New York Journal of Mathematics

New York J. Math. 15 (2009) 301–318.

Distal actions and ergodic actions on compact groups

C. Robinson Edward Raja

Abstract. Let K be a compact metrizable group and Γ be a group of automorphisms of K. We first show that each α ∈ Γ is distal on K implies Γ itself is distal on K, a local to global correspondence provided Γ is a generalized F C-group or K is a connected finite-dimensional group. We also prove a connection between distality and ergodicity which is used to show that ergodic actions of nilpotent groups on compact connected finite-dimensional abelian groups contains ergodic automorphisms.

Contents

1. Introduction 301 2. Distal and ergodic 305 3. Compact abelian groups 306 4. Distal actions 308 5. Finite-dimensional compact groups 309 6. Example 316 References 317

1. Introduction

We shall be considering actions on compact groups. By a compact group we shall mean a compact metrizable group and by an automorphism we shall mean a continuous automorphism. For a compact group K, Aut (K) denotes the group of all automorphisms of K. An action of a topological group Γ on a compact metrizable group K by automorphisms is a homomorphism

Received March 2, 2009. Mathematics Subject Classification. 22B05, 22C05, 37A15, 37B05. Key words and phrases. Compact groups, nilpotent groups, automorphisms, distal,

ergodic.

Partially supported by NSERC postdoctoral fellowship.

301^{ISSN 1076-9803/09}

ϕ: Γ → Aut (K) such that the map (α, x) ↦ ϕ(α)(x) is a continuous map: when only one action is studied or when there is no confusion instead of ϕ(α)(x) we may write α(x) for α ∈ Γ and x ∈ K. In such cases, the map ϕ is said to define the action of Γ on K and such actions are called algebraic actions.

We shall assume that a topological group Γ acts on a compact metrizable group K. For each α ∈ Γ, (n, a) ↦ α^{n}(a) defines a ℤ-action on K and this action on K is called the ℤ_{α}-action. Suppose K_{1} ⊃ K_{2} are closed Γ-invariant subgroups of K such that K_{2} is normal in K_{1}. By an action of Γ on K_{1}/K_{2}, we mean the canonical action of Γ on K_{1}/K_{2} defined by α(xK_{2}) = α(x)K_{2} for all x ∈ K_{1} and all α ∈ Γ.

Suppose Γ acts on the compact groups K and L. We say that K and L are Γ-isomorphic if there exists a continuous isomorphism Φ: K → L such that Φ(α(x)) = α(Φ(x)) for all α ∈ Γ and x ∈ K.

It is interesting to find properties of group actions that hold if the property holds for every ℤ_{α}-action. We term any such property a local to global correspondence as this property holds for the whole group Γ when it holds locally at every point of Γ. We first state the following well-known classical local to global correspondence for linear actions on vector spaces, a proof of which may be found in [6].

Burnside Theorem. Let V be a finite-dimensional vector space over the reals and let G be a finitely generated subgroup of GL(V), the group of linear transformations on V. If each element of G has finite order, then G itself is a finite group.

The main aim of the note is to exhibit such local to global correspondences for algebraic actions on compact groups.

Definition 1.1. We say that the action of Γ on K is distal if for any x ∈ K \ (e), e is not in the closure of the orbit Γ(x) = {α(x) | α ∈ Γ}. In such case, we say that Γ is distal (on K).

We now introduce a type of action which is obviously distal.

Definition 1.2. We say that the action of Γ on K is compact (respectively, finite) if the group ϕ(Γ) is contained in a compact (respectively, finite) subgroup of Aut (K) where ϕ is the map defining the action of Γ on K.

We now see the notion of ergodic action which is hereditarily antithetical to distal action (cf. Proposition 2.1).

Definition 1.3. Let K be a compact group and ω_{K} be the normalized Haar measure on K. We say that an (algebraic) action of Γ on K is ergodic if any Γ-invariant Borel set A of K satisfies ω_{K}(A) = 0 or ω_{K}(A) = 1.

Definition 1.4. Let K be a compact group and α be a continuous automorphism of K. If the action of ℤ_{α} on K is distal (respectively, ergodic), then we say that α is a distal (respectively, ergodic) automorphism of K or α is distal (respectively, ergodic) on K.

We now introduce a class of groups whose action is one of the main studies in this article.

Definition 1.5. A locally compact group G is called a generalized F Cgroup if G has a series G = G_{0} ⊃ G_{1} ⊃ ⋯ ⊃ G_{n} = {e} of closed normal subgroups such that G_{i}/G_{i+1} is a compactly generated group with relatively compact conjugacy classes for i = 0, 1, …, n − 1.

It follows from Theorem 2 of [14] that compactly generated locally compact groups of polynomial growth are generalized F C-groups and any polycyclic group is a generalized F C-group.

It can easily be seen that the class of generalized F C-groups is stable under forming continuous homomorphic images and closed subgroups. If H is a compact normal subgroup of a locally compact group G such that G/H is a generalized F C-group, then it is easy to see that G is also a generalized F C-group.

It is evident that Γ is distal implies each α ∈ Γ is distal. For actions on connected Lie groups [1] and for certain actions on p-adic Lie groups [17] the local to global correspondence for distality, that is passing from each α ∈ Γ being distal on K to the whole group Γ being distal on K, is valid: the distal notion has a canonical extension to actions on locally compact spaces (cf. [7]). In general each α ∈ Γ is distal need not imply Γ is distal (cf. Example 1, [19]). We will now closely examine a general form of Example 1 of [19].

Example 1.6. Let M be a compact group and Γ be a countably infinite group. Take K = M^{Γ}. The (left)-shift action of Γ on K is defined as follows: for α ∈ Γ and f ∈ M^{Γ}, αf is defined to be αf(β) = f(α^{−1}β) for all β ∈ Γ. For x ≠ e ∈ M, consider f_{x} ∈ M^{Γ} defined by f_{x}(α) = e if α ≠ 1 and f_{x}(α) = x if α = 1 where 1 is the identity in Γ. Choose a sequence (α_{n}) is Γ such that α_{n} ≠ α_{m} whenever n ≠ m. Then α_{n}(f_{x}) → e in M^{Γ} which may be seen as follows: for α ∈ Γ, α_{n}^{−1}α ≠ 1 for large n, so (α_{n}f_{x})(α) = f_{x}(α_{n}^{−1}α) = e for large n, hence α_{n}(f_{x}) → e in M^{Γ}. Thus, the shift action of Γ is not distal. Suppose Γ is a torsion group (for instance, Γ may be the group of all finite permutations). Then each α ∈ Γ is distal but we have seen that Γ is not distal.

