Theory and Applications of Categories, Vol. 25, No. 18, 2011, pp. 490–507.

FLOWS: COCYCLIC AND ALMOST COCYCLIC

MICHAEL BARR, JOHN F. KENNISON, AND R. RAPHAEL

Abstract. A flow on a compact Hausdorff space is an automorphism. Using the closed structure on the category of uniform spaces, a flow gives rise, by iteration, to an action of the integers on the topological group of automorphisms of the object. We study special classes of flows: periodic, cocyclic, and almost cocyclic, mainly in term of the possibility of extending this action continuously to various compactifications of the integers.

1. Introduction

In this paper we denote by Flowthe category whose objects are pairs (X, t) where X is a compact Hausdorff space and t : X [figure] X is an automorphism. The notion of flow is obviously much broader (any category, any endomorphism), but we are restricting to this special case here. If (X, t) and (Y, u) are flows, a continuous map f : X [figure] Y is a flow morphism if uf = ft. A flow (X, t) is called periodic if there is an integer n > 0 such that t^{n} = id.

We let C(X) denote the lattice-ordered ring of all continuous functions from X to R and, by abuse of notation, also denote by t : C(X) [figure] C(X) the automorphism induced by t (so that t(f) is the composite ft). Since t is an automorphism, it follows that t^{n} : X [figure] X and t^{n} : C(X) [figure] C(X) are defined for all n ∈ Z.

The ring C(X) has a norm ∥ ∥, defined by ∥f∥ = sup_{x∈X} |f(x)|. It satisfies ∥ft∥ = ∥f∥.

Section 2 surveys some mostly well-known definitions and facts on uniform spaces. Section 3.1 is a review of uniform completions of the group Z of integers. In section 4, we define the notion of a t-periodic element of C(X). We show that a flow is periodic if and only if every element of C(X) is t-periodic (Theorem 4.5). We define a flow to be cocyclic if the t-periodic elements of C(X) are dense. We then show that a flow is cocyclic if and only if it is a filtered inverse limit of periodic flows (Theorem 4.9). We also show that cocyclic flows are characterized by the fact that the action Z × X [figure] X, defined by (n, x) ↦ t^{n}x extends to a continuous action pf(Z) × X [figure] X, where pf(Z) is the profinite completion of Z (Theorem 4.10). In Section 5, we define the notion of an almost periodic element of C(X). In Section 6, we define and study almost cocyclic flows. These are similarly characterized by the possibility of extending the action of Z to a continuous action by b(Z), the Bohr compactification of Z, from which it is immediate that a cocyclic flow is almost cocyclic.

The first author would like to thank NSERC of Canada for its support of this research. We would all like to thank McGill and Concordia Universities for partial support of the second author’s visits to Montreal.

Received by the editors 2011-06-14 and, in revised form, 2011-10-16.

Transmitted by Susan Niefeld. Published on 2011-11-19.

2000 Mathematics Subject Classification: 18B30,37C55,54C30,54B30.

Key words and phrases: flow on compact spaces, periodic and cocyclic flows, almost cocyclic flows.

\begin{matrix}c \\ ◯ \end{matrix} Michael Barr, John F. Kennison, and R. Raphael, 2011. Permission to copy for private use granted.

2. On uniform spaces

We refer to [Isbell, 1964] for an introduction to uniform spaces and uniform maps. Every completely regular space has at least one uniformity that induces the same topology, but there are usually more than one. A uniform map is continuous, but the converse is false in general. An easy counter-example is given by the homeomorphism between the space of positive integers and the space of their reciprocals, each given the subspace metric of R.

However, there are two cases, each relevant here, in which the uniformity is unique and continuous functions are uniform. The first is a compact Hausdorff space, which has a unique uniformity, in which all covers are uniform; equivalently every neighbourhood of the the diagonal is an entourage. Every continuous function from a compact space to a uniform space that is continuous in the uniform topologies, is also uniform. The second case is that of an abelian topological group. Such a group does not, as a space, have a unique uniformity, but it does as a group. A basic uniform cover is the set of all translates of a neighbourhood of 0 by all the group elements. Should the neighbourhood be a subgroup, this is just the cover by its cosets. All continuous homomorphisms between abelian topological groups are uniform. Non-abelian groups have two uniformities, one in which the left translates of neighbourhoods of the identity are the uniform covers and one using the right translates. A continuous homomorphism between topological groups is uniform so long as you use left translates for both or right translates for both. If one or both groups are abelian (or if the domain is compact) then such a homomorphism is uniform regardless. One point that should be made is that a non-abelian topological group is not necessarily a group in the category of uniform spaces since neither the inverse map nor the multiplication need be uniform.

On these two classes of objects, we will not distinguish between topological and uniform structures or between continuous and uniform morphisms.

If X and Y are uniform spaces, then hom(X, Y) denotes the set of all uniform functions X [figure] Y equipped with the uniformity of uniform convergence on all of X. This means that if V ⊆ Y × Y is an entourage on Y, then the set

{(f, g) | (fx, gx) ∈ V for all x ∈ X }

is an entourage for the uniform structure on hom(X, Y). This is an obvious generalization of the sup norm when Y is metric. As usual, we denote by Hom(X, Y) the set of all uniform functions X [figure] Y. It is, of course, the underlying set of hom(X, Y).

For later use, we record the following property of compact spaces. It follows immediately from [Isbell, 1964, Theorem 31].

2.1. Proposition. Suppose X and Y are compact Hausdorff spaces. Then hom(X, Y) is complete.

2.2. Definition. A uniform space is called totally bounded if it can be uniformly embedded in a compact space. This is equivalent to the fact that every uniform cover has a finite subcover, see [Isbell, 1964].

The following result is implicit in [Isbell, 1964, Theorem III.24] but we find the proof somewhat opaque and therefore we include our own proof.

2.3. Theorem. When the space Z is totally bounded, then for any uniform spaces X and Y, Hom(Z × X, Y) ≅ Hom(Z, hom(X, Y)).

This is true also for the internal hom, but we have no need of it. The proof is based on the following lemma in which all entourages used in both the statement and proof will be understood to be symmetric.

2.4. Lemma. Suppose Z, X, Y are uniform spaces with Z totally bounded and suppose θ : Z [figure] hom(X, Y) is a uniform map. Then for each entourage V ⊆ Y × Y there are entourages U ⊆ X × X and W ⊆ Z × Z such that for all (z, z′) ∈ W and (x, x′) ∈ U, we have (θz(x), θz′(x′)) ∈ V.

