) n enumerate the tiles of T t , and for each n let τ n = S n c n for some S n ∈ S and c n ∈ G. Recall {τ n : n ∈ N} is a partition of G.
Let n ∈ N, and consider c = c n and S = S n . Observe that, because t encodes the tiling T t , we have (σ c t) s = (S, s) for each s ∈ S. Then by (R2), there exists a block b = b n ∈ P(S, X) such that (σ cx ) s = b s for all s ∈ ∂S and (σ cx ) s = Φ(b s ) for all s ∈ int S.
Define a point x ∈ A G X by setting x(τ n ) = b n for each n ∈ N. We claim that x is an allowed point of X and that φ t (x) =x. Toward this, let g ∈ G be arbitrary, and consider the translate Kg (recall that K specifies X as an SFT).
If Kg intersects the interior of any tile τ n = S n c n , then Kg ⊂ τ n by Lemma 2.6. In this case, the pattern (σ g x)(K) is a subpattern of b n , and must therefore be allowed in X as b n ∈ P(S n , X). The alternative is that Kg is disjoint from the interior of every tile, in which case Kg ⊂ n ∂τ n . By (R2), we also havex g ∈ A X for every g ∈ n int τ n . In this case we have (σ g x)(K) = (σ gx )(K), which is again allowed in X by (R1).
In either case we have that (σ g x)(K) is allowed in X for any g ∈ G, and hence x ∈ X. Then by the definition of φ t , we see that φ t (x) =x. Thus, we have found a point (x, t) ∈ X × T such that ϕ(x, t) = (φ t (x), t) = (x, t), and hence (x, t) ∈X. We conclude thatX =X 1 , and thereforeX is an SFT.
Let F ⊂ G be a finite subset such that the following conditions are met.
(F1) F is (U U −1 , ϑ)-invariant, where U = S and ϑ > 0 is a constant such that ϑ|U ||U U −1 | < δ (recall δ was selected at the beginning of this proof). Note this implies that F may be well approximated by tiles from any exact tiling of G over S, in the sense of Lemma 3.8. (F2) F is large enough to δ-approximate (Definition 2.24) the entropy of the shifts X , W and T (recall that h(T ) < δ, in which case h(F, T ) < 2δ).
Such a set exists by Proposition 2.25. This set is fixed for the remainder of this proof. Recall thatφ is a one-block code, and therefore there is a well defined map Φ F : P(F,X ) → P(F, W ) which takes a pattern p ∈ P(F,X ) and applies the one-block code to p (at each element of F ).
Recall also that a pattern p ∈ P(F,X ) is of the form p = (φ t (x), t)(F ) for some points x ∈ X and t ∈ T . The point t encodes an exact tiling T t of G over S. For each tile τ ∈ T t , the definition of φ t implies that (5) φ t (x)(int τ ) ∈ A * W , and φ t (x)(∂τ ) ∈ A * X .
Let q =Φ F (p) ∈ P(F, W ). Recall that every element g ∈ F belongs to a unique tile τ = T t (g) ∈ T × t (F ), where T × t (F ) ⊂ T t is the outer approximation of F by the tiling T t (Definition 3.7). By (5), we have that In particular, we have q g = pX g whenever g belongs to the set where the union is taken over all τ ∈ T × t (F ). In light of these observations, we are ready to estimate |P (F,X )| in terms of |P (F, W )|. We first useΦ F to split over P(F, W ), and then we split again over all possible T -layers. Indeed, we have where the sums are taken over all patterns q ∈ P(F, W ) and t ∈ P(F, T ). Choose and fix patterns q and t. If p ∈ P(F,X ) is a pattern such thatΦ F (p) = q and p T = t, then the observations above imply that p is uniquely determined by where the union is taken over all tiles τ ∈ T × t (F ). Moreover, our choice of S and the property (F1) of F together yield that Therefore, there are at most |A X | δ(1+δ)|F | patterns p such thatΦ F (p) = q and p T = t. From this and Equation (6), we have By taking logs and dividing through by |F |, we obtain the following: where we have used property (F2) of the set F , the above inequality, property (F2) again, and our choice of δ respectively. SinceX ⊂X was arbitrary, we have that which completes the proof.

Subsystem entropies for sofic shifts.
Here we present our main result concerning subsystem entropies for sofic shifts. The proof follows easily by combining our extension result (Theorem 5.1) with our result for SFTs (Theorem 4.2).
Theorem 5.2. Let G be a countable amenable group, let W be a sofic G-shift and let V ⊂ W be any subsystem such that h(V ) < h(W ). Then Proof. We prove the density directly.
. By Theorem 5.1, there exists an SFT X and a factor map φ : Thus h(U ) ∈ (a, b), which completes the proof.