If Γ is assumed to be finitely generated nilpotent or finitely generated solvable, then situation as in Example 1.6 does not arise as Γ is torsion implies Γ is finite. Recently Theorem 2.9 of [12] showed that if K is a zerodimensional compact group and Γ is a generalized F C-group, then each α ∈ Γ is distal and the whole group Γ is distal are equivalent to the action being equicontinuous (that is, having invariant neighborhoods). Motivated by this, here we prove that each α ∈ Γ is distal on a compact group K if and only if Γ is distal on K provided Γ is a generalized F C-group: the fact that generalized F C-groups have a normal series of compactly generated subgroups plays a crucial in the proof of our results.

It can easily be observed that the compact group K in Example 1.6 can not be a connected finite-dimensional group and so we in fact prove that if K is a compact connected finite-dimensional group, then (with no restriction on Γ) each α ∈ Γ is distal on K if and only if Γ is distal on K.

The study of ergodic actions on compact groups is a key tool in proving the afore-stated results. We first establish a connection between distal actions and nonergodic actions. This makes us to ponder if there is any local to global correspondence for nonergodic actions and leads us to the question of determining K and Γ so that action of Γ on K is ergodic if and only if Γ contains an ergodic automorphism: this is a local to global correspondence for nonergodicity. The following example is useful in determining conditions on K and Γ to obtain a local to global correspondence for nonergodicity.

Example 1.7. Let Γ be a countable infinite group and M be a compact abelian group. Let K = M^{Γ}. We consider the shift action of Γ on K defined as in Example 1.6. We now claim that the shift action of Γ on K is ergodic. Let M̂ be the (dual) group of characters on M. Then the dual K̂ of K consists of functions f: Γ → M̂ such that f(b) is the trivial character for all but finitely many b ∈ Γ (see Theorem 17 of [15]). Then the dual action of Γ on the dual K̂ is given by af(b) = f(a^{−1}b) for all f ∈ K̂ and all a, b ∈ Γ. Let f ∈ K̂. Then define F = {b ∈ Γ | f(b) ≠ 1} where 1 is the trivial character on M, the identity in M̂. If af = f, then a^{−1}F ⊂ F and hence aF = F. Since Γ is infinite, if the orbit Γ(f) is finite, then for infinitely many a ∈ Γ, af = f and aF = F, hence F is empty or infinite. Thus, the orbit Γ(f) is infinite for any nontrivial f ∈ K̂. This implies that the action of Γ on K is ergodic. Suppose Γ is a torsion group (one may take F_{n} = ∏_{k=1}^{n} ℤ/kℤ and Γ = ∪F_{n}). We get that the action of Γ on K is ergodic. Since any a ∈ Γ has finite order, the action of ℤ_{a} is never ergodic for any a ∈ Γ. Using the counterexamples to the Burnside problem we get finitely generated infinite (nonsolvable) torsion groups and so such groups act ergodically but no element of which is ergodic.

If K is a compact connected finite-dimensional (abelian) group, then situation as in Example 1.7 does not arise. In this aspect Berend [2] proved that an ergodic action of commuting epimorphisms on a compact connected finite-dimensional abelian group contains ergodic epimorphisms. Recently [3] proved that certain hereditarily ergodic actions of solvable groups on compact connected finite-dimensional abelian groups contain ergodic automorphisms. In this article we apply our study of distal actions and ergodic actions to prove that an ergodic action of a nilpotent group on a compact connected finite-dimensional abelian group admits ergodic automorphisms and we provide examples to show that this type of result is limited to nilpotent actions (cf. Example 5.16).

Having explained our results, it is easy to see that only ϕ(Γ) matters and not all of Γ. So, we may assume that Γ is a group of automorphisms of K.

2. Distal and ergodic

We now explore the connection between distal actions and ergodic actions on compact (metrizable) groups using the dual structure of compact groups.

Let K be a compact group and Γ be a group acting on K. Let K̂ be the equivalence classes of continuous irreducible unitary representations of K. If π is a continuous irreducible unitary representation of K, then [π] ∈ K̂ denotes the set of all continuous irreducible unitary representations of K that are unitarily equivalent to π. We write π_{1} ∼ π_{2} if π_{1}, π_{2} ∈ [π] for some [π] ∈ K̂. For a continuous irreducible unitary representation π of K and α ∈ Γ, α(π) is defined by

α(π)(x) = π(α^{−1}(x))

for all x ∈ K and it can easily be verified that α(π) is also a continuous irreducible unitary representation of K. If α ∈ Γ and π_{1}, π_{2} ∈ [π], then α(π_{1}) ∼ α(π_{2}). Thus, the map (α, [π]) ↦ α[π] = [α(π)] is well-defined and is known as the dual of action of Γ on the dual K̂ of K. For k ≥ 1, let U_{k}(ℂ) be the group of unitaries on ℂ^{k} and I_{k} denote the identity matrix in U_{k}(ℂ). Then U_{k}(ℂ) is a compact group and for each [π] ∈ K̂, there exists a k ≥ 1 such that π(x) ∈ U_{k}(ℂ) for all x ∈ K: see [8] for details on representations of compact groups.

Proposition 2.1. Let K be a compact group and Γ be a group of automorphisms of K. Then the following are equivalent:

(1) Γ is distal on K.

(2) For each Γ-invariant nontrivial closed subgroup L of K, action of Γ on L is not ergodic.

(3) For each Γ-invariant nontrivial closed subgroup L of K, there exists a nontrivial continuous irreducible unitary representation π of L such that the orbit Γ[π] = {α[π] | α ∈ Γ} is finite in L̂.

Proof. Let L be a nontrivial Γ-invariant closed subgroup of K. If the action of Γ on L is ergodic, then by Theorem 2.1 of [2], Γ(x) = {α(x) | α ∈ Γ} is dense in L for some x ∈ L. Since L is nontrivial, x ≠ e and hence e is in the closure of Γ(x) for x ≠ e. Thus, we get that (1) ⇒ (2) and that (2) ⇒ (3) follows from Theorem 2.1 of [2].

Now assume that (3) holds. Let x ≠ e be in K and L be the closed subgroup generated by Γ(x). Then L is a nontrivial Γ-invariant closed subgroup of K. Then by assumption there exists a nontrivial [π_{1}] ∈ L̂ such that Γ([π_{1}]) is finite. Let Γ_{0} = {α ∈ Γ | α(π_{1}) ∼ π_{1}}. Then Γ_{0} is a closed subgroup of Γ of finite index. Let Γ_{1} = ∩_{α∈Γ}αΓ_{0}α^{−1}. Then Γ_{1} is a normal subgroup of Γ of finite index and Γ_{1} is contained in Γ_{0}. Let A = {[π] ∈ L̂ | Γ_{1}[π] = [π]}. Then A contains π_{1}. Since Γ_{1} is normal in Γ, A is Γ-invariant. Let L_{1} = ∩_{[π]∈A}{g ∈ L | π(g) = π(e)}. Then L_{1} is a Γ-invariant closed normal subgroup of L and L_{1} is a proper subgroup of L as A is nontrivial. If e is in the closure of Γ(x), then since Γ/Γ_{1} is finite, e is in the closure of Γ_{1}(x). Let α_{n} ∈ Γ_{1} be such that α_{n}(x) → e and [π] ∈ A. Then there exist u_{n} ∈ U_{k}(ℂ) (k may depend on π) such that

u_{n}^{−1}π(g)u_{n} = π(α_{n}(g))

for all g ∈ L. This implies that

u_{n}^{−1}π(x)u_{n} = π(α_{n}(x)) → π(e) = I_{k}

as n → ∞. Since U_{k}(ℂ) is compact, π(x) = I_{k}. This implies that x ∈ L_{1} which is a contradiction as L_{1} is a proper Γ-invariant subgroup of L and L is the closed subgroup generated by Γ(x). Thus, e is not in the closure of Γ(x). Hence (3) ⇒ (1). □