Proof. Let V̂ ⊆ Y × Y be an entourage such that V̂ ^{∘} V̂ ^{∘} V̂ ^{∘} V̂ ⊆ V. Let

H = {(f, g) ∈ Hom(X, Y) × Hom(X, Y) | (fx, gx) ∈ V̂ for all x ∈ X }

Then H is an entourage in hom(X, Y). Since θ is a uniform map, there is an entourage Ŵ ⊆ Z × Z such that (θ × θ)(Ŵ) ⊆ H. Let W ⊆ Ŵ be an entourage such that W ^{∘} W ^{∘} W ⊆ Ŵ.

Since Z is totally bounded, there is a finite subset z_{1}, z_{2}, …, z_{n} ∈ Z such that {W [z_{i}]} covers Z. For each i = 1, …, n, we have that θz_{i} is a uniform map X [figure] Y and so there is an entourage U_{i} ⊆ X × X such that whenever (x, x′) ∈ U_{i}, we have (θz_{i}(x), θz_{i}(x′)) ∈ V̂. If U = ⋂ U_{i}, then we have that for all i = 1, …, n and all (x, x′) ∈ U, (θz_{i}(x), θz_{i}(x′)) ∈ V̂.

Now suppose that (z, z′) ∈ W. There are indices i and j such that (z_{i}, z) ∈ W and (z′, z_{j}) ∈ W. This implies that (z_{i}, z_{j}) ∈ W ^{∘} W ^{∘} W ⊆ Ŵ. We can now infer that all of (θz, θz_{i}), (θz_{i}, θz_{j}) and (θz_{j}, θz′) belong to H. Using the fact that (θz_{j}(x), θz_{j}(x′)) ∈ V̂, we see that all of (θz(x), θz_{i}(x)), (θz_{i}(x), θz_{j}(x)), (θz_{j}(x), θz_{j}(x′)), and (θz_{j}(x′), θz(x′)) belong to V̂, whence (θz(x), θz′(x′)) ∈ V.

Proof of the theorem. The lemma implies immediately that if θ : Z [figure] hom(X, Y) is uniform, so is the transpose Z × X [figure] Y. For the converse, simply observe that if we have a uniform map Z × X [figure] Y, the conclusion of the lemma is valid from which it is immediate that the transpose Z [figure] hom(X, Y) is uniform.

In general, the closed structure on the category of uniform spaces is not symmetric, so this condition is specific to Z and it is not sufficient that X be totally bounded. It is sufficient that X be discrete, but that will not be the case in our applications.

2.5. Proposition. The supremum of a family of totally bounded uniformities on a set is totally bounded.

Proof. Let X be a set and let {U_{α}} be a family of totally bounded uniformities on X. For each α let f_{α} : X [figure] C_{α} be a uniform embedding of X, equipped with the U_{α} uniformity, to a compact space C_{α}. Then the map X [figure]∏ C_{α} whose α coordinate is f_{α} is an injection of X into a compact space for which the induced uniformity on X is the supremum of the U_{α}.

3. Completions and compactifications of Z

A group compactification of Z is a compact group \overline{Z} and a homomorphism Z [figure] \overline{Z} whose image is dense. We do not require the map to be injective, so that the canonical map Z [figure] Z/nZ, for any positive integer n, is a compactification. It is not hard to see that these are the only finite compactifications of Z and that in all other cases, the map Z [figure] \overline{Z} is injective. If G is a topological group, a map Z [figure] G will be said to lift to \overline{Z} when there is a commutative diagram

[figure: \overline{Z}; G; Z; \overline{Z}; Z; G]

Although this terminology is not entirely reasonable when Z [figure] \overline{Z} fails to be injective, we will use it anyway. When we talk of compactifications of Z we will always mean group compactifications. Similarly, when we talk of topologies on Z, we always mean group topologies.

3.1. Uniform completions of Z. We will denote by Z_{pf} the topological group whose underlying group is the integers, but whose topology has the non-zero subgroups as its basic neighbourhoods at 0. In the discrete topology, the integers are complete, but in this topology they are not. We denote by pf(Z) the completion in the uniformity defined by the topology just described. There are at least two good ways of describing it. One is as the inverse limit of all Z/nZ as n ranges over the positive integers. When n|m, we have a commutative diagram

[figure: Z/mZ; Z/nZ; Z; Z/mZ; Z; Z/nZ]

More generally, if m and n are arbitrary and l their least common multiple, we have the commutative diagram

[figure: Z/mZ; Z/lZ; Z/lZ; Z/nZ; Z; Z/mZ; Z; Z/lZ; Z; Z/nZ]

which shows that the diagram is cofiltered.

Since for non-zero n ∈ Z a finite number of translates of the subgoup nZ covers Z, it follows that the topology is totally bounded and the completion is compact.

Here is the second way to construct pf(Z). If p is a prime, let pf(Z)_{p} denote p-adic integers, which is the inverse limit of the sequence

Z/pZ [figure] Z/p^{2}Z [figure] Z/p^{3}Z [figure] ⋯

Then pf(Z) = ∏ pf(Z)_{p}, the product over all primes. An element of pf(Z)_{p} is an infinite series α = a_{0} +a_{1}p+⋯+a_{n}p^{n} +⋯ with coefficients between 0 and p−1. Note, however, that addition and multiplication are not mod p, but involve carries into higher powers of p. The usual argument shows that such an element is invertible if and only if a_{0} ≠ 0. We say that the element above has order n if a_{0} = ⋯ = a_{n−1} = 0 and a_{n} ≠ 0. It will simplify the exposition below if we let p^{∞} denote the 0 element of pf(Z)_{p} and write ord_{p}0 = ∞.

We let ρ_{n} : pf(Z) [figure] Z/nZ and σ_{p} : pf(Z) [figure] pf(Z)_{p} be the canonical projections. Let us denote by ord_{p}λ the order of σ_{p}λ. If λ, μ ∈ pf(Z), it is clear that λ|μ if and only if, for each prime p, ord_{p}λ ≤ ord_{p}μ.

3.2. Proposition. A closed ideal in pf(Z) has the form I = ∏ I_{p}, where I_{p} is an ideal in pf(Z)_{p}. Every non-zero ideal in pf(Z)_{p} is principal, generated by a power of p.

Proof. Let I be a closed ideal in pf(Z). Let e_{p} denote the element of ∏ pf(Z)_{q} whose pth coordinate is 1 and all other coordinates are 0. Let I_{p} = e_{p}I. Clearly ∑ I_{p} ⊆ I ⊆ ∏ I_{p}. But since the sum is dense in the product and I is closed, it is clear that ∏ I_{p} = I.

For a non-zero ideal I_{p} ⊆ pf(Z)_{p}, let p^{n}(a_{0} + a_{1}p + ⋯), a_{0} ≠ 0, be an element of I of least degree. Since the element in parentheses is invertible, we see that p^{n} ∈ I_{p}. Since n was chosen as small as possible, it is clear the I_{p} is generated by p^{n}.

