MIXING RANK-ONE ACTIONS OF

LOCALLY COMPACT ABELIAN GROUPS

Alexandre I. Danilenko and Cesar E. Silva

Abstract. Using techniques related to the (C, F)-actions we construct explicitly mixing rank-one (by cubes) actions T of G = ℝ^{d_{1}} × ℤ^{d_{2}} for any non-negative d_{1} and d_{2} with d_{1} ≠ 1. It is also shown that h(T_{g}) = 0 for each g ∈ G.

0. Introduction

Mixing rank-one transformations (and actions of more general groups) have been of interest in ergodic theory since Ornstein’s original construction in 1970 [Or]. It is now well known that they possess strong ergodic properties: they are mixing of all orders and have minimal self-joinings of all orders. This implies in turn that they are prime and have trivial centralizer (see, for example, the surveys in [Fe] and [Th]). The first family of mixing ℤ-actions, due to Ornstein [Or], and the more recent constructions of mixing rank-one ℝ-actions in [Pr], were obtained via cutting-and-stacking techniques using “random spacers”. While demonstrating the existence of mixing rank-one actions, these works do not exhibit a specific such transformation or action—it is shown only that the objects under question form a subset of full measure in a space of “parameters”. The first explicit cuttingand-stacking transformation, now called the staircase, was shown to be mixing by Adams [Ad] in 1998. Higher dimensional mixing staircase ℤ^{d}-actions were later constructed in [AdS]. We note that the complete proof was given there only in dimension d = 2. As one of the consequences of our work, we complete the proof for all d > 2. Recently, a more general family of mixing “polynomial” staircase ℤ-actions was constructed in [CrS]. While not directly related to our work, another construction that we note is the recent work [Fa], where smooth models of mixing rank-one flows on the 3-torus are presented.

Our main purpose here is to construct explicitly mixing rank-one actions of ℝ^{d_{1}} × ℤ^{d_{2}} for all non-negative d_{1} and d_{2} with d_{1} ≠ 1. Our approach is based on ideas that first appeared in [Ad], [AdS], and [CrS]. However, in this paper we proceed entirely in the framework of (C, F)-actions for locally compact second countable (l.c.s.c.) Abelian groups, and in fact we develop a large part of the theory in the more general context of these actions. In particular, we encounter here some new problems that are specific to higher dimensions and the continuity of the groups. The (C, F)-construction of finite measure-preserving actions of discrete countable amenable groups appeared in [Ju] as an algebraic counterpart of the

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 37A40.

Key words and phrases. Ergodic action, mixing, rank-one action, entropy.

“geometrical” cutting-and-stacking method developed for ℤ-actions. Later it was used (in a modified form) by the authors in the framework of infinite measurepreserving and non-singular countable Abelian group actions, as a convenient tool for modeling examples and counterexamples with various properties of weak mixing and multiple recurrence (see [Da1], [Da2], [DaS]).

Let G be a non-compact l.c.s.c. Abelian group and T = (T_{g})_{g∈G} a measurable action of G on a standard probability space (X, 𝔅, μ).

Definition 0.1. T is said to be mixing if for all subsets A, B ∈ 𝔅 we have

(0-1)  lim_{g→∞} μ(T_{g}A ∩ B) = μ(A)μ(B).

A sequence g_{n} → ∞ in G is called mixing if (0-1) holds along g_{n} as n → ∞.

Notice that an action is mixing whenever each sequence converging to infinity in G contains a mixing subsequence.

Definition 0.2.

(i) A Rokhlin tower or column for T is a triple (Y, f, F), where Y ∈ 𝔅, F is a relatively compact subset of G and f : Y → F is a measurable map such that for any Borel subset H ⊂ F and an element g ∈ G with g + H ⊂ F, one has f^{−1}(g + H) = T_{g}f^{−1}(H).

(ii) We say that T is of funny rank-one if there exists a sequence of Rokhlin towers (Y_{n}, f_{n}, F_{n}) such that lim_{n→∞} μ(Y_{n}) = 1 and for any subset B ∈ 𝔅, there is a sequence of Borel subsets H_{n} ⊂ F_{n} such that

lim_{n→∞} μ(B△f_{n}^{−1}(H_{n})) = 0.

(iii) If G = ℝ^{d_{1}} × ℤ^{d_{2}}, T is of funny rank-one and, in addition, the subsets F_{n} from (ii) are as follows

F_{n} = {(t_{1}, …, t_{d_{1}+t_{2}}) ∈ G | 0 ≤ t_{i} < a_{n} for all i = 1, …, d_{1} + d_{2}}

for some a_{n} ∈ ℝ, n = 1, 2, …, then we say that T is of rank-one (or rank-one by cubes).

It is easy to see that any funny rank-one action is ergodic.

The outline of the paper is as follows. In Section 1 we extend the concept of (C, F)-action introduced for countable discrete groups (see [Ju], [Da1]) to the class of l.c.s.c. Abelian ones. It is distinguished a special family of such actions whose mixing properties will be under investigation. In Section 2 we first prove a fundamental lemma (Lemma 2.2) linking the mixing with the averaging along the ‘spacer mappings’. Then we find a sufficient condition for the total ergodicity of the actions under considerations. We also start to check the mixing property for some special sequences. In particular, we show that if a sequence is of ‘moderate growth’ relative to a fixed Følner sequence in G then (under some extra conditions on G and the action) it is uniformly mixing (Lemma 2.8). Section 3 is devoted to the actions with restricted growth—the property which was introduced explicitly in [CrS] for G = ℤ and considered in [Ad] and [AdS] in an implicit form. Theorem 3.5 provides a sufficient condition for such actions to be mixing. It is used in Section 4 to prove the main results of the paper—Theorems 4.9, 4.10 and 4.12—on the existence of mixing rank-one actions of ℝ^{d_{1}}, ℤ^{d_{2}} and ℝ^{d_{1}} × ℤ^{d_{2}} respectively. In the final section (Section 5) we show that the actions constructed in § 4 have ‘very weak’ stochastic properties—the entropy of any individual transformation from such actions is trivial. This fact holds for any rank-one (by cubes) action. However, it is no longer true for a more general class of action of rank-one ‘by rectangles’ (see [Ru] for a counterexample).

Acknowledgements. This project was supported in part by the NSF under the Collaboration in Basic Science and Engineering Program (COBASE), contract INT-0002341. Also, the first named author was supported in part by Civilian Research and Development Foundation (CRDF), grant UM1-2546-KH-03.

1. (C, F)-actions of locally compact Abelian groups

In this section we introduce the (C, F)-actions of l.c.s.c. Abelian groups and specify a subclass of them. (The aim of the paper is to show that this subclass contains mixing actions.)

Let G be a l.c.s.c. Abelian group. Denote by λ_{G} a (σ-finite) Haar measure on it. Given two subsets E, F ⊂ G, by E + F we mean their algebraic sum, i.e. E + F = {e + f | e ∈ E, f ∈ F }. The algebraic difference E − F is defined in a similar way. We hope that the reader will not confuse it with the set theoretical difference E \ F. If E is a singleton, say E = {e}, then we will write e + F for E + F. If (E − E) ∩ (F − F) = {0} then E and F are called independent. For an element g ∈ G and a subset E ⊂ G, we set E(g) = E ∩ (E − g).

To define a (C, F)-action of G we need two sequences (F_{n})_{n≥0} and (C_{n})_{n>0} of subsets in G such that the following hold

(1-1) F_{0} ⊂ F_{1} ⊂ F_{2} ⊂ ⋯ is a Følner sequence in G, (1-2) C_{n} is finite and #C_{n} > 1, F_{n} + C_{n+1} ⊂ F_{n+1}, (1-3) (1-4) F_{n} and C_{n+1} are independent.

We put X_{n} := F_{n} × ∏_{k>n} C_{k}, endow X_{n} with the standard product Borel σalgebra and define a Borel embedding X_{n} → X_{n+1} by setting

(1-5)  (f_{n}, c_{n+1}, c_{n+2}, …) ↦ (f_{n} + c_{n+1}, c_{n+2}, …).

Then we have X_{1} ⊂ X_{2} ⊂ · · ·. Hence X := ⋃_{n} X_{n} endowed with the natural Borel σ-algebra, say 𝔅, is a standard Borel space. Given a Borel subset A ⊂ F_{n}, we denote the set

{x ∈ X | x = (f_{n}, c_{n+1}, c_{n+2} …) ∈ X_{n} and f_{n} ∈ A}

by [A]_{n} and call it an n-cylinder. It is clear that the σ-algebra 𝔅 is generated by the family of all cylinders.

Now we are going to define a measure on (X, 𝔅). Let κ_{n} stand for the equidistribution on C_{n} and ν_{n} := (#C_{1} ⋯ #C_{n})^{−1}λ_{G} ↾ F_{n} on F_{n}. We define an infinite product measure μ_{n} on X_{n} by setting

μ_{n} = ν_{n} × κ_{n+1} × κ_{n+2} × ⋯,

n ∈ ℕ. Then the embeddings (1-5) are all measure preserving. Hence a σ-finite measure μ on X is well defined by the restrictions μ ↾ X_{n} = μ_{n}, n ∈ ℕ. To put it in other way, (X, μ) = inj lim_{n}(X_{n}, μ_{n}). Since

μ_{n+1}(X_{n+1}) = \frac{ν_{n+1}(F_{n+1})}{ν_{n+1}(F_{n} + C_{n+1})}μ_{n}(X_{n}) = \frac{λ_{G}(F_{n+1})}{λ_{G}(F_{n})#C_{n+1}} μ_{n}(X_{n}),

it follows that μ is finite if and only if

(1-6)  ∏_{n=0}^{∞} \frac{λ_{G}(F_{n+1})}{λ_{G}(F_{n})#C_{n+1}} < ∞, i.e. ∑_{n=0}^{∞} \frac{λ_{G}(F_{n+1} \ (F_{n} + C_{n+1}))}{λ_{G}(F_{n})#C_{n+1}} < ∞.

For the rest of the paper we will assume that (1-6) is satisfied. Moreover, we choose (i.e. normalize) λ_{G} in such a way that μ(X) = 1.

To construct a measure preserving action of G on (X, μ), we fix a filtration K_{1} ⊂ K_{2} ⊂ ⋯ of G by compact subsets. Thus ⋃_{m=1}^{∞} K_{m} = G. Given n, m ∈ ℕ, we set

D_{m}^{(n)} := ((F_{n} − K_{m}) ∩ F_{n}) × ∏_{k>n} C_{k} ⊂ X_{n} and
R_{m}^{(n)} := ((F_{n} + K_{m}) ∩ F_{n}) × ∏_{k>n} C_{k} ⊂ X_{n}.

It is easy to verify that

D_{m+1}^{(n)} ⊂ D_{m}^{(n)} ⊂ D_{m}^{(n+1)} and R_{m+1}^{(n)} ⊂ R_{m}^{(n)} ⊂ R_{m}^{(n+1)}.

We define a Borel mapping

K_{m} × D_{m}^{(n)} ∋ (g, x) ↦ T_{m,g}^{(n)}x ∈ R_{m}^{(n)}

by setting for x = (f_{n}, c_{n+1}, c_{n+2}, …),

T_{m,g}^{(n)}(f_{n}, c_{n+1}, c_{n+2} …) := (g + f_{n}, c_{n+1}, c_{n+2}, …).

