SPECTRAL RIGIDITY OF GROUP ACTIONS: APPLICATIONS TO THE CASE gr⟨t, s; ts = st^{2}⟩

OLEG N. AGEEV

Abstract. We apply a technique to study the notion of spectral rigidity of group actions to a group gr⟨t, s; ts = st^{2}⟩. As an application, we prove that there exist rank one weakly mixing transformations conjugate its square, thereby giving a positive answer to a well-known question.

Introduction

In [1] we introduced a notion of spectral rigidity of group actions. This notion is essential only for noncommutative groups (see Remark 8 in [1]). We say that a property holds for a typical element from a topological space D (or a typical element of a topological space D has some property) if the set of elements from D with this property contains a dense G_{δ} subset of D. Following [1], we say that an element h of a group H has spectral rigidity if for a typical H-action T the set of essential values of the spectral multiplicity function of T_{h}, named M(T_{h}), is constant. If every element of H has this property, we say that the group H has spectral rigidity. It is easy to see that the notion of spectral rigidity can be considered as one of invariants (in general, not complete) with respect to group, metrical, or unitary isomorphisms.

It is well known that a typical transformation has rank one, and is weakly mixing. However, it is not conjugate to its square (see [9]). More information about these notations can be found in [5]. The main goal of this paper is to show how the technique to study the spectral rigidity works in the case of actions of a group G = gr⟨t, s; ts = st^{2}⟩. Let us recall that even in this simple case we have no useful spectral theorem, because G is wild (see [10]).

1. Preliminaries and the basic result

Let T be a transformation defined on a non-atomic standard Borel probability space (X, F, μ). The set of all transformations (automorphisms of the σ- algebra of measurable sets) of (X, μ) is a Polish (complete metrizable separable) topological group, noted Aut(μ), with respect to the weak (or coarse) topology (see [7]) defined by

T_{n} → T ⇔ μ(T_{n}^{−1}AΔT^{−1}A) → 0 for each measurable A

(we identify transformations if they are coincide up to a set of measure zero).

Date: November 24. 04.

1991 Mathematics Subject Classification. Primary 37A, 28D05, 28D15; Secondary 47A05, 47A35, 47D03.

Key words and phrases. Group actions, ergodic theory, conjugations to its squares.

The author was supported in part by Max Planck Institute of Mathematics, Bonn and a grant from the President of Russia for Support of Leading Scientific Schools #00-15-96107.

The spectral properties of T are those of the induced unitary operator on L_{2}(μ) defined by

T̂ : L_{2}(μ) → L_{2}(μ); T̂f(x) = f(T x).

This is Koopman’s representation of dynamical systems, therefore if T S = ST^{2}, then ŜT̂ = T̂^{2}Ŝ. Let us recall that a transformation T has rank n if n is the smallest number such that, for any k, there exist integers h_{k,i} and towers (columns)

A_{k,i}, T A_{k,i}, …, T^{h_{k,i}−1}A_{k,i}, i = 1, …, n

such that all levels T^{j}A_{k,i}, 0 ≤ j < h_{k,i}, and the remaining set form a measurable partition of X, say ξ_{k}, and ξ_{k} → ε, i.e. for any measurable set A there are ξ_{k}- measurable sets A_{k} such that μ(AΔA_{k}) → 0 as k → ∞. A transformation has infinite rank if there is no such number n.

In this paper we study metrical properties of some elements of typical G-actions, in particular, we answer a well-known question (see [4],[5],[6]) by proving the following theorem:

Theorem 1.1. There exists a weakly mixing rank one transformation conjugate to its square.

For the proof of Theorem 1.1, we apply a group action approach. Given a discrete group G, a G-action T_{g} is any homomorphism T : G → Aut(μ). The G-action T is called free if μ{x ∈ X : (∃g ≠ 1) T_{g}x = x} = 0. The set of all G-actions is equipped with the weak topology defined by

T (n) → T ⇔ (∀g ∈ G) T_{g}(n) → T_{g}.