The Bohr compactification. The Bohr compactifcation of an abelian group G can be defined as the value at G of the left adjoint to the underlying functor from compact abelian groups to topological abelian groups. It can be directly constructed, as Bohr did, using the construction from the adjoint functor theorem (it is a matter of historical fact that Bohr’s construction was one of the motivations behind Freyd’s construction). Another method constructs it as |G^{∗}|^{∗} where ^{∗} is the Pontrjagin dual. A third is as the completion under the finest totally bounded topology on G.

Let b(Z) denote the Bohr compactification of Z. We say that the topology on Z induced by the embedding Z [figure] b(Z) is the Bohr topology on Z and it will be denoted by Z_{b}. Although this topology is not well understood, we can give some idea of the open neighbourhoods of 0. Like all compact abelian groups, b(Z) can be embedded algebraically and topologically into a power of the circle group T = R/Z. Thus, for some set S, we have an embedding Z_{b} [figure] T^{S}. A standard argument shows that we can take S = Hom(Z_{b}, T) = Hom(Z, T) since every homomorphism from Z to a compact group extends to b(Z) and thus is continuous on Z_{b}. A homomorphism f : Z [figure] T is entirely determined by f(1) which can be any element of γ ∈ T and thus is multiplication mod Z by some real number γ, which can be taken to lie in the interval [0, 1) although any real number could be used; two reals that differ by an integer give the same homomorphism. For each ϵ > 0, let

U(γ, ϵ) = {n ∈ Z | nγ is within ϵ of an integer}

Then the U(γ, ϵ) form a subbase for the neighbourhood system at 0 in Z_{b}. Informally, these sets look like “almost subgroups” or “almost periodic sets”, although we will not attempt to make this notion precise. For example, when γ = π, and ϵ = 0.1, the numbers in U(π, ϵ) between 0 and 113 consist of the multiples of 7 up to 77 together with the numbers congruent to 1 mod 7 between 36 and 113. Since 355/113 is such a good approximation for π, this sequence of elements of U(π, ϵ) will continue to repeat as follows. Let A denote the set consisting of the 24 numbers {0, 7, 14, …, 77, 36, 43, 50, …, 13}. Then U(π, 0.1) continues with the numbers of the form a + 113k, for a ∈ A and k up to k = 87. After that, determined by a better continued fraction approximation for π, the sequence begins to veer off.

4. Periodic and cocyclic flows

Recall that in this paper, all spaces are assumed compact Hausdorff. Also recall that a flow (X, t) is called periodic if there is an n > 0 such that t^{n} = id. We say that n is a period of the flow. The least such positive n will be called the minimal period of the flow.

4.1. Action of Z on a flow. If (X, t) is a flow, then there is an action of Z on X given by (n, x) ↦ t^{n}(x). This gives rise to a group homomorphism φ : Z [figure] Aut(X) given by n ↦ t^{n}. When n is the minimal period of t, then the kernel of Z [figure] Aut(X) is exactly nZ.

4.2. Proposition. The group Aut(X) becomes a topological group if we equip it with the uniformity it inherits from hom(X, X) (see Section 2).

Proof. A neighbourhood of an automorphism s is determined by an entourage E as

E(s) = {r | (rx, sx) ∈ E for all x ∈ X}

For the remainder of this computation, treat all set builders as indexed over all the elements of X. Then we have

E(su) = {r | (rx, sux) ∈ E}
= {r | (ru^{−1}y, sy) ∈ E} (replace x by u^{−1}y)
= {vu | (vy, sy) ∈ E} (replace r by vu)
= E(s)u

In particular, E(s) = E(1)s so that the topology is given by right translates of neighbourhoods of the identity. Assume now that E is a symmetric entourage. Then

E(1) = {r | (rx, x) ∈ E} = {r | (x, rx) ∈ E} = {s^{−1} | (x, s^{−1}x) ∈ E}
= {s^{−1} | (sy, y) ∈ E} (replace x by sy)
= E(1)^{−1}

Finally, let F be an entourage such that F ^{∘} F^{op} ⊆ E. Then for r, s ∈ F (1), we have (rx, x), (sx, x) ∈ F so that (rx, sx) ∈ F ^{∘} F^{op} ⊆ E. But then for y = sx, (rs^{−1}y, y) ∈ E and so we see that F (1)F (1)^{−1} ⊆ E(1) and the conditions for being a topological group are satisfied.

Notation. We will denote by aut(X) the group Aut(X) of automorphisms of X equipped with this topology.

Since X is compact Hausdorff it is a subspace of R^{C(X)}. Thus for all f ∈ C(X), we define φ_{f} : Z [figure] C(X) as the composite:

Z [figure] aut(X) [figure] Hom(X, X) [figure] Hom(X, R^{C(X)}) [figure: p_{f}] Hom(X, R) = C(X)

Then φ_{f}(n) = ft^{n}.

We will say that an f ∈ C(X) is t-periodic if there is an n > 0 for which ft^{n} = f.

4.3. Theorem. Let (X, t) be a flow and f ∈ C(X). Of the following conditions, the first three are equivalent and imply the fourth:

(1) {ft^{n} | n ∈ Z} is finite;

(2) f is t-periodic;

(3) φ_{f} : Z [figure] C(X) extends to a finite compactification of Z;

(4) φ_{f} : Z [figure] C(X) extends continuously to pf(Z).

Proof. The proofs that (1) [figure](2) [figure](3) [figure](1) are trivial and left to the reader. The proof that (3) [figure](4) follows from the fact that Z/nZ is a quotient of pf(Z).

4.4. Example. Here is an example to show that (4) does not imply the other three. Let X = pf(Z) and let t : X [figure] X be given by t(x) = x + 1. Then for all n ∈ Z, t^{n}(x) = x + n and we can evidently extend this to all z ∈ pf(Z) by t^{z}(x) = x + z. Clearly t is not periodic and it follows from Theorem 4.5 below that some f ∈ C(X) is not t-periodic.

Clearly, if (X, t) is periodic, then every f ∈ C(X) is t-periodic. For the converse we have the following theorem. Note that the argument makes no use of the compactness of the space. Also note that all the functions used in the argument are bounded.

4.5. Theorem. Let (X, t) be a flow. If every f ∈ C^{∗}(X) is t-periodic, then t is periodic on X.

Proof. If t is not periodic, then there must be, for any m > 0, an infinite set S_{m} ⊆ X such that t^{m}x ≠ x for all x ∈ S_{m}. The set S_{m} might consist of a single infinite orbit, if there is one, or a set of periodic points whose orbit size does not divide m. An obvious argument using LCM would show that if t^{m}x = x for all but a finite set of periodic points, then t would be periodic.

