PUTNAM’S RESOLVING MAPS IN DIMENSION ZERO

MIKE BOYLE

Abstract. A block code from an irreducible shift of finite type can be lifted canonically through resolving maps to a resolving map. There is an application to Markovian maps.

Contents

1. Introduction 1 2. Background 2 3. The futures cover of a map 5 4. A resolving lift of a block code 8 5. The canonical mapping property 10 6. The maps of Nasu and Putnam 13 7. Resolving degree and Markovian maps 16 References 17

1. Introduction

Let S be the set of 1-to-1 almost everywhere factor maps between irreducible Smale spaces. Ian Putnam [16] proved the following theorem: given π in S, there are Smale spaces \overline{X} and \overline{Y}, and maps α, β and π̃ in S, such that α and β are u-resolving, π̃ is s-resolving and the following diagram commutes:

\xymatrix{\overline{X} \ar[r]^{π̃} \ar[d]_{α} & \overline{Y} \ar[d]^{β} \\ X \ar[r]^{π} & Y}

Putnam’s result was a surprise even in dimension zero, where resolving maps have been studied extensively. In the zero dimensional case (where a Smale space is a shift of finite type and a finitely presented system is a sofic shift), we will reinterpret and extend Putnam’s work. Because a lot is known about resolving maps, this adds to our understanding of general factor maps. We hope this work will also be relevant to progress in positive dimension.

We view Putnam’s construction in two stages: first construct the map β, then construct the diagram by a fiber product argument. For β, we construct a canonically associated map we call π_{+}, the futures cover of π. This map is itself a consequence of a more general construction, the full futures cover of π (Section 3). The fiber product argument is carried out in Section 4, and associates a canonical lift to π. For these sections, the map π can be any block code from an irreducible SFT to a sofic shift, even an infinite to one code. In the infinite to one case, the lift π̃ is only “eresolving”–resolving in the sense of existence (Sec. 2)–but this is meaningful.

2000 Mathematics Subject Classification. Primary: 37B10; Secondary: 37D20, 37D25.

Key words and phrases. resolving; shift of finite type; factor map; Smale space.

Partially supported by NSF Grant 0400493, the Max Planck Institute and ESF.

In Section 5, we show that our construction is the unique (up to topological conjugacy) minimal construction – any other such lift must factor through it. This, and the generalization to finitely presented range systems, answer (in the zero dimensional case) the two questions raised by Putnam in [16].

The futures cover arises from an effort to simplify Putnam’s construction. Where both the futures cover and the map β constructed by Putnam are defined, they and their associated diagrams are isomorphic (Section 6). The futures cover is also closely related to a canonical cover of Nasu (Section 6). The three constructions (futures cover, Putnam cover, Nasu cover) are all isomorphic where they are all defined (for 1-1 a.e. maps between irreducible SFTs).

An unexpected spinoff of this investigation is some new information about infinite to one maps. A magic word construction very familiar for finite to one maps turns out to be meaningful (7.1) and of some use for infinite to one maps. We also learn that any factor map between irreducible SFTs can after precomposition with a degree one resolving map be made Markovian (7.2).

This work was begun at the Max Planck Institute in Bonn during the 2004 Activity on Algebraic and Topological Dynamics, supported by the European Science Foundation. I thank MPI for a wonderful working environment, and (with regard to this paper) for the opportunity to interact with Ian Putnam during the Activity.

2. Background

In this section, we recall and slightly expand some background results and terminology. For further detail, see the texts [8, 11]. We will be concerned with subshifts (X, T) where X is a shift-invariant space of doubly infinite sequences over a finite alphabet, with the usual topology (relative of product of discrete, making X zero dimensional compact metrizable) and T is the restriction of the shift map σ to X. (We use the left shift, (σx)_{i} = x_{i+1}.) The alphabet for (X, T) may be denoted A(X) or A(T), and the set of words of length n occurring in points of X is denoted W_{n}(X) or W_{n}(T). A point x of a subshift (X, S) is left transitive (right transitive) if for every S-word W there exist negative (positive) integers i ≤ j such that W = x[i, j]. A point is doubly transitive if it both left transitive and right transitive. Points x and x′ in a subshift are left asymptotic if there exists n in ℤ such that x(−∞, n] = x′(−∞, n]. Systems (X, T) and (X′, T′) are topologically conjugate if there is a homeomorphism φ : X → X′ such that T′φ = φT. An SFT is a system (X, S) such that there exists n such that a point x is in X if and only if x[i + 1, i + n] ∈ W_{n}(S) for every i ∈ ℤ. An SFT is 1-step if this length n can be chosen to be 2. An edge SFT is a 1-step SFT for which the alphabet is the set of edges in a nondegenerate directed graph G and EE′ is an allowed word of length 2 if the terminal vertex of E equals the initial vertex E′. An SFT is irreducible if it has a dense forward orbit. A directed graph is nondegenerate if every vertex has at least one incoming edge and at least one outgoing edge. An edge SFT defined by a nondegenerate directed graph G is irreducible iff there is a path in G from any vertex to any other. An irreducible component of an SFT is a maximal irreducible subsystem. The entropy of an SFT is the maximum of the entropies of its irreducible components.

A block code is a map φ : (X, S) → (Y, T) between subshifts such that for some n there is a map φ : W_{2n+1}(S) → A(T) such that for all x in X and i ∈ ℤ, (φx)_{i} = φ(x[i − n, i + n]). When such an n exists, the block code has range n. If n = 0, then φ is a 1-block code. Every continuous shift commuting map (or code) between subshifts is given by a block code. If φ is a surjective block code, then it is a factor map. A factor map φ from an irreducible SFT onto a subshift Y has a well defined degree d: the cardinality of the preimage of every double transitive point in Y is d. In this case we also say that φ is d-to-1 almost everywhere. A surjective code respects entropy if it is finite to one. An irreducible SFT has no proper subsystem of equal entropy. A sofic shift is a subshift which is the image of an SFT under a block code. The SFTs are the zero dimensional Smale spaces and the sofic shifts are the zero dimensional finitely presented dynamical systems (see [7, 16]).