When G is countable, the weak topology makes the action space, noted Ω_{G}, a Polish space.

Theorem 1.1 is a natural corollary of the following main theorem:

Theorem 1.2. For a typical G-action T, T_{t} is a weakly mixing rank one transformation.

2. Natural properties and basic notation for G

Denote by Q_{2} the subgroup of G generated by all conjugations of t, i.e. g^{−1}tg, g ∈ G. Following [11], a subset of G, say F, tiles G if there is a set of centers C ⊆ G such that {F c : c ∈ C} is a partition of G.

Proposition 2.1. (1) Q_{2} is an abelian group consisting of elements s^{−m}t^{n}s^{m},

m, n ∈ Z.

(2) Q_{2} is isomorphic to a subgroup of R via an isomorphism ϕ defined by ϕ(s^{−m}t^{n}s^{m}) = n2^{m}, m, n ∈ Z.

(3) G′ = Q_{2}.

(4) G = ⊔_{i∈Z}s^{i}G′. Therefore G/G′ ≅ Z.

(5) There exists a Følner sequence of sets, say F_{k}, such that every F_{k} tiles G.

Proof. The proof is almost obvious. Indeed, part 1 is a consequence of the following equality:

(2.1)  s^{m}t^{p}s^{−m}s^{q}t^{r}s^{−q} = s^{q}t^{r+p2^{q−m}}s^{−q} = s^{q}t^{r}s^{−q}s^{m}t^{p}s^{−m},

where q ≥ m ≥ 0.

SPECTRAL RIGIDITY OF GROUP ACTIONS:APPLICATIONS TO THE CASE gr⟨t, s; ts = st^{2}⟩ 3

It is easy to see that the map ϕ is well defined. Using (2.1) once more, we prove part 2. The statement

(∀g ∈ Q_{2}) (∃g_{1}, g_{2} ∈ Q_{2}) [gs^{±1} = s^{±1}g_{1} & gt^{±1} = g_{2}]

is trivial. This implies that G = ⊔_{i∈Z}s^{i}Q_{2}, because s^{i} ∈ Q_{2} iff i = 0.

It is easy to check that [g_{1}, g_{2}] ∈ Q_{2} for g_{j} = s^{i_{j}}q_{j}, q_{j} ∈ Q_{2}. Thus G′ ⊆ Q_{2}. To prove both parts 3 and 4, it remains to mention that Q_{2} ⊆ G′, because

t = [s^{−1}t, s] ∈ G′.

Part 5, were we replaced G by Q_{2}, holds for some sequence of subsets , say F ′_{k}, of Q_{2}. Therefore ∪_{0≤i<k}s^{i}F ′_{k} is the sequence we need in part 5. Proposition 2.1 is proved.

□

3. Approximations

3.1. The density of conjugations to any fixed free action in Ω_{G}. For any countable group G, the set of free actions is dense in Ω_{G} (see [3]). Thus we only need to approximate a free action of G, say T ′, by conjugations of a fixed free action T of G. Given a finite set F ⊆ G, and a measurable set A ⊆ X with μ(A) > 0, F × A is said to form a (Rokhlin) tower if the sets T_{g}A, g ∈ F are mutually disjoint. We say that such a tower is an (ε)-tower if μ(⋃_{g∈F} T_{g}A) > ε. It said that the Rokhlin lemma is valid for F, and a free action T if for all ε > 0 there is a measurable set A such that F × A forms a (1 − ε)-tower of T. Consider the same sets F_{k} as in Proposition 2.1. The Rokhlin lemma is valid for every F_{k} and every free G-action, because F_{k} tiles G, and is finite (see [11], and [12] for the proofs of theorems announced in [11]). Therefore we can choose a measurable set A (A′) such that F_{k} × A (F_{k} × A′) forms a (1 − 1/k)-tower for T (T ′, respectively). Obviously, A and A′ can be chosen such that μ(A) = μ(A′). Therefore there exists a metrical isomorphism S_{k} : A′ → A, i.e. a measure-preserving map (onto) which is one-to-one up to zero measure subsets of A′ and A. It is clear that S_{k} can be extended to a metrical isomorphism of (X, μ) and (X, μ) (hence it is an automorphism) such that

S_{k}T ′_{g}x = T_{g}S_{k}x for any g ∈ F_{k}, x ∈ A′.