Suppose x_{1} ∈ S_{1}. Let f_{1} : X [figure] [0, 1] be a continuous function such that f_{1}(x_{1}) = 1 and f_{1}(tx_{1}) = 0. Let g_{1} = f_{1}/2. Clearly g_{1}(x_{1}) ≠ g_{1}(tx_{1}). Next choose x_{2} ∈ S_{2} such that {x_{2}, t^{2}x_{2}} is disjoint from {x_{1}, tx_{1}}. The fact that S_{2} is infinite and t a bijection allows such a choice to be made. Let f_{2} : X [figure] [0, 1] be a continuous function such that f_{2}(x_{1}) = f_{2}(tx_{1}) = 0 and

\{ \begin{matrix}f_{2}(x_{2}) = 1 and f_{2}(t^{2}x_{2}) = 0 & if g_{1}(x_{2}) > g_{1}(t^{2}x_{2}) \\ f_{2}(x_{2}) = 0 and f_{2}(t^{2}x_{2}) = 1 & if g_{1}(x_{2}) ≤ g_{1}(t^{2}x_{2}) \end{matrix}

Let g_{2} = g_{1} + f_{2}/4. Clearly g_{2}(x_{1}) = 1/2 while g_{2}(tx_{1}) = 0. If g_{1}(x_{2}) > g_{1}(t^{2}x_{2}), we have that g_{2}(x_{2}) = g_{1}(x_{2}) + 1/4 > g_{1}(t^{2}x_{2}) = g_{2}(t^{2}x_{2}), while if g_{1}(x_{2}) ≤ g_{1}(t^{2}x_{2}), we have g_{2}(x_{2}) = g_{1}(x_{2}) < g_{1}(t^{2}x_{2}) + 1/4 = g_{2}(t^{2}x_{2}). Thus we see that g_{2}(x_{i}) ≠ g_{2}(t^{i}x_{i}) for i = 1, 2.

Suppose that points x_{1} ∈ S_{1}, x_{2} ∈ S_{2}, …, x_{k} ∈ S_{k} and functions f_{1}, f_{2}, …, f_{k} : X [figure] [0, 1] have been chosen so that for all j < i, f_{i}(x_{j}) = f_{i}(t^{j}x_{j}) = 0 and such that the function g_{k} = ∑_{i=1}^{k} 2^{−i}f_{i} satisfies g_{k}(x_{i}) ≠ g_{k}(t^{i}x_{i}) for all i ≤ k. As above, we can choose x_{k+1} ∈ S_{k+1} so that {x_{k+1}, t^{k+1}x_{k+1}} is disjoint from {x_{1}, x_{2}, …, x_{k}, tx_{1}, t^{2}x_{2}, … t^{k}x_{k}}. Now let f_{k+1} : X [figure] [0, 1] be a continuous function that vanishes on all the elements of {x_{1}, x_{2}, …, x_{k}, tx_{1}, t^{2}x_{2}, … t^{k}x_{k}} and such that

\{ \begin{matrix}f_{k+1}(x_{k+1}) = 1 and f_{k+1}(t^{k+1}x_{k+1}) = 0 & if g_{k}(x_{k+1}) > g_{k}(t^{k+1}x_{k+1}) \\ f_{k+1}(x_{k+1}) = 0 and f_{k+1}(t^{k+1}x_{k+1}) = 1 & if g_{k}(x_{k+1}) ≤ g_{k}(t^{k+1}x_{k+1}) \end{matrix}

Let g_{k+1} = g_{k} + f_{k+1}/2^{k+1}. We have that for i < k + 1, g_{k+1}(x_{i}) = g_{k}(x_{i}) ≠ g_{k}(t^{i}x_{i}) = g_{k+1}(t^{i}x_{i}). If g_{k}(x_{k+1}) > g_{k}(t^{k+1}x_{k+1}), we have that g_{k+1}(x_{k+1}) = g_{k}(x_{k+1}) + 1/2^{k+1} > g_{k}(t^{k+1}x_{k+1}) = g_{k+1}(t^{k+1}x_{k+1}), while if g_{k}(x_{k+1}) ≤ g_{k}(t^{k+1}x_{k+1}), we have that g_{k+1}(x_{k+1}) = g_{k}(x_{k+1}) < g_{k}(t^{k+1}x_{k+1}) + 1/2^{k+1} = g_{k+1}(t^{k+1}x_{k+1}), which completes the induction step. We conclude by letting g = ∑_{i=1}^{∞} 2^{−i}f_{i} = lim g_{i}. We readily see that g = g_{k+1} on the set {x_{1}, x_{2}, …, x_{k}, tx_{1}, t^{2}x_{2}, … t^{k}x_{k}} so that g(x_{k}) ≠ g(t^{k}x_{k}), so that g ≠ gt^{k} for all k, which contradicts the assumption that every bounded function on X is t-periodic.

4.6. Definition. We say that a flow (X, t) is cocyclic if the set of t-periodic elements of C(X) is dense.

4.7. Notation. We denote by C_{n}(X) the subset of C(X) consisting of all f such that ft^{n} = f. In other words it consists of all the t-periodic elements whose minimal period divides n.

4.8. Proposition. Let (X, t) be a flow. Then C_{n}(X) is a complete t-invariant latticeordered subalgebra.

Proof. All but the completeness is completely obvious. If f_{1}, f_{2}, …, f_{k}, … is a sequence of functions that converges to f, then the sequence f_{1}t^{n}, f_{2}t^{n}, …, f_{k}t^{n} converges to ft^{n}. But that sequence is the original one.

4.9. Theorem. A flow is cocyclic if and only if it is a filtered inverse limit of periodic flows.

Proof. Assume (X, t) is cocyclic. It follows from Gelfand duality that C_{n}(X) ≅ C(X_{n}) for a quotient space X_{n} of X. The family of these quotients X_{n} is filtered, which together with compactness, implies that the induced map X [figure]lim X_{n} is surjective. To see that it is injective, let x ≠ y ∈ X. If f(x) = f(y) for all t-periodic f, the same would be true for all functions in the closure of the algebra of the t-periodic functions, which is all of C(X).

Conversely, suppose {(X_{α}, t_{α})} is a filtered diagram of periodic flows, such that X = lim X_{α} and t is the flow on X uniquely determined by {t_{α}}. By Gelfand duality, C(X) = colim C(X_{α}). Since the colimit is filtered, it is just the closure of the union of the corresponding subalgebras of C(X) so that the t-periodic functions are dense in the union, since they are dense in each C(X_{α}).