Let φ denote a factor map from a subshift (X, S) to a subshift (Y, T). φ is right resolving if it is a 1-block code with the following property: if a_{0} ∈ A(X), with image symbol b_{0} ∈ A(Y), and if b_{0}b_{1} is a Y -word, then there exists at most one a_{1} ∈ A(X) such that a_{1} maps to b_{1} and a_{0}a_{1} is an X word. φ is right closing, or u-resolving, if for all left asymptotic x and x′ ∈ X, φ(x) = φ(x′) implies x = x′. (The terminology “right closing” is standard in symbolic dynamics [8, 11]; Putnam’s more general terminology “u-resolving” means injective on unstable sets [16], which in this setting means right closing. We will generally favor Putnam’s terminology.)

Similarly, φ is left resolving if it is a 1-block code with the following property: if a_{1} ∈ A(X), with image symbol b_{1} ∈ A(Y), and if b_{0}b_{1} is a Y -word, then there exists at most one a_{0} ∈ A(X) such that a_{0} maps to b_{0} and a_{0}a_{1} is an X word. Likewise, φ is left closing, or s-resolving, if for all right asymptotic x and x′ ∈ X, φ(x) = φ(x′) implies x = x′. Here “s-resolving” means injective on stable sets [16]. We say a map is resolving if it is left, right, u- or s-resolving. A resolving map is finite to one and therefore its domain and image are subshifts of equal entropy.

We say φ is right eresolving if the right resolving condition holds, but with the uniqueness condition replaced by existence. That is, in the definition of right resolving, we replace “at most one” with “at least one”. We say φ is u-eresolving if given x in X and y ∈ Y such that φ(x) is left asymptotic to y, there exists at least one \overline{x} in X such that \overline{x} is left asymptotic to x and φ(\overline{x}) = y. For block codes, u-eresolving has been called right continuing [6, Sec. 5]. The terms left eresolving, left continuing and s-eresolving are defined analogously.

In the next proposition, we record some known facts relating these notions. Obviously the left/s analogues likewise hold.

Proposition 2.1. Let φ be a block code from a subshift (X, S) onto a subshift (Y, T).

(1) If φ is right resolving, then φ is u-resolving (i.e. right closing).

(2) If φ is u-resolving, then there is a topological conjugacy ψ and a right resolving φ′ such that φ′ is ψ followed by φ.

(3) If φ is right eresolving, then φ is u-eresolving (i.e. right continuing).

(4) If φ is u-eresolving, then there is a topological conjugacy ψ and a right eresolving φ′ such that φ′ is ψ followed by φ.

(5) If (X, S) and (Y, T) are irreducible SFTs of equal entropy, then φ is uresolving if and only if it is u-eresolving; if in addition these SFTs are 1-step and φ is 1-block, then φ is right resolving if and only if it is right eresolving.

(6) If φ is u-resolving and (X, S) is an irreducible SFT, then (Y, T) is an SFT if and only if φ is u-eresolving.

Proof. (1) and (3) are easy observations. (2) is proved in [8, Proposition 4.33]. (4) is proved in [6, Proposition 5.1]. (5) is partly the result [11, Proposition 8.2.2], and the rest of (5) and (6) are proved by a similar appeal to the Perron-Frobenius theory element [11, Theorem 4.4.7]. □

Remark 2.2. Resolving maps from irreducible SFTs onto strictly sofic shifts are important in the theory of sofic shifts, and such maps cannot be eresolving. On the other hand, eresolving maps between irreducible SFTs of unequal entropy are relevant to understanding Markovian codes [6]. In this paper we will encounter both types of map in the same setting, so for clarity we have introduced the new term eresolving. Caveat: the term “resolving” has sometimes been used to mean u-resolving; or, eresolving; or, both resolving and eresolving.

Lemma 2.3. Suppose φ : (X, S) → (Y, T) is a u-resolving block code such that (X, S) is SFT, (Y, T) is irreducible SFT, and (X_{0}, S_{0}) is an irreducible component of (X, S) of maximum entropy.

Then any point of (X, S) which is left asymptotic to X_{0} is contained in X_{0}.

Proof. This is [8, Lemma 5.1.4] in that case that φ is right resolving. The conclusion then follows from Proposition 2.1(2). □

“Magic words” are an important technical tool in the theory of codes from SFTs. Let φ : (X, S) → (Y, T) be a surjective one-block code. Above every Y -word W = W [1, n] of length |W | = n there is a set of X-words W′ = W′[1, n] of length n which by the coding rule are sent to W, i.e. φW′ = W. Given 1 ≤ i ≤ n, set d(W, i) = |{W′_{i} : φW′ = W }|. We define the resolving degree δ(φ) of φ to be the minimum of d(W, i) over all Y -words W and 1 ≤ i ≤ |W |. W is a magic word if, for some i, d(W, i) = δ(φ). A magic symbol for φ is a magic word of length 1, that is, an element μ of A(Y) such that whenever W is a Y -word and W_{i} = μ, then d(W, i) = δ(φ). (In other words, if μ occurs at a position in W, then “above” W we see every symbol which maps to μ, and this set of symbols has cardinality δ(φ).) For an irreducible SFT (X, S) and finite-to-one block code φ : (X, S) → (Y, T), the proof that φ has a well defined degree proceeds by showing every doubly transitive point has exactly δ(φ) preimages [8, 11].

We have defined magic words and symbols exactly as in Kitchens’ text [8, p.102 and p.113], except that we do not restrict to the case that φ is finite to one. In particular, our number δ(φ) is the number d in [8, p.101]. Two block codes α, β are topologically conjugate if there are topological conjugacies ψ_{1}, ψ_{2} such that ψ_{1}β = αψ_{2}. We introduce the term “resolving degree” because it turns out to be a conjugacy invariant of infinite to one maps as well as finite to one maps (7.1). Likewise, the next lemma will be useful to us in the general (not just finite to one) case.

Lemma 2.4. A surjective block code φ is topologically conjugate to a 1-block code with a magic symbol. If φ is u-resolving, then in addition this code can be required to be right resolving.

Proof. The first statement is proved in [8, Proposition 4.3.2]. The restriction in the statement to finite to one codes is not needed in the proof. For the second statement, after passing to a topologically conjugate code, we may assume φ is right resolving. The recoding construction of [8, Proposition 4.3.2] is easily adjusted to respect the right resolving property. □

The next lemma says that a u-resolving map from an irreducible SFT onto a sofic shift is (despite Proposition 2.1(6)) u-eresolving with respect to left transitive points in the image.

Lemma 2.5. Suppose (X, S) is SFT and φ : (X, S) → (Y, T) is a u-resolving block code. Suppose y is left transitive in Y, and x is a point in X such that φx is left asymptotic to y.