3. Compact abelian groups

We now consider compact abelian groups and prove preliminary results for actions on compact abelian groups using Pontryagin duality of locally compact abelian groups: cf. [15] and [20] for results on duality of locally compact abelian groups and for any unexplained notations.

If G is a group and A_{1}, A_{2}, …, A_{n} are subsets of G, then ⟨A_{1}, …, A_{n}⟩ is defined to be the subgroup generated by the union of the sets A_{1}, A_{2}, …, A_{n} and if any A_{i} = {g}, we may write g instead of {g}.

Lemma 3.1. Let K be a compact abelian group and Γ be a group of automorphisms of K. Let α be an automorphism of K such that αΓα^{−1} = Γ. Suppose the action of Γ is not ergodic on K and for each α-invariant proper closed subgroup L of K, the action of ℤ_{α} on K/L is not ergodic. Then the group generated by Γ and α is not ergodic on K or equivalently there exists a nontrivial character χ on K such that the orbit {β(χ) | β ∈ ⟨Γ, α⟩} is finite.

Proof. We first note that the assumption on α is equivalent to saying that for any α-invariant nontrivial subgroup A of K̂ there exists a nontrivial character χ ∈ A such that the orbit {α^{n}(χ) | n ∈ ℤ} is finite.

Let A = {χ ∈ K̂ | Γ(χ) is finite}. Since Γ is not ergodic on K, A is nontrivial. Since αΓα^{−1} = Γ, A is α-invariant. By assumption on α, there exists a nontrivial χ_{0} in A such that α^{k}(χ_{0}) = χ_{0} for some k ≥ 1. Then

Γα^{n}(χ_{0}) ⊂ ∪_{i=1}^{k}Γα^{i}(χ_{0})

for all n ∈ ℤ. Since χ_{0} ∈ A and A is α-invariant, we get that each Γα^{i}(χ_{0}) is finite for 1 ≤ i ≤ k and hence {β(χ_{0}) | β ∈ ⟨Γ, α⟩} is finite. □

Lemma 3.2. Let K be a (nontrivial) compact abelian group and Γ be a group of automorphisms of K. Suppose Γ is a generalized F C-group and for each α ∈ Γ and each α-invariant proper closed subgroup L of K, the action of ℤ_{α} on K/L is not ergodic. Then the action of Γ on K is not ergodic or equivalently there exists a nontrivial character χ on K such that the corresponding Γ-orbit {α(χ) | α ∈ Γ} is finite.

Proof. Since K is compact abelian, Aut (K) is totally disconnected and hence by Proposition 2.8 of [12], Γ contains a compact open normal subgroup Δ such that Γ/Δ contains a polycyclic subgroup of finite index. Let Λ be a closed normal subgroup of Γ of finite index containing Δ such that Λ/Δ is polycylic. Let Λ_{0} = Λ and Λ_{i} = [Λ_{i−1}, Λ_{i−1}] for i ≥ 1. Then there exists a k ≥ 0 such that Λ_{k}Δ ≠ Δ and Λ_{k+1}Δ = Δ. It can be easily seen that each Λ_{i}Δ is finitely generated modulo Δ. For 0 ≤ i ≤ k, let α_{i,1}, …, α_{i,m} be in Λ_{i} such that α_{i,1}, …, α_{i,m} and ΔΛ_{i+1} generate ΔΛ_{i}. It can be easily seen that α_{i,j} normalizes ⟨α_{i,1}, …, α_{i,j−1}, Λ_{i+1}, Δ⟩ for all i and j with α_{i,0} to be trivial. Then repeated application of Lemma 3.1 yields a nontrivial character χ ∈ K̂ such that the orbit Λ(χ) is finite. Since Λ is a normal subgroup of finite index in Γ, Γ(χ) is also finite. □

We next prove a lemma which shows that the (global) distal condition in Proposition 2.1, can be relaxed to the local distal condition provided Γ is a generalized F C-group.

Lemma 3.3. Let K be a (nontrivial) compact abelian group and Γ be a group of automorphisms of K. Suppose Γ is a generalized F C-group and each α ∈ Γ is distal on K. Then Γ is not ergodic on K or equivalently there exists a nontrivial character χ on K such that the corresponding Γ-orbit {α(χ) | α ∈ Γ} is finite.

Proof. Let α ∈ Γ and L be a α-invariant proper closed subgroup of K. Since α is distal on K, the action ℤ_{α} on K/L is also distal (Corollary 6.10 of [4]). This shows by Proposition 2.1 that the action of ℤ_{α} is not ergodic on K/L. Thus, the result follows from Lemma 3.2. □

We now prove the local to global correspondence for distal actions of generalized F C-groups on compact abelian groups.

Theorem 3.4. Let K be a compact abelian group and Γ be a group of automorphisms of K. Suppose Γ is a generalized F C-group. Then each α ∈ Γ is distal on K if and only if Γ is distal on K.

Proof. Suppose each α ∈ Γ is distal on K. Let L be a nontrivial Γ-invariant closed subgroup of K. Then each α ∈ Γ is distal on L also. It follows from Lemma 3.3 that Γ is not ergodic on L. Since L is arbitrary Γ-invariant nontrivial closed subgroup, by Proposition 2.1 we get that Γ is distal on K. □

4. Distal actions

We now consider distal actions on compact groups and prove that the distal condition has local to global correspondence for actions on compact groups provided the group of automorphisms is a generalized F C-group.

Theorem 4.1. Let K be a compact group and Γ be a group of automorphisms of K. Suppose Γ is a generalized F C-group. Then the following are equivalent:

(1) Each α ∈ Γ is distal on K.

(2) The action of Γ on K is distal.

Proof. Suppose each α ∈ Γ is distal on K. Let x ∈ K be such that e is in the closure of the orbit Γ(x). We now claim that x = e.