4.10. Theorem. Let (X, t) be a flow. Then the following are equivalent:

(1) (X, t) is cocyclic;

(2) The action φ̃ : Z × X [figure] X given by φ̃(n, x) = t^{n}x is continuous when Z is replaced by Z_{pf};

(3) The action φ̃: Z × X [figure] X extends to a continuous action by pf(Z).

Proof. (1) [figure] (2): Assume that (X, t) is cocyclic. Then X = lim X_{n}, as seen in the proof of Theorem 4.9. We claim that Z_{pf} × X [figure] X (see 3.1) is continuous. Since X = lim X_{n}, it is sufficient to show that the composite Z_{pf} × X [figure] X [figure] X_{n} is continuous for each n ∈ Z. Clearly the composite factors through Z/nZ × X [figure] X_{n}. That map is clearly continuous since Z/nZ is discrete and each power of t is continuous.

(2) [figure](3): Assume that φ̃ is continuous on Z_{pf}. It follows from Theorem 2.3 that Hom(Z_{pf} × X, X) ≅ Hom(Z_{pf}, hom(X, X)) so the transposed map φ : Z_{pf} [figure] hom(X, X) is continuous. But by Proposition 2.1, hom(X, X) is complete and so we can extend φ̃ to a map pf(Z) [figure] hom(X, X) which, by another application of 2.3 transposes to a uniform morphism pf(Z) × X [figure] X. The fact that φ̃(n + m, x) = φ̃(n, φ̃(m, x)) for all n, m ∈ Z, together with continuity and the fact that Z_{pf} is dense in pf(Z), implies that the extension remains a group action.

(3) [figure](1): Assume that the action of Z on X extends to an action of pf(Z) on X. We denote φ̃(λ, x) by t^{λ}x. For n > 0 in Z, let

E_{n} = {(x, y) ∈ X × X | y = t^{nℓ}x for some ℓ ∈ Z}
F_{n} = {(x, y) ∈ X × X | y = t^{nλ}x for some λ ∈ pf(Z)}

At this point, we insert:

4.11. Claim. F_{n} = cl(E_{n}) in X × X.

Proof. We will show that E_{n} is dense in F_{n} and that F_{n} is closed. For the first, suppose λ ∈ pf(Z). Let {ℓ_{α}} denote a net in Z that converges to λ. Then continuity of the action implies that for all x ∈ X, the net {t^{nℓ_{α}}x} converges to t^{nλ}x. This shows that E_{n} is dense in F_{n}. To see that F_{n} is closed, suppose that {(x_{α}, t^{nλ_{α}}x_{α})} is a net in F_{n} that converges to (x, y). Since npf(Z) is the continuous image of the compact space pf(Z) it is compact and hence the net {nλ_{α}} has a subnet that converges to an element nλ ∈ pf(Z). If we restrict to this subnet, the original net converges to (x, t^{nλ}x). But the original net converges and the space is Hausdorff, so it can only converge to (x, t^{nλ}x) and hence we conclude that y = t^{nλ}x which shows that (x, y) ∈ F_{n}.

We now return to the proof of the theorem. As before, let C_{n}(X) denote the subalgebra of C(X) consisting of all f such that f = ft^{n}. As in 4.9, there is quotient π_{n} : X [figure] X_{n} such that C(X_{n}) = C_{n}(X) ⊆ C(X). Let K_{n} [figure] X be the kernel pair of π_{n}. It is immediate that E_{n} ⊆ K_{n} from which we conclude that F_{n} ⊆ K_{n}. But then X/F_{n} maps surjectively onto X/K_{n} = X_{n}, whence C(X_{n}) is a subalgebra of C(X/F_{n}). But for every (x ∈ X), (x, t^{n}x) ∈ F_{n}, from which we see that every f ∈ C(X/F_{n}) is t-periodic of period n. Thus C(X/F_{n}) = C(X_{n}) so that X_{n} ≅ X/F_{n} and therefore F_{n} = K_{n}.

Now suppose that x and y are distinct elements of X. We want to show that there is an n ∈ Z such that π_{n}x ≠ π_{n}y. Assume that π_{n}x = π_{n}y for all n > 0. Then, since F_{n} = K_{n} for each n > 0, there is a λ ∈ pf(Z) for which y = t^{λ}x and n divides λ. Choose such a λ. For each prime p let λ_{p} be the projection of λ in pf(Z)_{p}. Let ℓ_{p} = ord_{p}λ_{p} (see 3.2 and the discussion that precedes it.) Let I[x] = {ζ ∈ pf(Z) | t^{ζ}x = x}. It is readily shown that I[x] is a closed ideal, so by Proposition 3.2, we can write I[x] = ∏ I_{p}, where I_{p} is an ideal of pf(Z)_{p} generated by a power, p^{k_{p}}. If ℓ_{p} ≥ k_{p} for each prime p, then λ ∈ I[x], which implies that y = t^{λ}x = x. Hence there is a prime p for which ℓ_{p} < k_{p} (which includes the possibility that k_{p} = ∞). Let n = p^{ℓ_{p}}. Suppose that π_{n}x = π_{n}y. Then there is a μ ∈ pf(Z) such that n|μ, whence ord_{p}μ ≥ ℓ_{p}, and also t^{μ}x = y. But then μ − λ ∈ I[x] which means that ord_{p}(μ − λ) ≥ k_{p} > ℓ_{p} which is impossible since ord_{p}μ > ord_{p}λ implies that ord_{p}(μ − λ) = ord_{p}λ = ℓ_{p}. We conclude from this that the canonical map X [figure]lim X_{n} is a monomorphism and as seen in the proof of 4.9 that it is surjective and therefore a homeomorphism.

4.12. Theorem. Let (X, t) be a flow. Then (X, t) is cocyclic if and only if the map φ : Z [figure] aut(X) is uniform on the topology of Z_{pf}.

Proof. Suppose that (X, t) is cocyclic. Then φ̃ : Z × X [figure] X extends continuously to pf(Z) × X [figure] X and thus restricts to Z_{pf} × X [figure] X.

Conversely, assume that φ : Z [figure] aut(X) is uniform on Z_{pf}. We claim that aut(X) is complete, since it is evidently closed in hom(X, X) and the latter is complete by Proposition 2.1. Thus φ : Z_{pf} [figure] aut(X) extends continuously to pf(Z) [figure] aut(X) and then, by Theorem 2.3 (which is also [Isbell, 1964, Theorem 24]) it transposes to pf(Z) × X [figure] X. The conclusion follows from the preceding theorem.