Then there exists a point x′ in X such that x′ is left asymptotic to x and φx′ = y. Also, if φ is right resolving and (φx)(−∞, 0] = y(−∞, 0], then x′ can be chosen such that x′(−∞, 0] = x(−∞, 0].

Proof. After passing to a topologically conjugate map, we can assume without loss of generality that φ is right resolving with a magic symbol μ. Choose n < 0 such that y_{n} = μ. Because μ is a magic symbol, for all k > 0 there exists an X-word a_{n} ⋯ a_{n+k} such that φ(a_{n} ⋯ a_{n+k}) = y_{n} ⋯ y_{n+k} and a_{n} = x_{n}. Because φ is right resolving, the word a_{n} ⋯ a_{n+k} is uniquely determined by a_{n} and y. Define x′ by setting x′_{i} = x_{i} if i ≤ n and x′_{i} = a_{i} if i > n. □

3. The futures cover of a map

To begin, let π : (X, S) → (Y, T) be a 1-block code from a 1-step SFT onto a sofic shift. We will associate to π a surjective right resolving 1-block code π_{++} : (Y_{++}, T_{++}) → (Y, T), where (Y_{++}, T_{++}) is a 1-step SFT. Given y ∈ Y and m ∈ ℤ, define the following subset of A(S):

(3.1)  U_{m}(y) = {x_{m} : x ∈ X and π(x_{j}) = y_{j} if j ≤ m} .

Define the alphabet of T_{++} to be

A(T_{++}) = {U_{0}(y) : y ∈ Y } = {U_{m}(y) : y ∈ Y and m ∈ ℤ} .

Define the allowed T_{++}-words of length two (and thus determine the 1-step SFT T_{++}) as follows: UU′ is allowed if U ∈ A(T_{++}) and there exists j in A(T) such that

U′ = {i ∈ A(S) : π(i) = j and i follows some element of U} .

Note, given U, j and U′ as above, we do have U′ in A(T_{++}), since there exists a y in Y with U_{0}(y) = U and y_{1} = j, and then U′ = U_{1}(y). Similarly, we see that the set of all Y_{++}-words is

{U_{0}(y)U_{1}(y) ⋯ U_{n}(y) : n ≥ 0, y ∈ Y } .

The one-block code π_{++} : (Y_{++}, T_{++}) → (Y, T) is defined by sending a Y_{++}-symbol U to the Y -symbol which is π(i) for every i in U. The map π_{++} is clearly welldefined, surjective and right resolving.

Finally, we associate to π_{++} the shift-commuting Borel injection τ_{++} : Y ↪ Y_{++} defined by setting (τ_{++}y)_{i} = U_{i}(y). The map τ_{++} is a section to π (πτ_{++} is the identity map on Y).

Definition 3.2. The code π_{++} (and the SFT (Y_{++}, T_{++})) is called the full futures cover of π, and τ_{++} is called its canonical section.

The next result adapts arguments of Krieger [10].

Proposition 3.3. Suppose π : (X, S) → (Y, T) and π′ : (X′, S′) → (Y′, T′) are 1-block codes from 1-step SFTs onto sofic shifts. Let η : (X, S) → (X′, S′) and κ : (Y, T) → (Y′, T′) be topological conjugacies such that κπ = π′η. Let the full futures covers and sections for π and π′ be as in Definition 3.2. Then there is a unique topological conjugacy κ̃ such that the following diagram commutes:

\xymatrix{Y_{++} \ar@{-->}[rrr]^{κ̃} \ar[dd]_{π_{++}} &  &  & Y′_{++} \ar[dd]^{π′_{++}} \\  & X \ar[r]^{η} \ar[dl]^{π} & X′ \ar[dr]_{π′} \\ Y \ar[rrr]_{κ} \ar[uu]_{τ_{++}} &  &  & Y′ \ar[uu]^{τ′_{++}}}

Proof. The images of the sections τ_{++} and τ′_{++} are dense, and between these images κ̃ is determined. Thus the uniqueness claim is clear. Let N be a positive integer such that the block codes η, κ, η^{−1} and κ^{−1} have range at most N.

Given y ∈ Y and y′ = κy ∈ Y′, we claim that U_{0}(y′) is the set of symbols (ηx)_{0} such that x ∈ X and x satisfies the following two conditions:

(1) x_{i} ∈ U_{i}(y), −3N ≤ i ≤ −N,

(2) x[−3N ⋯ N] is an X-word which determines (κπx)[−2N, 0] = y′[−2N, 0]. To prove the claim, first suppose x′ ∈ X′ and (π′x′)(−∞, 0] = y′(−∞, 0]. Set x = η^{−1}x′. Then (x′)_{0} = (ηx)_{0}, and (κπx)(−∞, 0] = y′(−∞, 0]. This shows x[−3N, N] satisfies (2). Because y′(−∞, 0] determines (κ^{−1}y′)(−∞, −N], and κ^{−1}y′ = πx, we have x_{i} ∈ U_{i}(y) for i ≤ −N, and x satisfies (1).

Conversely, suppose x ∈ X satisfying (1) and (2). Because x_{−3N} ∈ U_{−3N}(y), there exists \overline{x} in X such that (π\overline{x})_{i} = y_{i} if i ≤ −3N, and \overline{x}_{−3N} = x_{−3N}. Because X is 1-step SFT, without loss of generality we choose x such that in addition to (1) and (2) we have \overline{x}_{i} = x_{i}, i ≤ −3N.

Now (κπx)(−∞, −2N] = y′(−∞, −2N] (because (πx)_{i} = y_{i}, i ≤ −N), and (κπx)[−2N, 0] = y′[−2N, 0] (by (2)). Since (κπx)(−∞, 0] = y′(−∞, 0], we have (ηx)_{0} ∈ U_{0}(y′). This completes the proof of the claim.

The claim shows that for any y in Y, the Y_{++}-word (τy)[−3N, N] determines the Y′_{++} symbol τy′[0]. Because the image of τ is dense in Y_{++}, this provides a function K from Y_{++}-words of length 4N + 1 to A(Y′_{++}). Define the block code κ̃ on Y_{++} by setting (κ̃w)_{0} = K(w[−3N, N]). Clearly now κ̃τ = τ′κ. It follows easily that κ̃ maps Y_{++} onto Y′_{++}, and κπ_{++} = π′_{++}κ̃.

