Factoring strongly irreducible group shift actions onto full shifts of lower entropy

We show that if $G$ is a a countable amenable group with the comparison property, and $X$ is a strongly irreducible $G$-shift satisfying certain aperiodicity conditions, then $X$ factors onto the full $G$-shift over $N$ symbols, so long as the logarithm of $N$ is less than the topological entropy of $G$.


Introduction
A well-known result in the study of symbolic dynamical systems states that any subshift of finite type (SFT) with the action of Z and entropy greater or equal than log N factors onto the full shift over N symbols -this was proven in [9] and [1] for the cases of equal and unequal entropy respectively.Extending these results for actions of other groups has been difficult, and it is known that a factor map onto a full shift of equal entropy may not exist in this case (see [3]).Johnson and Madden showed in [7] that any SFT with the action of Z d , which has entropy greater than log N and satisfies an additional mixing condition (known as corner gluing), has an extension which is finite-to-one (hence of equal entropy) and maps onto the full shift over N symbols.This result was later improved by Desai in [4] to show that such a system factors directly onto the full shift, without the intermediate extension, and then by Boyle, Pavlov and Schraudner ( [2]) to replace the corner gluing by a weaker mixing condition (block gluing).
In this paper we use similar methods to adapt these constructions to symbolic dynamical systems with actions of amenable groups.Our approach requires three assumptions: that the group G has the comparison property (satisfied, for instance, for all countable amenable groups with subexponential growth), that the system X is strongly irreducible (which replaces the corner gluing condition, and allows the construction to be valid without assuming that the underlying system be a subshift of finite type), and that it has non-periodic blocks for all possible sets of periods (this condition seems reminiscent of faithfulness but we were not able to directly derive it from such an assumption).With these assumptions we prove that X can be factored onto any full shift of smaller entropy (i.e. a full shift over N symbols, where log N < h(X)).We note that our method does not apply for the case of equal entropy (log N = h(X)); as we mention earlier, Boyle and Schraudner have shown in [3] that there exist Z d -shifts of finite type which do not factor onto full shifts of equal entropy.

Preliminaries
In this section we establish the definitions, notation and standard facts we will use in this paper.Since this is mainly standard material, we omit most proofs and references.
2.1.Amenability, Følner sets and invariance.Throughout this paper, G we will denote a countable amenable group, and (F n ) will denote a fixed Følner sequence, i.e. a sequence of finite subsets of G such that for every g ∈ G the sequence |gFn△Fn| |Fn| tends to 0 as n goes to infinity.Multiplication involving sets will always be understood element-wise, so gF n is the set {gf : f ∈ F n }, and KF in the following definition denotes the set {kf The defining property of the Følner sequence can be equivalently (and usefully) restated thus: for every finite K ⊂ G and every ε > 0 there exists an N such that for every n N the set F n is (K, ε) invariant.Definition 2.2.Let D, F be finite subsets of G. Let F D = {g ∈ F : Dg ⊂ F }. We will refer to F D as the D-interior of F .
A straightforward computation shows that for any 2.2.Symbolic dynamical systems.Let Λ be a finite set (referred to as the alphabet).The full G-shift over Λ is the product set Λ G with the product topology (induced by the discrete topology on Λ) endowed with the right-shift action of G: By a symbolic dynamical system, a shift space, or subshift, we understand any closed, shiftinvariant subset X of Λ G .Definition 2.3.For a finite T ⊂ G, by a block with domain T we understand a function B : T → Λ.If X is a symbolic dynamical system over Λ, and x ∈ X, then we say that B occurs in x (at position g) if for every t ∈ T we have x(tg) = B(t), which we denote more concisely as x(T g) = B. We say that B occurs in X if it occurs in some x ∈ G. Definition 2.4.We say that a symbolic dynamical system X is strongly irreducible (with irreducibility distance D) if for any blocks B 1 , B 2 (with domains T 1 , T 2 ) which occur in X, and any g 1 , g 2 ∈ G such that DT 1 g 1 ∩ T 2 g 2 = ∅, there exists an x ∈ X such that x(T 1 g 1 ) = B 1 and x(T 2 g 2 ) = B 2 .Definition 2.5.If P is a finite subset of G, we say that a block B with domain T is P -aperiodic if for every p ∈ P there exists a t ∈ T such that tp is also in T , and B(tp) = B(t).We will say that a symbolic dynamical system X is aperiodic if it has a P −aperiodic block for every finite P ⊂ G. Definition 2.6.For a fixed Følner sequence (F n ), let N Fn (X) be the number of different blocks with domain F n which occur in X.The topological entropy of a symbolic dynamical system X is defined as (In this paper we use logarithms with base 2, although as usual the theorems remain true if one defines entropy using any other base, so long as the choice remains consistent throughout).It is a standard fact that the obtained value of h(X) does not depend on the choice of the Følner sequence.In fact, the relation between entropy and the number of blocks holds for any sufficiently invariant domain, as per the following theorem ([8]) Theorem 2.7.For any ε > 0 there exists an N > 0 and δ > 0 such that if T is an (F n , δ)invariant set for some n > N, then N T (X) > 2 (h(X)−ε)|T | .

