ON ASYMMETRY OF THE FUTURE AND THE PAST FOR LIMIT SELF-JOININGS

OLEG N. AGEEV

Abstract. Let △_{T} be an off-diagonal joining of a transformation T. We construct a non-typical transformation having asymmetry between limit sets of △_{T^{n}} for positive and negative powers of T. It follows from a correspondence between subpolymorphisms and positive operators, and from structure of limit polynomial operators. We apply this technique to find all polynomial operators of degree 1 from a weak closure (in the space of positive operators on L_{2}) of powers for Chacon’s automorphism and for its generalizations.

Introduction

In [1], D. Rudolph introduced the notion of joinings. This notion turned out to be very fruitful (see [2]-[6]). It is known that every automorphism or endomorphism can be characterized both as a measure on a graph of this map, i. e. a joining or, more general, a polymorphism [7], and as an operator on L_{2}. This gives a way to study structure of joinings by operator methods, and automorphisms by joinings.

It is not difficult to show that limit sets for positive and negative powers of a transformation T are equal in the space of all automorphisms if T is rigid. If T is not rigid, then these sets are empty. Moreover, for rigid or mixing transformations (see Section 1) these limit sets also are equal in the space of all linear operators on L_{2}.

Nevertheless we prove (Theorem 2.4) that there exist rank-one transformations T with different limit sets of off-diagonal joinings for positive and negative powers of T or, in terms of operators, the sets of limit operators are different.

We apply a new approach, via limit polynomials. This approach gave recently a possibility to solve([8],[9]) old Rokhlin’s problem (see [10]-[13]) and to answer some well-known questions, for example, the paper [8] contains an answer to Katok’s question.

In Section 1 we introduce subpolymorphisms and a natural homeomorphism between the space of such measures and a subspace of positive operators on L_{2}. Note that, for any transformation T, the space of its self-joinings lies in the space of subpolymorphisms.

Theorem 2.1 describes all possible limit linear polynomials in T_{(k)} for powers of transformations T_{(k)}, where T_{(k)} are constructed in Section 1. In particular, this implies that only one linear polynomial 1/2E+1/2T is a limit of powers for classical Chacon’s transformation T (see [3],[14],[15]).

Date: April 10. 01.

1991 Mathematics Subject Classification. Primary 37A, 28D05, 28D15, 20M14, 47B65; Secondary 47A05, 47A15, 47D, 60G.

Key words and phrases. Joinings, Chacon’s automorphism, weak operator convergence.

The author was supported in part by Max Planck Institute of Mathematics, Bonn and RFBR Grants #100-15-96107, #99-01-01104.

In Section 3 we show that the future and the past for limit self-joinings can not be completely different for any automorphism. Moreover, their intersection contains an abelian semigroup that is trivial (i.e. {μ × μ}) only for mixing transformations.

1. Basic definitions and notations

Let T be a transformation defined on a non-atomic standard Borel probability space (X, F, μ). A transformation and a unitary operator on L_{2}(μ) : T f(x) = f(T x) are often called automorphisms and denoted by the same symbol T. It is clear that T is contained in the space of positive operators on L_{2}(μ) L^{+} = {U : (Uf ∈ L_{2} if f ∈ L_{2})&(Ug ≥ 0 if g ≥ 0)} equipped with the weak operator convergence. Everywhere below the identity automorphism will be denoted by E. The group of all automorphisms Aut(μ) of (X, F, μ) becomes a completely metrizable topological group when endowed with the weak convergence of transformations (T_{n} → T iff for any measurable A μ(T_{n}(A)△T (A)) → 0 as n → ∞). Note that this topology is a restriction of the weak operator topology in L^{+} to the non-closed Aut(μ). Denote by C(T) the commutant of T, i.e. the set {S ∈ Aut(μ) : ST = T S}.

Let ν be a finite Borel measure on X × X with marginal measures, say π_{1}ν and π_{2}ν, such that

(1.1) π_{i}ν(A) ≤ μ(A) for anyμ-measurable set A.