In the same way, construct a block code κ̃^{−1} from Y′_{++} to Y_{++}. The compositions of κ̃ and κ̃^{−1} give the identity map on a dense set, hence everywhere. Therefore κ̃ is a conjugacy. □

Given a block code π : X → Y from an SFT onto a sofic system, let φ be a conjugacy from X to a higher block presentation, (φx)_{0} = x[−k ⋯ k], where k is sufficiently large that the composition πφ^{−1} is a 1-block map from a 1-step SFT, so its full futures cover (πφ^{−1})_{++} has been defined. We also say that (πφ^{−1})_{++} is a full futures cover of π, and (suppressing k) denote it as π_{++}. More generally, we make the following definition.

Definition 3.4. If p is any u-resolving map from an SFT onto Y and there exists a conjugacy φ such that p = πφ, where π is a full futures cover as defined above, then we say that p is a full futures cover of π.

By Proposition 3.3, if p and p′ are two different full futures covers of π, then there is a conjugacy φ such that p′ = pφ. Thus given π we have a distinguished class of u-resolving maps from SFTs onto Y.

Next we consider the special case which is our main interest.

Definition 3.5. Suppose π is a one block code from a 1-step irreducible SFT (X, S) onto a sofic shift (Y, T). Let (Y_{+}, T_{+}) be the unique maximum entropy irreducible component of the SFT (Y_{++}, T_{++}), and let π_{+} be the restriction of π_{++} to this component. We call π_{+} (or (Y_{+}, T_{+})) the futures cover of π. The restriction of τ_{++} to the left transitive points of Y is denoted τ_{+}.

The next proposition shows the definition makes sense and lists some basic properties of the futures cover.

Proposition 3.6. Suppose π : (X, S) → (Y, T) is a 1-block code from an irreducible 1-step SFT onto a sofic shift.

(1) The full futures cover of π has a unique irreducible component of maximal entropy (denoted Y_{+}).

(2) The futures cover π_{+} is surjective right resolving with degree 1, and τ_{+} : Y ↪ Y_{+}.

(3) Y_{+} is a 1-step SFT and the set of allowed Y_{+}-words is

{U_{0}(y)U_{1}(y) ⋯ U_{n}(y) : n ≥ 0, y is left transitive in Y }

(4) After removal of the section maps τ_{++} and τ′_{++}, Proposition 3.3 remains true if each ++ is replaced by +.

Proof. The claim (1) and the rest of (2) do not depend on the conjugacy class of π_{++}; so, on account of Proposition 3.3 and Lemma 2.4, to prove them we may assume that π has a magic symbol, μ. Let μ̃ denote π^{−1}{μ}. If π_{++}x = y and y_{n} = μ, then x_{n} must equal μ̃. Because π_{++} is right resolving, for any n in ℤ the conditions y = π_{++}x and x_{n} = μ̃ determine x[n, ∞). Consequently, if y is left transitive in Y, then y has a unique preimage under π_{++}. Because π_{++} must map any irreducible component of Y_{++} of maximum entropy onto Y, it follows that there is only one such component. We let Y_{+} denote this component and we let π_{+} denote the restriction of π_{++} to Y_{+}.

The claim (2) is now clear.

The image of τ_{+} is a dense subset of Y_{+}, and we have already seen that the set of all Y_{++}-words is

{U_{0}(y)U_{1}(y) ⋯ U_{n}(y) : n ≥ 0, y ∈ Y } ,

i.e., it is the set of words (τ_{++}y)[0, n], y ∈ Y. This proves (3).

The existence claim of (4) follows from (1) and Proposition 3.3. The uniqueness claim holds because κ̃ is uniquely determined on a dense set, the image of τ_{+}. □

As with the full futures cover, we make a definition which gives us a canonical conjugacy class.

Definition 3.7. Given a block code π from an irreducible SFT onto a sofic shift, we say that a map p is a futures cover of π if it is conjugate to the restriction of a full futures cover of π to its unique irreducible component of maximum entropy.

4. A resolving lift of a block code

To begin, let π be a one block code from an irreducible 1-step SFT (X, S) onto a sofic shift (Y, T). Let π_{+} be the futures cover Y_{+} → Y of Definition 3.5. We will apply a fiber product construction to produce (in the zero dimensional case) a generalization of Putnam’s resolving lift.

The fiber product (a.k.a. fibered product, or pull-back) of π : (X, S) → (Y, T) and π_{+} : (Y_{+}, T_{+}) → (Y, T) is the subsystem of (X × Y_{+}, S × T_{+}) with domain F = {(x, w) : π(x) = π_{+}(w)}. The coordinate projections p_{1} : (x, w) ↦ x and p_{2} : (x, w) ↦ w give a commuting diagram of surjective maps

\xymatrix{ & F \ar[dl]_{p_{1}} \ar[dr]^{p_{2}} \\ X \ar[dr]_{π} &  & Y_{+} \ar[dl]^{π_{+}} \\  & Y}

We view F as a subshift, with alphabet

A(F) = {(x_{j}, w_{j}) : (x, w) ∈ F }
⊂ {(i, U) ∈ A(X) × A(Y_{+}) : π(i) = π_{+}(U)} .

The subshift F is a 1-step SFT, because X and Y_{+} are 1-step SFTs, and π and π_{+} are one-block codes. Because π_{+} is finite to one, so is p_{1}; so an irreducible component of F has maximum entropy (which is equal to that of X) if and only its image under p_{1} is all of X. If there were two such components, then there would be a left transitive point x in X with two preimages in F, so the left transitive point πx would have two preimages under π_{+} in Y_{+}, which is impossible. Thus F has a unique irreducible component of maximum entropy in F, which we denote F_{0}; and the restriction ρ_{1} of p_{1} to F_{0} has degree one. Let ρ_{2} similarly denote the restriction of p_{2} to F_{0}. We have a commuting diagram of surjective maps

\xymatrix{ & F_{0} \ar[dl]_{ρ_{1}} \ar[dr]^{ρ_{2}} \\ X \ar[dr]_{π} &  & Y_{+} \ar[dl]^{π_{+}} \\  & Y}

Because π_{+} is right resolving, so is p_{1}, and therefore so is ρ_{1}.

Lemma 4.1. A(F_{0}) = {(i, U) ∈ A(X) × A(Y_{+}) : i ∈ U} .