Quasitilings and tilings.
Definition 2.8.A quasitiling of a countable amenable group G is any collection T of finite subsets of G (referred to as tiles) such that there exists a finite collection S of finite subsets of G (referred to as shapes) such that every T ∈ T has a unique representation T = Sc for some S ∈ S and some c ∈ G.We refer to such a c as the center of T .If the tiles of T are disjoint and their union is all of G, then we refer to T as a tiling.Remark 2.9.If we enumerate the set of shapes of a quasitiling T as {S 1 , S 2 , . . ., S r }, then we can identify T with an element x T of the set {0, 1, . . ., r} G defined as x T (g) = j if S j g is a shape of T , and x T (g) = 0 otherwise.This in turn induces (via orbit closure) a subshift X T , and any element of X T in turn corresponds to a quasitiling of G which has the same disjointness, invariance and density properties as T .This allows us to discuss some properties of quasitilings using the notions of topological dynamics (in particular, it makes sense to talk of entropy and of factorizations), by interpreting these notions as applied to the corresponding subshifts.We will use several theorems which guarantee the existence of quasitilings and/or tilings satisfying certain properties.The first is proven in [5] and we will invoke it when constructing subsystems with specified entropy (in section 3) and marker blocks (in section 4).
Theorem 2.10.For any K ⊂ G and any ε > 0 there exists a tiling T of G such that all tiles of T are (K, ε)-invariant and h(T ) = 0.
When we construct the factor map onto the full shift, we will rely on combining two other results.The first one originally appears in the seminal paper by Ornstein and Weiss ( [10]), although in [5] it is stated and proven in an equivalent form closer to the one stated here (the main difference being that the original version does not explicitly use the notion of lower Banach density): Theorem 2.11.For any δ > 0 and N > 0 there exists a quasitiling T of G such that: (1) the shapes of T are all Følner sets, i.e. S = {F n 1 , F n 2 , . . ., F nr }, where N n 1 < n 2 < . . .< n r , and r depends only on δ.
The last theorem we will need is proven in [6], although the authors do not state it as a stand-alone fact, but rather establish it as a step in proving Theorem 7.5.The cited paper also contains a definition and a lengthy discussion of the comparison property; here we will just recall that the class of groups with comparison property includes all amenable groups of subexponential growth and it remains an open question whether there are any countable amenable groups without the comparison property.
Theorem 2.12.Suppose G is a countable amenable group with the comparison property, and K is a finite subset of G containing the neutral element e.For every ε > 0 there exist δ > 0 and N > 0 such that every (1 − δ)-covering quasitiling T whose tiles are pairwise disjoint and shapes are (K, δ)-invariant and have cardinality larger than N can be modified to a tiling T • whose shapes are (K, ε)-invariant.Moreover, the sets of centres of T and T • are identical, and there exists a finite H ⊂ G such that for any g ∈ G we can determine the tile of T • to which it belongs, so long as we know the set Hg ∩ T (i.e., using the language of topological dynamics, T • is a factor of T ).