4.13. Theorem. A quotient of a cocyclic flow is cocyclic.

Proof. Let q : (X, t) [figure] (Y, s) be a surjective flow map in compact Hausdorff spaces

and assume that (X, t) is cocyclic. The continuous action pf(Z) × X [figure: φ̃] X [figure: q] Y gives, by Theorem 2.3, a map pf(Z) [figure] hom(X, Y) that takes z ∈ pf(Z) to the function x ↦ q(t^{z}x). Since the category of compact Hausdorff spaces is equational, q is a regular epimorphism. If E is the kernel pair of q, the coequalizer diagram E [figure] X [figure] Y gives an equalizer hom(Y, Y) [figure] hom(X, Y) [figure] hom(E, Y), since hom(−, Y) has a left adjoint. In particular, hom(Y, Y) is closed in hom(X, Y). We then have a commutative square

[figure: hom(Y, Y); hom(X, Y); Z; hom(Y, Y); Z; pf(Z); pf(Z); hom(X, Y)]

in which the top arrow is dense and the bottom one is a closed inclusion, so that the diagonal fill-in gives the map pf(Z) [figure] hom(Y, Y). The conclusion follows from Theorem 4.10.

4.14. Theorem. The full subcategory of cocyclic flows is a reflective subcategory of the category of flows.

Proof. Let (X, t) be a flow. Let P (X) be the full subalgebra of C(X) consisting of the t-periodic elements. The sum and product of t-periodic elements as well as all constants are t-periodic so that P (X) is a subalgebra, evidently t-invariant (that is, invariant under the actions of both t and t^{−1} on C(X)). The topological closure cl(P (X)) has by [Gillman & Jerison (1960), Theorem 5.14] the form C(Y) for some quotient Y of X. Moreover, by Gelfand duality, there is an action of t on Y that is compatible with the quotient mapping X [figure] Y. If (X, t) [figure] (Z, t) is a flow map to a cocyclic flow, then it is clear that the image of P (Z) [figure] C(X) lies in P (X) = P (Y) and hence the image of C(Z) lies in C(Y) so that the map X [figure] Z factors through Y.

4.15. Corollary. An arbitrary limit of cocyclic flows is cocyclic.

4.16. Proposition. A finite limit or finite colimit of periodic flows is periodic. A finite colimit of cocyclic flows is cocyclic.

Proof. Periodic flows are obviously closed under finite products, finite sums, t-invariant subobjects, and quotients. As for cocyclic flows, closure under finite sums is evident from Gelfand duality: if X = ∑ X_{α} is a finite sum, then C(X) = ∏ C(X_{α}) and if the periodic elements are dense in each factor, they are dense in the product. Closure under quotients is given by Theorem 4.13.

4.17. Examples: Two flows that are cocyclic, but not periodic. Let [n] denote the finite set {0, 1, …, n − 1} with the discrete topology and the action that takes i to i + 1 (mod n). The first example is the one point compactification of ∑[n]. We extend the action to fix the point at infinity. The second example is ∏[n]. It is easy to see that neither flow is periodic because the periods get too large, while each is the inverse limit of a chain of periodic flows and hence cocyclic.

5. Almost t-periodic functions

5.1. Definition. Let (X, t) be a flow. Suppose that f ∈ C(X). For each ϵ > 0, let U(f, ϵ) denote {n ∈ Z | ∥ft^{n} − f∥ < ϵ}.

Recall that when (X, t) is a flow and f ∈ C(X), we have defined φ_{f} : Z [figure] C(X) by φ_{f}(n) = ft^{n}.

The following observation is routine and left to the reader, (The weak topology is the coarsest topology on the domain that renders a function continuous.)

5.2. Proposition. The topology generated by the U(f, ϵ) for all ϵ > 0 is just the weak topology from φ_{f}.

5.3. Proposition. For any f ∈ C(X), the weak topology Z gets from φ_{f} is a group topology on Z.

Proof. For any m ∈ Z, we have ∥ft^{m+n} − ft^{m}∥ = ∥ft^{n} − f∥ since as x ranges over all of X, so does t^{m}x. Clearly U(f, ϵ/2) − U(f, ϵ/2) ⊆ U(f, ϵ).

We will call this the topology on Z induced by f.

5.4. Remark. This topology is Hausdorff if and only if f is not t-periodic.

5.5. Definition. We say that f ∈ C(X) is almost t-periodic if the topology on Z induced by f is totally bounded.

This means that for every ϵ > 0, a finite number of translates of U(f, ϵ) covers Z. If we unwind that, it turns out to mean that for all ϵ > 0, there is a finite set S ⊆ Z such that for all n ∈ Z there is an s ∈ S with ∥ft^{n} − ft^{s}∥ < ϵ. We will call such a finite set S an ϵ-span for U(f, ϵ). Although the phrase “almost periodic” has been used in many not entirely compatible ways, our definition seems to capture their spirit. See the first item in the following theorem.

5.6. Theorem. Let (X, t) be a flow and f ∈ C(X). Then the following are equivalent:

(1) The closure of {ft^{n} | n ∈ Z} is compact in the sup norm on C(X);

(2) f is almost t-periodic;

(3) The topology on Z induced by φ_{f} : Z [figure] C(X) is totally bounded and hence the uniform completion is compact;

(4) φ_{f} : Z [figure] C(X) extends continuously to the Bohr compactification b(Z) of Z.

Proof. (1) [figure](2): Let A = {ft^{n} | n ∈ Z}. For any ϵ > 0, every element of cl(A) is within ϵ of an element of A. Thus the cover of A by ϵ-spheres around each element of A also covers cl(A) and hence has a finite refinement, say the ϵ-spheres around the elements ft^{s_{1}}, ft^{s_{2}}, …, ft^{s_{m}}. Clearly S = {s_{1}, s_{2}, … s_{m}} is an ϵ-span.

(2) [figure](3): This is just the definition.

(3) [figure] (4): Assuming the uniform completion Z^{♯} is compact, the map Z [figure] C(X) extends, since C(X) is complete, to a continuous map Z^{♯} [figure] C(X). But the topology on Z is a group topology and hence Z^{♯} is a group. Since every group compactification of Z is a quotient of b(Z), we have b(Z) [figure] Z^{♯} [figure] C(X).

(4) [figure](1): Trivial since the image of b(Z) [figure] C(X), which is compact, includes the image of φ_{f} : Z [figure] C(X).

Recall from 4.1 that, given a flow (X, t), we get a homomorphism φ : Z [figure] aut(X) defined by φ(n) = t^{n}.

5.7. Definition. Let (X, t) be a flow. The t-induced topology on Z is the weak topology induced by φ. This is the coarsest topology on Z for which φ is continuous and is automatically a group topology since aut(X) is a topological group and φ is a group homomorphism.

5.8. Proposition. A topology on Z makes φ continuous (and therefore uniform) if and only if for each f ∈ C(X), and each ϵ > 0, the set U(f, ϵ) is a neighbourhood of 0.