Proof. First, suppose (i, U) ∈ A(F_{0}). Then (i, U) must occur as z_{0} in some point z of F_{0} which maps (via πρ_{1} = π_{+}ρ_{2}) to a left transitive point y in Y. Because (ρ_{1}z)_{0} = i and π(ρ_{1}z) = y, we have i ∈ U_{0}(y). Because (ρ_{2}z)_{0} = U and π_{+}(ρ_{2}z) = y and y has a unique preimage under π_{+}, which is the sequence (U_{m}(y))_{m∈ℤ}, we have U = U_{0}(y). Thus i ∈ U.

Next, suppose U ∈ A(Y_{+}) and i ∈ U. Then there exists a left transitive y in Y and an x in X such that π(x_{j}) = y_{j} for j ≤ 0, and U_{0}(y) = U and x_{0} = i. Thus (x, y)_{i} = (x_{i}, U_{i}(y)) for i ≤ 0, and for some n ≤ 0 the sequence (x, y)(−∞, n] is a sequence from F_{0}. By Lemma 2.3, there are no paths exiting from F_{0}, so (i, U) ∈ A(F_{0}). □

Proposition 4.2. Let (X, S) be an irreducible 1-step SFT. Let π : (X, S) → (Y, T) be a surjective one-block code with a magic symbol. In the associated fiber product diagram

\xymatrix{ & F_{0} \ar[dl]_{ρ_{1}} \ar[dr]^{ρ_{2}} \\ X \ar[dr]_{π} &  & Y_{+} \ar[dl]^{π_{+}} \\  & Y}

all maps are surjective one-block; ρ_{1} and π_{+} are right resolving; and ρ_{2} is left eresolving. If π is finite to one, then ρ_{2} is left resolving.

Proof. The claims before “ρ_{2} is left eresolving” are already established. Now suppose (i_{1}, U_{1}) ∈ A(F_{0}) and U_{0}U_{1} is a word of length two in Y_{+}. To show ρ_{2} is left eresolving, we must show there exists (\overline{i_{0}}, \overline{U_{0}}) in A(F_{0}) which can precede (i_{1}, U_{1}) and which maps to U_{0}. Because (i_{1}, U_{1}) ∈ A(F_{0}), we have i_{1} ∈ U_{1}; because U_{1} follows U_{0}, we may choose i_{0} in U_{0} such that i_{1} follows i_{0} in X. Because i_{0} ∈ U_{0}, we have (i_{0}, U_{0}) ∈ A(F_{0}). Let (\overline{i_{0}}, \overline{U_{0}}) = (i_{0}, U_{0}). We have shown ρ_{2} is left eresolving.

The final claim follows from Proposition 2.1(5): a map between equal entropy irreducible SFTs is left resolving if and only if it is left eresolving. □

We can now assemble a presentation-invariant version of our zero dimensional generalization of Putnam’s theorem.

Theorem 4.3. Suppose π : (X, S) → (Y, T) is a block code from an irreducible SFT onto a sofic shift. Then there are surjective block codes α, β, γ from irreducible SFTs which give a commutative diagram

[figure: γ; α; β; π]

such that α and β are 1-1 a.e. u-resolving and γ is s-eresolving. If π is finite to one, then γ is also s-resolving.

Such a diagram will be produced if β is a version of the futures cover π_{+} for π and α, γ are the restrictions to the unique maximal entropy component of the projections from the fiber product of π and β. In this case, the constructed diagram is canonically associated to π: if (α, β, γ, π) and (α′, β′, γ′, π′) are two such diagrams and subshift conjugacies η, κ effect a conjugacy of π and π′, then there will be a unique pair of subshift conjugacies η̃, κ̃ such that the following diagram commutes:

[figure: γ′; α′; β′; α; γ; η̃; β; κ̃; π′; η; π; κ]

Proof. The conclusions of Theorem 4.3 are invariant under replacement of π by a topologically conjugate map. By Lemma 2.4, the map π is topologically conjugate to a map which satisfies the conditions of Proposition 4.2, and this proves the existence claims of Theorem 4.3. That the construction is canonical then follows from Proposition 3.6(4), because the fiber product construction is canonical for a given pair π, π_{+}. □

Remark 4.4. A (mostly) weaker version of Putnam’s result in zero dimension was stated some time ago in [9, before Theorem A]. Given π bounded to one between irreducible SFTs, [9] provides a factorization πα = βπ̃, with all maps between irreducible SFTs, with π̃ left resolving and β right resolving. However, here the map α need not be in any way resolving, and the maps α, β need not have degree one.

The overlooked resolving possibilities for α probably had to do, at least in part, with the nature of fiber products. In Proposition 4.2, if F_{0} is replaced by the full fiber product F (π, π_{+}), then the projection F (π, π_{+}) → Y_{+} will be u-resolving (sresolving) if and only if π is already u-resolving (s-resolving). Thus it is surprising (at least, without the guidance of Putnam’s theorem) that the restriction to F_{0} has any resolving property independent of π, and this is perhaps part of the reason that Putnam’s theorem had no zero dimensional precursor.

5. The canonical mapping property

In this section we will restrict attention to maps in the category C of surjective finite to one block codes from irreducible SFTs to sofic shifts. For c_{1}, c_{2} in C, the composition c_{1}c_{2} is left/right/u/s -resolving if and only if both c_{1} and c_{2} are [5, Propositions 4.10 and 4.11].

Definition 5.1. Given π ∈ C, we define D(π) to be the category of commutative diagrams D = (a, b, c, π) of the form

[figure: c; a; b; π]

which satisfy the following conditions:

(1) a, b and c are in C;

(2) a and b are u-resolving;

(3) c is s-resolving.

Note that the domains of a, π and c are irreducible SFTs.

For D = (a, b, c, π) and D′ = (a′, b′, c′, π′) in D(π), a morphism D′ → D is a pair (φ, ψ) such that {φ, ψ} ⊂ C, and the following diagram commutes (i.e., D′ factors through D).

[figure: c′; φ; a′; ψ; b′; c; a; b; π]

We may write such a morphism as (φ, ψ) : D′ → D. Note that φ and ψ here are forced to be u-resolving. Diagrams D, D′ are isomorphic in this category (they are “conjugate over π”) if there are topological conjugacies φ, ψ such that (φ, ψ) : D′ → D. The conjugacy class of D is the set of all D′ in D(π) which are conjugate over π to D. A diagram D is minimal in a subset E of D(π) if D ∈ E and D′ → D for every D′ in E. For technical reasons we will be particularly concerned with the following subcategory of D(π).