We will combine these two results into the following form (which will be used in the main construction): Theorem 2.13.If G is a countable amenable group with the comparison property, then for every ε > 0, N > 0 and every finite K ⊂ G there exists a quasitiling T ′ of G such that (1) the shapes of T ′ are all Følner sets, i.e. S = {F n 1 , F n 2 , . . ., F nr }, where N n 1 < n 2 < . . .< n r , and r depends only on ε.

Subsystem entropy
Theorem 3.1.If X is a strongly irreducible subshift of positive entropy h, then the set of topological entropies of subshifts of X is dense in (0, h).
Proof.Let a and b be such that 0 < a < b < h.We will show that there exists a subshift Y ⊂ X such that a h(Y ) b (which is an equivalent formulation of the theorem).Fix a positive η smaller than (b − a)/2 and note that for every sufficiently large L there exists a positive integer n such that 1 L log n is in the interval (a + η, b − η).Combined with theorem 2.7 this lets us state the following: Fact.There exists an N > 0, δ > 0 and M > 0 such that if T is an (F n , δ)-invariant set for some n > N, and Let D denote the irreducibility distance of X.By theorem 2.13 there exists a tiling T of G such that h(T ) = 0 and the shapes of T can have arbitrarily good invariance properties, which we will specify within the next few sentences.Enumerate the shapes of T by S 1 , S 2 , . . ., S J .There exist some δ * , δ * * and n such that if the shapes of T are (D, δ * ) and (F n , δ * * )-invariant, then for every j we can choose a family of blocks B j ⊂ B (S j ) D (X) such that if we denote In addition, since for small enough δ * the relative difference between |(S j ) D | and |S j | can be arbitrarily small, we can even write Now, let Y be the orbit closure of the set of all points x ∈ X such that for every T ∈ T x(T D ) ∈ B j , where j is such that S j is the shape of T .Observe that strong irreducibility, combined with the fact that T D is disjoint from all other tiles of T , means (via a standard compactness argument) that we can choose the blocks x(T D ) independently of each other, and any such choice will yield a valid element of Y .
We will estimate the entropy of Y by considering the number of blocks with domain F n as n increases to infinity, beginning with estimating this number of blocks from above.Fix some large n and note that every block B with domain F n that occurs in Y has the property that there exists some right-translate T ′ of T such that for every tile T of T ′ such that T ⊂ F n the block B(T D ) belongs to B j , where j is such that S j is the shape of T .Let H n be the number of ways in which the right-translates of T can intersect F n , or equivalently, the number of ways in which the right-translates of F n can intersect T .Since T has entropy 0, for large enough n we have 1  |Fn| log H n < η 2 .For any such right-translate T ′ let l j be the number of tiles of T ′ with shape S j that are subsets of F n .Note that if n is large enough, then J j=1 l j |S j | is almost equal to |F n |.Since for every tile of T ′ the block B(T D ) is a block from B j , and thus one of N j possible blocks, the D-interiors of the tiles of T ′ can be filled in at most J j=1 N l j j ways.We have no control over the symbols outside these interiors, but we know that there are at most |F n | − J j=1 l j |(S j ) D | such symbols.It follows that the number of blocks associated with T ′ is at most Taking logarithms and dividing by |F n |, we obtain If δ * was small enough so that the D-interiors of the tiles of T ′ form a (1 − η 2|Λ| )-covering quasitiling (which we can safely assume since D, η and Λ were known before we started the construction), then for large enough n the second term will not exceed η 2 .As for the first term, we can further estimate it as follows: It follows that for any right-translate T ′ of T , the number of blocks in Y with domain F n that have blocks from B j in every tile of T ′ with shape S j does not exceed 2 (b− η 2 )|Fn| , and thus the cardinality of B Fn (Y ) does not exceed H n 2 (b− η 2 )|Fn| .This lets us estimate that 1 and hence h(Y ) < b.
The lower bound for entropy is much simpler: for any n the number of blocks with domain F n which occur in Y is equal to at least , where l j is the number of tiles of T with shape S j which are subsets of F n (this time we do not even need to consider possible translates of T ).Consequently, which for large enough n will be greater than a, and therefore h(Y ) > a, which concludes the proof.