If one selects V = ∅ for the above theorem, then one recovers the statement that the entropies of the sofic subsystems of W are dense in [0, h(W )]. Next, we present our result concerning the entropies of arbitrary subsystems of sofic shifts. Proof. If h(W ) = 0, then r = 0, in which case one may simply select R = W . If h(W ) > 0, then let W 0 = W and let (ε n ) n be a sequence of positive real numbers converging to zero. We have that W 0 is sofic and r ≤ h(W 0 ), and without loss of generality we assume that h(W 0 ) < r + ε 0 .
Inductively construct a descending sequence of sofic shifts as follows. If W n ⊂ W is a sofic shift such that r ≤ h(W n ) < r + ε n , then by Theorem 5.2 there exists a sofic shift W n+1 ⊂ W n for which r ≤ h(W n+1 ) < r + ε n+1 .
Then R = n W n ⊂ W is a subshift such that h(R) = lim n h(W n ) = r by Proposition 2.26.

A counter-example
Theorem 5.2 implies that the entropies of the sofic subsystems of a sofic shift space W are dense in [0, h(W )]. One may wonder if this can be somehow "sharpened"; that is, one may wonder whether {h(W ) : W ⊂ W and W is an SFT} is dense in [0, h(W )]. However, this statement is nowhere close to true in general, as we illustrate in this section by counterexample. This example is an adaptation of a construction of Boyle, Pavlov, and Schraudner [5]. Proposition 6.1. There exists a sofic Z 2 -shift with positive entropy whose only SFT subsystem is a singleton.
Proof. We first construct a certain point in {0, 1} Z as the limit of a sequence of finite words, then consider the subshift it generates. Let δ = 0.1 and let (T n ) n be the sequence of natural numbers given by T n = 2n · 2 n · δ −1 + 1 for each n. Let w 1 = 010 ∈ {0, 1} 3 , and for each n define the word (7) w n+1 = w n w n · · · w n w n 0 n 10 n , where the w n term is repeated exactly T n times. The limit word w ∞ ∈ {0, 1} N0 is an infinite one-sided sequence. Define a two-sided sequence (P3) For each x ∈ X, either x = 0 Z or x contains the word 0 n 10 n for every n. For (P1), let N be arbitrary. Note that because X = O(x * ), any word of length N occurring in any point x ∈ X is also a word occurring in x * . By the recursive definition (7) and the fact that the sequence {T n } ∞ n=1 is recursive, there is an algorithm which, upon input N , prints all the words of length N that do not appear as subwords of x * . The shift X is therefore effective.
For (P2), we argue that x * satisfies the condition. For each n, let L n be the length of the word w n . Note that by the recurrence (7), we have L n+1 = T n L n + 2n + 1 ∀n.
For each n, let f n be the frequency of 1s in w n , given by Observe that f n ≤ 1 for each n and f 1 = 1 3 . It follows from the recurrence (7) that f n+1 = f n T n L n + 1 T n L n + 2n + 1 for each n. This implies that Tn < δ 2 n for each n. Hence, we have that 1 3 − δ < f n for each n. By the recurrence (7), we therefore have that and the subsequence along (L n ) n is a witness. For (P3), let n be given. First, observe that the infinite sequence w ∞ is the concatenation of a sequence of blocks, where each block is either the word w n or 0 m 10 m for some m ≥ n. Moreover, each w n begins with 0 and ends with 0 n . This implies that 1s in distinct blocks are separated by at least n + 1 appearances of the symbol 0. Therefore, if for any k ≤ n we have that 10 k 1 appears anywhere in w ∞ , then it must appear as a subword of a single block (rather than overlapping two distinct blocks), and that block must be w n .
Next, let x ∈ X be arbitrary. If the symbol 1 appears in x at most one time, then (P3) trivially holds. Otherwise, assume that 10 k 1 appears somewhere in x for some k ≥ 1. Without loss of generality, suppose x 0 = x k+1 = 1 and x i = 0 for i ∈ [1, k]. Now consider the subword ω = x([−L n , L n ]) for any n such that k < L n . Because X = O(x * ), the word ω must be a subword of x * . Then, either ω is a subword of x * ([−2L n , 2L n ]), or ω is a subword of w ∞ or a mirror reflection of one. In the first case, the definitions of x * and w ∞ imply that ω contains the word w n or its mirror. In the latter two cases, the observation of the previous paragraph implies that ω must contain w n or its mirror. In any case, 0 n 10 n is a subword of x. As n can be made arbitrarily large, this proves (P3).