Definition 5.2. D^{(1)}(π) is the full subcategory of D(π) whose objects are the diagrams D = (a, b, c, π) in D(π) such that the SFT F (b, π) (the fiber product of b and π) has a unique irreducible component of maximum entropy.

The condition of Definition (5.2) is satisfied whenever the degree of b is 1. A “full” subcategory of D(π) is simply one whose morphisms are all the D(π) morphisms between its objects. When a fiber product F (b, π) contains a unique irreducible component of maximum entropy, we will denote that component as F_{0}(b, π).

The next proposition shows that much of the structure of a diagram D = (a, b, c, π) in D(π) is dictated by the pair (b, π).

Proposition 5.3. Suppose D is the following diagram in D(π).

\xymatrix{V \ar[r]^{c} \ar[d]_{a} & W \ar[d]^{b} \\ X \ar[r]^{π} & Y}

Then there is an irreducible component V′ of maximum entropy in F (π, b) and a constant to one map φ : V → V′, such that the following diagram D′

\xymatrix{V′ \ar[r]^{p_{2}} \ar[d]_{p_{1}} & W \ar[d]^{b} \\ X \ar[r]^{π} & Y}

is in D(π), and (φ, Id) : D → D′. Here p_{1} and p_{2} denote the restrictions to V′ of the usual projections (x, w) ↦ x, (x, w) ↦ w from F (π, b) to X and W.

Proof. For v ∈ V, define φ(v) = (av, cv). Then φ maps V into F (π, b). Because V is irreducible, the image of φ must be contained in an irreducible component V′ of F (π, b). Then h(V) = h(φV) ≤ h(F (π, b)) = h(X) = h(V), so V′ has full entropy in F (π, b) and πV = V′.

Because p_{1}φ = a is u-resolving, so are p_{1} and φ; because p_{2}φ = c is s-resolving, so are p_{2} and φ. The resolving properties for p_{1} and p_{2} show that D′ = (p_{2}, p_{1}, b, π) is in D(π). The map φ is constant to one because it is a u-resolving and s-resolving map between irreducible SFTs [8, Proposition 4.3.4]. Clearly (φ, Id) : D → D′. □

Given π ∈ C, let D_{π} be the element of D(π) constructed as for Proposition 4.2, i.e. D_{π} is the diagram

\xymatrix{F_{0} \ar[r]^{ρ_{2}} \ar[d]_{ρ_{1}} & Y_{+} \ar[d]^{π_{+}} \\ X \ar[r]^{π} & Y}

where F_{0} = F_{0}(π, π_{+}), all maps are 1-block codes, and the three SFTs F_{0}, Y_{+} and X are irreducible 1-step. We will characterize the conjugacy class of D_{π} with a mapping property (answering a question in [16]).

Theorem 5.4. Suppose π ∈ C.

(1) If D ∈ D^{(1)}(π), then there exists a unique morphism (φ_{1}, φ_{2}) : D → D_{π}.

(2) D is minimal in D^{(1)}(π) if and only if D is conjugate over π to D_{π}.

Question 5.5. Is 5.4(1) true with D(π) in place of D^{(1)}(π)?

Proof. To prove (1), suppose we have D ∈ D(π) as below.

\xymatrix{V \ar[r]^{c} \ar[d]_{a} & W \ar[d]^{b} \\ X \ar[r]^{π} & Y}

Without loss of generality, we suppose the SFTs are 1-step and the maps in the diagram are 1-block codes.

By Proposition 5.3, without loss of generality we may assume V = F_{0}(π, b), with (a, c) = (p_{1}, p_{2}) where p_{1}, p_{2} are the projection maps from F_{0}(π, b) to X and W. Again without loss of generality, by Proposition 2.1(2), we may suppose that b is right resolving. Because a then arises from the fiber product construction, a is also right resolving. The map c is at this point still only s-resolving (i.e. left closing).

We will find (φ, ψ) : D → D_{π}. We first give the rule for ψ. For w left transitive in W, define ψw by

(ψw)_{m} = U_{m}(bw) , m ∈ ℤ ,

where U_{m}(y) was defined in (3.1). This is a shift-commuting rule and the left transitive points are dense; so, to show the rule determines a block code, it suffices to show the map w ↦ U_{0}(bw) is uniformly continuous on left transitive points w.

Because c : V → W is s-eresolving (by Proposition 2.1(5)), we may choose N ≥ 0 such that if v ∈ V and (cv)_{i} = (w′)_{i} for i ≥ −N, then there exists ṽ in V such that cṽ = w′ and ṽ_{i} = v_{i} for i ≥ 0 [6, Proposition 5.1]. Now, suppose \overline{w} and w are left transitive points in W, \overline{w}[−N, 0] = w[−N, 0] and i ∈ U_{0}(bw). We will show i ∈ U_{0}(b\overline{w}). For a point (biinfinite sequence) z, we will use the notations z_{−} = z(−∞, 0] and z_{+} = z[0, +∞).

Because i ∈ U_{0}(bw), we may choose x in X such that x_{0} = i and (πx)_{−} = (bw)_{−}. Because bw is left transitive and b is right resolving and (bw)_{−} = (πx)_{−}, by Lemma 2.5 there is a w′ in W such that (w′)_{−} = w_{−} and bw′ = πx. Therefore (x, w′) is a point in the fiber product F (π, b). Because w′ is left transitive, the point (x, w′) must be in the irreducible component of maximum entropy, i.e. (x, w′) ∈ V. Define w′′ in W by w′′(−∞, 0] = \overline{w}(−∞, 0] and w′′[−N, ∞) = w′[−N, ∞). Note, the overlapping definition is consistent. Now

(c(x, w′))[−N, ∞) = w′′[−N, ∞) ,

so there exists ṽ = (x̃, w̃) such that cṽ = w′′ and ṽ[0, ∞) = (x, w′)[0, ∞). Therefore (πx̃)(−∞, 0] = (b\overline{w})(−∞, 0], x̃_{0} ∈ U_{0}(b\overline{w}) and x̃_{0} = x_{0}. We have shown that our rule gives a block code ψ : W → Y_{+}.