Proof. Using the fact that every compact Hausdorff space is homeomorphic to a subspace of a power of the unit interval, indexed by its maps to the interval, we have a sequence of embeddings

aut(X) ⊆ hom(X, X) [figure] hom(X, R^{C(X)}) ≅ hom(X, R)^{C(X)}

so that φ is continuous if and only if, for each f ∈ C(X), the composite

Z [figure] aut(X) ⊆ hom(X, X) [figure] hom(X, R^{C(X)}) ≅ hom(X, R)^{C(X)} [figure: p_{f}] C(X)

is continuous, where p_{f} is the projection on the f coordinate. When the identifications are sorted out, the composite map takes n ∈ Z to ft^{n}. In particular the inverse image of an ϵ-neighbourhood in R is

{m ∈ Z | ∥ft^{n} − ft^{m}∥ < ϵ} = {m ∈ Z | ∥ft^{n−m} − f∥ < ϵ}

and it follows that φ is continuous if and only if for every f ∈ C(X) and every ϵ > 0, U(f, ϵ) is a neighbourhood of 0.

This result has two interesting consequences:

5.9. Corollary. The t-induced topology is the sup of the topologies induced by all the f ∈ C(X) (see 5.3).

5.10. Corollary. Suppose (X, t) is a flow and suppose that U(f, ϵ) contains a non-zero subgroup of Z for each f ∈ C(X) and each ϵ > 0. Then (X, t) is cocyclic.

Proof. In that case φ : Z [figure] aut(X) will be continuous in the topology generated by those subgroups. But that topology is coarser than that of the Z_{pf} and hence φ is also continuous on Z_{pf}. Now Theorems 4.10 and 2.3 give the desired conclusion.

Simultaneous almost t-periodicity. Recall that an f ∈ C(X) is almost t-periodic if for all ϵ > 0 a finite number of translates of the set U(f, ϵ) covers Z. A set F of functions is simultaneously almost t-periodic if a finite number of translates of ⋂_{f∈F} U(f, ϵ) covers Z.

5.11. Proposition. A finite set of almost t-periodic functions is simultaneously almost t-periodic.

Proof. Suppose F = {f_{1}, …, f_{k}}. The set ⋂_{i=1}^{k} U(f_{i}, ϵ) is open in the supremum of the topologies induced by the f_{i} ∈ F. Since each of those topologies is totally bounded it follows from 2.5 that the supremum is also and hence a finite number of translates covers Z.

5.12. Proposition. Let F be any set of almost t-periodic functions. Then every element of the complete t-invariant subalgebra generated by F is almost t-periodic.

Proof. If f, g ∈ F, Then U(f ± g, ϵ) ⊇ U(f, ϵ/2) ∩ U(g, ϵ/2) and the preceding argument implies that {f, g} is simultaneously almost t-periodic so that a finite number of translates of the the right-hand side covers Z. A similar argument works for f ∧ g and f ∨ g. For the product, it is sufficient to show that fg is almost t-periodic when ∥f∥ = ∥g∥ = 1. But in that case we also get that U(fg, ϵ) ⊇ U(f, ϵ/2) ∩ U(g, ϵ/2). Since U(f, ϵ) = U(ft, ϵ) we conclude that every element of the smallest t-invariant subalgebra generated by F is almost t-periodic. Finally, suppose that f is in the closure of that subalgebra. Given ϵ > 0, there is a g in the subalgebra such that ∥f − g∥ ≤ ϵ/3. But then a standard argument shows that U(f, ϵ) ⊇ U(g, ϵ/3).

6. Almost cocyclic flows

We say that a flow is almost cocyclic if every f ∈ C(X) is almost t-periodic. We might have defined this to mean that the almost t-periodic functions were dense, in parallel with the definition of cocyclic, but 5.12 shows that the conditions are equivalent.

6.1. Example of an almost cocyclic flow that is not cocyclic. Let X = R/Z, the circle group. Define t : X [figure] X as addition mod Z of an irrational number γ. It is well known that the orbit of any point is dense. Fix an x ∈ X. For any n > 0, the orbit of x under t^{n} is dense. If ft^{n} = f, then for any y ∈ X, the set {t^{kn}x} comes arbitrarily close to y so that for some k ∈ Z, we have f(x) = ft^{kn}(x), which is arbitrarily close to f(y). Thus f(x) = f(y) and we conclude that the only t-periodic functions in C(X) are the constants. Hence the flow is not cocyclic.

Now let f ∈ C(X) and let ϵ > 0 be given. Since X is compact, there is a δ > 0 such that |x_{1} − x_{2}| < δ implies |f(x_{1}) − f(x_{2})| < ϵ. In the following, all arithmetic is carried out mod Z. Since the integer multiples of γ are dense, the intervals {(nγ − δ, nγ + δ)} cover X and hence a finite set of them, say {(sγ − δ, sγ + δ) | s ∈ S}, S finite, also covers X. Now given n ∈ Z, let s ∈ S so that |nγ − sγ| < δ, whence for all x ∈ X,

|t^{s}x − t^{n}x| = |(sγ + x) − (nγ + x)| = |sγ − nγ| < δ

so that |ft^{s}x − ft^{n}x| < ϵ. Since this is independent of x, it follows that ∥ft^{s} − ft^{n}∥ < ϵ so that S is an ϵ-span.

6.2. Theorem. Let (X, t) be a flow. Then the following are equivalent:

(1) (X, t) is almost cocyclic;

(2) The t-induced topology is totally bounded;

(3) The action of Z extends continuously to b(Z);

Proof. That (1) holds if and only if (2) does is obvious. That (2) holds if and only if (3) does follows from Proposition 5.8.

6.3. Proposition. A quotient of an almost cocyclic flow is almost cocyclic.

Proof. Suppose (X, t) [figure] (Y, s) is a quotient mapping between flows. Then C(Y) is a subalgebra of C(X). If the elements of C(X) are almost t-periodic, the same is true of C(Y).

6.4. Proposition. Let (X, t) be almost cocyclic. For each x ∈ X, the map θ_{x} : Z [figure] X, defined by θ_{x}(n) = t^{n}(x) is uniform when Z is topologized by all U(f, ϵ), f ∈ C(X) and ϵ > 0.

Proof. The map Z [figure] aut(X) is uniform by 5.8 and the map aut(X) [figure] X that evaluates at x is induced by x : 1 [figure] X and hence is uniform by [Isbell, 1964, III.2] and therefore so is the composite.

Recall that b(Z) is the Bohr compactification of Z. Define t : b(Z) [figure] b(Z) as the extension to b(Z) of the map from Z [figure] b(Z) defined by t(ζ) = ζ + 1, which is obviously uniform.