It is easy to see that for any g ∈ G

S_{k}^{−1}T_{g}S_{k}x = T ′_{g}x if x ∈ T ′_{g_{1}}A′ for some g_{1} ∈ F_{k} ∩ g^{−1}F_{k}.

Therefore,

(∀g ∈ G) [μ\{x ∈ X : S_{k}^{−1}T_{g}S_{k}x ≠ T ′_{g}x\} < \frac{1}{k} + \frac{|F_{k}\g^{−1}F_{k}|}{|F_{k}|} → 0 as k → ∞].

This implies that

S_{k}^{−1}T S_{k} → T ′ as k → ∞.

3.2. An example of a free action of G such that T_{t} is weakly mixing. Let W be a measurable flow on X (i.e. some continuous homomorphism W : R → Aut(X, μ)) such that the transformation W^{1} is weakly mixing. Define a transformation T_{0} : Ω → Ω by

T_{0} = ×_{i=−∞}^{∞}W^{2^{i}},

where Ω = ∏_{i=−∞}^{∞} X. Also consider the right shift map, say S, acting on Ω by

(Sx)_{i} = x_{i−1}, i ∈ Z,

where y_{i} is the ith coordinate of y ∈ Ω. Clearly, maps T_{0}, S are weakly mixing transformations of the space (Ω, ν), where ν = ×_{i=−∞}^{∞}μ, and the map

t → T_{0}, s → S

can be naturally extended to a homomorphism from G to a group generated by T_{0} and S, so a pair (T_{0}, S) defines an action of G, say T. Using Proposition 2.1, to prove the freeness of T, we only need to show that

|k| + |l| ≠ 0 ⇒ ν{x ∈ Ω : T_{0}^{k}x = S^{l}x} = 0.

It is trivial in the case l = 0, because W is the aperiodic transformation. To have the same in the case l ≠ 0, it is remain to mention that for any transformation R of a non-atomic standard Borel probability space (X, F, μ) we have μ × μ{(x, Rx) : x ∈ X} = 0. This implies that T is a free action.

3.3. An example of a free action of G such that T_{t} has rank 1. Let Λ be a countable discrete subgroup of T such that the restriction of the map ϕ : Λ → Λ, ϕ(λ) = λ^{2}, to this subgroup is a group automorphism, and Λ has no roots of unity. By the theorem of Pontryagin, the group of characters of Λ, noted Λ̂, is an abelian compact group. It is clear that ϕ^{∗} is an automorphism of Λ̂, where ϕ^{∗} is defined by (ϕ^{∗}χ)(λ) = χ(ϕ(λ)) for any character χ(λ). Consider a rotation S : Λ̂ → Λ̂, S(χ) = χ_{0}χ, where χ_{0}(λ) = λ is a character of Λ. Maps ϕ^{∗}, S are transformations of a Lebesgue space (Λ̂, μ), where μ is a normalized Haar measure on Λ̂. Clearly, ϕ^{∗}(χ) = χ^{2}, and ϕ^{∗} ∘ S = S^{2} ∘ ϕ^{∗}. All characters of Λ̂ form an orthonormal basis of L_{2}(Λ̂, μ). By the theorem of Pontryagin, every character of Λ̂ has the following form

λ(χ) = χ(λ)

for some λ ∈ Λ. Clearly,

Ŝλ(χ) = λ(Sχ) = λ(χ_{0})λ(χ) = λλ(χ),
ϕ̂^{∗}λ(χ) = λ(χ^{2}) = λ^{2}(χ).