We summarize our situation with the following commuting diagram

\xymatrix{V \ar[rrr]^{c} \ar[dd]_{a} \ar@{.>}[dr] &  &  & W \ar[dl]_{ψ} \ar[dd]^{b} \\  & F_{0} \ar[r]^{ρ_{2}} \ar[dl]_{ρ_{1}} & Y_{+} \ar[dr]^{π_{+}} \\ X \ar[rrr]_{π} &  &  & Y}

in which F_{0} denotes F_{0}(π, π_{+}) and V = F_{0}(π, b). The dotted arrow V → F_{0} is the map φ we need but have not yet defined. The maps a, ψc factor through π, π_{+} and therefore can be factored through F (π, π_{+}) by some map φ : V → F (π, π_{+}). The image of φ must be F_{0}; φ defines the dotted arrow. Now (φ, ψ) : D → D_{π}. There can be only one morphism (φ, ψ) : D → D_{π} because it is uniquely determined on the dense set of transitive points, because π_{+} and ρ_{1} are one to one on transitive points. This proves (1).

On to (2). It is obvious that a diagram conjugate over π to a minimal element of D^{(1)}(π) (such as D_{π}) will also be a minimal element of D^{(1)}(π). Now suppose D = (a, b, c, π) is a minimal element of D^{(1)}(π). We have (φ_{1}, ψ_{1}) : D_{π} → D, which forces a and b to have degree 1. We also have (φ_{2}, ψ_{2}) : D → D_{π}, so (φ_{1}φ_{2}, ψ_{1}ψ_{2}) : D → D. By uniqueness of the morphism, φ_{2}φ_{1} and ψ_{2}ψ_{1} must be identity maps, as must φ_{1}φ_{2} and ψ_{1}ψ_{2}. Thus D and D_{π} are conjugate over π. □

Remark 5.6. Suppose π ∈ C. The characterization of the conjugacy class of π_{+} by a minimal mapping property is analogous to the characterization of the right Fisher cover of a sofic system by a minimal mapping property [4]. Similarly, the relation of the futures cover of π to the full futures cover is analogous to the relation of the Fisher cover to the Krieger cover [10].

6. The maps of Nasu and Putnam

Let π : X → Y be a block code from an irreducible SFT onto a sofic shift. Without loss of generality (after recoding if necessary), we take the code to be oneblock and the SFT to be 1-step. In Proposition 4.2, we associated to π a certain diagram D_{π} = (ρ_{1}, ρ_{2}, π, π_{+}). In this section we will explain the relationship of D_{π} to the construction of Putnam, and the relation of π_{+} to a canonical cover defined by Nasu.

Putnam. Given a surjective 1-1 a.e. morphism π between irreducible Smale spaces, Putnam constructed a commuting diagram D̃ = (α, π̃, β, π),

\xymatrix{\overline{X} \ar[r]^{π̃} \ar[d]_{α} & \overline{Y} \ar[d]^{β} \\ X \ar[r]^{π} & Y}

with all of the maps 1-1 a.e. surjective morphisms π between irreducible Smale spaces, with α and β u-resolving, and with π̃ s-resolving.

Theorem 4.3 and Putnam’s theorem overlap in the case that π is a surjective, 1-1 a.e. block code between irreducible SFTs. We claim in this case that the two diagrams D and D̃ are conjugate over π. To show this, we first find a conjugacy of subshifts ψ : Y_{+} → \overline{Y} such that π_{+} = βψ. Without loss of generality, we may presume π is a one block map and X is a 1-step SFT.

Putnam constructed \overline{Y} as the completion in a certain metric δ_{Y} of the set π(W) of points left asymptotic to the orbit of a chosen periodic point y_{0}, where y_{0} has a unique preimage x_{0} under π, and W is the set of points in X left asymptotic to the orbit of x_{0}. Our desired map ψ must agree with the natural injection ι from (π_{+})^{−1}(π(W)) to π(W). We claim that ι is uniformly continuous (with respect in the domain to the restriction of a metric giving the usual topology on Y_{+}, and with respect in the range to the metric δ_{Y}).

For brevity, we will not define all the terms and facts we use from the subtle construction of [16]. Let y and y′ be in π(W), with z = π_{+}^{−1}(y) and z′ = π_{+}^{−1}(y′) in Y_{+}. Now suppose that N > 0 and z[−N, N] = z′[−N, N]. By definition of Y_{+}, we have y[−N, N] = y′[−N, N] and moreover there exists a bijection ν : π^{−1}(y) → π^{−1}(y′) such that (νw)[−N, N] = w[−N, N], for all w in π^{−1}(y). Define a point y′′ in Y by y′′(−∞, ∞) = y(−∞, −N]y′[−N + 1, ∞). Then for large enough N, the path (y, y′′, y′) is a rectangular path from y to y′′ ([16, Definition 2.13]), whose length is the sum dist(y, y′′) + dist(y′′, y′), which is at most 2^{−N+1}, and which is an upper bound for δ_{Y}^{0} (y, y′) ([16, Definition 2.14]). Then for a number r with 0 < r < 1 and the uniform constant C = max δ_{Y}^{0} , we get

δ_{Y} (y, y′) ≤ ∑_{k=0}^{N} r^{k}2^{−N+k} + ∑_{k=N+1}^{∞} Cr^{k} .

This quantity goes to zero with N.

Thus ι is uniformly continuous, and extends to a continuous map ψ : Y_{+} → \overline{Y}. This map commutes with the shift on a dense set, hence everywhere, and is thus a block code. There is also by Theorem 5.4 a block code ψ′ : \overline{Y} → Y_{+} such that β = π_{+}ψ′. Clearly ψ and ψ′ are inverse to each other, so ψ is a conjugacy, and β is a version of π_{+}. Thus by Proposition 5.3, the map α is the map ρ_{1} preceded by some constant to one factor map; because the degree of α is one, the constant to one map is a conjugacy of subshifts. This completes the proof.

Nasu.

In [12], given a one-block code π : X → Y from an edge SFT onto a sofic system, Masakazu Nasu associated to π a map he called the induced right resolving cover of π. (This cover appears as an important tool in [9].) This cover is a one-block code we will denote η_{+} : N_{+} → Y. Here N_{+} is another edge SFT.