Obviously, marginal measures are μ-absolutely continuous and ∥ν∥ = ν(X×X) ≤ 1. The set of all such measures, say M(μ), is a convex compact metrizable space with respect to the topology determined by

μ_{n} → μ_{0} ⇔ μ_{n}(A × B) → μ_{0}(A × B)

for any μ-measurable sets A and B. Now we shall give the following definition.

Definition 1.1. Each measure from M(μ) is called a subpolymorphism.

Fix T ∈ Aut(μ). The set J_{c}(T, T) of c-self-joinings, i.e. T ×T -invariant elements ν of M(μ), with ∥ν∥ = c, is a closed subspace of M(μ) for any c ∈ [0, 1]. If T is ergodic, then each ν from J_{c}(T, T) has cμ as marginal measures. This gives that J(T, T) = J_{1}(T, T) is exactly the set of well-known self-joinings. Let S ∈ C(T). As usual, by an off-diagonal joining △_{S} we mean a measure from J(T, T) completely defined by △_{S}(A × B) = μ(S^{−1}A ∩ B), where A, B are any μ-measurable sets. Denote by LJ_{+}(T) and LJ_{−}(T) limit sets of △_{T^{n}} in M(μ) for all positive n and negative n respectively. It is clear that LJ_{±}(T) ⊆ J(T, T). If T is mixing, then LJ_{±}(T) consist of only one point μ × μ. It is well known that T is weakly mixing iff LJ_{±}(T) contain at least μ × μ. We say T is rigid if T^{n_{k}} → E for some sequence n_{k} → ∞.

Proposition 1.2. LJ_{−}(T) = LJ_{+}(T) for rigid T.

Proposition 1.3. LJ_{+}(T^{∗}) = LJ_{−}(T) = σLJ_{+}(T),

where σ is a flip map (i.e. σ(x, y) = (y, x)), and S^{∗} denotes the operator adjoint to S.

Corollary 1.4. LJ_{−}(T) = LJ_{+}(T) iff LJ_{+}(T) is invariant with respect to σ.

We leave simple proofs of statements above to the reader. It seems to be well known that the set of rigid transformations is a dense G_{δ}-set in Aut(μ) (see [16] about close results in different subclasses of the set of non-singular transformations). It turns out that LJ_{−}(T) = LJ_{+}(T) for a typical transformation T.

1.1. Construction of T_{(k)}. We consider the following “generalized” Chacon’s automorphisms. For each k ≥ 3, let T_{(k)} be a rank-one transformation, where each column C_{n+1} is obtained by cutting C_{n} into k subcolumns, say C_{n}(i), of equal width, placing a spacer only on the subcolumn C_{n}(k − 1) , and then stacking the subcolumn C_{n}(i + 1) on top of C_{n}(i) for 1 ≤ i < k. It is clear that T_{(3)} is exactly Chacon’s automorphism. For the column C_{n}, let h_{n} be its height and let d_{n} be a measure of its one level, where n ≥ 1.

1.2. Correspondence between positive operators and M(μ). Consider

L_{μ}^{+} = {U ∈ L^{+} : ∫_{A} U(1)dμ ≤ μ(A)

& ∫_{A} U^{∗}(1)dμ ≤ μ(A) for anyμ-measurable set A}

with a restriction of the weak operator topology to this set. Obviously,

L_{μ}^{+} = {U ∈ L^{+} : U(1) ≤ 1
&U^{∗}(1) ≤ 1 for a.e. x with respect to μ}.

Proposition 1.5. Natural correspondence given by

⟨U_{ν}f, g⟩ = ∫ f ⊗ ḡdν, (1.2)

for any f, g ∈ L_{2}(μ), defines a linear homeomorphism, say ϕ, between topological spaces M(μ) and L_{μ}^{+}.

Note that in (1.2) f and U_{ν}f are from different spaces L_{2}(μ) , but we naturally identify these spaces.