Therefore S is an ergodic transformation having discrete spectrum (moreover, Λ_{d}(S) = Λ, where Λ_{d}(S) is a set of all eigenvalues of Ŝ). This implies that S has rank 1 (see [8]).

It is clear that a pair (ϕ^{∗}, S) defines a G-action T via an identification T_{s} = ϕ^{∗}^{−1}, T_{t} = S. To prove a freeness of T, it is enough to show that

(3.1) (∀f ∈ L_{2})((∀k, l ∈ Z) [[ϕ̂^{∗}^{k}f = Ŝ^{l}f] ⇒ [either k = l = 0 or f = const]].

Moreover, we can assume that k ≥ 0. The transformation S is totally ergodic, because Λ has no roots of unity. Therefore (3.1) is true in the case k = 0. Let k > 0. Using a uniqueness of a representation of f in the form ∑_{λ} c_{λ}λ(χ), where ∑_{λ} |c_{λ}|^{2} < ∞, we have

[ϕ̂^{∗}^{k}f = Ŝ^{l}f] ⇔ [∑_{λ} c_{λ}λ^{2^{k}}(χ) = ∑_{λ} c_{λ}λ^{l}λ(χ)],

and |c_{λ}| = |c_{λ^{2^{k}}} | = |c_{λ^{2^{k+1}}} | = ....

If λ ∈ Λ\{1}, then λ, λ^{2^{k}}, λ^{2^{k+1}}, … are pairwise different. Thus c_{λ} = 0 if λ ≠ 1. Therefore f ≡ c_{1}. It follows both (3.1) and the freeness of the G-action T.

We need the following modification of Theorem 5 in [1] Theorem 3.1. Suppose D is a dense subset of Ω_{G}, and g ∈ G; then

(1) If T_{g} is weakly mixing for any T ∈ D, then T_{g} is weakly mixing for a typical T from Ω_{G}.

(2) If T_{g} has rank 1 for any T ∈ D, then T_{g} has rank 1 for a typical T from Ω_{G}.

Remark 3.2. Let us mention that Theorem 3.1 is also true in the case of any discrete group G such that Ω_{G} is a Polish space with respect to the weak topology. Moreover part 2 is true if we replace the property to have rank 1 by the property to have at most rank k. The proof of this remark is a certain modification of the proofs of Theorem 3.1 and Theorem 5 in [1].

Proof. Choose a countable subset of D, say {T (k) ∈ D : k ∈ N}, which is dense in Ω_{G}. Let ε_{i} → 0 as i → ∞ for some sequence ε_{i}. Consider

B(ε_{1}, ε_{2}, …) = ⋂_{k} ⋃_{i≥k} U_{ε_{i}}(T (i)),

where U_{ε}(T) = {S ∈ Ω_{G} : ρ(T, S) < ε}, ρ is a metric of the space Ω_{G}. It is easy to check that Ω_{G} has no isolated points. This implies that ⋃_{i≥k} T (i) is also dense for any k. Therefore B(ε_{1}, ε_{2}, …) is a dense G_{δ} set.

Obviously, to prove Theorem 3.1, it is enough to find a sequence ε_{i} such that B(ε_{1}, ε_{2}, …) consists of group actions having properties as required in parts 1,2.

Fix a dense sequence of elements in L_{2}(X, μ), f_{i}, i = 1, 2, . . ..

To prove part 1, we consider the weak operator topology. Next, we use the wellknown fact that a transformation S is weakly mixing if and only if Ŝ^{n_{i}} → ∫ for some sequence n_{i}, and ∫ is the operator of orthogonal projection onto the space of constants. Let 0 < δ_{n} → 0 as n → ∞. Choose a sequence k_{i}(n) such that T̂_{g}^{k_{i}(n)}(n) → ∫ as i → ∞. Fix a number m_{n} = k_{i(n)}(n) such that

(3.2)  (∀p, q ≤ n) |⟨T̂_{g}^{m_{n}}(n)f_{p}, f_{q}⟩ − ⟨∫ f_{p}dμ, f_{q}⟩| < \frac{δ_{n}}{2} .