Now let D_{m} := ⋃_{n=1}^{∞} D_{m}^{(n)} and R_{m} := ⋃_{n=1}^{∞} R_{m}^{(n)}. Then a Borel mapping

K_{m} × D_{m} ∋ (g, x) ↦ T_{m,g}x ∈ R_{m}

is well defined by the restrictions T_{m,g} ↾ D_{m}^{(n)} = T_{m,g}^{(n)} for g ∈ K_{m} and n ≥ 1. It is easy to see that D_{m} ⊃ D_{m+1}, R_{m} ⊃ R_{m+1} and T_{m,g} ↾ D_{m+1} = T_{m+1,g} for all m. It follows from (1-1) that μ_{n}(D_{m}^{(n)}) → 1 and μ_{n}(R_{m}^{(n)}) → 1 as n → ∞. Hence μ(D_{m}) = μ(R_{m}) = 1 for all m ∈ ℕ. Finally we set X̂ := ⋂_{m=1}^{∞} D_{m} ∩ ⋂_{m=1}^{∞} R_{m} and define a Borel mapping

T : G × X̂ ∋ (g, x) → T_{g}x ∈ X̂

by setting T_{g}x := T_{m,g}x for some (and hence any) m such that g ∈ K_{m}. It is clear that μ(X̂) = 1.

Proposition 1.1. T = (T_{g})_{g∈G} is a free Borel measure preserving action of G on a conull subset of the standard probability space (X, 𝔅, μ). It is of funny rank-one.

Proof. It suffices to verify only the latter claim. According to Definition 0.2 we have to find a sequence of Rokhlin towers ‘appoximating’ the dynamical system. Let s_{n} denote the projection of X_{n} = F_{n} × C_{n+1} × ⋯ onto the first coordinate. It is easy to see that the sequence (X_{n}, s_{n}, F_{n}) is as desired. □

Throughout the paper we will not distinguish between two measurable sets (or mappings) which agree almost everywhere. It is easy to see that T does not depend on the choice of filtration (K_{m})_{m=1}^{∞}.

Definition 1.2. T is called the (C, F)-action of G associated with (C_{n}, F_{n})_{n}.

We will often use the following simple properties of (X, μ, T): for Borel subsets A, B ⊂ F_{n},

[A ∩ B]_{n} = [A]_{n} ∩ [B]_{n}, (1-7)

[A]_{n} = [A + C_{n+1}]_{n+1} = ⨆_{c∈C_{n+1}}[A + c]_{n+1}, (1-8)

T_{g}[A]_{n} = [A + g]_{n} if A + g ⊂ F_{n}, (1-9) (1-10) μ([A]_{n}) = #C_{n+1} · μ([A + c]_{n+1}) for every c ∈ C_{n+1},

μ([A]_{n}) ≤ \frac{λ_{G}(A)}{λ_{G}(F_{n})}, (1-11)

where the sign ⊔ means the union of mutually disjoint sets.

Recall that an action T of G on (X, 𝔅, μ) is partially rigid if there exists δ > 0 with

lim inf_{g→∞} μ(T_{g}B ∩ B) ≥ δμ(B) for all B ∈ 𝔅.

It is clear that partial rigidity is incompatible with the mixing. For the (C, F)- actions, there is a simple condition that implies the rigidity.

Proposition 1.3. If lim inf_{n→∞} #C_{n} < ∞ then T is partially rigid and hence is not mixing.

Proof. Let n_{i} < n_{2} < ⋯ be a sequence of indices with #C_{n_{1}} = #C_{n_{2}} = · · ·. Select c_{i} ≠ c′_{i} in C_{i} and set g_{i} := c_{i} − c′_{i}. Then g_{i} ∉ F_{n_{i}} − F_{n_{i}} by (1-4). On the other hand, it follows from (1-1) that ⋃_{i=1}^{∞}(F_{n_{i}} − F_{n_{i}}) = G. Hence g_{i} → ∞ as i → ∞. Take a cylinder B ∈ 𝔅. We can represent it eventually (i.e. for all large enough i) as B = [B_{i}]_{n_{i}}, where B_{i} is a Borel subset of F_{n_{i}}. If follows from (1-8)—(1-10) that

μ(T_{g_{i}}B ∩ B) = μ(T_{g_{i}}[B_{n_{i−1}} + C_{n_{i}}]_{n_{i}} ∩ [B_{n_{i−1}} + C_{n_{i}}]_{n_{i}})

≥ μ(T_{g_{i}}[B_{n_{i−1}} + c′_{i}]_{n_{i}} ∩ [B_{n_{i−1}} + c_{i}]_{n_{i}})

= μ([B_{n_{i−1}} + c_{i}]_{n_{i}}) = \frac{1}{#C_{n_{i}}} μ([B_{n_{i−1}}])

= \frac{1}{#C_{n_{1}}} μ(B).

Since the cylinders generate a dense subalgebra in 𝔅, we are done. □

Now we isolate a special subfamily of (C, F)-actions to show in the sequel that it contains mixing actions.

Let H be a discrete countable group, and let ϕ_{n}, s_{n} and c_{n+1} be three mappings from H to G such that ϕ_{n} is a homomorphism, s_{n}(0) = 0 and c_{n+1} := ϕ_{n} + s_{n}, n ∈ ℕ. Suppose that

(1-12) (H_{n})_{n≥0} is a Følner sequence in H, 0 ∈ H_{n} and (1-13) ϕ_{n}(H) is a lattice in G.

Now we define F_{n} ⊂ G to be a Borel fundamental domain for ϕ_{n}(H) (i.e. a subset which meets every ϕ_{n}(H)-coset exactly once) and put C_{n+1} := c_{n+1}(H_{n}), n ≥ 0. Assume that (1-1)–(1-5) are all satisfied.

Definition 1.4. We call the corresponding (C, F)-action T of G on the probability space (X, 𝔅, μ) the action associated with (H_{n}, ϕ_{n}, s_{n}, F_{n})_{n}.

In view of Proposition 1.3, we will always assume that lim_{n→∞} #H_{n} = ∞.

Notice also that if s_{n} are all trivial, i.e. s_{n}(h) = 0 for all h ∈ H_{n} then the action of G associated with (H_{n}, ϕ_{n}, s_{n}, F_{n}) has pure point spectrum with rational eigenvalues only. This simple fact will not be used in this paper. We leave its proof to the reader.

In the statements of our main results here it will be assumed that the mappings s_{n} are polynomials of degree > 1.

Definition 1.5 [Le]. For any h ∈ H, the h-derivative of s is a mapping ∂_{h}s : H → G given by ∂_{h}s(k) = s(k + h) − s(k). Let d be a nonnegative integer. Then s is called a polynomial of degree ≤ d if for any h_{1}, …, h_{d+1} ∈ H \ {0}, we have ∂_{h_{1}} ⋯ ∂_{h_{d+1}}s = 0. The minimal d with this property is called the degree of s.

It is easy to see that every polynomial of degree 0 is constant. As was shown in [Le], a polynomial of degree one is a non-constant affine mapping (i.e. a homomorphism plus a constant). A polynomial from ℤ^{d} to ℝ^{l} is an l-tuple of usual polynomials in d variables with real coefficients. A polynomial from ℤ^{d} to ℤ^{l} is an l-tuple (p_{1}, …, p_{l}) of usual polynomials in d variables with rational coefficients such that p_{i}(ℤ^{d}) ⊂ ℤ for all i = 1, …, l.

2. Uniformly mixing sequences in G

For the remaining of the paper (X, 𝔅, μ, T) will stand for the (C, F)-action of G associated to a sequence (H_{n}, ϕ_{n}, s_{n}, F_{n})_{n}.

In this section we prove a fundamental Lemma 2.2 and use it to show that some special sequences in G are uniformly mixing. As an auxiliary result for that we exhibit a sufficient condition for the total ergodicity of T. The connection between the uniform mixing and total ergodicity is established in Corollary 2.6.

Definition 2.1. A sequence (g_{n})_{n=1}^{∞} of elements from G is called uniformly mixing for T if

(2-1)  sup_{A⊂F_{n}} |μ(T_{g_{n}}[A]_{n} ∩ B) − μ([A]_{n})μ(B)| → 0 as n → ∞

for every subset B ∈ 𝔅.

It is obvious that if a sequence is uniformly mixing then it is mixing.

Let L be a finite set and a : L → G a map. We define a linear operator M_{a,L} in L^{2}(X, μ) by setting

M_{a,L}(f) := \frac{1}{#L} ∑_{l∈L} f ∘ T_{a(l)}.

If L is a subset of G, we write M_{L} for M_{id,L}. Let P_{0} stand for the projection onto the subspace of constant functions, i.e. P_{0}(f) = ∫_{X} f dμ. The inner product in L^{2}(X, μ) will be denoted by ⟨·, ·⟩.

Fix a sequence (h_{n})_{n≥1} of elements from H. For brevity, we will denote ∂_{h_{n}}s_{n} by s′_{n}, n ∈ ℕ.

Lemma 2.2. If the following conditions are satisfied

\frac{#H_{n}(h_{n})}{#H_{n}} → 1, (2-2)

\frac{1}{#H_{n}} ∑_{h∈H_{n}(h_{n})} \frac{λ_{G}(F_{n} \ F_{n}(s′_{n}(h)))}{λ_{G}(F_{n})} → 0 (2-3)

then for all n-cylinders A, B ⊂ X, we have

μ(T_{ϕ_{n}(h_{n})}A ∩ B) = \frac{1}{#H_{n}} ∑_{h∈H_{n}} μ(A ∩ T_{s′_{n}(h)}B) + \overline{o}(1) = ⟨χ_{A}, M_{s′_{n},H_{n}}(χ_{B})⟩ + \overline{o}(1),

where χ_{A} and χ_{B} are the indicators of A and B respectively and \overline{o}(1) denotes a sequence that tends to 0 and that does not depend on A and B. The same formula holds as well for

(i) an arbitrary subset B ∈ 𝔅 and A as above with \overline{o}(1) depending on B only and

(ii) arbitrary subsets A, B ∈ 𝔅 with \overline{o}(1) depending on both A and B.

Proof. Let A_{n} and B_{n} be the Borel subsets of F_{n} such that A = [A_{n}]_{n} and B = [B_{n}]_{n}. For h ∈ H_{n}(h_{n}), we put A_{n,h} := A_{n} ∩ F_{n}(−s′_{n}(h)). Then

(2-4)  A_{n,h} − s′_{n}(h) ⊂ F_{n}.

We also make a simple but important observation that

(2-5)  \begin{aligned}ϕ_{n}(h_{n}) + c_{n+1}(h) & = ϕ_{n}(h_{n}) + ϕ_{n}(h) + s_{n}(h) \\ & = ϕ_{n}(h + h_{n}) + s_{n}(h + h_{n}) − s′_{n}(h) \\ & = c_{n+1}(h + h_{n}) − s′_{n}(h). \end{aligned}

It follows from (1-8), (1-9), (2-4), (1-3), (2-5) and (1-10) that

μ(T_{ϕ_{n}(h_{n})}A ∩ B) = ∑_{h∈H_{n}} μ(T_{ϕ_{n}(h_{n})}[A_{n} + c_{n+1}(h)]_{n+1} ∩ [B_{n}]_{n})

= ∑_{h∈H_{n}(h_{n})}(μ(T_{ϕ_{n}(h_{n})}[A_{n,h} + c_{n+1}(h)]_{n+1} ∩ [B_{n}]_{n})

± μ([(A_{n} \ A_{n,h}) + c_{n+1}(h)]_{n+1}))

± ∑_{h∈H_{n}\H_{n}(h_{n})} μ([F_{n} + c_{n+1}(h)]_{n+1})

= ∑_{h∈H_{n}(h_{n})}(μ([A_{n,h} − s′_{n}(h) + c_{n+1}(h + h_{n})]_{n+1} ∩ [B_{n}]_{n})

± \frac{1}{#H_{n}} μ([A_{n} \ A_{n,h}]_{n})) ± (1 − \frac{#H_{n}(h_{n})}{#H_{n}}).