Suppose K is connected. Let T be a maximal compact connected abelian subgroup of K containing x (see Theorem 9.32 of [11]). Then Aut(K) = Inn(K)Ω where Ω = {α ∈ Aut(K) | α(T) = T } and Inn(K) is the group of inner-automorphisms of K (cf. Corollary 9.87 of [11]). Let Γ′ = ΓInn(K) and Ω′ = (Γ′ ∩ Ω). Then Γ′ and Ω′ are also generalized F C-groups. Since Inn(K) is a compact normal subgroup, each α ∈ Γ′ is distal on K. Since Aut(K) = Inn(K)Ω, Γ′ = Inn(K)Ω′. Since e is in the closure of Γ(x), e is in the closure of Γ′(x). Since Inn (K) is compact, e is in the closure of Ω′(x). As x ∈ T, applying Theorem 3.4, we get that x = e.

We now consider any compact group K. Let K_{0} be the connected component of e in K. Then K_{0} is Γ-invariant and by Corollary 6.10 of [4], each α ∈ Γ is distal on K/K_{0}. Since K/K_{0} is totally disconnected, by Proposition 2.8 and Lemma 2.3 of [12], K/K_{0} has arbitrarily small compact open subgroups invariant under Γ. This shows that x ∈ K_{0}. Now x = e follows from the connected case. □

Example 1.7 showed that an ergodic action of a general, even a commutative group Γ on a compact abelian group need not imply the existence of a nontrivial subgroup or a nontrivial quotient that admits an ergodic ℤ_{α}-action for some α ∈ Γ but we now prove that this can not happen if Γ is assumed to be a generalized F C-group. Thich may be viewed as an initial result on the existence of ergodic automorphisms for ergodic actions on general compact groups.

Proposition 4.2. Let K be a (nontrivial) compact group and Γ be a group of automorphisms of K. Suppose Γ is a generalized F C-group and the action of Γ on K is ergodic. Then we have the following:

(1) There exist a β ∈ Γ and a β-invariant nontrivial closed subgroup L of K such that the action of ℤ_{β} on L is ergodic.

(2) In addition if K is abelian, there exist an α ∈ Γ and an α-invariant proper closed subgroup L of K such that the action of ℤ_{α} on K/L is ergodic.

Proof. Suppose for each α ∈ Γ and each α-invariant nontrivial closed subgroup L of K, the action of ℤ_{α} on L is not ergodic. Then by Proposition 2.1, each α ∈ Γ is distal on K. By Theorem 4.1, the action of Γ on K is distal and hence by Proposition 2.1, the action of Γ on K is not ergodic unless K is trivial. Thus, (1) is proved.

We now assume that K is abelian. Suppose for every α ∈ Γ and for every proper closed α-invariant subgroup L of K, the action of ℤ_{α} on K/L is not ergodic. By Lemma 3.2, the action of Γ on K is not ergodic. Thus, (2) is proved. □

5. Finite-dimensional compact groups

We now consider finite-dimensional compact groups. Let ℚ_{d}^{r} be the additive group ℚ^{r} with discrete topology (r > 0). We may regard ℚ_{d}^{r} as a finite-dimensional vector space over ℚ. Let B_{r} denote the dual of ℚ_{d}^{r}. Then B_{r} is a compact connected group of finite-dimension and any compact connected finite-dimensional abelian group is a quotient of B_{r} for some r: see [15].

We first show that distal condition for algebraic actions on B_{r} has local to global correspondence with no restriction on the acting group Γ. The dual of any automorphism of B_{r} is a ℚ-linear transformation of ℚ_{d}^{r} onto ℚ_{d}^{r}. It can be easily seen that any group of unipotent transformations of ℚ_{d}^{r} is distal on B_{r} and the following shows that up to finite extensions these are the only distal actions on B_{r} which may be proved along the lines of Proposition 2.3 of [13] (with some minor modifications).

Proposition 5.1. Let Γ be a group of automorphisms of B_{r}. Suppose Γ is distal on B_{r}. Then B_{r} has a series of closed connected Γ-invariant subgroups

B_{r} = K_{0} ⊃ K_{1} ⊃ K_{2} ⊃ ⋯ ⊃ K_{n−1} ⊃ K_{n} = (e)

such that the action of Γ on K_{i}/K_{i+1} is finite for any i ≥ 0. In particular, Γ is a finite extension of a group of unipotent transformations of ℚ_{d}^{r}.

Theorem 5.2. Let Γ be group of automorphisms of B_{r}. Suppose each α ∈ Γ is distal on B_{r}. Then Γ is distal on B_{r}. In addition if the dual action of Γ on ℚ_{d}^{r} is irreducible, then Γ is finite.

Proof. By considering the dual action of Γ, we may view Γ as a group of linear maps on ℚ_{d}^{r}. Then by Proposition 5.1, eigenvalues of elements of Γ are of absolute value one. Let V = ℚ_{d}^{r} ⊗ ℝ. Then by [5], there exist Γ-invariant ℝ-subspaces (0) = V_{1} ⊂ V_{2} ⊂ ⋯ ⊂ V_{m} = V such that the action of Γ on V_{i+1}/V_{i} is isometric. Thus, there exists a Γ-invariant ℝ-subspace W of V such that W ≠ V and the action of Γ on V/W is isometric.

Assume that the dual action of Γ on ℚ_{d}^{r} is irreducible. Since V = ⟨ℚ_{d}^{r}⟩ and W ≠ V, W ∩ ℚ_{d}^{r} ≠ ℚ_{d}^{r} and is Γ-invariant. Since action of Γ on ℚ_{d}^{r} is irreducible, W ∩ ℚ_{d}^{r} = (0).

Let α ∈ Γ. Then by Proposition 5.1, α^{k} is unipotent for some k ≥ 1. This implies that α^{k} is unipotent and isometric on V/W and hence α^{k}(v) ∈ v+W for all v ∈ V. This implies that for v ∈ ℚ_{d}^{r}, α^{k}(v) − v ∈ W ∩ ℚ_{d}^{r} = (0) and hence α^{k} is identity. Thus, every element of Γ has finite order. It follows from Lemma 4.3 of [2] that Γ is finite. □

We now proceed to show that ergodic action of Γ on a finite-dimensional compact connected abelian group yields an ergodic automorphism in Γ provided Γ is nilpotent.

Lemma 5.3. Let Γ be a group of automorphisms of a compact abelian group K and α be an automorphism of K. Suppose Γ and α are distal on K and αΓα^{−1} = Γ. Then the group generated by Γ and α is distal on K.

Proof. Let Δ be the group generated by Γ and α. Let L be a nontrivial closed subgroup of K invariant under Δ. Let A = {χ ∈ L̂ | Γ(χ) is finite}. Since Γ is normalized by α, A is α-invariant. Since α and Γ are distal on K, it follows from Proposition 2.1 that there exists a nontrivial χ_{0} ∈ A such that α^{k}(χ_{0}) = χ_{0} for some k ≥ 1. Now, Γα^{i}(χ_{0}) ⊂ ∪_{j=1}^{k}Γ(α^{j}(χ_{0})) for any i ∈ ℤ. This implies that the orbit Δ(χ_{0}) is finite. This shows by Proposition 2.1 that Δ is distal on K. □

Lemma 5.4. Let α be an ergodic automorphism of B_{r} and L be a closed connected α-invariant subgroup of B_{r}. Then α is ergodic on L.