Obviously, for any k ∈ Z and h_{1}, h_{2} ∈ L_{2}(X, μ), the map ϕ : Ω_{G} → R defined by ϕ(T) = ⟨T̂_{g}^{k}h_{1}, h_{2}⟩ is continuous.

Thus there exists ε_{n} > 0 such that

ρ(T (n), T) < ε_{n} ⇒
(3.3)  (∀p, q ≤ n) |⟨T̂_{g}^{m_{n}}(n)f_{p}, f_{q}⟩ − ⟨T̂_{g}^{m_{n}}f_{p}, f_{q}⟩| < \frac{δ_{n}}{2} .

Let T ∈ B(ε_{1}, ε_{2}, …), where ε_{i} were chosen above. Then, for some n_{j} → ∞, we have T ∈ U_{ε_{n_{j}}} (T (n_{j})). Using (3.2) and (3.3), we obtain that

|⟨T̂_{g}^{m_{n_{j}}} f_{p}, f_{q}⟩ − ⟨∫ f_{p}dμ, f_{q}⟩| < δ_{n_{j}} for p, q = 1, …, n_{j}.

Hence

(∀p, q) lim_{j}⟨T̂_{g}^{m_{n_{j}}} f_{p}, f_{q}⟩ = ⟨∫ f_{p}dμ, f_{q}⟩.

Part 1 is proved.

Let us prove part 2. Let 0 < δ_{n} → 0 as n → ∞. Since T_{g}(n) has rank 1, there exist integers k_{i}(n), towers

A_{i}(n), T_{g}(n)A_{i}(n), …, T_{g}^{k_{i}(n)−1}(n)A_{i}(n)

and corresponding partitions ξ_{i}(n) of X such that ξ_{i}(n) → ε as i → ∞. Choose a sequence i(n) > n such that ξ_{i(n)}(n) → ε as n → ∞. Obviously, for any k ∈ Z the map ϕ : Ω_{G} → Aut(μ) defined by ϕ(T) = T̂_{g}^{k} is continuous. Thus there exists ε_{n} > 0 such that

ρ(T (n), T) < ε_{n} ⇒
(3.4)  max_{0<j<k_{i(n)}(n)} μ(T_{g}^{j}(n)A_{i(n)}(n)△T_{g}^{j}A_{i(n)}(n)) < \frac{δ_{n}}{k_{i(n)}(n)}μ(A_{i(n)}(n)).

Let T ∈ B(ε_{1}, ε_{2}, …), where ε_{i} were chosen above. Then, for some n_{j} → ∞, we have T ∈ U_{ε_{n_{j}}} (T (n_{j})).

Consider the following set

A′_{j} = A_{i(n_{j})}(n_{j}) ∩ T_{g}^{−1}T_{g}(n_{j})A_{i(n_{j})}(n_{j}) ∩ …
∩T_{g}^{1−k_{i(n_{j})}(n_{j})}T_{g}^{k_{i(n_{j})}(n_{j})−1}(n_{j})A_{i(n_{j})}(n_{j}).

Obviously,

T_{g}^{l}A′_{j} ⊆ T_{g}^{l}(n_{j})A_{i(n_{j})}(n_{j}), l = 0, …, k_{i(n_{j})}(n_{j}) − 1.

Therefore the sets T_{g}^{l}A′_{j} ( l = 0, …, k_{i(n_{j})}(n_{j}) − 1) and the remaining part of X form a partition of X, say η_{j}. Using (3.4), we obtain that

\frac{μ(A′_{j})}{μ(A_{i(n_{j})}(n_{j}))} > 1 − \frac{δ_{n_{j}}}{2} → 1 as j → ∞.