Proof. Indeed, the right part of (1.2) is a linear bounded functional for every f ∈ L_{2}(μ) because

| ∫ f ⊗ ḡdν| ≤ ∥f∥_{ν}∥g∥_{ν} = ∥f∥_{π_{1}ν}∥g∥_{π_{2}ν} ≤ ∥f∥_{μ}∥g∥_{μ}, (1.3)

where ∥h∥_{l} means a norm of h ∈ L_{2}(l). Thus U_{ν}f ∈ L_{2}(μ) for each f ∈ L_{2}(μ). Remain properties of the map ϕ are obvious.

Some properties of operators U_{ν} were considered in [17] due to Vershik for polymorphisms ν, i. e. for elements of M(μ) with exact equality in (1.1) for each A.

Remark 1.6. Clearly, ϕ(△_{T}) = T, where T ∈ Aut(μ). Also, U_{ν} commutes with T if and only if ν ∈ J_{∥ν∥}(T, T) .

Note that

∫ f ⊗ ḡdν = ∫_{X} ḡ(y)dπ_{2}ν ∫_{X} f(x)dν_{y}(x) = ∫_{X} ḡ(y)ρ_{2}(y)dμ(y) ∫_{X} f(x)dν_{y}(x),

where ρ_{2}(y) is a density of π_{2}ν with respect to μ, and ν_{y}(x) is a canonical system of conditional measures corresponding to ν. Hence, changing g to U_{ν}f in (1.2) and (1.3), we get

Corollary 1.7. L_{μ}^{+} is a compact convex metrizable space. For any subpolymorphism ν ∈ M(μ), ∥U_{ν}∥ ≤ 1, and

U_{ν}(f) = ρ_{2}(y) ∫_{X} f(x)dν_{y}(x).

Remark 1.8. The space L_{μ}^{+} is a semigroup, and closed with respect to taking parts of operators, i.e. if 0 ≤ V ≤ U and U ∈ L_{μ}^{+}, then V ∈ L_{μ}^{+}.

Remark 1.9. It is readily seen that ϕ(LJ_{+}(T)) and ϕ(LJ_{−}(T)) are T -invariant closed semigroups.

2. Limit polynomials

The following theorem completely determines the simplest limit polynomial.

Theorem 2.1. Let P_{1}[x] be the set of polynomials of degree at most 1. Then

ϕ(LJ_{+}(T_{(k)})) ∩ P_{1}[T_{(k)}] = \{\frac{1}{k − 1}E + \frac{k − 2}{k − 1}T_{(k)}\}.

Proof. In order to prove Theorem 2.1 we need for some definitions and a technical lemma. For some clarity we restrict our attention on the case k = 4. The proof for k ≠ 4 is analogous.

Let m_{i} → +∞ as i → +∞. Fix i and choose n such that h_{n} ≤ m_{i} < h_{n+1}. Consider the (n + 1)st column. Number its levels by 1, 2, ..., 4h_{n} + 1 from the base consequently. There exists i_{0} such that (3h_{n} − m_{i})th(modh_{n+1}) level belongs to C_{n}(i_{0}). Let p_{i} be a number of higher levels in C_{n}(i_{0}). Denote by B_{i}(j) the set of the top p_{i} levels of C_{n}(j) and A_{i}(j) = C_{n}(j)\B_{i}(j) (A_{i}(j) or B_{i}(j) can be empty). Define O_{i} = {x ∈ C_{n+1} : T^{r}x ∈ C_{n+1}for r = 1, 2, ..., h_{n+1}}. It is clear that on each level a measure of x from O_{i} is 2/3d_{n+1}.

Define operators

Q_{i} = χ_{A_{i}(i_{0})}T_{(4)}^{m_{i}}, R_{i} = χ_{B_{i}(i_{0})}T_{(4)}^{m_{i}},
Q′_{i} = χ_{O_{i}}Q_{i}, R′_{i} = χ_{O_{i}}R_{i}.

Note that Q_{i}, Q′_{i}R_{i}, R′_{i} are in L_{μ}^{+} because they less than T_{(4)}^{m_{i}}.

Lemma 2.2.