We now use the shift X to construct the Z 2 -shift which is desired for the theorem. For each point x ∈ X, let x Z ∈ {0, 1} Z 2 denote the Z 2 -labelling given by x Z (i,j) = x i for each (i, j) ∈ Z 2 . That is, x Z is a Z 2 -labelling such that the symbols along each column are constant, and each row is equal to x itself. We shall also denote It is a theorem of Aubrun and Sablik [1] that if X is effective, then X Z is sofic. Next, consider the alphabet {0, 1, 1 }, where we have artificially created two independent 1 symbols. Let π : {0, 1, 1 } Z 2 → {0, 1} Z 2 be the one-block code which collapses 1 and 1 . Let Y = π −1 (X Z ) ⊂ {0, 1, 1 } Z 2 . The shift Y is a copy of the shift X Z , in which the 1 symbols of every point have been replaced either by 1 or 1 in every possible combination.
We claim that the shift Y is the desired subshift for the theorem. Specifically, we claim that Y is sofic, that Y has positive entropy, and that the only nonempty SFT subsystem of Y is the singleton {0 Z 2 }.
To prove that Y is sofic, we construct an SFT S and a factor map φ : S → Y to witness the soficity of Y . Since X Z is sofic, there is an SFT S ⊂ A Z 2 and a factor map φ : S → X Z . Without loss of generality, assume that φ is a one-block code induced by the function Φ : A → {0, 1}. Let S = S × {1, 1 } Z 2 , which we regard as a subshift of (A × {1, 1 }) Z 2 . Note that S is an SFT, because both S and {1, 1 } Z 2 (the full Z 2 -shift on two symbols) are SFTs. A point s ∈ S is of the form s = (s, ι), where s is a point of S and ι ∈ {1, 1 } Z 2 is an arbitrary 2-coloring of Z 2 . The reader may easily check that (π • φ )(s, ι) = φ(s) ∈ X Z , from which it follows that φ (S ) = π −1 (X Z ) = Y . Then φ : S → Y is a factor map. Since S is an SFT, we conclude that Y is sofic.
Next, we will show that h(Y ) > 0. From property (P2), the point x * ∈ X exhibits 1s in more than 10% of the positions in each of infinitely many symmetric intervals, say of the form [− n , n ] for an increasing sequence of natural numbers ( n ) n . Therefore, the point (x * ) Z exhibits 1s in more than 10% of the positions in each square F n = [− n , n ] 2 . Each 1 in the pattern (x * ) Z (F n ) may be replaced by 1 or 1 independently to yield an allowed pattern of Y , which implies that |P(F n , Y )| ≥ 2 0.1|Fn| ∀n.
As (F n ) n is a Følner sequence for Z 2 , we then have h(Y ) ≥ 0.1 log 2 > 0.
It remains to show that the only nonempty SFT subsystem of Y is the singleton {0 Z 2 }. Suppose to the contrary that Z ⊂ Y is an SFT subsystem of Y which contains a nonzero point. Since Z is an SFT, we may find a constant k ∈ N such that the allowed patterns of Z are specified by the shape K = [0, k) 2 ⊂ Z 2 .
Let z ∈ Z be a point different from 0 Z 2 and note π(z) = x Z ∈ X Z for some x ∈ X with x = 0 Z . By property (P1), the string 0 n 10 n appears in x for every n. Let n > k be fixed. Suppose without loss of generality that 0 n 10 n appears centered at the origin of x (with x 0 = 1 and x i = 0 for 0 < |i| ≤ n). Thus we have z (0,0) = 1 or 1 . In fact, by the definition of Y , we have z (0,j) ∈ {1, 1 } for every j ∈ Z.
Consider the i = 0 column of the point z. Starting with each index ∈ Z and looking up, there is a corresponding vertically oriented word ω ∈ {1, 1 } n given by ω j = z (0, +j) for each j ∈ [0, n). By the pigeonhole principle, there must exist a word ς ∈ {1, 1 } n such that ς = ω for infinitely many choices of . That is, for infinitely many choices of , we have z (0, +j) = ς j for each j ∈ [0, n).
Let 1 < 2 be two such indices where a repetition occurs, with 2 − 1 > n. That is, we have z (0, 1 +j) = z (0, 2+j) = ς j for every j ∈ [0, n). Now consider the rectangle r = z [−n, n] × [ 1 , 2 ) . Tile Z 2 with infinitely many translated copies of r to obtain a new point z ∈ {0, 1, 1 } Z 2 . Figure 4 illustrates the construction. Every pattern of shape K = [0, k) 2 which occurs in z is a pattern which occurs in z (including the pattern of all zeroes), hence they are all allowed in Z. Because Z is an SFT specified by K, it then follows that z ∈ Z. Because Z ⊂ Y = π −1 (X Z ), there must exist a point x ∈ X such that π(z ) = (x ) Z . We obtain a contradiction, as the point x cannot satisfy the property (P3) of X. For instance, the word 0 3n 10 3n cannot appear in x (as each row of z is periodic in the horizontal direction with period 2n + 1 < 3n). This demonstrates that if Z is an SFT, then it contains no nonzero point. Therefore, the only nonempty SFT subsystem of Y is {0 Z 2 }.