This implies that η_{j} → ε as j → ∞, because ξ_{i(n_{j})}(n_{j}) → ε as j → ∞. Thus T_{g} has rank 1. Theorem 3.1 is proved. □

4. The proof of Theorem 1.2

Proof. To prove Theorem 1.2, it is enough to collect all remarks of Section 3. Namely, denote by A the property of a G-action T to have a weakly mixing T_{t}. Properties we consider are invariant with respect to any metrical isomorphism (in particular, any conjugation). Therefore the set of actions having the property A is dense. Finally, Theorem 3.1 implies that A holds for a typical G-action. It is remain to repeat the same arguments for the set of G-actions T satisfying the rank T_{t} = 1 condition. Theorem 1.2 is proved. □

5. Closing remarks

There are many observations about metrical properties of dynamical systems conjugated to their composition squares (see [5], [6]). We restrict our attention to the ”typical” case, i.e. we consider metrical properties of T_{t} for a typical G-action T. It is not difficult to prove that in our case T_{t} is rigid, i.e. T^{k_{i}} → E for some sequence k_{i}, where E means the identity transformation. Therefore the typical transformation is not mixing. It is also easy to see that the typical transformation T_{t} has a simple singular spectrum. Moreover, the maximal spectral type of T̂_{t} is invariant and ergodic with respect to a map ϕ : T → T, ϕ(λ) = λ^{2} , where the ergodicity means that every invariant Borel set has zero measure or is the torus up to a set of zero measure.

All statements of this paper can be naturally extended to the cases both maps T_{t}^{q}, where q = k2^{l}, k ≠ 0, l ∈ Z, and G_{n} = gr⟨t, s; ts = st^{n}⟩, where n ≠ 0.

However let us mention that the case n = −1 is essentially different, because (see [1], Theorem 7) for a typical G_{−1}-action T, transformation T_{t} has a homogeneous spectrum of multiplicity 2 in the orthogonal complement of the constant functions.

References

1. O. Ageev, The homogeneous spectrum problem in ergodic theory, to appear in Invent. Math.

2. I.P. Cornfel’d , S.V. Fomin, Ya. G. Sinai, Ergodic Theory (Springer-Verlag, 1980).

3. E. Glasner, J.L. King, A zero-one law for dynamical properties, In Topol. Dyn. and Appl. 215 (1998), 231–242

4. G. Goodson, A survey of recent results in the spectral theory of ergodic dynamical systems, J. Dynam. Control Systems 5 (1999), 173–226.

5. G. Goodson, Ergodic dynamical systems conjugate to their composition squares, Acta Math. Univ. Comenian. (N.S.) 71 (2002), 201–210.

6. G. Goodson, Spectral properties of ergodic dynamical systems conjugate to their composition squares, preprint.

7. P.R. Halmos, Lectures on ergodic theory (N.Y.: Chelsea Publ. Comp., 1956).

8. A. del Junco, Transformations with discrete spectrum are stacking transformations, Canad. J. Math. 24 (1976), 836–839.

9. A. del Junco, Disjointness of measure-preserving transformations, self-joinings and category, Prog. Math. 10 (1981), 81–89.

10. A.A. Kirillov, Elements of the theory of representations (Springer-Verlag, 1976).

11. D. Ornstein, B. Weiss, Ergodic theory of amenable group actions. I. The Rohlin lemma, Bull. Amer. Math. Soc. (N.S.) 2 (1980), 161–164.

12. D. Ornstein, B. Weiss, Entropy and isomorphism theorems for actions of amenable groups, J. Analyse Math. 48 (1987), 1–141.

Department of Mathematics, Moscow State Technical University, 2nd Baumanscaya St. 5, 105005 Moscow, Russia

  E-mail address: ageev@mx.bmstu.ru

Current address: Max Planck Institute of Mathematics, P.O.Box 7280, D-53072 Bonn, Germany

  E-mail address: ageev@mpim-bonn.mpg.de