T_{(4)}^{m_{i}} − P_{i} → 0 as i → +∞, (2.1)

where 1. P_{i} = (4/3T_{(4)}^{−1} + 8/3E)Q_{i} + (1/3T_{(4)}^{−1} + 2E + 5/3T_{(4)})R_{i} if h_{n} ≤ m_{i} < 2h_{n}.

2. P_{i} = (1/3T_{(4)}^{−2} + 2T_{(4)}^{−1} + 5/3E)Q_{i} + (2/3T_{(4)}^{−1} + 8/3E + 2/3T_{(4)})R_{i} if 2h_{n} ≤ m_{i} < 3h_{n}.

3. P_{i} = (2/3T_{(4)}^{−2} + 8/3T_{(4)}^{−1} + 2/3E)3/2Q′_{i} + (4/3T_{(4)}^{−1} + 8/3E)3/2R′_{i} if 3h_{n} ≤ m_{i} < h_{n+1}.

Proof. It is enough to show (2.1) on pairs of functions from some dense set in L_{2}(μ). Therefore we can assume that f and g are constant, say f_{n}(j) and g_{n}(j), on each jth level of C_{n} for sufficiently large n.

Obviously, if μ(D_{i}) → 0, then χ_{D_{i}}T^{m_{i}} → 0. Thus

T_{(4)}^{m_{i}} − ∑_{j} χ_{A_{i}(j)}T_{(4)}^{m_{i}} − ∑_{j} χ_{B_{i}(j)}T_{(4)}^{m_{i}} → 0. (2.2)

Next we will calculate the connection between components of each sum in (2.2), using that T_{(4)}^{m_{i}}f has a “regular” structure on sets A_{i}(j) and B_{i}(j) and g is inde-

pendent of j. Consider the (n + 1)st column.

1. If m_{i} < 2h_{n}, then i_{0} = 2. Clearly,

⟨χ_{A_{i}(1)}T_{(4)}^{m_{i}}f, g⟩ = d_{n+1} ∑_{l=1}^{h_{n}−p_{i}} f_{n}(l + p_{i})ḡ_{n}(l) = ⟨Q_{i}f, g⟩, (2.3)

⟨χ_{B_{i}(1)}T_{(4)}^{m_{i}}f, g⟩ = d_{n+1} ∑_{l=h_{n}−p_{i}+1}^{h_{n}} f_{n}(l + p_{i} − h_{n})ḡ_{n}(l) = ⟨T_{(4)}R_{i}f, g⟩.

Here and next values of f and g can be written wrong on bases and tops of A_{i}(j) and B_{i}(j), but this fact does not essential for convergence of such operators. It is clear that

χ_{A_{i}(1)}T_{(4)}^{m_{i}} − Q_{i} → 0,
χ_{B_{i}(1)}T_{(4)}^{m_{i}} − T_{(4)}R_{i} → 0.

Analogously,

χ_{A_{i}(3)}T_{(4)}^{m_{i}} − T_{(4)}^{−1}Q_{i} → 0.

By construction of T_{(4)}, the function f(T_{(4)}^{m_{i}}x) has two values on each level from B_{i}(3), A_{i}(4), B_{i}(4). The first one is exactly at x from O_{i}, and the second one is elsewhere when one point from {x, T_{(4)}x, ..., T_{(4)}^{m_{i}}x} is outside of C_{n+1}. Thus

⟨χ_{B_{i}(3)}T_{(4)}^{m_{i}}f, g⟩ = ⟨χ_{B_{i}(3)∩O_{i}}T_{(4)}^{m_{i}}f, g⟩ + ⟨χ_{B_{i}(3)\O_{i}}T_{(4)}^{m_{i}}f, g⟩ =

\frac{2}{3}d_{n+1} ∑_{l=h_{n}−p_{i}+2}^{h_{n}} f_{n}(l + p_{i} − h_{n} − 1)ḡ_{n}(l)+

\frac{1}{3}d_{n+1} ∑_{l=h_{n}−p_{i}+3}^{h_{n}} f_{n}(l + p_{i} − h_{n} − 2)ḡ_{n}(l) = \frac{2}{3}⟨R_{i}f, g⟩ + \frac{1}{3}⟨T_{(4)}^{−1}R_{i}f, g⟩.

