TRANSITIVE SYSTEMS WITH ZERO SEQUENCE

ENTROPY AND SEQUENCE ENTROPY PAIRS

Wen Huang^{1}, Simin Li^{2}, Song Shao^{3} and Xiangdong Ye^{4}

Abstract.

A measure-preserving transformation (resp. a topological system) is null if the metric (resp. topological) sequence entropy is zero for any sequence. Kushnirenko showed that an ergodic measure-preserving transformation T has discrete spectrum if and only if it is null. We prove that for a minimal system the above statement remains true modulo an almost one-to-one extension, i.e. if a minimal system (X, T) is null, then (X, T) is an almost one-to-one extension of an equicontinuous system. It allows us to show that a scattering system is disjoint from any null minimal system. Moreover, we show that if a transitive non-minimal system (X, T) is null then there are non-empty open sets U and V of X such that N(U, V) = {n ∈ ℤ_{+} : U ∩T^{−n}V ≠ ∅} has zero upper Banach density. Examples of null minimal systems which are not equicontinuous exist.

Localizing the notion of sequence entropy, we define sequence entropy pairs and show that there is a maximal null factor for any system. Meanwhile, we define a weaker notion, namely weak mixing pairs. It turns out that a system is weakly mixing if and only if any pair not in the diagonal is a sequence entropy pair if and only if the same holds for a weak mixing pair, answering Question 3.9 in [BHM] by Blanchard, Host and Maass. For a group action we show that the factor induced by the smallest invariant equivalence relation containing weak mixing pairs is equicontinuous, supplying another proof concerning regionally proximal relation. Furthermore, for a minimal distal system the set of sequence entropy pairs coincides with the regionally proximal relation and thus a non-equicontinuous minimal distal system is not null.

§1 Introduction

By a (topological) dynamical system we mean a compact metric space X with a continuous map T from X onto X. Kushnirenko showed that an ergodic measurepreserving transformation T has discrete spectrum if and only if it is null. Thus a natural question is what the situation is in the topological setting?

By the known Halmos and von Neumann theorem a (topologically) transitive system has topological discrete spectrum if and only if it is minimal and equicontinuous. Note that it is easy to show that an equicontinuous system is null. It is natural to

Key words and phrases. Minimality, sequence entropy, weak mixing, null system. 1991 Mathematics Subject Classification. Primary 54H20, 58F03.

Project Supported by National Key Project (973) and One Hundrad Talents Project (Academy Sinica)

conjecture that if a minimal system is null then it is equicontinuous. Unfortunately it is not the case. In [G] Goodman obtained a minimal non-equicontinuous system which is null, and the question in what extend Kushinirenko’s result is valid remains open. We try to answer the question in the present paper.

We prove that for a minimal system the statement of Kushnirenko remains true modulo an almost one to one extension, i.e. if a minimal system (X, T) is null, then (X, T) is an almost one to one extension of an equicontinuous system, which is uniquely ergodic and has discrete spectrum with respect to the unique measure (in view of the general structure theorem of minimal systems, it is the best result we can get). It allows us to show that a scattering system is disjoint from any null minimal system. Moreover, we show that if a transitive non-minimal system (X, T) is null then there are non-empty open sets U and V of X such that

N(U, V) = {n ∈ ℤ_{+} : U ∩ T^{−n}V ≠ ∅} has zero upper Banach density.

Localizing the notion of sequence entropy, we define sequence entropy pairs and show that there is a maximal null factor for each system. Meanwhile, we define a weaker notion, namely weak mixing pairs. It turns out that a system is weakly mixing if and only if any pair not in the diagonal is a sequence entropy pair if and only if the same holds for a weak mixing pair, answering Question 3.9 by Blanchard, Host and Maass in [BHM]. For a group action we show that the factor induced by the smallest invariant equivalence relation containing weak mixing pairs is equicontinuous, supplying another proof concerning regionally proximal relation. Furthermore, for a minimal distal system the set of sequence entropy pairs coincides with the regionally proximal relation and thus a non-equicontinuous minimal distal system is not null.

The text is organized as follows. In Section 2, we give necessary definitions and some general results. To study null minimal systems we find that we need to study null transitive non-minimal systems and this is carried out in Section 3. In Section 4, we obtain the results on null minimal systems. Finally in section 5 we discuss some properties of null systems.

Acknowledgement: A pre-final version of the paper was written when the first and the fourth authors attended the dynamical year activity in ICTP. We thank ICTP (Italy) and NNSF (China) for the financial support. The second author is supported in part by the JSPS Postdoctoral Fellowship. We thank Francois Blanchard for some interesting questions which gave the first motivation of the research and Sergiy Kolyada for supplying reference [KST]. We also thank Alejandro Maass for useful comments relating section 5. Main results of the paper were presented in the spring conference in Maryland (2002) by the fourth author who thanks Joe Auslander and Tomasz Downarowicz for clarifying some points.