Proof. It can easily be seen that α is ergodic on B_{r} if and only if no power of α on ℚ_{d}^{r} has a nonzero fixed point. Let V be the ℚ-subspace of ℚ_{d}^{r} such that the dual of L is ℚ_{d}^{r}/V. Since α is ergodic, no power of α on ℚ_{d}^{r} has a nonzero fixed point and hence no power of α on ℚ_{d}^{r}/V has a nonzero fixed point. Thus, α is ergodic on L. □

Lemma 5.5. Let α and β be automorphisms of B_{r}. Suppose α is contained in a group Γ of automorphisms of B_{r} such that Γ is distal and β is ergodic and normalizes Γ. Then α^{i}β^{j} and β^{j}α^{i} are ergodic for all i and j in ℤ with j ≠ 0.

Proof. It is enough to show that αβ and βα are ergodic. We first consider the case when Γ is finite. Assume Γ is finite. Let χ be a character such that the orbit {(αβ)^{n}(χ) | n ∈ ℤ} is finite. Since Γ is finite and Γ is normalized by αβ, the orbit Γ̃(χ) is also finite where Γ̃ is the group generated by αβ and Γ. Since β ∈ Γ̃ and β is ergodic, we get that χ is trivial. Thus, αβ is ergodic.

We now consider the general case. Let V = {χ ∈ ℚ_{d}^{r} | Γ(χ) is finite}. Since Γ is distal, V is a nontrivial ℚ-subspace and V is invariant under β as Γ is normalized by β. Let L be the closed connected subgroup of B_{r} such that the dual B_{r}/L is V. Then L is a proper closed connected subgroup invariant under Γ and β and Γ is finite on B_{r}/L. Then αβ is ergodic on B_{r}/L. Since the dual of L is ℚ_{d}^{r}/V, L ≃ B_{s} for some s < r. By Lemma 5.4, β is ergodic on L and hence by induction on dimension of B_{r}, αβ is ergodic on L. Thus, αβ is ergodic on B_{r}. Similarly we may show that βα is also ergodic on B_{r}. □

Lemma 5.6. Let Δ be a group of automorphisms of B_{r}. Suppose there exists a Δ-invariant closed connected subgroup K of B_{r} such that Δ is ergodic on K and Δ is distal on B_{r}/K. Then K is N(Δ)-invariant where N(Δ) is the normalizer of Δ in Aut(B_{r}).

Proof. Let V_{1} be the ℚ-subspace of ℚ_{d}^{r} defined by

V_{1} = {χ ∈ ℚ_{d}^{r} | Δ(χ) is finite}

and define V_{i} inductively by

V_{i} = {χ ∈ ℚ_{d}^{r} | Δ(χ) + V_{i−1} is finite in ℚ_{d}^{r}/V_{i−1}}

for any i > 1. Then each V_{i} is a Δ-invariant ℚ-subspace. Since ℚ_{d}^{r} has finite-dimension over ℚ, there exists a n such that V_{n} = V_{n+i} for all i ≥ 0 and for any nontrivial χ ∈ ℚ_{d}^{r}/V_{n}, the orbit Δ(χ) + V_{n} is infinite. Let S be a closed subgroup of B_{r} such that the dual of S is ℚ_{d}^{r}/V_{n}. Then S is Δ-invariant and connected. The choice of V_{n} shows that Δ is ergodic on S and Δ is distal on B_{r}/S. This implies that Δ is distal on KS/S ≃ K/K ∩ S and also on KS/K ≃ S/K ∩ S but Δ is ergodic on K and also on S. Thus, S = K ∩ S = K.

If βΔ = Δβ, then for any χ ∈ V_{1}, Δ(β(χ)) = β(Δ(χ)) is finite. Thus, V_{1} is β-invariant. Since each V_{i}/V_{i−1} is the space of all characters whose Δ-orbit is finite in ℚ_{d}^{r}/V_{i−1}, we get that V_{i} is β-invariant for any i ≥ 1. Thus, K is β-invariant. □

Lemma 5.7. Let Δ be a nilpotent group of automorphisms of B_{r} generated by α and a subgroup Γ such that αΓα^{−1} = Γ. Suppose Γ is distal on B_{r}. Then there exists a closed connected Δ-invariant subgroup K of B_{r} such that α is ergodic on K and α is distal on B_{r}/K.

Proof. Let V_{1} be the ℚ-subspace of ℚ_{d}^{r} defined by

V_{1} = {χ ∈ ℚ_{d}^{r} | (α^{n}(χ)) is finite}

and define V_{i} inductively by

V_{i} = {χ ∈ ℚ_{d}^{r} | (α^{n}(χ) + V_{i−1}) is finite in ℚ_{d}^{r}/V_{i−1}}

for any i > 1. Then each V_{i} is a α-invariant ℚ-subspace. Since ℚ_{d}^{r} has finite-dimension over ℚ, there exists a n such that V_{n} = V_{n+i} for all i ≥ 0 and for any nontrivial χ ∈ ℚ_{d}^{r}/V_{n}, the orbit (α^{n}(χ) + V_{n}) is infinite. Let K be a closed subgroup of B_{r} such that the dual of K is ℚ_{d}^{r}/V_{n}. Then K is α-invariant and connected. The choice of V_{n} shows that α is ergodic on K and α is distal on B_{r}/K.

Let Δ_{0} = Δ and Δ_{i} = [Δ, Δ_{i−1}] for i > 0. Since Δ is nilpotent, there exists a k ≥ 0 such that Δ_{k} is nontrivial and Δ_{k+1} is trivial. Since Δ/Γ is abelian, Δ_{i} ⊂ Γ for i ≥ 1, hence each Δ_{i} is distal on B_{r} (i ≥ 1). Now α commutes with elements of Δ_{k} and hence V_{1} as well as V_{i} is Δ_{k}-invariant. Thus, K is Δ_{k}-invariant. Since α normalizes Δ_{k}, it follows from Lemma 5.3 that ⟨α, Δ_{k}⟩ is distal on B_{r}/K and ergodic on K. Now for β ∈ Δ_{k−1}, α^{−1}β^{−1}αβ ∈ Δ_{k} and hence β^{−1}αβ ∈ ⟨α, Δ_{k}⟩. By Lemma 5.6, K is Δ_{k−1}- invariant. Repeating this argument we can get that K is Δ-invariant. □

Lemma 5.8. Let α be an automorphism of B_{r}. Then there exists a compact connected subgroup K of B_{r} isomorphic to B_{s} for some s > 0 such that α is ergodic on K and α is distal on B_{r}/K. Moreover, if Γ is a nilpotent group of automorphisms of B_{r} with Γ = Γ_{0} and Γ_{k} = [Γ, Γ_{k−1}] for k ≥ 1 and if α ∈ Γ_{i} \ Γ_{i+1} and Γ_{i+1} is distal on B_{r}, then K is Γ-invariant.