Notice that for all c ∈ C_{n+1} and h ∈ H_{n}(h_{n}), we have by (1-7), (1-8) and (1-11),

[A_{n,h} − s′_{n}(h) + c]_{n+1} ∩ [B_{n}]_{n} = [((A_{n,h} − s′_{n}(h)) ∩ B_{n}) + c]_{n+1} and (2-6)

μ([(A_{n} \ A_{n,h})]_{n}) ≤ \frac{λ_{G}(A_{n} \ A_{n,h})}{λ_{G}(F_{n})} ≤ \frac{λ_{G}(F_{n} \ F_{n}(s′_{n}(h)))}{λ_{G}(F_{n})}. (2-7)

Hence it follows from (1-10), (2-2), (2-3), (2-6) and (2-7) that

μ(T_{ϕ_{n}(h_{n})}A ∩ B) = \frac{1}{#H_{n}} ∑_{h∈H_{n}(h_{n})} μ([(A_{n,h} − s′_{n}(h)) ∩ B_{n}]_{n}) + \overline{o}(1).

Applying (1-7) and (1-9) we obtain

\begin{aligned}μ(T_{ϕ_{n}(h_{n})}A ∩ B) & = \frac{1}{#H_{n}} ∑_{h∈H_{n}(h_{n})} μ(T_{−s′_{n}(h)}[A_{n,h}]_{n} ∩ [B_{n}]_{n}) + \overline{o}(1) \\ & = \frac{1}{#H_{n}} ∑_{h∈H_{n}(h_{n})}(μ(A ∩ T_{s′_{n}(h)}B) ± μ([A_{n} \ A_{n,h}]_{n})) + \overline{o}(1). \end{aligned}

It remains to make use of (2-7), (2-3) and (2-2).

The final claim of Lemma 2.2 follows from the fact that the cylinders generate a dense subalgebra in 𝔅. □

Corollary 2.3. Let (2-2) and (2-3) hold. Then the following are satisfied:

(i) The sequence (ϕ_{n}(h_{n}))_{n=1}^{∞} is mixing for T if and only if M_{s′_{n},H_{n}} → P_{0} in the weak operator topology.

(ii) If M_{s′_{n},H_{n}} → P_{0} in the strong operator topology then (ϕ_{n}(h_{n}))_{n=1}^{∞} is uniformly mixing.

We first examine when T is totally ergodic. Recall some standard definitions.

Definition 2.4.

(i) Given a subset B ∈ 𝔅, we denote by G_{B} the stabilizer of B, i.e. G_{B} := {g ∈ G | T_{g}B = B}.

(ii) T is called totally ergodic if for any co-compact subgroup K ⊂ G, the action

(T_{g})_{g∈K} is ergodic. (iii) T is called weakly mixing if the diagonal action (T_{g} × T_{g})_{g∈G} of G is er-

godic. Equivalently, if there exist a function f ∈ L^{2}(X, μ) and a continuous character χ of G such that f ∘ T_{g} = χ(g)f a.e. then f is constant.

It is easy to see that T is totally ergodic if and only if the stabilizer of any subset B ∈ 𝔅 with 0 < μ(B) < 1 is not co-compact. Moreover, if an action is weakly mixing then it is totally ergodic. The converse is true for G = ℝ but it does not hold for general groups.

Proposition 2.5. Let (2-2) and (2-3) hold. Let K be a co-compact subgroup of G and π : G → G/K stand for the corresponding quotient map. Denote by κ_{n} the image of the equidistributed probability on H_{n} under the mapping (π ∘ s′_{n})_{∗}, n ∈ ℕ. If κ_{n} does not ∗-weakly converge to a Dirac δ-measure on G/K then K is not the stabilizer of any measurable subset B ∈ 𝔅 with 0 < μ(B) < 1.

Proof. Suppose that the contrary holds, i.e. there exists B ∈ 𝔅 with 0 < μ(B) < 1 and K = G_{B}. Then the quotient compact group G/K acts naturally on the sub-σ-algebra F of (T_{g})_{g∈K}-invariant subsets. Denote this action by T̂. Then T̂_{π(g)}A := T_{g}A for all g ∈ G and A ∈ 𝔉. It is clear that T̂ is free. We set a_{n} := π(ϕ_{n}(h_{n})). Passing to a subsequence, if necessary, we may assume without loss of generality that a_{n} converges to some a ∈ G. Then μ(T̂_{a_{n}}B△T̂_{a}B) → 0 as n → ∞. Hence

μ(B) = μ(T_{ϕ_{n}(h)}B ∩ T̃_{a_{n}}B) = μ(T_{ϕ_{n}(h)}B ∩ T̃_{a}B) + \overline{o}(1).

We then deduce from this formula and Lemma 2.2 that

(2-8)  \begin{aligned}μ(B) & = \frac{1}{#H_{n}} ∑_{g∈s′_{n}(H_{n})} μ(B ∩ T_{g}T̂_{a}B) + \overline{o}(1) \\ & = \frac{1}{#H_{n}} ∑_{g∈s′_{n}(H_{n})} μ(T̂_{π(g)+a}B ∩ B) + \overline{o}(1) \\ & = ∫_{G/K} μ(T̂_{a+b}B ∩ B) dκ_{n}(b) + \overline{o}(1). \end{aligned}

Since G/K is compact, we may assume (passing to a subsequence, if necessary) that κ_{n} converges ∗-weakly to a probability κ which is not a Dirac δ-measure by the condition of the proposition. Hence passing to a limit in (2-8) we obtain

μ(B) = ∫_{G/H} μ(T̂_{a+b}B ∩ B) dκ(b)

Hence μ(B) = μ(T̂_{a+b}B ∩ B), i.e. B = T̂_{a+b}B, for κ-a.a. b ∈ G/K. Since T̂ is free, we deduce that Supp κ = {−a}. Hence Supp κ is a singleton, a contradiction. □

Now we are interested in the following particular case. There exist non-zero k ∈ H and a polynomial s : H → G of degree 2 such that h_{n} = k and s_{n} = s for all n ∈ ℕ. Then, of course, (2-2) is satisfied. Moreover, for any non-zero t ∈ H, we have

∂_{t}s(h) = ψ_{t}(h) + a_{t}, at all h ∈ H

for some nontrivial homomorphism ψ_{t} : H → G and an element a_{t} ∈ G (see the text following Definition 1.5). Hence s′_{n}(h) = ψ_{k}(h) + a_{k} for all h ∈ H and n ∈ ℕ.

Corollary 2.6. Let

(2-9)  max_{h∈H_{n}} \frac{λ_{G}(F_{n} \ (F_{n} + ψ_{k}(h)))}{λ_{G}(F_{n})} → 0 as n → ∞.

Then the following are satisfied:

(i) If the action (T_{ψ_{t}(h)})_{t∈H} is ergodic then the sequence (ϕ_{n}(t))_{n=1}^{∞} is uniformly mixing.

(ii) If the subgroup generated by ⋃_{t∈H} ψ_{t}(H) is dense in G then T is totally

ergodic. (iii) If the condition of (ii) holds and ψ_{t}(H) is a lattice in G then the sequence

(ϕ_{n}(t))_{n=1}^{∞} is uniformly mixing.

Proof. It is obvious that (2-2) holds. Notice also that (2-9) implies (2-3).

(i) follows from Corollary 2.3 and the von Neumann mean ergodic theorem for (T_{ψ_{t}(h)})_{t∈H}.

(ii) Suppose that the contrary holds, i.e. there exists a co-compact subgroup K ⊂ G and a subset B ⊂ X such that 0 < μ(B) < 1 and K = G_{B}. Fix t ∈ H, t ≠ 0. Then κ_{n} is the translation of (π ∘ ψ_{t})_{∗}ν_{n} by π(a_{t}), where ν_{n} stands for the equidistribution on H_{n}, n ∈ ℕ. Denote by G_{t} the closure of the subgroup (π ∘ ψ_{t})(H) in G. Since H_{n} is a Følner sequence in H and π ∘ ψ_{t} : H → G is a homomorphism, it is easy to verify that (π ∘ ψ_{t})_{∗}ν_{n} converges ∗-weakly to the Haar measure λ_{G_{t}} which is considered now as a measure on G supported on G_{t}. It follows from Proposition 2.4 that Supp λ_{G_{t}} is a singleton, i.e. G_{t} ⊂ K. By the condition of the corollary, the subgroup generated by ⋃_{t∈H} G_{t} is dense in G. Hence K = G which contradicts to the ergodicity of T.

(iii) follows directly from (i) and (ii). □

Now we return to the general case and prove two lemmas.

Lemma 2.7. Suppose that there exists a finite subset Q ⊂ H such that

(2-10)  F_{n} + F_{n} ⊂ ϕ_{n}(Q) + F_{n}, for all n ∈ ℕ.

Let g_{n} = ϕ_{n}(k)+f_{n} for some k ∈ H and f_{n} ∈ F_{n}, n ∈ ℕ. If the sequence (ϕ_{n}(t))_{n=1}^{∞} is uniformly mixing for every t ∈ k + Q then so is the sequence (g_{n})_{n=1}^{∞}.

Proof. Since f_{n} + F_{n} ⊂ ϕ_{n}(Q) + F_{n} and F_{n} is a fundamental domain for ϕ_{n}(H), a finite partition of F_{n} is well defined as follows: F_{n} = ⨆_{q∈Q} F_{n}(f_{n} − ϕ_{n}(q)). Now let A be an n-cylinder in X and A = [A_{n}]_{n} for some Borel subset A_{n} ⊂ F_{n}. Consider the induced partition ⨆_{q∈Q} A_{n,q} of A_{n}, where A_{n,q} := A_{n} ∩ F (f_{n} − ϕ_{n}(q)). Then by (1-9),

T_{g_{n}}[A_{n,q}]_{n} = T_{ϕ_{n}(k+q)}T_{f_{n}−ϕ_{n}(q)}[A_{n,q}]_{n} = T_{ϕ_{n}(k+q)}[f_{n} − ϕ_{n}(q) + A_{q}]_{n}.

Hence for any subset B ∈ 𝔅,

μ(T_{g_{n}}A ∩ B) − μ(A)μ(B) = ∑_{q∈Q}(μ(T_{ϕ_{n}(k+q)}[A_{n,q}]_{n} ∩ B) − μ([A_{n,q}]_{n} ∩ B)).

It remains to use the uniform mixing of (ϕ_{n}(t))_{n=1}^{∞}, t ∈ k + Q. □

Suppose that—in addition to (2-10)—there is a finite subset Q_{−} ⊂ H such that

(2-11)  −F_{n} ⊂ ϕ_{n}(Q_{−}) + F_{n} for all n ∈ ℕ.