Marker blocks
Our main objective in this section will be to prove the following theorem: Theorem 4.1.Let X be an aperiodic, strongly irreducible symbolic dynamical system (with the irreducibility distance D) over the alphabet Λ, with action of a countable amenable group G.
Let Y be a proper subshift of X.Then there exists a block M with shape K, and a tiling T (with a family of shapes S), satisfying the following conditions: (1) For every T ∈ T , any t ∈ T , and any x ∈ X, if x(Kt) = M, then the set T D ∩ Kt is not empty, and the block x(T D ∩ Kt) does not occur in Y .(2) For two different g 1 , g 2 ∈ G and any x ∈ X, if x(Kg 1 ) = x(Kg 2 ) = M, then Kg 2 \ DKg 1 = ∅, and x(Kg 2 \ DKg 1 ) is a block which does not appear in Y .
Proof.For clarity, we will separate the proof into stages (1) Since Y is a proper subshift of X, there exists some block B which occurs in X but not in Y .Let Z denote the shape of B.
(2) Let P e = Z −1 DZ and for any g ∈ G lest P g = g −1 P e g.Observe that for any g, P g is a finite set (and all of these sets have equal cardinality), and e ∈ P g (because we assumed that e ∈ D).Let P ′ g be the largest (in terms of cardinality) subset of P g such that P ′ g ⊂ P h for infinitely many h (if there is more than one such subset of equal cardinality, we can choose any of them).Choose g 0 so that P ′ g 0 has maximal cardinality among all the P g 's.Set P = P ′ g 0 and G P = {g ∈ G : P ⊂ P g }.Observe that if any h ∈ G occurs in infinitely many P g for g ∈ G P , then due to the maximality of P we necessarily have h ∈ P .It follows that for any E ⊂ G we have E ∩ P g ⊂ P for all but finitely many g ∈ G P .
(3) By our assumption, there exists in X a block M 0 with domain K 0 which is P -aperiodic, i.e. for every g ∈ P there exists a k ∈ K 0 such that k and kg are both elements of K 0 , and M 0 (kg) = M 0 (k).By strong irreducibility, we can also require that M 0 has B as as subblock, and thus M 0 cannot occur in Y .
(4) Let T ′ be a tiling of G, such that the shapes of T ′ are supersets of DZ, let S ′ denote the union of all shapes of T ′ , and let T be another tiling, whose shapes are (S ′ , δ)-invariant, where δ is so small that for every shape S of T , and every s ∈ G the set S D s −1 includes an entire tile of T ′ .Let S be the union of all shapes of T .By the invariance property established earlier, for any s ∈ S \ S DK 0 the set S D s −1 includes an entire tile of T ′ , which allow us to choose a finite set C ′ (consisting of centers of such tiles) such that for every s ∈ S \ S DK 0 the set Strong irreducibility yields the existence of a block M 1 with domain K 1 , such that M 1 (K 0 ) = M 0 , and for every c ′ ∈ C ′ we have M 1 (Zc ′ ) = B. Observe that for any tile T ∈ T , if g ∈ T , then either K 0 g ⊂ T D , or for at least one c ′ ∈ C ′ we have Zc ′ g ⊂ T D .Indeed, T has the form Sc, where S is one of the shapes of T .Let s = gc −1 .If s ∈ S DK 0 , then DK 0 s ⊂ S, and thus DK 0 g = DK 0 sc ⊂ Sc = T .Otherwise, we know that for some c ′ ∈ C ′ we have That way we have obtained a block M 1 with domain K 1 that satisfies the first property stated in our theorem.Note that any block that has M 1 as a subblock will retain this property.
(5) Now choose c 2 as an element of G P which satisfies the following conditions (each of them is satisfied for all but finitely many elements of the group, and G P is infinite, which makes such a choice possible).The significance of these conditions is certainly not obvious at first, but will become clear later.