6.5. Proposition. The flow (b(Z), t) is almost cocyclic but is not cocyclic.

Proof. For any f ∈ C(b(Z)), the restriction of f to Z is continuous in the topology on Z induced by its inclusion in b(Z). Hence, for any ϵ > 0, U(f, ϵ) is open in that topology. But Z is totally bounded in that topology, which comes from the inclusion into a compact group. Hence finitely many translates of U(f, ϵ) cover Z, which implies that f is almost t-periodic. On the other hand, the orbit of 0 under any power of t is all of Z, which is dense in b(Z). The same argument used in 6.1 shows that only constant functions are t-periodic and thus the flow is not cocyclic.

6.6. Definition. Let Z-Cmp denote the category of compact Hausdorff spaces equipped with an action of Z. This name is chosen by analogy with G-set. We similarly let Z-C denote the category of complete lattice-ordered rings equipped with an action by Z.

In each of these categories, an object C with a Z-action is determined by an automorphism t : C [figure] C, with t being the value of the action at the integer 1. It must be an automorphism since the value of the action at −1 must be t^{−1}. Thus a Z-action is the same thing as an automorphic flow.

6.7. Proposition. The Gelfand duality between Cmp and C extends to a duality between Z-Cmp and Z-C.

Proof. An automorphism t : X [figure] X in Cmp gives a morphism C(t) : C(X) [figure] C(X), which must be an isomorphism since C is a functor and so C(t^{−1}) = C(t)^{−1}. Similarly, if R is a complete lattice-ordered ring and t : R [figure] R is an automorphism, then Max(t) : Max(R) [figure] Max(R) is an automorphism of the maximal ideal spaces.

Now suppose that (X, t) is a flow and that R is the ring of almost t-periodic functions with respect to t. Define an equivalence relation E on X by (x, y) ∈ E if f(x) = f(y) for all f ∈ R. Then it follows from Gelfand duality that R = C(X/E). From the preceding discussion it also follows that X/E has a flow we call t/E and that (X, t) [figure] (X/E, t/E) is a flow morphism. It is trivial that (X/E, t/E) is an almost cocyclic flow.

6.8. Theorem. The category of almost cocyclic flows is a reflective subcategory of the category of all flows, given by (X, t) [figure] (X/E, t/E).

Proof. We have done everything except to exhibit the adjunction. Suppose we have a flow map p : (X, t) [figure] (Y, s) in which (Y, s) is almost cocyclic. The category C of C^{∗}-algebras has as morphisms the “norm-reducing” (which is to say, non-increasing) ring homomorphisms. It follows that if f ∈ C(Y), U(f, ϵ) ⊆ U(fp, ϵ) and if a finite number of translates of U(f, ϵ) covers Z, then the same must be true of U(fp, ϵ). It follows that the image of C(Y) in C(X) actually lies in C(X/E). That is, we have a commutative triangle

[figure: C(Y); C(X); C(Y); C(X/E); C(X/E); C(X)]

which, upon dualizing, gives

[figure: X/E; Y; X; X/E; X; Y]

as required.

6.9. Example: A flow that is periodic at each element, but is not cocyclic or even almost cocyclic. Let X be the subset of R^{2} consisting of circles centered at the origin of radius 1 − 1/n for n = 2, 3, 4, …, ∞. Define t : X [figure] X in polar coordinates by t(r, θ) = (r, θ + π/n) when r = 1 − 1/n. This function is continuous on the circles of radius < 1 as well as on the outer circle where it is the identity. But as you get nearer the outer circle the function gets closer and closer to the identity and hence the function is continuous on the union. We use Theorem 6.2 to show the action is not almost cocyclic. Suppose the action α : Z × X [figure] X that takes (n, x) to t^{n}x extended to b(Z) × X [figure] X. Let ζ ∈ b(Z) be any accumulation point of the sequence 2, 3, 4, . . .. Then (ζ, (1, 0)) is an accumulation point of (2, (1/2, 0)), (3, (2/3, 0)), (4, (3/4, 0)), . . .. Apply α to the sequence to get (1/2, π), (2/3, π), (3/4, π), …, which approaches (1, π), while clearly t(ζ, (1, 0)) = lim t(n, (1, 0)) = (1, 0), which shows that the extended action cannot be continuous.

References

L. Gillman and M. Jerison (1960), Rings of Continuous Functions. D. Van Nostrand, Princeton.

John R. Isbell (1964), Uniform Spaces. Amer. Math. Soc. Colloquium Publications 13.

John F. Kennison (2006), Spectra of finitely generated Boolean flows. Theory Appl. Categories 16, 434–459.

Department of Mathematics and Statistics McGill University, Montreal, QC, H3A 2K6 Department of Mathematics and Computer Science Clark University, Worcester, MA 01610

Department of Mathematics and Statistics Concordia University, Montreal, QC, H4B 1R6

Email: barr@math.mcgill.ca
      jkennison@clarku.edu
      raphael@alcor.concordia.ca

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Clemens Berger, Université de Nice-Sophia Antipolis, cberger@math.unice.fr Richard Blute, Université d’ Ottawa: rblute@uottawa.ca Lawrence Breen, Université de Paris 13: breen@math.univ-paris13.fr Ronald Brown, University of North Wales: ronnie.profbrown(at)btinternet.com

Valeria de Paiva: valeria.depaiva@gmail.com

Ezra Getzler, Northwestern University: getzler(at)northwestern(dot)edu Martin Hyland, University of Cambridge: M.Hyland@dpmms.cam.ac.uk P. T. Johnstone, University of Cambridge: ptj@dpmms.cam.ac.uk Anders Kock, University of Aarhus: kock@imf.au.dk Stephen Lack, Macquarie University: steve.lack@mq.edu.au F. William Lawvere, State University of New York at Buffalo: wlawvere@buffalo.edu Tom Leinster, University of Glasgow, Tom.Leinster@glasgow.ac.uk Jean-Louis Loday, Université de Strasbourg: loday@math.u-strasbg.fr Ieke Moerdijk, University of Utrecht: moerdijk@math.uu.nl Susan Niefield, Union College: niefiels@union.edu Robert Paré, Dalhousie University: pare@mathstat.dal.ca Jiri Rosicky, Masaryk University: rosicky@math.muni.cz James Stasheff, University of North Carolina: jds@math.upenn.edu Ross Street, Macquarie University: street@math.mq.edu.au Walter Tholen, York University: tholen@mathstat.yorku.ca Myles Tierney, Rutgers University: tierney@math.rutgers.edu Robert F. C. Walters, University of Insubria: robert.walters@uninsubria.it R. J. Wood, Dalhousie University: rjwood@mathstat.dal.ca