Hence

χ_{B_{i}(3)}T_{(4)}^{m_{i}} − (\frac{2}{3}E + \frac{1}{3}T_{(4)}^{−1})R_{i} → 0.

In the same way, we get

χ_{A_{i}(4)}T_{(4)}^{m_{i}} − (\frac{2}{3}E + \frac{1}{3}T_{(4)}^{−1})Q_{i} → 0,

χ_{B_{i}(4)}T_{(4)}^{m_{i}} − (\frac{2}{3}T_{(4)} + \frac{1}{3}E)R_{i} → 0.

6 OLEG N. AGEEV

This completes the calculation in the case 1.

2. Here i_{0} = 1. As above, we have

χ_{A_{i}(2)}T_{(4)}^{m_{i}} − T_{(4)}^{−1}Q_{i} → 0, χ_{B_{i}(2)}T_{(4)}^{m_{i}} − (\frac{2}{3}E + \frac{1}{3}T_{(4)}^{−1})R_{i} → 0,

χ_{A_{i}(3)}T_{(4)}^{m_{i}} − (\frac{2}{3}T_{(4)}^{−1} + \frac{1}{3}T_{(4)}^{−2})Q_{i} → 0, χ_{B_{i}(3)}T_{(4)}^{m_{i}} − (\frac{2}{3}E + \frac{1}{3}T_{(4)}^{−1})R_{i} → 0,

χ_{A_{i}(4)}T_{(4)}^{m_{i}} − (\frac{2}{3}E + \frac{1}{3}T_{(4)}^{−1})Q_{i} → 0, χ_{B_{i}(4)}T_{(4)}^{m_{i}} − (\frac{2}{3}T_{(4)} + \frac{1}{3}E)R_{i} → 0.

3. In this case, i_{0} = 4, and f(T_{(4)}^{m_{i}}x) has two values on each level, except for from A_{i}(1). Using

⟨Q′_{i}f, g⟩ = \frac{2}{3}d_{n+1} ∑_{l=1}^{h_{n}−p_{i}} f_{n}(l + p_{i})ḡ_{n}(l),

calculate

⟨χ_{A_{i}(1)}T_{(4)}^{m_{i}}f, g⟩ = d_{n+1} ∑_{l=1}^{h_{n}−p_{i}} f_{n}(l + p_{i} − 1)ḡ_{n}(l) = \frac{3}{2}⟨T_{(4)}^{−1}Q′_{i}f, g⟩,

⟨χ_{A_{i}(4)}T_{(4)}^{m_{i}}f, g⟩ = ⟨χ_{A_{i}(4)∩O_{i}}T_{(4)}^{m_{i}}f, g⟩ + ⟨χ_{A_{i}(4)\O_{i}}T_{(4)}^{m_{i}}f, g⟩ =

⟨Q′_{i}f, g⟩ + \frac{1}{3}d_{n+1} ∑_{l=1}^{h_{n}−p_{i}} f_{n}(l + p_{i} − 1)ḡ_{n}(l) =

⟨Q′_{i}f, g⟩ + \frac{1}{2}⟨T_{(4)}^{−1}Q′_{i}f, g⟩.

For j = 2, 3, we get

⟨χ_{A_{i}(j)}T_{(4)}^{m_{i}}f, g⟩ = \frac{2}{3}d_{n+1} ∑_{l=1}^{h_{n}−p_{i}} f_{n}(l + p_{i} − 1)ḡ_{n}(l)+

\frac{1}{3}d_{n+1} ∑_{l=1}^{h_{n}−p_{i}} f_{n}(l + p_{i} − 2)ḡ_{n}(l) = ⟨T_{(4)}^{−1}Q′_{i}f, g⟩ + \frac{1}{2}⟨T_{(4)}^{−2}Q′_{i}f, g⟩.