Proof. Suppose Γ is a nilpotent group containing α with α ∈ Γ_{i} \ Γ_{i+1} and Γ_{i+1} is distal on B_{r}. Let Δ be the group generated by Γ_{i+1} and α. By Lemma 5.7, B_{r} contains a closed connected Δ-invariant subgroup K such that α is ergodic on K and α is distal on B_{r}/K. Since Γ_{i+1} is distal and α normalizes Γ_{i+1}, Δ is distal on B_{r}/K (cf. Lemma 5.3). Since α ∈ Δ, Δ is ergodic on K. Since Δ is a normal subgroup, it follows from Lemma 5.6 that K is Γ-invariant. □

Lemma 5.9. Let Γ be a nilpotent group of automorphisms of B_{r} and α, β ∈ Γ. Let Γ_{0} = Γ and Γ_{i} = [Γ, Γ_{i−1}] for i ≥ 1. Let k ≥ 1 be such that α ∈ Γ_{k−1} \ Γ_{k}. Suppose α is ergodic on B_{r} and Γ_{k} is distal on B_{r}. Then there exists a i ≥ 0 such that α^{i}β is ergodic on B_{r}.

Proof. We prove the result by induction on the dimension of B_{r}. If r = 1, then we have nothing to prove. So, we may assume that r > 1. If αβ is ergodic on B_{r}, then we are done. Hence we may assume that αβ is not ergodic on B_{r}. Let Δ be the group generated by αβ and Γ_{k}. Then Δ is a nilpotent group and [Δ, Δ] ⊂ Γ_{k}. So, we may assume by Lemma 5.7 that there exists a closed connected Δ-invariant subgroup K of B_{r} such that αβ is ergodic on K and αβ is distal on B_{r}/K. Since αβ is not ergodic on B_{r}, K ≠ B_{r}. Then by Lemma 5.3, Δ is distal on B_{r}/K. Since α and β commute modulo Γ_{k}, we get that α normalizes Δ. By Lemma 5.6, K is α-invariant and hence by Lemma 5.5, α^{i}β is ergodic on B_{r}/K for all i ≥ 2. Since α is ergodic on K, induction hypothesis applied to K in place of B_{r} and α^{2}β in place of β, we get that α^{j}β is ergodic on K for some j ≥ 2. Thus, α^{j}β is ergodic on B_{r} for some j ≥ 2. □

Lemma 5.10. Let Γ be a nilpotent group of automorphisms of B_{r} and α, β ∈ Γ. Let Γ_{0} = Γ and Γ_{i} = [Γ, Γ_{i−1}] for i ≥ 1 and k ≥ 1 be such that α ∈ Γ_{k−1} \ Γ_{k}. Let K be a closed Γ-invariant subgroup of B_{r} isomorphic to B_{s} for some s ≥ 0 such that α is ergodic on K and α is distal on B_{r}/K. If β is ergodic on B_{r}/K and Γ_{k} is distal on B_{r}, then there exists j ≥ 0 such that α^{j}β is ergodic on B_{r}.

Proof. Since α normalizes Γ_{k}, by Lemma 5.3, the group generated by α and Γ_{k} is distal on B_{r}/K. Since β centralizes α modulo Γ_{k}, it follows from Lemma 5.5 that α^{i}β is ergodic on B_{r}/K for all i ≥ 0. By Lemma 5.9, α^{j}β is ergodic on K for some j ≥ 0. This shows that for some j ≥ 0, α^{j}β is ergodic on B_{r}. □

Lemma 5.11. Let Γ be a nilpotent group of automorphisms of B_{r}. Let Γ_{0} = Γ and Γ_{i} = [Γ, Γ_{i−1}] for i ≥ 1. Suppose that the action of Γ on B_{r} is ergodic. Then there exist a series

(e) = K_{0} ⊂ K_{1} ⊂ K_{2} ⊂ ⋯ ⊂ K_{m−1} ⊂ K_{m} = B_{r}

of closed connected Γ-invariant subgroups with each K_{i} ≃ B_{r_{i}} for some r_{i} ≥ 0 and automorphisms α_{1}, α_{2}, …, α_{m} in Γ with the following properties for each i = 1, 2, …, m:

(1) If k_{i} is the smallest integer k for which α_{i} ∉ Γ_{k}, then the action of Γ_{k_{i}} on B_{r}/K_{i−1} is distal.

(2) The action of ℤ_{α_{i}} on K_{i}/K_{i−1} is ergodic.

(3) The action of ℤ_{α_{i}} on B_{r}/K_{i} is distal.

Proof. For each α ∈ Γ, if the action of ℤ_{α} is distal on B_{r}, then by Theorem 5.2, the action of Γ is distal. This is a contradiction to the ergodicity of Γ by Proposition 2.1. Thus, the action of ℤ_{α} is not distal for some α ∈ Γ.

Since Γ is nilpotent, there exists a k such that Γ_{k} ≠ (e) and Γ_{k+1} = (e). Now, choose α_{1} ∈ Γ_{k_{1}−1} \ Γ_{k_{1}} such that the action of ℤ_{α_{1}} is not distal on B_{r} but the action of Γ_{k_{1}} is distal on B_{r}. By Lemma 5.8, there exists a nontrivial Γ-invariant closed connected subgroup K_{1} of B_{r} isomorphic to B_{r_{1}} for some r_{1} > 0 such that α_{1} is ergodic on K_{1} and α_{1} is distal on B_{r}/K_{1}.

Let L = B_{r}/K_{1}. Then the action of Γ on L is ergodic and L ≃ B_{s_{1}} for s_{1} < r as K_{1} is nontrivial. By applying induction on the dimension of B_{r}, we get Γ-invariant closed connected subgroups (e) = K_{0} ⊂ K_{1} ⊂ K_{2} ⊂ ⋯ ⊂ K_{m−1} ⊂ K_{m} = B_{r} and automorphisms α_{2}, …, α_{m} satisfying (1)–(3) for 2 ≤ i ≤ n. □

Theorem 5.12. Let Γ be a nilpotent group of automorphisms of B_{r}. If Γ is ergodic on B_{r}, then Γ contains ergodic automorphisms of B_{r}.