For a subset A ⊂ G, let A^{•} stand for the difference A − A. Then there exists a finite subset Q_{+} ⊂ H such that

(F_{n}^{•} + F_{n}^{•}) \ F_{n}^{•} ⊂ ϕ_{n}(Q_{+}) + F_{n} for all n ∈ ℕ.

We will assume that Q_{+} is the minimal subset with this property. It is clear that 0 ∉ Q_{+}.

Lemma 2.8. Let (2-10) and (2-11) hold and for any t ∈ H \ {0}, and let the sequence (ϕ_{n}(t))_{n=1}^{∞} be uniformly mixing. Take g_{n} ∈ (F_{n}^{•} + F_{n}^{•}) \ F_{n}^{•} for all n ∈ ℕ. If for an integer l > 0, we have

−lQ_{+} ∩ (\underbrace{Q + ⋯ + Q}_{l times}) = ∅

then the sequence (lg_{n})_{n=1}^{∞} is uniformly mixing.

Proof. Since Q_{+} is finite, we can partition the sequence g_{n} into finitely many subsequences of the form ϕ_{n}(q) + f_{n} for some f_{n} ∈ F_{n}, n ∈ ℕ, and a fixed q ∈ Q_{+}. Therefore it is enough to assume that g_{n} itself enjoys this property. Then condition (2-10) applied l − 1 times yields

lg_{n} = ϕ_{n}(lq) + \underbrace{f_{n} + ⋯ + f_{n}}_{l times} = ϕ_{n}(lq) + ϕ_{n}(q_{1} + ⋯ + q_{l−1}) + f̃_{n}

for some q_{1}, …, q_{l−1} ∈ Q_{+} and f̃_{n} ∈ F_{n}. Since 0 ∉ lq + q_{1} + ⋯ + q_{l−1} + Q by the condition of the lemma, we may apply Lemma 2.7 to complete the proof. □

3. Restricted growth condition

In this section we introduce a restricted growth condition for the (C, F)-actions specified in Section 2. It is an analogue of a concept considered in [CrS] for G = ℤ. The concept, in turn, is a counterpart of a sufficient condition introduced in [Ad] for a classical staircase to be mixing. The main result here is Theorem 3.5, which provides a sufficient condition for mixing in the class of (C, F)-actions with restricted growth. We conclude the section with some technical statements needed to verify this condition for some special ℝ^{d_{1}} × ℤ^{d_{2}}-actions that will be constructed in the next section.

Definition 3.1. We say that T satisfies the restricted growth condition if (2-3) holds for any sequence h_{n} ∈ H such that there exists lim_{n→∞} #H_{n}(h_{n})/#H_{n} > 0.

Definition 3.2. Given two sequences A_{n}, B_{n} ⊂ G, we write A_{n} ∼_{n} B_{n} if

\frac{λ_{G}(A_{n}△B_{n})}{λ_{G}(F_{n})} → 0 as n → ∞.

It follows from (1-11) that μ([A_{n}]_{n}△[B_{n}]_{n}) → 0 whenever A_{n} ∼_{n} B_{n} and A_{n}, B_{n} ⊂ F_{n}.

Lemma 3.3. Suppose that (2-10) and the following hold

(3-1)  \frac{1}{#H_{n}} ∑_{h∈H_{n}} \frac{λ_{G}(F_{n} \ F_{n}(s_{n}(h)))}{λ_{G}(F_{n})} → 0 as n → ∞.

Let g_{n} := ϕ_{n}(h_{n}) + f_{n} for some h_{n} ∈ H and f_{n} ∈ F_{n}, n ∈ ℕ. Then

F_{n+1}(g_{n}) ∼_{n+1} ⨆_{h∈H_{n}(h_{n})}(c_{n+1}(h) + F_{n}).

Proof. Since

λ_{G}(F_{n+1}△ ⨆_{h∈H_{n}}(ϕ_{n}(h) + F_{n})) ≤ ∑_{h∈H_{n}} λ_{G}((c_{n+1}(h) + F_{n})△(ϕ_{n}(h) + F_{n}))

= ∑_{h∈H_{n}} λ_{G}((s_{n}(h) + F_{n})△F_{n}),

it follows from (3-1) that F_{n+1} ∼_{n+1} (ϕ_{n}(H_{n}) + F_{n}). For q ∈ Q, we set F_{n,q} := F_{n}(f_{n} − ϕ_{n}(q)) and F′_{n,q} := F_{n}(−f_{n} + ϕ_{n}(q)). Then (2-10) yields

(3-2)  F_{n} = ⨆_{q∈Q} F_{n,q} = ⨆_{q∈Q} F′_{n,q}.

It follows that

F_{n+1} − g_{n} ∼_{n+1} ⨆_{h∈H_{n}} ⨆_{q∈Q}(ϕ_{n}(h) − ϕ_{n}(h_{n}) − f_{n} + F′_{n,q})

= ⨆_{q∈Q} ⨆_{h∈H_{n}}(ϕ_{n}(h − h_{n} − q) + F_{n,q}).

Therefore

(3-3)  F_{n+1} ∩ (F_{n+1} − g_{n}) ∼_{n+1} ⨆_{q∈Q} ⨆_{h∈H_{n}(h_{n}+q)}(ϕ_{n}(h) + F_{n,q}).

Next, since λ_{G}(F_{n+1}) ≥ #H_{n}λ_{G}(F_{n}) and

\frac{λ_{G}(⨆_{q∈Q} ⨆_{h∈H_{n}(h_{n}+q)△H_{n}(h_{n})}(ϕ_{n}(h) + F_{n,q}))}{λ_{G}(F_{n+1})}

≤ \frac{∑_{q∈Q} #(H_{n}(h_{n} + q)△H_{n}(h_{n}))λ_{G}(F_{n,q})}{#H_{n}λ_{G}(F_{n})}

≤ ∑_{q∈Q} \frac{#(H_{n}(h_{n} + q)△H_{n}(h_{n}))}{#H_{n}} → 0,

it follows from (3-3) and (3-2) that

F_{n+1}(g_{n}) ∼_{n+1} ⨆_{q∈Q} ⨆_{h∈H_{n}(h_{n})}(ϕ_{n}(h) + F′_{n,q}) = ⨆_{h∈H_{n}(h_{n})}(ϕ_{n}(h) + F_{n}).

It remains to use (3-1) once more to conclude that

⨆_{h∈H_{n}(h_{n})}(ϕ_{n}(h) + F_{n}) ∼_{n+1} ⨆_{h∈H_{n}(h_{n})}(c_{n+1}(h) + F_{n}).

□

Corollary 3.4. We have

(A + C_{n+1}) ∩ F_{n+1}(g_{n}) ∼_{n+1} A + c_{n+1}(H_{n}(h_{n}))

uniformly in A ⊂ F_{n}. Hence

sup_{A⊂F_{n}} μ(([A_{n}]_{n} ∩ [F_{n+1}(g_{n})]_{n+1})△[A_{n} + c_{n+1}(H_{n}(h_{n}))]_{n+1}) = \overline{o}(1).

Now we are going to prove the main result of the section.

Theorem 3.5. Let T be a (C, F)-action associated with (H_{n}, ϕ_{n}, s_{n}, F_{n})_{n} and satisfying (2-10), (2-11) and the restricted growth condition. If M_{s′_{n},H_{n}(h_{n})} → P_{0} strongly for any sequence h_{n} ∈ H \ {0} such that lim inf_{n→∞} #H_{n}(h_{n})/#H_{n} > 0 then T is mixing. (Here s′_{n} stands for the h_{n}-derivative of s_{n}.)

Proof. Take any sequence g_{n} → ∞ in G. We are going to prove that it contains a mixing subsequence. Since F_{1}^{•} ⊂ F_{2}^{•} ⊂ ⋯ and ⋃_{n=1}^{∞} F_{n}^{•} = G, we can find p_{n} such that g_{n} ∈ F_{p_{n}+1}^{•} \ F_{p_{n}}^{•} for all n ∈ ℕ. It is clear that p_{n} → ∞ in G. Notice that

F_{n+1}^{•} + F_{n+1} ⊂ ϕ_{n+1}(Q + Q + Q_{−}) + F_{n+1}.

Then using a similar idea as in the proof of Lemma 2.7, we partition F_{p_{n}+1} as follows

F_{p_{n}+1} = ⨆_{q∈Q+Q+Q_{−}} F_{p_{n}+1}(g_{n} − ϕ_{p_{n}+1}(q)).

Take two p_{n}-cylinders A, B ⊂ X. Let A_{m} and B_{m} be the Borel subsets of F_{m} such that A = [A_{m}]_{m} and B = [B_{m}]_{m} for all m ≥ p_{n}. For q ∈ Q + Q + Q_{−}, we put A_{p_{n}+1,q} := A_{p_{n}+1} ∩ F_{p_{n}+1}(g_{n} − ϕ_{p_{n}}(q)). We have now

(3-4)  T_{g_{n}}[A_{p_{n}+1,q}]_{p_{n}+1} = T_{ϕ_{pn+1}(q)}[A_{p_{n}+1,q} + g_{n} − ϕ_{p_{n}+1}(q)]_{p_{n}+1}

and A_{p_{n}+1} = ⨆_{q∈Q+Q+Q_{−}} A_{p_{n}+1,q}. For any q ∈ Q+Q+Q_{−}, the constant sequence h_{n} := q, n ∈ ℕ satisfies (2-2). Moreover, it satisfies (2-3) since T has restricted growth (we need this to apply Corollary 2.3 later). By the condition of the theorem, M_{∂_{q}s_{n},H_{n}(q)} → P_{0} strongly whenever q ≠ 0. Hence M_{∂_{q}s_{n},H_{n}} → P_{0} strongly. Therefore by Corollary 2.3, the sequence (ϕ_{n}(q))_{n=1}^{∞} is uniformly mixing. Then it follows from (3-4) that

(3-5) sup_{A is a p_{n}-cylinder} |μ(T_{g_{n}}[A_{p_{n}+1,q}]_{p_{n}+1} ∩ B) − μ([A_{p_{n}+1,q}]_{p_{n}+1})μ(B)| → 0

for every q ∈ (Q + Q + Q_{−}) \ {0}.

It remains to consider the case q = 0 which is more involved. Let us represent g_{n} as g_{n} = ϕ_{p_{n}}(h_{n}) + f_{n} for some (uniquely determined) h_{n} ∈ H and f_{n} ∈ F_{p_{n}}. Since A_{p_{n}+1,0} = A_{p_{n}+1} ∩ F_{p_{n}+1}(g_{n}) and the restricted growth condition implies (3-1), it follows from Corollary 3.4 that

(3-6)  sup_{A in a p_{n}-cylinder} |μ([A_{p_{n}+1,0}]_{p_{n}+1}) − μ(A)\frac{#H_{p_{n}}(h_{n})}{#H_{p_{n}}} | = \overline{o}(1).

Hence, if lim inf_{n→∞} #H_{p_{n}}(h_{n})/#H_{p_{n}} = 0,

lim inf_{n→∞} sup_{A in a p_{n}-cylinder} μ([A_{p_{n}+1,0}]_{p_{n}+1}) = 0.