As before, for any j

⟨χ_{B_{i}(j)}T_{(4)}^{m_{i}}f, g⟩ = \frac{2}{3}d_{n+1} ∑_{l=h_{n}−p_{i}+2}^{h_{n}} f_{n}(l + p_{i} − h_{n} − 1)ḡ_{n}(l)+

\frac{1}{3}d_{n+1} ∑_{l=h_{n}−p_{i}+3}^{h_{n}} f_{n}(l + p_{i} − h_{n} − 2)ḡ_{n}(l) = ⟨R′_{i}f, g⟩ + \frac{1}{2}⟨T_{(4)}^{−1}R′_{i}f, g⟩.

1. Fix P = aE + bT_{(4)}. Let T_{(4)}^{m_{i}} → P for some m_{i} → +∞. It is clear that a, b ≥ 0, because P ∈ L_{μ}^{+}. Choose a subsequence of m_{i} (if necessary) such that

Q_{i}^{(x_{i})} → Q, (2.4)

R_{i}^{(x_{i})} → R,

where (x_{i}) means nothing or prime for each i, and (x_{i}) is completely defined by m_{i} as in Lemma 2.2. Obviously, Q, R ∈ L_{μ}^{+}. By construction of T_{(4)},

∥T_{(4)}χ_{D_{i}} − χ_{D_{i}}∥_{μ}^{2} ≤ 2d_{n+1} → 0,

where D_{i} is A_{i}(i_{0}) or A_{i}(i_{0}) ∩ O_{i}. Thus

∥(T_{(4)}Q_{i}^{(x_{i})} − Q_{i}^{(x_{i})}T_{(4)})f∥_{μ}^{2} ≤ ∥T_{(4)}χ_{D_{i}} − χ_{D_{i}}∥_{μ}∥T_{(4)}^{2}f∥_{μ} → 0,

for any f ∈ L_{2}(μ). Therefore Q commutes with T_{(4)}. Analogously, we have that R commutes with T_{(4)}. Denote by P_{1}^{+}[x] the subset of P_{1}[x] with non-negative coefficients.

Next we will show that if P = ∑_{i=1}^{l} S_{i}, where S_{i} ∈ L_{μ}^{+}, and S_{i} commute with T_{(4)}, then S_{i} ∈ P_{1}^{+}[T_{(4)}]. Indeed, measures ϕ^{−1}S_{i} are T_{(4)} × T_{(4)}- invariant and absolutely continuous with respect to the subpolymorphism ϕ^{−1}P = a△_{E} +b△_{T_{(4)}}. The transformation T_{(4)} × T_{(4)} is ergodic for measures △_{E} and △_{T_{(4)}}. Hence every T_{(4)}×T_{(4)}-invariant part of the measure △_{E} (△_{T_{(4)}}) is c△_{E} (c△_{T_{(4)}}) for some c > 0. This gives ϕ^{−1}S_{i} = a_{i}△_{E} + b_{i}△_{T_{(4)}} for some a_{i}, b_{i} ≥ 0.

¿From Lemma 2.2 and (2.4) we have that

P = UQ + V R, (2.5)

where (U, V) is at least one of the following pairs (4/3T_{(4)}^{−1} + 8/3E, 1/3T_{(4)}^{−1} + 2E + 5/3T_{(4)}), (1/3T_{(4)}^{−2}+2T_{(4)}^{−1}+5/3E, 2/3T_{(4)}^{−1}+8/3E+2/3T_{(4)}), (T_{(4)}^{−2}+4T_{(4)}^{−1}+E, 2T_{(4)}^{−1}+ 4E). In any case P contains c_{1}Q and c_{2}R as parts. Thus Q, R ∈ P_{1}^{+}[T_{(4)}]. Obviously,

1 = ⟨T_{(4)}^{m_{i}}1, 1⟩ → ⟨P 1, 1⟩. (2.6)

This implies that P ≠ 0. Therefore (2.5) is possible only when P = (4/3T_{(4)}^{−1} + 8/3E)cT_{(4)}. It remains to mention, using (2.6), that c = 1/4.