Proof. We now prove the result by induction on r. If r = 1, we are done. By Lemma 5.11, there are Γ-invariant closed connected subgroups (e) = K_{0} ⊂ K_{1} ⊂ K_{2} ⊂ ⋯ ⊂ K_{m−1} ⊂ K_{m} = B_{r} with each K_{i} ≃ B_{r_{i}} for some r_{i} ≥ 0 and automorphisms α_{1}, α_{2}, …, α_{m} in Γ satisfying (1)–(3) of Lemma 5.11. We may assume that K_{i} ≠ K_{i−1} for 1 ≤ i ≤ m. Induction hypothesis applied to the action of Γ on B_{r}/K_{1} ≃ B_{r−r_{1}} yields β ∈ Γ such that β is ergodic on B_{r}/K_{1}. By Lemma 5.10, there exists α ∈ Γ such that α is ergodic on B_{r}. □

We now consider compact connected finite-dimensional abelian groups. Let K be a compact connected finite-dimensional abelian group. Then

ℤ^{r} ⊂ K̂ ⊂ ℚ_{d}^{r}

and K is a quotient of B_{r} for some r ≥ 1. Let α be an automorphism of K. Then α is an automorphism of K̂. Since ℤ^{r} ⊂ K̂, α has a canonical extension to an invertible ℚ-linear map on ℚ_{d}^{r}, say α̃. Thus, any automorphism α of K can be lifted to a unique automorphism α̃ of B_{r}. Let Γ be a group of automorphisms of K and Γ̃ be the group consisting of lifts α̃ of automorphisms α ∈ Γ. We consider Γ and Γ̃ as topological groups with their respective compact-open topologies as automorphism groups of K and B_{r}. By looking at the dual action, we can see that the topological groups Γ and Γ̃ are isomorphic. If ϕ: B_{r} → K is the canonical quotient map, then for α ∈ Γ and x ∈ B_{r}, we have

ϕ(α̃(x)) = α(ϕ(x))

where α̃ is the lift of α on B_{r}.

Proposition 5.13. Let K be a compact connected finite-dimensional abelian group. Let Γ be a group of automorphisms of K and Γ̃ be the corresponding group of automorphisms of B_{r}. Then Γ is distal (respectively, ergodic) on K if and only if Γ̃ is distal (respectively, ergodic) on B_{r}.

Proof. For χ ∈ ℚ_{d}^{r}, there exists n ≥ 1 such that nχ ∈ K̂ and since K̂ ⊂ ℚ_{d}^{r}, Γ is ergodic on K if and only if Γ̃ is ergodic on B_{r}. Since K is a quotient of B_{r}, Γ̃ is distal on B_{r} implies Γ is distal on K (see [4], Corollary 6.10).

Suppose Γ is distal on K. By Proposition 2.1, there exists a nontrivial character χ_{1} in K̂ ⊂ ℚ_{d}^{r} such that Γ(χ_{1}) is finite. Let

V_{1} = {χ ∈ ℚ_{d}^{r} | Γ̃(χ) is finite}.

Then V_{1} is a nontrivial Γ̃-invariant ℚ-subspace as χ_{1} ∈ V_{1}. Let M be a closed subgroup of B_{r} such that the dual of M is ℚ_{d}^{r}/V_{1}. Then M is Γ̃- invariant and M ≃ B_{s} for s < r. Let A = V_{1} ∩ K̂ and L be a closed subgroup of K such that the dual of L is K̂/A. Then L is Γ-invariant. Since K̂/A ⊂ ℚ_{d}^{r}/V_{1}, K̂/A has no element of finite order and hence L is connected (see Theorem 30 of [15]). It can be verified that the dimension of L is same as the dimension of M. Hence by induction on the dimension of K we get that Γ̃ is distal on M. Since the action of Γ̃ on B_{r}/M is finite, Γ̃ is distal on B_{r}. □

Theorem 5.14. Let K be a compact connected finite-dimensional abelian group and Γ be a nilpotent group of automorphisms of K. Suppose Γ is ergodic on K. Then there exists an α ∈ Γ such that α is ergodic on K.

Proof. Now let K be a compact connected finite-dimensional abelian group and r be the dimension of K. Then K is a quotient of B_{r}. Let Γ̃ be the group of lifts of automorphisms of Γ. Then by Proposition 5.13, Γ̃ is ergodic on B_{r}. It follows from Theorem 5.12 that there exists α ∈ Γ such that the lift α̃ of α is ergodic on B_{r}. Another application of Proposition 5.13 shows that α itself is ergodic on K. □

We now show that the distal condition for algebraic actions on connected finite-dimensional compact groups has a local to global correspondence with no restriction on the acting group Γ.

Theorem 5.15. Let Γ be a group of automorphisms of a compact connected finite-dimensional group K. Suppose each α ∈ Γ is distal on K. Then the action of Γ on K is distal.

Proof. If K is abelian, then the result follows from Proposition 5.13 and Theorem 5.2. Suppose K is any finite-dimensional compact connected group. Let x ∈ K and (α_{n}) be a sequence in Γ. Suppose α_{n}(x) → e.

Let T be a maximal compact connected abelian subgroup of K containing x: cf. Theorem 9.32 of [11] for existence of such T. Since K is a connected group, Aut(K) = Inn(K)Ω where Ω = {α ∈ Aut(K) | α(T) = T } and Inn(K) is the group of inner automorphisms of K (see Corollary 9.87 of [11]). Let Λ = Inn(K)Γ. Since Inn(K) is a compact normal subgroup, each α ∈ Λ is distal on K. Let α_{n} = a_{n}β_{n} where a_{n} ∈ Inn(K) and β_{n} ∈ Ω ∩ Λ for all n ≥ 1. Since Inn(K) is compact, by passing to a subsequence, if necessary, we may assume that β_{n}(x) → e. Since T is closed in K which is of finite-dimension, T is also of finite-dimension ([16]). It follows from the abelian case that Ω ∩ Λ is distal on T and hence x = e as x ∈ T and β_{n} ∈ Ω ∩ Λ. Thus, the action of Γ is distal on K. □

We now provide an example to show that the nilpotency assumption on the acting group Γ in Theorem 5.14 can not be relaxed: it may be noted that Theorem 5.14 is true with no restriction on the acting group Γ if the compact group K is the two-dimensional torus.

Example 5.16. Let Γ be a subgroup of GL(n, ℚ). Let Γ^{+} be the semidirect product of Γ and ℚ_{d}^{n} with the canonical action of Γ on ℚ_{d}^{n}. We define an action of Γ^{+} on ℚ_{d}^{n+1} by

(α, w)(q_{1}, …, q_{n}, q_{n+1}) = α(q_{1}, …, q_{n}) + wq_{n+1} + (0, …, 0, q_{n+1})

for all (α, w) ∈ Γ^{+} and (q_{1}, …, q_{n}, q_{n+1}) ∈ ℚ_{d}^{n+1}: ℚ_{d}^{n} is identified as a subset of ℚ_{d}^{n+1} via the canonical map (q_{1}, …, q_{n}) ↦ (q_{1}, …, q_{n}, 0). It may be useful to note that (α, w) has the following matrix form

(\begin{matrix}α & w^{T} \\ 0 & 1 \end{matrix} )

where w^{T} is the transpose of w. Considering the dual action, we get that Γ^{+} ⊂ Aut(B_{n+1}). For z ∈ ℚ_{d}^{n}, Γ^{+}(z) = Γ(z) and for z ∈ ℚ_{d}^{n+1} \ ℚ_{d}^{n}, Γ^{+}(z) can easily be seen to be infinite. Thus, Γ is ergodic on B_{n} if and only if Γ^{+} is ergodic on B_{n+1}. For any Γ ⊂ GL(n, ℚ), no (α, w) ∈ Γ^{+} is ergodic on B_{n+1}. For n ≥ 1, take Γ to be the group generated by α ∈ GL(n, ℚ) that is ergodic on B_{n}. Then Γ^{+} is a solvable group and is ergodic on B_{n+1} but no automorphism in Γ^{+} is ergodic on B_{n+1}.