Therefore passing to a subsequence in (p_{n})_{n=1}^{∞} we conclude that (3-5) is also true for q = 0. This implies that the corresponding subsequence of (g_{n})_{n=1}^{∞} is mixing. Thus from now on we may assume that lim inf_{n→∞} #H_{p_{n}}(h_{n})/#H_{p_{n}} > 0. This implies h_{n} ≠ 0 eventually. It follows from Corollary 3.4 that

μ(T_{ϕ_{pn} (h_{n})}[A_{p_{n}+1,0}]_{p_{n}+1} ∩ B) = μ(T_{ϕ_{pn} (h_{n})}[A_{n} + c_{p_{n}+1}(H_{p_{n}}(h_{n}))]_{p_{n}+1} ∩ B) + \overline{o}(1),

where \overline{o}(1) does not depend on A and B. Using that together with the restricted growth condition and arguing almost literally as in the proof of Lemma 2.2 we obtain

\begin{aligned}μ(T_{ϕ_{pn} (h_{n})}[A_{p_{n}+1,0}]_{p_{n}+1} ∩ B) & = \frac{1}{#H_{p_{n}}} ∑_{h∈H_{pn} (h_{n})} μ(T_{−s′_{n}(h)}[A_{p_{n}}]_{p_{n}} ∩ B) + \overline{o}(1) \\ & = \frac{#H_{p_{n}}(h_{n})}{#H_{p_{n}}}⟨χ_{A}, M_{s′_{n},H_{pn} (h_{n})}(χ_{B})⟩ + \overline{o}(1), \end{aligned}

where s′_{n} is the h_{p_{n}}-derivative of s_{n} and \overline{o}(1) does not depend on A and B. It is easy to find a sequence (ĥ_{n})_{n=1}^{∞} with lim inf_{n→∞} #H_{n}(ĥ_{n})/#H_{n} > 0, ĥ_{p_{n}} ≠ 0 and ĥ_{p_{n}} = h_{n} for all n ∈ ℕ. Let s′_{n} denote the ĥ_{n}-derivative of s_{n}. Then s′_{n} = s′_{p_{n}}. By the condition of the theorem, M_{s′_{n},H_{n}(ĥ_{n})} → P_{0} strongly. Hence M_{s′_{n},H_{pn} (h_{n})} → P_{0} strongly. Therefore we deduce from the above and (3-6) that

(3-7)  \begin{aligned}μ(T_{ϕ_{pn} (h_{n})}[A_{p_{n}+1,0}]_{p_{n}+1} ∩ B) & = \frac{#H_{p_{n}}(h_{n})}{#H_{p_{n}}} · μ(A)μ(B) + \overline{o}(1) \\ & = μ([A_{n+1,0}]_{p_{n}+1})μ(B) + \overline{o}(1), \end{aligned}

where \overline{o}(1) does not depend on A. To complete the proof, we will use once more the ‘partition trick’. Let A_{p_{n},q} := A_{p_{n}} ∩ F_{p_{n}}(f_{n} − ϕ_{p_{n}}(q)) for q ∈ Q. Then A_{p_{n}} = ⨆_{q∈Q} A_{p_{n},q}. This partition induces a partition of A_{p_{n}+1,0}:

A_{p_{n}+1,0} = ⨆_{q∈Q}([A_{p_{n},q}]_{p_{n}})_{p_{n}+1,0}.

Notice that (A_{p_{n},q})_{p_{n}+1,0} = (A_{p_{n},q} + C_{p_{n}+1}) ∩ F_{p_{n}+1}(g_{n}) and f_{n} − ϕ_{p_{n}}(q) + A_{p_{n},q} ⊂ F_{p_{n}}. From this and (3-7) we deduce

μ(T_{g_{n}}[A_{p_{n}+1,0}]_{p_{n}+1} ∩ B) = ∑_{q∈Q} μ(T_{g_{n}}([A_{p_{n},q}]_{n})_{p_{n}+1,0} ∩ B)

= ∑_{q∈Q} μ(T_{ϕ_{pn} (h_{n})}([f_{n} − ϕ_{p_{n}}(q) + A_{p_{n},q}]_{p_{n}})_{p_{n}+1,0} ∩ B)

= ∑_{q∈Q} μ(([f_{n} − ϕ_{p_{n}}(q) + A_{p_{n},q}]_{p_{n}})_{p_{n}+1,0})μ(B) + \overline{o}(1)

= ∑_{q∈Q} μ(([A_{p_{n},q}]_{p_{n}})_{p_{n}+1,0})μ(B) + \overline{o}(1)

= μ([A_{p_{n}+1,0}]_{p_{n}+1})μ(B) + \overline{o}(1).

This plus (3-5) imply that the sequence (g_{n})_{n=1}^{∞} is mixing. □

The following material will be used in the next section to check the condition on the strong convergence from Theorem 3.5.

Definition 3.6. Let I, J be finite subsets in G and let ϵ be a non-negative number. We say that I is ϵ-tiled by J if there exists a finite set F ⊂ G such that the following are satisfied:

(i) J + F ⊂ I, (ii) J and F are independent and (iii) #(I \ (J + F)) ≤ ϵ#I.

Lemma 3.7. Let I be ϵ-tiled by J. Then for any f ∈ L^{2}(X, μ),

∥M_{I}(f)∥_{2} ≤ ∥M_{J}(f)∥_{2} + ϵ∥f∥_{2}.

Proof. Since T preserves μ, it follows that

∥M_{J+g}(f)∥_{2} = ∥M_{J}(f) ∘ T_{g}∥_{2} = ∥M_{J}(f)∥_{2}

for every g ∈ G. Let F be as in Definition 3.1. Then

∥M_{I}(f)∥_{2} ≤ \frac{#J}{#I} ∑_{g∈F} ∥M_{J+g}(f)∥_{2} + \frac{#(I \ (J + F))}{#I}∥M_{I\(J+F)}(f)∥_{2}
≤ ∥M_{J}(f)∥_{2} + ϵ∥f∥_{2}.

□

Let g ∈ G, i, j ∈ ℤ and 0 ≤ i < j. The subset {ig, (i + 1)g, …, jg} is called a g-interval.

Remark 3.8. Let V ⊂ G be a finite subset, I, J two g-intervals in G and l ∈ ℕ. If V is tiled (i.e. 0-tiled) by I then V is also \frac{l#J}{#I} -tiled by the lg-interval l · J.

15 Lemma 3.9. Let (g_{n})_{n≥1} be a sequence in G such that for some subset A ⊂ X and every l ∈ ℕ, we have μ(T_{lg_{n}}A ∩ A) → μ(A)^{2} as n → ∞. Then for any sequence of g_{n}-intervals I_{n} whose cardinality is constant, say L, the following enequality holds eventually (i.e. for all large enough n)

∥M_{I_{n}}(χ_{A}) − μ(A)∥_{2}^{2} ≤ \frac{2}{L}.

Proof. Without loss of generality we may assume that I_{n} = {0, g, …, (L − 1)g}. Now the inequality follows from the formula

∥\frac{1}{L} ∑_{i=0}^{L−1} χ_{A} ∘ T_{ig_{n}} − μ(A)∥_{2}^{2} = \frac{μ(A)}{L} + \frac{1}{L^{2}} ∑_{i≠j}(μ(T_{|i−j|g_{n}}A ∩ A) − μ(A)^{2}),

which is established by a straightforward calculation. □

Lemma 3.10. Let V_{n} be a finite subset of G which is tiled by a g_{n}-interval I_{n} and m_{n} ∈ ℕ for all positive integers n. If m_{n}/#I_{n} → 0 and μ(T_{lm_{n}g_{n}}A ∩ A) → μ(A)^{2} as n → ∞ for any integer l > 0 then M_{V_{n}}(χ_{A}) → μ(A) in L^{2}(X, μ).

Proof. For an ϵ > 0, fix an integer r > ϵ^{−1}. Let J_{n} be a g_{n}-interval of cardinality r. By Lemma 3.9, ∥M_{m_{n}J_{n}}(χ_{A}) − μ(A)∥_{2}^{2} < 2ϵ eventually in n. It follows from Remark 3.8 that V_{n} is \frac{m_{n}r}{#I_{n}} -tiled by the m_{n}g_{n}-interval m_{n}J_{n}. By Lemma 3.7,

∥M_{V_{n}}(χ_{A}) − μ(A)∥_{2} ≤ ∥M_{m_{n}J_{n}}(χ_{A}) − μ(A)∥_{2} + \frac{m_{n}r}{#I_{n}} < 3ϵ

eventually in n. □

Lemma 3.11 (Hilbertian van der Corput trick). Let (v_{h})_{h∈H} stand for a bounded family of vectors in a Hilbert space. If for any k ∈ H \ {0}, we have lim_{n→∞} \frac{1}{#H_{n}} ∑_{h∈H_{n}}⟨v_{h+k}, v_{h}⟩ = 0 then lim_{n→∞} ∥\frac{1}{#H_{n}} ∑_{h∈H_{n}} v_{h}∥ = 0.

Proof. In case H = ℤ, we refer the reader to [Be] for a short proof. Only a slight and obvious modification of that proof is needed to adapt it to the general case. □

Corollary 3.12. Given a map t : H → G, let ⟨M_{∂_{k}t,H_{n}}χ_{B}, χ_{B}⟩ → μ(B)^{2} for every nontrivial k ∈ H and all B ∈ 𝔅. Then M_{t,H_{n}} → P_{0} strongly as operators in L^{2}(X, μ).

Proof. Fix a subset B ∈ 𝔅. For any h ∈ H, we set v_{h} := χ_{B} ∘ T_{t(h)} − μ(B) ∈ L^{2}(X, μ). Then ⟨v_{h+k}, v_{h}⟩ = ⟨χ_{B} ∘ T_{∂_{k}t(h)}, χ_{B}⟩ − μ(B)^{2}. It remains to apply Lemma 3.11. □

4. Mixing rank-one actions of ℝ^{d_{1}} × ℤ^{d_{2}}

In this section we prove the main results of the paper—Theorems 4.9, 4.10 and 4.12.

Let d_{1} and d_{2} be non-negative integers such that d := d_{1} + d_{2} ≠ 0 and d_{1} ≠ 1. We set G = ℝ^{d_{1}} × ℤ^{d_{2}} and H = ℤ^{d}. For g = (g_{1}, …, g_{d}) ∈ G, we let ∥g∥_{∞} := max_{1≤i≤d} |g_{i}|. If g_{i} ≥ 0 for all i = 1, …, d we write g ≥ 0. In a similar way we define ∥h∥_{∞} and h ≥ 0 for h ∈ H.

To define a mixing (C, F)-action of G we first fix a sequence of positive integers r_{n} > 2 which goes to infinity as n → ∞. Some restrictions on its growth will be imposed later. Let s_{n}^{(1)} : H → ℝ^{d_{1}} be a usual polynomial with real coefficients and s_{n}^{(2)} : H → ℤ^{d_{2}} a usual polynomial with rational coefficients such that s_{n}^{(1)}(0) = 0 and s_{n}^{(2)}(0) = 0. Then s_{n} := (s_{n}^{(1)}, s_{n}^{(2)}) is a polynomial mapping from H to G in the sense of Definition 1.5. Let H_{n} := {h ≥ 0 | ∥h∥_{∞} < r_{n}}. We define a sequence of positive reals (a_{n})_{n≥0} recurrently by setting

a_{n+1} := the integer part of a_{n}r_{n} + max_{h∈H_{n}} ∥s_{n}(h)∥_{∞}

and choosing a_{0} arbitrarily. It is clear that a_{n+1} ≥ a_{0}r_{1} ⋯ r_{n} for all n ∈ ℕ. Since r_{n} → ∞, the sequence (a_{n})_{n=1}^{∞} grows faster than any exponent. We finally let

F_{n} := {g ≥ 0 | ∥g∥_{∞} < a_{n}},
ϕ_{n}(h) := a_{n}h for h ∈ H and
C_{n+1} := (ϕ_{n} + s_{n})(H_{n}),

for n = 0, 1, . . .. It is easy to see that (1-1)–(1-3), (1-12) and (1-13) are satisfied. In the case d = 1, the condition (1-4) is satisfied too if s_{n} increases on the half-line [0, +∞), for instance if the coefficients of s_{n} are all non-negative. The situation is more difficult when d > 1. That is why we introduce the following definition.