2. The proof that 1/3E + 2/3T_{(4)} ∈ ϕ(LJ_{+}(T_{(4)})) is almost obvious. Namely, consider m_{i} = h_{n}. We obtain i_{0} = 2, p_{i} = 0, R_{i} = 0. Thus (2.3) gives

⟨Q_{i}f, g⟩ = d_{n+1} ∑_{l=1}^{h_{n}} f_{n}(l)ḡ_{n}(l) = \frac{d_{n}}{4} ∑_{l=1}^{h_{n}} f_{n}(l)ḡ_{n}(l) = \frac{1}{4}⟨f, g⟩.

This implies that

⟨P_{i}f, g⟩ = ⟨(\frac{1}{3}T_{(4)}^{−1} + \frac{2}{3}E)f, g⟩.

Thus Theorem 2.1 follows from Lemma 2.2 and Remark 1.9.

Remark 2.3. By the same argument as in [3], it is not difficult to show that T_{(k)} have minimal self-joinings. Then Q, R can be written in the following form

α ∫ + ∑_{j} a_{j}T_{(k)}^{j},

where ∫ is an orthogonal projection onto the space of constants, and 0 ≤ α, 0 ≤ a_{i}. This gives that the first part of Theorem 2.1 also follows directly from (2.5).

Our main result is the following.

Theorem 2.4. LJ_{+}(T_{(k)}) ≠ LJ_{−}(T_{(k)}) for k > 3.

Proof. Indeed, it is clear that

ϕ(LJ_{+}(T^{∗})) = {U_{ν}^{∗} : ν ∈ LJ_{+}(T)}.

Therefore, using Proposition 1.3, Remark 1.9, and Theorem 2.1, we have

ϕ(LJ_{−}(T_{(k)})) ∩ P_{1}[T_{(k)}] = \{\frac{k − 2}{k − 1}E + \frac{1}{k − 1}T_{(k)}\},

and Theorem 2.4 is proved.

3. Closing remarks

Proposition 3.1. For any T ∈ Aut(μ)

LJ_{+}(T) ∩ LJ_{−}(T) ≠ ∅.

This is an immediate consequence of the next proposition.

Proposition 3.2.

ϕ(LJ_{+}(T)) ∩ ϕ(LJ_{−}(T)) ⊇ {T_{+}T_{−} : T_{±} ∈ ϕ(LJ_{±}(T))}, (3.1)

and

{T_{+}T_{−} : T_{±} ∈ ϕ(LJ_{±}(T))} = {∫ } ⇔ T is mixing,

where ∫ is defined as in Remark 2.3.

Proof. Fix T_{±} ∈ ϕ(LJ_{±}(T)), and n_{i}, k_{i} → +∞ such that T^{k_{i}} → T_{+}, T^{−n_{i}} → T_{−}. Consider also a dense set of functions from L_{2}(μ), say f_{l}. For each ϵ > 0 and m ∈ ℕ, choose i such that

|⟨T^{−n_{i}}f_{l_{1}}, T_{+}^{∗}f_{l_{2}}⟩ − ⟨T_{−}f_{l_{1}}, T_{+}^{∗}f_{l_{2}}⟩| < ϵ,

for all l_{1}, l_{2} ≤ m. Finally choose j = j(i) such that k_{j(i)} − n_{i} > m and

|⟨T^{k_{j(i)}}T^{−n_{i}}f_{l_{1}}, f_{l_{2}}⟩ − ⟨T_{+}T^{−n_{i}}f_{l_{1}}, f_{l_{2}}⟩| < ϵ,

for all l_{1}, l_{2} ≤ m . Thus

|⟨T^{k_{j(i)}−n_{i}}f_{l_{1}}, f_{l_{2}}⟩ − ⟨T_{−}f_{l_{1}}, T_{+}^{∗}f_{l_{2}}⟩| < 2ϵ.

Therefore T^{k_{j(i)}−n_{i}} → T_{+}T_{−}, where k_{j(i)} − n_{i} → +∞.