. Also choose c 3 as an element of G P satisfying the following conditions (which is again possible because each of the following is true for all but finitely many elements of Such an M exists, because conditions (5a) and (5d) imply that DZc 2 is disjoint from K 1 , and DZc 3 is disjoint from K 1 ∪ Zc 2 , which allows us to invoke the strong irreducibility of X.Now, assume that for some x ∈ X and g ∈ G we have x(Kg) = x(K) = M.We will show that in such a situation, at least one of the sets K 1 g, Zc 2 g, Zc 3 g is disjoint with DK.This will require some laborious and repetitive computations, which we will separate into the following steps.(a) x(Kg) = X(K) = M implies that for every k ∈ K 1 such that kg is also in K 1 , we have M 1 (k) = M 1 (kg).Since M 1 is P -aperiodic, g cannot belong to P .(b) Observe that for any g ∈ G each of the sets DK 1 , DZc 2 , DZc 3 is intersected by at most one of the sets K 1 g and Zc 2 g.Indeed: • Suppose that DK 1 ∩ K 1 g and DK 1 ∩ Zc 2 g are both nonempty.This implies We have shown that if x(Kg) = x(K) = M, then Kg \ DK is a superset of at least one out of K 1 g, Zc 2 g, and Zc 3 g.Thus the block x(Kg \ DK) has either M 1 or B as a subblock, but neither of those blocks occur in Y , and therefore x(Kg \ DK) also does not occur in Y .More generally, if x(Kg , so this block also does not occur in Y , as required. Note that the tiling T constructed above can (and probably will) have shapes smaller than the domain of M. This will not be an obstacle in our construction, but nevertheless we note that one can replace T it by a larger, congruent tiling (i.e one whose tiles are unions of tiles of T ), the shapes of which can be arbitrarily large and have arbitrarily good invariance properties, and such a replacement we will retain the properties specified in the theorem.Thus we can make the following remark: Remark 4.2.The tiling T in Theorem 4.1 can be chosen to be (F, ε) invariant for any ε > 0 and any finite F ⊂ G.

Constructing the extension
Theorem 5.1.If G is a countable amenable group with the comparison property, and X is an aperiodic, strongly irreducible symbolic dynamical system with the shift action of G, then for every N ∈ N such that log(N) < h(X) there exists a factor map from X onto the full G-shift over N symbols.Proof.By theorem 3.1, X has a subsystem Y such that log N < h(Y ) < h(X).Before we delve into the technical details, here is a rough outline of the construction: (1) The factor map will determine if an element of X can be tiled (at least within some finite area around the neutral element of G) using a certain collection of large shapes.
(2) This decision will be made based on the occurrences of a marker block within a certain window.(3) If such a local tiling can be found, there is a correspondence between blocks over its tiles and blocks in the full shift, which induces the image under the factor map (in other words, we define the map as a sliding block code).If the contents of x does not induce a local tiling (which is entirely possible, and in fact more likely than not), the code just assigns 0 to the image a the "problematic" coordinates.(4) To show that this map is a surjection, we tile every element of the full shift using large shapes, and construct an element in X that only has marker blocks at the centres of such shapes; and blocks from Y elsewhere.Since h(Y ) > log N, the space not taken up by markers is enough to encode the entire element of the full shift.We can apply theorem 4.1 to obtain a block M with domain K, and tiling T with the following properties: (1) If T is a tile of T and for some x ∈ X we have x(Kt) = M, then the (nonempty) block x(T D ∩ Kt) does not occur in Y (2) For any two different g 1 , g 2 ∈ G, if x(Kg 1 ) = x(Kg 2 ) = M, then the (nonempty) block x(Kg 2 \ DKg 1 ) does not occur in Y .Let ε = h(Y ) − log N and let E denote the union of the shapes of T .There exists a δ such that if S is any (E, δ)-invariant set, then the union of tiles of T which are contained in S is a (1 − ε 2 )-subset of S.
We are about to apply theorem 2.13 (note: we are not yet applying this theorem; we are just discussing one of its parameters) with the parameter δ, so we know it will yield a quasitiling with r δ shapes S 1 , S 2 , . . ., S r δ , where r δ will depends only on δ.Let K * ⊂ G be a finite set such that |B K * (Y )| > r δ • |E|, and thus there exists a surjective function ψ : B K * (Y ) → {1, . . ., r δ } × E. We can also assume that K * is disjoint from DK.