Definition 4.1. A mapping s : H → G is monotonic if for all non-negative x, y ∈ H with ∥x − y∥_{∞} = 1,

max\{\frac{s(x)_{i} − s(y)_{i}}{x_{i} − y_{i}} | for all 1 ≤ i ≤ d such that x_{i} ≠ y_{i}\} ≥ 0.

Example 4.2. Let s(x)_{i} = (α_{i}x_{i} + γ_{i})(x_{1} + ⋯ + x_{d}) + β_{i}x_{i}^{2} + δ_{i}x_{i} for some reals α_{i}, β_{i}, γ_{i} and δ_{i} such that α_{i} > 0, γ_{i} ≥ 0, α_{i} + 2β_{i} ≥ 0 and α_{i} + β_{i} + δ_{i} ≥ 0 for i = 1, …, d. Suppose that the mapping

H ∋ x ↦ s(x) = (s(x)_{1}, …, s(x)_{d})

takes values in G. Then it is monotonic. To show this, consider two non-negative elements x ≠ y ∈ H such that ∥x − y∥_{∞} = 1 and set z := y − x. Without loss of generality we may assume that z_{1} + ⋯ + z_{d} ≥ 0. Then there exists a coordinate j such that z_{j} = 1. By a straightforward calculation,

s(x + z)_{j} − s(x)_{j} =(α_{j}x_{j} + γ_{j}) ∑_{i=1}^{d} z_{i} + α_{j} · ∑_{i≠j}(x_{i} + z_{i})
+(α_{j} + 2β_{j})x_{j} + α_{j} + β_{j} + δ_{j} ≥ 0.

We remark that in [AdS], the following monotonic polynomial was under consideration: s(x)_{i} = x_{i}(x_{1} + ⋯ + x_{d}) − (x_{i}^{2} + x_{i})/2, i = 1, …, d in the case when G = ℤ^{d}.

Lemma 4.3. If s_{n} is monotonic and

(4-1)  max_{x∈H_{n}} ∥s_{n}(x)∥_{∞} < a_{n}/2

then (1-4) holds.

Proof. Suppose that the contrary holds. Then there exist x ≠ y ∈ H_{n} and f, f′ ∈ F_{n} such that a_{n}x + ϕ_{n}(x) + s_{n}(x) + f = a_{n}y + ϕ_{n}(y) + s_{n}(y) + f′. Since

F_{n} − F_{n} = {g ∈ G | ∥g∥_{∞} < a_{n}},

it follows that ∥a_{n}(x − y) + s_{n}(x) − s_{n}(y)∥_{∞} < a_{n}. If ∥x − y∥_{∞} ≥ 2 then we get a contradiction with (4-1). If ∥x − y∥_{∞} = 1 then we get a contradiction with the fact that s_{n} is monotonic. □

Now suppose that s_{n} = s for all n ≥ 0, where s is a monotonic polynomial of degree l > 1. Let λ_{G} stand for the direct product of the Lebesgue measure on ℝ^{d_{1}} and the ‘counting’ measure on ℤ^{d_{2}}. Let us check (1-6). Since λ_{G}(F_{n}) = a_{n}^{d} and

λ_{G}(F_{n+1} \ (F_{n} + C_{n+1})) = (r_{n}a_{n} + max_{h∈H_{n}} ∥s(h)∥_{∞})^{d} − (r_{n}a_{n})^{d},

we see that (1-6) is satisfied if and only if

∑_{n=1}^{∞} \frac{max_{h∈H_{n}} ∥s(h)∥_{∞}}{r_{n}a_{n}} < ∞.

Of course, there exists a strictly positive limit lim_{n→∞} max_{h∈H_{n}} ∥s(h)∥_{∞}/r_{n}^{l}. Thus (1-6) holds if and only if

(4-2)  ∑_{n=1}^{∞} \frac{r_{n}^{l−1}}{a_{n}} < ∞.

It is easy to see that (2-10) and (2-11) are satisfied if we set

(4-3)  Q := {h ∈ H | h ≥ 0 and ∥h∥_{∞} ≤ 1} and Q_{−} := {(−1, …, −1)}.

Lemma 4.4. Let s be monotonic and assume that (4-2) is satisfied. If

(4-4)  r_{n}^{l}/a_{n} → 0

then the (C, F)-action T associated with (H_{n}, ϕ_{n}, s, F_{n})_{n} satisfies the restricted growth condition.

Proof. Notice first that (4-4) implies (4-1) eventually. Hence (1-4) holds eventually by Lemma 4.3. It follows that the associated (C, F)-action T of G is well defined. Take a sequence (h_{n})_{n=1}^{∞} in H with lim inf_{n→∞} #H_{n}(h_{n})/#H_{n} > 0 (in fact, it suffices to have only H_{n} ∩ (H_{n} + h_{n}) ≠ ∅ eventually). It follows from (4-4) that

max_{h∈H_{n}(h_{n})} \frac{∥∂_{h_{n}}s(h)∥_{∞}}{a_{n}} ≤ max_{h,k∈H_{n}} \frac{∥s(h) − s(k)∥_{∞}}{a_{n}} → 0.

18 On the other hand, for any b ∈ G with ∥b∥_{∞} < a_{n}, we have

\frac{λ_{G}(F_{n} \ (F_{n} + b))}{λ_{G}} ≤ \frac{a_{n}^{d} − (a_{n} − ∥b∥_{∞})^{d}}{a_{n}^{d}} ≤ \frac{d∥b∥_{∞}}{a_{n}} .

Hence

\frac{1}{#H_{n}} ∑_{h∈H_{n}(h_{n})} \frac{λ_{G}(F_{n} \ F_{n}(∂_{h_{n}}s(h)))}{λ_{G}(F_{n})} ≤ max_{h∈H_{n}(h_{n})} \frac{λ_{G}(F_{n} \ (F_{n} + ∂_{h_{n}}s(h)))}{λ_{G}(F_{n})} → 0,

i.e. (2-3) holds. □

We see, in particular, that if r_{n} is of sub-exponential growth, i.e. r_{n}/ξ^{n} → 0 for some ξ > 0, then both (4-2) and (4-4) hold.

Now consider in more detail the case where s : H → G is a polynomial of degree 2. Given t ∈ H, we have ∂_{t}s(h) = ψ_{t}(h) + a_{t} for some homomorphism ψ_{t} : H → G and an element a_{t} ∈ G. It is easy to verify that ψ_{t_{1}+t_{2}}(h) = ψ_{t_{1}}(h) + ψ_{t_{2}}(h) for all t_{1}, t_{2} ∈ H. Hence the map H × H ∋ (t, h) ↦ ψ_{t}(h) ∈ G is a ‘bihomomorphic’ form. Then there is γ > 0 such that

(4-5)  ∥ψ_{t}(h)∥_{∞} ≤ γ∥t∥_{∞}∥h∥_{∞} for all t, h ∈ H.

Fix a standard ‘basis’ in G (and H):

e_{1} := (1, 0, …, 0), e_{2} := (0, 1, 0, …, 0), …, e_{d} := (0, …, 0, 1).

Then we can identify ψ_{h} with the corresponding (d × d)-matrix with real entries for any h ∈ H.

Proposition 4.5. Let the polynomial s be monotonic and (4-2) and (4-4) hold with l = 2. Moreover, suppose that

(i) the action (T_{ψ_{k}(h)})_{h∈H} is ergodic for any k ∈ H \ {0} and (ii) max_{1≤i≤d} ∥ψ_{k}(e_{i})∥_{∞} ≥ δ∥k∥_{∞} for some δ > 0 and every k ∈ H.

Then T is mixing.

Proof. By Lemma 4.4, T has restricted growth. Take a sequence (h_{n})_{n=1}^{∞} in H such that h_{n} ≠ 0 for all n ∈ ℕ and

(4-6)  lim inf_{n→∞} #H_{n}(h_{n})/#H_{n} > 0.

In view of Theorem 3.5, it suffices to show that M_{ψ_{hn} ,H_{n}(h_{n})} → P_{0} strongly. Suppose first that h_{n} = k for some k ∈ H and all n. Then just use (i) and apply the mean ergodic theorem to the action (T_{ψ_{k}(h)})_{h∈H} to conclude that M_{ψ_{k},H_{n}(k)} → P_{0}. Hence it remains to consider only the sequences (h_{n})_{n=1}^{∞} with h_{n} → ∞ in H. Let i(k) stand for an index at which the maximum in (ii) is attained. Then

(4-7)  ∥ψ_{h_{n}}(e_{i(h_{n})})∥_{∞} ≥ δ∥h_{n}∥_{∞}

Since H_{n}(h_{n}) is a parallelepiped in ℤ_{+}^{d}, there exist an element h′_{n} ∈ H, a positive integer r′_{n} ≤ r_{n} and two parallelepipeds J_{n} ⊂ ℤ_{+}^{i(h_{n})−1} and J′_{n} ⊂ ℤ_{+}^{d−i(h_{n})} such that

H_{n}(h_{n}) + h′_{n} = J_{n} × {0, 1, …, r′_{n} − 1} × J′_{n}.

It follows from (4-6) that there exist two reals δ′ > 0 and δ′′ ≤ 1 such that

(4-8)  r′_{n}/r_{n} ≥ δ′ and ∥h_{n}∥_{∞}/r_{n} ≤ δ′′ eventually.

We set V_{n} := ψ_{h_{n}}(H_{n}(h_{n})+h′_{n}), g_{n} := ψ_{h_{n}}(e_{i(h_{n})}) and I_{n} := {0, g_{n}, …, (r′_{n}−1)g_{n}}. By (4-7), ∥g_{n}∥_{∞} → ∞ (recall that h_{n} → ∞). Find p_{n} ∈ ℕ with g_{n} ∈ F_{p_{n}+1}^{•} \ F_{p_{n}}^{•}. Then, of course, p_{n} → ∞. Let m_{n} be the smallest positive integer such that m_{n}g_{n} ∉ F_{p_{n}+1}^{•}. Then

m_{n}g_{n} = (m_{n} − 1)g_{n} + g_{n} ∈ F_{p_{n}+1}^{•} + F_{p_{n}+1}^{•}.

Since F_{m}^{•} = {g ∈ G | ∥g∥_{∞} < a_{m}} for all m ∈ ℕ, we have

(4-9)  a_{p_{n}} ≤ ∥g_{n}∥_{∞} < a_{p_{n}+1} and a_{p_{n}+1} ≤ m_{n}∥g_{n}∥_{∞} < 2a_{p_{n}+1}.

Moreover, it is straightforward (see (4-3)) that

\underbrace{Q + ⋯ + Q}_{l times} = \{h ∈ H | h ≥ 0 and ∥h∥_{∞} ≤ l\} and

Q_{+} ∩ {h ∈ H | ∥h∥_{∞} = 1 and − h ≥ 0} = ∅.