Operators T_{+} and T_{−} belong to the von Neumann algebra generated by T. Thus T_{+}T_{−} = T_{−}T_{+}. This implies that arguing as above, we see that T^{−n_{j′(i)}+k_{i}} → T_{+}T_{−}, where −n_{j′(i)} + k_{i} → −∞.

The second part of the proof is more or less standard. Indeed, if T is not mixing, then there exists T_{+} from ϕ(LJ_{+}(T))\{∫ }. Thus T_{−} = T_{+}^{∗} ∈ ϕ(LJ_{−}(T))\{∫ }. Next for S^{∗}S = ∫, where S ∈ ϕ(J(T, T)), we have

∫ fdμ = 0 ⇒ ⟨Sf, Sf⟩ = ⟨S^{∗}Sf, f⟩ = ⟨∫ f, f⟩ = 0.

This means that Sf = 0, and finally S = ∫. Therefore the operator T_{+}T_{−} = T_{−}T_{+} = T_{+}^{∗}T_{+} is not ∫.

Remark 3.3. Obviously, in (3.1) we have exact equality if T is rigid or mixing. However, taking into account Remark 2.3 , the operator 1/2E + 1/2T can not be represented as T_{+}T_{−} for Chacon’s transformation T. This yields that, in general, the left part of (3.1) is different from the right part.

References

1. D. Rudolph, An example of a measure preserving map with minimal self-joinings, and applications, J. Analyse Math. 35 (1979), 97–122.

2. M. Ratner, Horocycle flows, joinings and rigidity of products, Ann. of Math.(2) 118 (1983), 277–313.

3. A del Junco, M. Rahe, and L. Swanson, Chacon’s automorphism has minimal self-joinings, J. Analyse Math. 37 (1980), 276–284.

4. J. King, The commutant is the weak closure of the powers, for rank-1 transformations, Erg. Theory and Dyn. Sys. 6 (1986), 363–384.

5. B. Host, Mixing of all orders and pairwise independent joinings of systems with singular spectrum, Isr. J. Math. 76 (1991), 289–298.

6. A del Junco and D. Rudolph, On ergodic actions whose self-joinings are graphs, Erg. Theory and Dyn. Sys. 7 (1987), 531–557.

7. A. Vershik, Dynamic theory of growth in groups: entropy, boundaries, examples, Uspekhi Mat. Nauk 55 (2000), 59–128; English transl., Russian Math. Surv. 55 (2000), 667–733.

8. O. Ageev, On ergodic transformations with homogeneous spectrum, J. Dynam. Control Systems 5 (1999), 149–152.

9. V. Ryzhikov, Transformations having homogeneous spectra, J. Dynam. Control Systems 5 (1999), 145–148.

10. A. Katok, Ja. Sinai, and A. Stepin, The theory of dynamical systems and general transformation groups with invariant measure, Mathematical analysis, Vol. 13 (Russian), pp. 129–262. (errata insert) Akad. Nauk SSSR VINITI, Moscow, 1975.

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

12. E. Robinson, Jr. Ergodic measure preserving transformations with arbitrary finite spectral multiplicities, Invent. Math. 72 (1983), 299–314.

13. O. Ageev, The spectral multiplicity function and geometric representations of interval exchange transformations, Math. Sb. 190 (1999), 3–28; English transl., Sb. Math. 190 (1999), 1–28.

14. D. Rudolph, Fundamentals of measurable dynamics. Ergodic theory on Lebesgue spaces, The Clarendon Press, Oxford University Press, New York, 1990.

15. O. Ageev, The spectrum of Cartesian powers for the classical automorphisms, Mat. Zametky 68 (2000), 643–647, in Russian.

16. O. Ageev, C. Silva, Genericity of rigidity and multiple recurrence for infinite measure preserving and nonsingular transformations, preprint.

17. M. Lemańczyk, B. Host, J.-P. Thouvenot, Gaussian automorphisms whose ergodic selfjoinings are Gaussian, Fund. Math. 164 (2000), 253–293.

Department of Mathematics, Moscow State Technical University, 2nd Baumanscaya St. 5, 107005 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