We can now apply theorem 2.13 to obtain a quasitiling T ′ such that: (1) T ′ has r δ shapes.
(2) T ′ factors onto a tiling T ′′ with the same set of centres, and such that all shapes of T ′′ are (E, δ)-invariant.We now have all the objects we need to define the factor map π from X onto the full G-shift over N symbols.We will do so by describing a procedure to determine the symbol π(x)(e) based on x(H ′ ) for a certain finite H ′ ⊂ G.
Let H ′ be large enough that the knowledge of how the set of centers of T ′ intersects H ′ allows us to determine the tile T ′′ of T ′′ such that e ∈ T ′′ .Consider the set of all c ∈ H ′ such that x(Kc) = M.For every such c, if we have ψ(K * c) = (j, g) (for 1 j r δ and g ∈ E), we obtain a set of the form S j gc.There are now two possibilities: • If the set of these shapes is equal to the intersection of H ′ with some right-translate of T ′ , then it determines the intersection of H with the same right-translate of T ′′ , and in particular it uniquely assigns e to a set of the form Sh for some shape S of T ′′ and some h ∈ H.If x(S D h \ D(K ∪ K * )h) is some block A which occurs in Y , then let x(e) = Π(A)(h −1 ).• If uniquely determining x(e) as above is not possible (or the resulting block A does not occur in Y ), set π(x)(e) = 0.
The map defined above is a sliding block code (with window H ′ ), and thus it is a factor map from X onto some subset of the full G-shift over N symbols, and the only non-trivial property left to verify is surjectivity.In other words, we need to show that for every z ∈ {0, 1, . . ., N − 1} G there exists some x ∈ X such that π(x) = z.For such an z, we will define its preimage x by prescribing the content of x within D-interiors of disjoint sets (mostly tiles of T ); strong irreducibility means that such an x will exist provided the individual blocks do occur in X.
Enumerate the tiles of T ′ as T ′ 1 , T ′ 2 , . ... For every k, the tile T ′ k has the form S ′ j(k) c ′ k , where j(k) ∈ {1, 2, . . ., r δ }.The center c ′ k belongs to some tile S k c k of T , and thus c ′ k = g k c k for some g k ∈ E. The set K * c k is a subset of the D-interior of some union of finitely many tiles of T ; denote this interior by K * c k .Let x(Kc k ) = M, let x(((S k ) D \ K)c k ) be any block from Y , and let x( K * c k ) be a block from Y such that ψ(x(K * c k )) = (j(k), g k ).Let Ŝk c k be the union of all tiles of T that are disjoint from D(Kc k ∪ K * c k ) and contained within (T ′ k ) D .There exists a block A in Y , with domain (S k ) D \ D(K ∪ K * ), such that Π(A) = z(T ′′ ), where T ′′ is the tile of T ′′ whose center is c k (we know there is exactly one such tile), and we can extend A to a block A k (also occurring with Y ) with domain Ŝk .Set x( Ŝk c k ) = A k .In addition, for any tile T of T which is not a subset of (T ′ k ) D for any k, we can set x(T D ) to be any block in Y .The above construction, together with the properties of M, means that x(Kc) = M if and only if c = c k for some k (this is because all D-interiors of tiles of T , except the places where we explicitly put the marker, were chosen to be blocks from Y ).This means that for every g we can uniquely determine the tile of T ′′ to which g belongs, based on the contents of x(H ′ g), and hence π(x) = z.

( 3 )
Every shape S of T ′′ is a superset of D(K ∪ K * ), and in fact the number of blocks with domain S D \ D(K ∪ K * ) that occurs in Y is greater than N |S| , hence there exists a surjective map Π :B S D \D(K∪K * ) (Y ) → {0, 1, . . ., N − 1} S .(Thelatter property easily follows from the fact that if the shapes of T ′ are sufficiently large Følner sets, then S D \ D(K ∪ K * ) can have arbitrarily large relative cardinality in S, and the entropy of Y exceeds log N).