Hence

−lQ_{+} ∩ (\underbrace{Q + ⋯ + Q}_{l times}) = ∅.

Furthermore, by (i) and Corollary 2.6(i), the sequence (ϕ_{n}(k))_{n=1}^{∞} is uniformly mixing for every 0 ≠ k ∈ H. Thus we may apply Lemma 2.8 and conclude that the sequence (lm_{n}g_{n})_{n=1}^{∞} is uniformly mixing for every l ∈ ℕ. Notice that

M_{ψ_{hn} ,H_{n}(h_{n})} = M_{ψ_{hn} (H_{n}(h_{n}))} = M_{V_{n}}U_{h′_{n}} = U_{h′_{n}}M_{V_{n}},

where U_{h′_{n}} in the unitary given by U_{h′_{n}}f := f ∘T_{−h_{n}}. Therefore M_{ψ_{hn} ,H_{n}(h_{n})} → P_{0} if and only if M_{V_{n}} → P_{0} strongly as n → ∞. Hence to complete the proof it remains only to establish that m_{n}/#I_{n} → 0 and apply Lemma 3.10.

By (4-5) and (4-8),

∥g_{n}∥_{∞} = ∥ψ_{h_{n}}(e_{i(h_{n})})∥_{∞} ≤ γ∥h_{n}∥_{∞}

eventually in n. Using that and (4-9) we obtain

\frac{m_{n}}{r_{n}} ≤ \frac{4γδ′′a_{p_{n}+1}}{∥g_{n}∥_{∞}^{2}} ≤ 2γδ′′ \frac{a_{p_{n}+1}}{a_{p_{n}}^{2}}

for all large enough n. It follows (use also (4-8) plus (4-2))

lim_{n→∞} \frac{m_{n}}{r′_{n}} ≤ \frac{1}{δ′} lim_{n→∞} \frac{m_{n}}{r_{n}} ≤ \frac{2γδ′′}{δ′} lim_{n→∞} \frac{a_{p_{n}+1}}{a_{p_{n}}^{2}} = \frac{2γδ′′}{δ′} lim_{n→∞} \frac{r_{p_{n}}}{a_{p_{n}}} = 0.

(Recall that lim_{n→∞} a_{n+1}/(a_{n}r_{n}) = 1.) □

Fix a family of reals ξ_{1}, …, ξ_{m}. For a nonempty subset J ⊂ {1, …, m}, we let ξ_{J} := ∏_{i∈J} ξ_{i}. We also let ξ_{∅} := 1.

Definition 4.6. If the family of reals ξ_{J}, J runs all the subsets of {1, …, m}, is independent over ℚ then we say that ξ_{1}, …, ξ_{m} is good.

It is clear that any subfamily of a good family is good. Moreover, given non-zero rationals q_{1}, …, q_{m}, the family q_{1}ξ_{1}, …, q_{m}ξ_{m} is good if and only if so is ξ_{1}, …, ξ_{m}.

We also let ℝ^{J} := {g = (g_{1}, …, g_{d}) ∈ ℝ^{d} | g_{i} = 0 for all i ∉ J}. In a similar way we define ℤ^{J}.

Lemma 4.7. Let ξ_{1}, …, ξ_{m} be a good family. Then

det ( \begin{matrix}q_{11} + ξ_{1} & q_{12} & … & q_{1m} \\ q_{21} & q_{22} + ξ_{2} & … & q_{2m} \\ ⋮ & ⋮ & ... & ⋮ \\ q_{1m} & q_{2m} & … & q_{mm} + ξ_{m} \end{matrix} ) ≠ 0

for any q_{i,j} ∈ ℚ, i, j = 1, …, m.

Proof. It is enough to notice that the determinant equals to ∑_{J⊂{1,...,m}} r_{J}ξ_{J} with some coefficients r_{J} ∈ ℚ. If the determinant vanishes then r_{J} = 0 for all J by the definition of a good family. However, it is easy to see that ξ_{{1,...,m}} = 1, a contradiction. □

We also state without proof a couple of well known facts.

Lemma 4.8.

(i) A weakly mixing action of a l.c.s.c. Abelian group is totally ergodic.

(ii) Let V and V′ be two mutually commuting actions of l.c.s.c. groups F and F′ respectively. If V is weakly mixing and V′ is ergodic then V′ is weakly mixing.

Now we are ready to prove the main results of the paper. We consider first the case G = ℝ^{d}.

Theorem 4.9. Let G = ℝ^{d}, d > 1, and (4-2) and (4-3) hold. Let s be taken from Example 4.2 and the parameters α_{i}, β_{i} satisfy the following additional conditions: α_{i} ∈ ℚ for all i = 1, …, d and the family α_{1} + 2β_{2}, …, α_{d} + 2β_{d} is good. Then the corresponding rank-one action T of G is mixing.

Proof. It is easy to calculate that

(4-10)  ψ_{e_{i}} = ( \begin{matrix}α_{1} \\ . & . & . \\ & & α_{i−1} \\ α_{i} & … & α_{i} & 2(α_{i} + β_{i}) & α_{i} & … & α_{i} \\ & & & & α_{i+1} \\ . & . & . \\ & & & & & & α_{d} \end{matrix} ) ,

where the entries outside the main diagonal and the i-th line are zero, i = 1, …, d.

We first claim that the group generated by ⋃_{i,j=1}^{d} ψ_{e_{i}}(e_{j}) is dense in G. For a real x ∈ ℝ, denote the fractional part of x by ⟨x⟩. Identify 𝕋 with the interval [0, 1) endowed with addition mod 1. Since ψ_{e_{i}}(e_{i}) = 2(α_{i} + β_{i})e_{i} and α_{i} + β_{i} ≠ 0, the map

π : G ∋ (x_{1}, …, x_{d}) ↦ (⟨\frac{x_{1}}{2(α_{1} + β_{1})}⟩, …, ⟨\frac{x_{d}}{2(α_{d} + β_{d})}⟩) ∈ 𝕋^{d}

is a quotient of G by the lattice generated by ⋃_{i=1}^{d} ψ_{e_{i}}(e_{i}). We see that

π(ψ_{e_{1}}(e_{i})) = ( \underbrace{⟨\frac{α_{1}}{2(α_{1} + β_{1})}⟩, 0, …, 0, ⟨\frac{α_{i}}{2(α_{i} + β_{i})}⟩}_{i}, 0, …, 0).

Since α_{1} + β_{1} ∉ ℚ and (α_{i} + β_{i})/(α_{1} + β_{1}) ∉ ℚ for i = 2, …, d, the group generated

by π(ψ_{e_{1}}(e_{i})) is dense in \underbrace{𝕋 × \{0\} × ⋯ × \{0\} × 𝕋}_{i} ×\{0\} × ⋯ × \{0\} ⊂ 𝕋. Thus our

first claim follows.

Now let us verify conditions (i) and (ii) from Proposition 4.5.

It follows from (4-10) that det ψ_{e_{i}} ≠ 0. Hence ψ_{e_{i}}(H) is a lattice in G. Then by Corollary 2.6(iii), the sequence (ϕ_{n}(e_{i}))_{n=1}^{∞} is uniformly mixing. Since ϕ_{n}(e_{i}) = a_{n}e_{i}, it follows that the transformation T_{e_{i}} is weakly mixing for every i = 1, …, m. By Lemma 4.8(ii), the action (T_{g})_{g∈ℝ^{J}} is also weakly mixing for any non-empty subset J ⊂ {1, …, m}. Now take any 0 ≠ t = (t_{1}, …, t_{d}) ∈ H. If det ψ_{t} ≠ 0 then ψ_{t}(H) is a lattice in G. By Corollary 2.6(ii), (T_{ψ_{t}(h)})_{h∈H} is ergodic, as desired. Consider now the second case when det ψ_{t} = 0. Since ψ_{t} = t_{1}ψ_{e_{1}} + ⋯ + t_{d}ψ_{e_{d}}, it follows from (4-10) that

ψ_{t} = ( \begin{matrix}(α_{1} + 2β_{1})t_{1} + α_{1}u & α_{1}t_{1} & … & α_{1}t_{1} \\ α_{2}t_{2} & (α_{2} + 2β_{2})t_{2} + α_{2}u & … & α_{2}t_{2} \\ ⋮ & ⋮ & ... & ⋮ \\ α_{d}t_{d} & α_{d}t_{d} & … & (α_{d} + 2β_{d})t_{d} + α_{d}u \end{matrix} ) ,

where u := ∑_{i=1}^{d} t_{i}. Let J := {i | t_{i} ≠ 0}. It follows immediately from Lemma 4.7 that there exists i ∉ J. Then for any j ≠ i, the j-th element of the i-th line of ψ_{t} is 0. Crossing out the i-th line and the i-column for all i ∉ J we obtain a (#J × #J)- matrix M and det ψ_{t} = u^{d−#J} ∏_{i∉J} α_{i} det M. It is easy to see that Lemma 4.7 is applicable to M. Hence det M ≠ 0. Since det ψ_{t} = 0, it follows that u = 0. Hence for any j ∉ J, the entire i-th line in ψ_{t} vanishes. Therefore ψ_{t}(H) ⊂ ℝ^{J}. Moreover, ψ_{t}(H) is a lattice in ℝ^{J} since det M ≠ 0. By Lemma 4.8(i), the action (T_{ψ_{t}(h)})_{h∈H} is ergodic. Thus, Proposition 4.5(i) holds. The other condition of that theorem is much easier to verify. Let ∥t∥_{∞} = |t_{j}| for some j. Take any l ≠ j. Then we have

∥ψ_{t}(e_{l})∥_{∞} ≥ α_{j}|t_{j}| ≥ min_{1≤i≤d} α_{i} · ∥t∥_{∞}.

□

Now we pass to the case G = ℤ^{d}.

Theorem 4.10. Let G = ℤ^{d} and (4-2) and (4-4) hold. Let s be taken from Example 4.2 with α_{1} = ⋯ = α_{d} = 1 and β_{1} = ⋯ = β_{d} = −0.5. Then the corresponding rank-one action T of G is mixing.

Proof. As in the proof of the previous theorem it suffices to show that the group generated by ⋃_{i,j=1}^{d} ψ_{e_{i}}(e_{j}) is (= dense in) G and the conditions (i) and (ii) of Proposition 4.5 are satisfied. The first claim is trivial since ψ_{e_{i}}(e_{i}) = e_{i} for all i = 1, …, d (see (4-10)). Now take any t = (t_{1}, …, t_{d}) ∈ G and set J(t) := {i | t_{i} ≠ 0}. We will check Proposition 4.5(i) by induction in #J(t).

If #J(t) = 1 then h = t_{i}e_{i} for some 1 ≤ i ≤ d and a non-zero integer t_{i}. Hence det ψ_{t} = t_{i}^{d}det ψ_{e_{i}} ≠ 0. It remains to apply Corollary 2.6(ii).