6. Example

We now provide an example to show that the existence of a finite sequence as in Proposition 5.1 need not be true for connected infinite-dimensional compact abelian groups. We first state a general form of Proposition 5.1 which can be proved as in Proposition 2.3 of [13].

Proposition 6.1. Let K be a compact abelian group and Γ be a group of automorphisms of K. Suppose Γ is distal. Then there exists a collection (K_{i}) of Γ-invariant closed subgroups of K such that:

(1) K_{0} = K.

(2) For i ≥ 0 either K_{i+1} = (e) or K_{i+1} is a proper subgroup of K_{i}.

(3) The action of Γ on the dual of K_{i}/K_{i+1} has only finite orbits for any i ≥ 0.

In contrast to the finite-dimensional case we now show by an example that the sequence (K_{i}) in Proposition 6.1 need not be finite. Let T_{k} be the k-dimensional torus, a product of k copies of the circle group. Let α_{k} be the automorphism of T_{k} defined by

α_{k}(x_{1}, x_{2}, …, x_{k}) = (x_{1}x_{2} … x_{k}, x_{2}x_{3} … x_{k}, …, x_{k−1}x_{k}, x_{k})

for all (x_{1}, x_{2}, …, x_{k}) ∈ T_{k}. For 0 ≤ j ≤ k, let

M_{k,j} = {(x_{1}, x_{2}, …, x_{k−j}, e, …, e) ∈ T_{k}}.

Then each M_{k,j} is α_{k}-invariant and α_{k} is trivial on M_{k,j}/M_{k,j+1} for j ≥ 0. We first prove the following fact about T_{k} and α_{k}.

Lemma 6.2. Let T_{k} and α_{k} be as above. Suppose there exists a series

T_{k} = M_{0} ⊃ M_{1} ⊃ ⋯ ⊃ M_{n−1} ⊃ M_{n} = (e)

of α_{k}-invariant closed subgroups such that for i ≥ 0, the action of ℤ_{α_{k}} on M_{i}/M_{i+1} is finite and M_{i}/M_{i+1} is not finite. Then n = k.

Proof. Let V be the Lie algebra of T_{k}. We first show that M_{n−1} is onedimensional. For 0 ≤ i < n, let V_{i} be the Lie subalgebra of V corresponding to the Lie subgroup M_{i}. Now, there exists a m such that α_{k}^{m} is trivial on M_{n−1}. Suppose (u_{1}, u_{2}, …, u_{k}) ∈ V_{n−1}. Then α_{k}^{m}(u_{1}, u_{2}, …, u_{k}) = (u_{1}, u_{2}, …, u_{k}). This implies that for 1 ≤ i ≤ k − 1, u_{i} = u_{i} + ∑_{j>i}^{k} m_{i,j}u_{j} where m_{i,j} ∈ ℕ. For i = k − 1, u_{k−1} = u_{k−1} + m_{k−1,k}u_{k} and hence u_{k} = 0. If u_{p} = 0 for all p > q > 1, then for i = q − 1,

u_{q−1} = u_{q−1} + ∑_{j≥q} m_{q−1,j}u_{j} = u_{q−1} + m_{q−1,q}u_{q}

and hence u_{q} = 0. Thus, V_{n−1} is at most one-dimensional. Since M_{n−1}/M_{n} = M_{n−1} is not finite, M_{n−1} has dimension one and

V_{n−1} = {(u_{1}, 0, …, 0) | u_{1} ∈ ℝ}.

It can be seen that T_{k}/M_{n−1} ≃ T_{k−1} and the action of ℤ_{α_{k}} on T_{k}/M_{n−1} is same as the action of ℤ_{α_{k−1}} on T_{k−1}. Moreover, M_{i+1}/M_{n−1} ⊂ M_{i}/M_{n−1} and \frac{M_{i}/M_{n−1}}{M_{i+1}/M_{n−1}} ≃ M_{i}/M_{i+1} for 0 ≤ i < n − 1 with M_{0}/M_{n−1} = T_{k}/M_{n−1} and M_{n−1}/M_{n−1} = (e). By induction on k, we get that n − 1 = k − 1. □

Let K = ∏_{k∈ℕ} T_{k}. Let α: K → K be the automorphism defined by α(f)(k) = α_{k}(f(k)) for all f ∈ K and all k ∈ ℕ. Then α is a continuous automorphism and the ℤ-action defined by α is distal on K.

If there is a finite sequence

(e) = K_{n} ⊂ K_{n−1} ⊂ ⋯ ⊂ K_{1} ⊂ K_{0} = K

of α-invariant closed subgroups such that the action of ℤ_{α} on K_{i}/K_{i+1} is finite for i ≥ 0. This implies that each T_{k} has a finite series

(e) = K_{n,k} ⊂ K_{n−1,k} ⊂ ⋯ ⊂ K_{1,k} ⊂ K_{0,k} = T_{k}

of α_{k}-invariant closed subgroups such that the action of ℤ_{α_{k}} on K_{i,k}/K_{i+1,k} is finite for i ≥ 0.

It follows from Lemma 6.2 that k ≤ n. Since k ≥ 1 is arbitrary, this is a contradiction. Thus, the sequence (K_{i}) of closed subgroups as in Proposition 6.1 for K and α is not finite.

Addendum. Recently [18] proved Theorem 5.14 for ℤ^{d}-actions on compact groups K provided the Z(ℤ^{d}), centralizer of ℤ^{d} in Aut (K) has DCC, that is any decreasing sequence of Z(ℤ^{d})-invariant closed subgroups is finite.

Acknowledgement. A part of the work was done when I was visiting W. Jaworski. I would like to thank Prof. W. Jaworski for offering a postdoctoral fellowship of NSERC and many helpful suggestions and discussions. I would also like to thank Department of Mathematics and Statistics, Carleton University for their hospitality during my stay.

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Stat-Math Unit, Indian Statistical Institute, 8th Mile Mysore Road, Bangalore 560 059. India creraja@isibang.ac.in

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