The given map π is presented by a labeled graph G, where the unlabeled graph defines the edge SFT and the label (by symbols from A(Y)) defines the one-block code. Nasu defined his cover by constructing another labeled graph, G_{+}. For a subset V of vertices of G, and a Y -word x, let S_{+}(V, x) denote the set of terminal vertices S_{+}(V, x) of the paths labeled x that go from some vertex in V. A right compatible set of G is a set S_{+}(V, x) where V is a singleton. Nasu defined the vertex set of G_{+} to be the set of all maximal right compatible sets and their nonempty sucessors. For each vertex S in G_{+}, a unique edge labeled a goes from W to S_{+}(W, a) if S_{+}(W, a) is nonempty. This defines the edge set of G_{+}, and thereby the edge SFT N_{+} and the one block map η_{+} : N_{+} → Y. Clearly η_{+} is surjective and right resolving. Nasu showed [12, Theorem 3.1] that η_{+} is canonically associated to π, in the sense that any conjugacy from π to another one-block code π′ : X′ → Y′ (such a conjugacy by definition is given by conjugacies X → X′, Y → Y′ which intertwine π and π′) lifts to an isomorphism of η_{+} and η′_{+}:

\xymatrix{N_{+} \ar@{.>}[rrr] \ar[dd]_{η_{+}} &  &  & X′_{+} \ar[dd]^{η′_{+}} \\  & X \ar[r] \ar[dl]_{π} & X′ \ar[dr]^{π′} \\ Y \ar[rrr] &  &  & Y′}

Nasu’s cover η_{+} (like the futures cover π_{+}) is well defined even if the domain SFT is not irreducible or the map π is not finite to one. If X is irreducible, then both N_{+} and Y_{+} will be irreducible, and the degrees of π_{+} and η_{+} are well defined. Here the map π_{+} will always have degree one, but η_{+} might have greater degree. For example, if π is itself right resolving, then clearly (as noted by Nasu [12, p.569]) η_{+} will equal π.

Now restrict to the case that X is irreducible and the degree of π is one. In this case, there is a conjugacy ψ between the domains of π_{+} and η_{+} such that η_{+} = π_{+}ψ. To see this we simply note that in the degree one case, the symbols U_{0}(y) in A(Y_{+}) are maximal compatible sets from the unique preimage symbol of a magic symbol for π. Thus the identity map embeds Y_{+} into N_{+}, and because the irreducible SFT N_{+} has no proper subsystem of equal entropy, this embedding is surjective. In particular, our diagram D_{π}, produced using F_{0}(π, π_{+}) will be conjugate over π to the diagram produced in the same way using F_{0}(π, η_{+}). Each of the diagrams is another version of Putnam’s diagram in dimension zero.

It is naturally to ask whether Nasu’s right resolving cover must factor through the futures cover. This is not always the case, as we see in the next example.

Example 6.1. Let π be the map given in [2, Example 1 on p. 491]. This map π can be defined by the following labeled graph G.

[figure: β; a; b; α; b; a; γ; a; b]

Forgetting the labels in the graph gives an unlabeled graph, with adjacency matrix

(\begin{matrix}1 & 1 & 0 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{matrix}), which defines an irreducible edge SFT (X, S). The edge labeling defines a 1-block code π from (X, S) to a shift (Y, T), which is the full 2-shift on symbols a, b. A labeled graph G_{++} defining a full futures cover (Y_{++}, T_{++}) of π as an edge SFT, and the 1-block code π_{++}, can be computed as follows. Vertices of G_{++} are sets of vertices of G. Begin with the vertex v = {α, β, γ}. The set of G-paths labeled a starting from v end in the set {α, β}. Add this set as another vertex and draw an edge labeled a from v to v′. Continue in this fashion until reaching a graph such that no new vertices can be added. Reduce to the maximal subgraph such that each vertex has an incoming edge and an outgoing edge. This produces the following G_{++}.

[figure: α, β, γ; b; a; α, β; a; b; α, γ; b; a]

The futures cover (Y_{+}, T_{+}) is obtained by restricting this graph to the irreducible component of maximum entropy, which as an unlabeled graph has adjacency matrix

(\begin{matrix}1 & 1 \\ 2 & 0 \end{matrix}). Because π is right resolving, the associated right resolving cover η_{+} of

Nasu is equal to π. Consequently, in this example there is no factor map φ such that η = π_{+}φ, for the following reason. There is a fixed point x such that π(x) is the fixed point b^{∞}, and φ could only map x to a fixed point y such that π_{+}(y) = b^{∞}; but there is no such fixed point y in the domain of π_{+}.

7. Resolving degree and Markovian maps

In this section we point out that the resolving degree is an invariant of topology conjugacy for block codes from irreducible SFTs. (This is well known when the code is finite to one; then resolving degree equals degree.) We also give an application of Theorem 4.3 to Markovian codes.

Proposition 7.1. Suppose π and π′ are topologically conjugate maps from irreducible SFTs. Then they have the same resolving degree.

Proof. Suppose that π is a one block code with a magic symbol μ. Let η_{+} and π_{+} be the Nasu and futures covers associated to π. Form the fiber product diagram

\xymatrix{ & F_{0} \ar[dl]_{p_{1}} \ar[dr]^{p_{2}} \\ N_{+} \ar[dr]_{η_{+}} &  & Y_{+} \ar[dl]^{π_{+}} \\  & Y}

Here F_{0} denotes the unique irreducible component of maximum entropy in the fiber product (it is unique because π_{+} has degree 1). The projections p_{1} and p_{2} are surjective. Clearly {μ} is a magic symbol for the map p_{2} and the cardinality of its preimage, which is the degree of p_{2}, is the cardinality of π^{−1}(μ). Because η_{+} and π_{+} are canonically associated to π, so is the fiber product diagram and in particular the degree of p_{2}. Therefore this number will be the same for π and π′. □

A surjective block code between irreducible SFTs is Markovian if any (equivalently every) fully supported ergodic Markov measure on the range space lifts to a fully supported ergodic Markov measure on the domain [6, 15]. From the current paper, we find that any block code between irreducible SFTs is in a certain sense not far from Markovian.

Proposition 7.2. Suppose φ : (X, S) → (Y, T) is a surjective block code between irreducible SFTs. Then there is a u-resolving degree 1 block code α from an irreducible SFT onto (X, S) such that φα is Markovian.

Proof. By Theorem 4.3, there is an α of the desired form such that φα equals an s-eresolving code followed by a u-resolving code, both between irreducible SFTs. Such codes (and their compositions) are Markovian [6]. □

Question 7.3. It is an open problem since [6] to provide a decision procedure to determine whether a surjective block code between irreducible SFTs is Markovian. Do the constructs around the futures cover provide clues?

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Mike Boyle, Department of Mathematics, University of Maryland, College Park, MD 20742-4015, U.S.A.

E-mail address: mmb@math.umd.edu
URL: www.math.umd.edu/∼mmb