Suppose now that there exists p < d such that the action (T_{ψ_{t}(h)})_{h∈H} is ergodic for any t ∈ H with 1 ≤ #J(t) ≤ p. Take any t ∈ H with #J(t) = p + 1. We are going to show that the action (T_{ψ_{t}(h)})_{h∈H} is ergodic. For convenience, let us assume that J(t) = {1, …, p + 1}. (In the general case one can argue in a similar way.) Then

ψ_{t} = ( \begin{matrix}∑_{i=1}^{p+1} t_{i} & t_{1} & … & t_{1} & t_{1} & … & t_{1} \\ t_{2} & ∑_{i=1}^{p+1} t_{i} & … & t_{2} & t_{2} & … & t_{2} \\ ⋮ & ⋮ & ... & ⋮ & ⋮ & ... & ⋮ \\ t_{p+1} & t_{p+1} & … & ∑_{i=1}^{p+1} t_{i} & t_{p+1} & … & t_{p+1} \\ 0 & 0 & … & 0 & ∑_{i=1}^{p+1} t_{i} & … & 0 \\ ⋮ & ⋮ & ... & ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & … & 0 & 0 & … & ∑_{i=1}^{p+1} t_{i} \end{matrix} ) .

We consider now three possible cases. Denote by A the (p + 1) × (p + 1)-submatrix of ψ_{t} standing at the upper left corner.

Case 1. If ∑_{i=1}^{p+1} t_{i} ≠ 0 and det A ≠ 0 then det ψ_{t} = (∑_{i=1}^{p+1} t_{i})^{d−p−1}det A ≠ 0. Hence ψ_{t}(H) is of finite index in G and we are done (just apply Corollary 2.6(ii)).

Case 2. If ∑_{i=1}^{p+1} t_{i} ≠ 0 but det A = 0 then it is easy to verify that rk A ≥ 2. Hence rk ψ_{t} ≥ d − (p + 1) + 2 > d − p. Therefore

ψ_{t}(H) ∩ ℤ^{J(t)} ≠ {0}.

On the other hand, it follows from the inductive assumption and Corollary 2.6(i) that the sequence (ϕ_{n}(h))_{n=1}^{∞} is mixing for T for any non-zero h ∈ ℤe_{1} + ⋯ + ℤe_{p}. Hence the transformation T_{h} is weakly mixing for all such h (recall that ϕ_{n}(h) = a_{n}h). Thus the transformation group (T_{ψ_{t}(h)})_{h∈H} contains a weakly mixing transformation and therefore it is ergodic.

Case 3. If ∑_{i=1}^{p+1} t_{i} = 0 then it is easy to see that ψ_{t}(H) ⊂ ℤ^{J(t)}. Moreover,

det A = t_{1} ⋯ t_{p+1} det ( \begin{matrix}0 & 1 & 1 & … & 1 \\ 1 & 0 & 1 & … & 1 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ 1 & 1 & 1 & … & 0 \end{matrix} ) ≠ 0.

Hence ψ_{t}(H) is of finite index in ℤ^{J(h)}. Since the transformation T_{e_{1}} is weakly mixing, the action (T_{g})_{g∈ℤ^{J(h)}} is weakly mixing by Lemma 4.8(ii). It remains to make use of Lemma 4.8(i).

The condition (ii) of Proposition 4.5 is checked in a straightforward way (as in the proof of Theorem 4.9). □

Remark 4.11. Note that Theorem 4.10 in case γ_{i} = δ_{i} = 0 for all i (see the definition of s in Example 4.2) is exactly the main result of [AdS]. However, it was proved there completely only for G = ℤ^{2}. Roughly speaking, it was assumed implicitly in [AdS] that det ψ_{t} ≠ 0 for all t ∈ ℤ^{d} \ {0}. While being the case for d = 2, it is no longer true for d > 2 (see Cases 2 and 3 in the proof of Theorem 4.10).

Combining the arguments from Theorems 4.9 and 4.10, one can prove the following.

Theorem 4.12. Let G = ℝ^{d_{1}} × ℤ^{d_{2}} with d_{1} > 1 and (4-2) and (4-4) hold. Let s be taken from Example 4.2 and α_{i}, β_{i} satisfy the condition of Theorem 4.9 for i = 1, …, d_{1} and the condition of Theorem 4.10 for i = d_{1} + 1, …, d_{1} + d_{2}. Then the corresponding rank-one action T of G is mixing.

Now we consider a more general case where the ‘spacer mapping’ s : H → G is a polynomial of arbitrary degree l > 2. Then for any t_{1}, …, t_{l−1} ∈ H, there exist a homomorphism ψ_{t_{1},...,t_{l−1}} : H → G and an element a_{t_{1},...,t_{l−1}} ∈ G such that

∂_{t_{1}} ⋯ ∂_{t_{l−1}}s(h) = ψ_{t_{1},...,t_{l−1}}(h) + a_{t_{1},...,t_{l−1}} for all h ∈ H.

It is easy to verify that the map

\underbrace{H × ⋯ × H}_{l times} ∋ (t_{1}, …, t_{l}) ↦ ψ_{t_{1},...,t_{l−1}}(t_{l}) ∈ G

is a ‘polyhomomorphic’ form and there exists γ > 0 such that

∥ψ_{t_{1},...,t_{l−1}}∥_{∞} ≤ γ∥t_{1}∥_{∞} ⋯ ∥t_{l}∥_{∞} for all t_{1}, …, t_{l} ∈ H.

The following statement is a higher dimensional analogue of Proposition 4.5.

Proposition 4.13. Let s be a monotonic polynomial of degree l > 2 and (4-2) and (4-4) hold. Moreover, suppose that

(i) the action (T_{ψ_{t_{1},...,t_{l−1}} (h)})_{h∈H} is ergodic for any t_{1}, …, t_{l−1} ∈ H \ {0} and (ii) max_{1≤i≤d} ∥ψ_{t_{1},...,t_{l−1}}(e_{i})∥_{∞} ≥ δ_{t_{1},...,t_{l−2}}∥t_{l−1}∥_{∞} for some δ_{t_{1},...,t_{l−2}} > 0 and

all t_{1}, …, t_{l−1} ∈ H \ {0}. Then T is mixing.

Proof. (Cf. the proof of Proposition 4.5.) By Lemma 4.4, T has restricted growth. Take a sequence (h_{n})_{n=1}^{∞} in H \ {0} such that (4-6) holds. In view of Theorem 3.5, it suffices too show that M_{s′_{n},H_{n}(h_{n})} → P_{0} strongly, where s′_{n} is the h_{n}-derivative of s. Applying Corollary 3.12 l − 2 times, we obtain that the latter holds whenever

(4-11)  M_{ψ_{t_{1},...,t_{l−2},hn} ,H_{n}(h_{n})} → P_{0} strongly for any t_{1}, …, t_{l−2} ∈ H \ {0}.

Suppose first that h_{n} = k for some k ∈ H and all n. Then just use (i) and apply the mean ergodic theorem to the action (T_{ψ_{t_{1},...,t_{l−2},k}(h)})_{h∈H} to deduce (4-11).

It remains to consider the second case where h_{n} → ∞ in H. Notice first that the since M_{∂_{k}s,H_{n}(k)} → P_{0} strongly, it follows from Corollary 2.3(ii) that the sequence (ϕ_{n}(k))_{n=1}^{∞} is uniformly mixing for every k ∈ H \ {0}. The rest of the argument is almost a literal repetition of that from the proof of Theorem 4.5. □

5. Entropic properties of rank-one actions

Let G = ℝ^{d_{1}} × ℤ^{d_{2}} with arbitrary d_{1}, d_{2} ≥ 0 and T a (C, F)-action of G on a probability space (X, 𝔅, μ). We will assume that

F_{n} = {g ∈ G | g ≥ 0 and ∥g∥_{∞} ≤ a_{n}}

for a sequence a_{n} ∈ ℝ_{+}. This implies that T is rank-one by cubes. It is well known that the entropy of any rank-one ℤ-action is zero. This fact extends easily to the rank-one (by cubes) actions of any group G. However in case of higher dimensional groups, say G = ℤ^{2}, there exist rank-one (by rectangles) actions S = (S_{g})_{g∈G} of G such that the transformation S_{(1,0)} is Bernoulli (see [Ru]). Then one has h(S) = 0 but h(S_{(1,0)}) > 0. We show now that this is impossible for rank-one (by cubes) actions.

Theorem 5.1. Let T be a rank-one (by cubes) action of G. Then h(T_{g}) = 0 for each g ∈ G.

Proof. We consider only the case G = ℝ^{d} (in the general case one can argue in a similar way). Let (M_{n})_{n=1}^{∞} be a sequence of positive reals such that

(5-1)  lim_{n→∞} \frac{M_{n}}{a_{n}} = 0 and lim_{n→∞} \frac{log a_{n}}{M_{n}} = 0.

Fix g ∈ G \ {0} and take a family v_{1}, …, v_{d} of mutually orthogonal (with respect to Eucleadian inner product in ℝ^{d}) vectors such that v_{1} = g. Consider the finest partition F of G into parellelepipeds whose vertices belong to the lattice generated by v_{1}, …, v_{d}. Let F_{n} be a maximal subset of F such that for any atom A ∈ F and 0 ≤ i < M_{n}, the atom ig + A of F is contained in F_{n}. It follows from the first limit in (5-1) that

(5-2)  \frac{#F_{n} · M_{n} · λ_{G}(A)}{λ_{G}(F_{n})} → 1.

Let P_{n} be a finite partition of X consisting of the n-cylinders [ig + A]_{n}, where A runs F_{n} and i runs {0, 1, …, M_{n} − 1}, and the complement B_{n} to the union of these n-cylinders. Then

(5-3)  H(P_{n}) = −#F_{n}μ([A]_{n}) log μ([A]_{n}) − μ(B_{n}) log μ(B_{n}),

where A is an atom of F_{n}. Since μ(X \ X_{n}) → 0 and (5-2) holds, it is easy to show that the second term in the right hand side of (5-3) goes to 0. Substituting (1-11) to (5-3) and making use of (5-2) we now obtain

H(P_{n}) ≤ #F_{n} \frac{λ_{G}(A)}{λ_{G}(F_{n})} log \frac{λ_{G}(F_{n})}{λ_{G}(A)} + \overline{o}(1) ≤ \frac{2d}{M_{n}} log \frac{a_{n}}{λ_{G}(A)} + \overline{o}(1)

(notice that λ_{G}(F_{n}) = a_{n}^{d}). It follows from the second limit in (5-1) that H(P_{n}) → 0 as n → ∞. Denote by Q_{n} the finite partition ⋁_{i=0}^{M_{n}−1} T_{g}^{i}P_{n} of X. It is easy to deduce from the definition of P_{n} that the sequence (Q_{n})_{n=1}^{∞} generates 𝔅, i.e. for any B ∈ 𝔅 and n ∈ ℕ, there exists a subset Q_{n} ∈ Q_{n} such that μ(B△Q_{n}) → 0 as n → ∞. Hence

h(T_{g}) = lim_{n→∞} h(T_{g}, Q_{n}) = lim_{n→∞} h(T_{g}, P_{n}) ≤ lim_{n→∞} H(P_{n}) = 0.

□

We see, in particular, that for all mixing actions T constructed in Theorems 4.9, 4.10 and 4.12, h(T_{g}) = 0 for all g ∈ G.

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Max Planck Institute of Mathematics, Vivatsgasse 7, D53111 Bonn, GERMANY and Institute for Low Temperature Physics & Engineering of Ukrainian National Academy of Sciences, 47 Lenin Ave., Kharkov, 61164, UKRAINE

   E-mail address: danilenko@ilt.kharkov.ua

Department of Mathematics, Williams College, Williamstown, MA01267, USA

   E-mail address: csilva@williams.edu