§2 Preliminary and basic properties

The topological entropy of T with respect to a finite open cover α is denoted by h(T, α). For an increasing sequence A = {t_{1}, t_{2}, … } of ℤ_{+} the sequence entropy of T with respect to A and a finite open cover α is

h_{A}(T, α) = lim sup_{n⟶+∞} \frac{1}{n} log N(∨_{i=1}^{n}T^{−t_{i}}α)

and the sequence entropy of T with respect to A is h_{A}(T) = sup_{α} h_{A}(T, α).

Definition 2.1. Let (X, T) be a dynamical system and X^{(2)} = X × X.

(1) We say that (x_{1}, x_{2}) ∈ X^{(2)} is a sequence entropy pair if x_{1} ≠ x_{2} and if whenever U_{i} are closed mutually disjoint neighborhoods of points x_{i}, i = 1, 2, there exists a sequence A ⊂ ℤ_{+} such that h_{A}(T, {U_{1}^{c}, U_{2}^{c}}) > 0.

(2) We say (X, T) has sequence uniform positive entropy (for short s.u.p.e.), if every point (x_{1}, x_{2}) ∈ X^{(2)}, not on the diagonal △_{X} = {(x, x) : x ∈ X} is a sequence entropy pair.

Remark 2.2. It is easy to see that (X, T) has s.u.p.e. if and only if for any cover R = {U, V } of X by two non-dense open sets, one has h_{A}(T, R) > 0 for some sequence A ⊂ ℤ_{+}.

Denote by SE(X, T) the set of all sequence entropy pairs, and by SE′(X, T) its closure. The notion of entropy pairs (denoted by E(X, T) the set of entropy pairs) was introduced by Blanchard in [B2] and a uniform positive entropy (u.p.e.) system, i.e. any pair which is not on the diagonal is an entropy pair, was first studied in [B1], which was used to study the topological analogue of K-system. In [BHM] the authors introduced the notion of complexity pairs (denoted by Com(X, T) the set of complexity pairs). Recall that the regionally proximal relation RP (X, T) is the set of all points (x_{1}, x_{2}) ∈ X^{(2)} such that for each ϵ > 0 and each open neighborhood U_{i} of x_{i}, i = 1, 2 there are x′_{i} ∈ U_{i}, i = 1, 2 and n ∈ ℕ with d(T^{n}(x′_{1}), T^{n}(x′_{2})) < ϵ. By [BHM], [HY2] and easy observations we have

E(X, T) ⊂ SE(X, T) ⊂ Com(X, T) ⊂ RP (X, T).

Moreover, the sequence entropy pairs share several important properties as entropy pairs and complexity pairs (the proofs are similar, see [B2]). Thus we have

Proposition 2.3. Let (X, T) be a dynamical system.

(1) If there is an open cover R = {U, V } of X with h_{A}(T, R) > 0 for some sequence A ⊂ ℤ_{+}, then there are points x ∈ U^{c} and y ∈ V^{c} such that (x, y) is a sequence entropy pair.

(2) If h_{A}(T) > 0 for some sequence A ⊂ ℤ_{+}, then SE′(X, T) is a non-empty closed T × T -invariant subset of X^{(2)} containing only sequence entropy pairs and points of Δ_{X}.

(3) Let π : (Y, S) ⟶ (X, T) be a factor map.

(a) If (x_{1}, x_{2}) ∈ SE(X, T), then there exists (y_{1}, y_{2}) ∈ SE(Y, S) with π(y_{1}) = x_{1} and π(y_{2}) = x_{2};

(b) If (y_{1}, y_{2}) ∈ SE(Y, S) and π(y_{1}) ≠ π(y_{2}), then (π(y_{1}), π(y_{2})) ∈ SE(X, T).

(4) Suppose W is a closed T -invariant subset of (X, T). Then if (x_{1}, x_{2}) is a sequence entropy pair of (W, T |_{W}), it is also a sequence entropy pair of (X, T).

By the above proposition we know that a system (X, T) is null if and only if SE(X, T) = ∅.

For a dynamical system (X, T), we say (X, T) is transitive if for each pair of nonempty open subsets U and V of X there is n ∈ ℕ such that U ∩ T^{−n}(V) ≠ ∅; (X, T) is weakly mixing if (X^{(2)}, T × T) is transitive. For a transitive system, x ∈ X is a transitive point if the forward orbit of x is dense and the set of transitive points forms a dense G_{δ} set. If the set of transitive points is the whole space X, we then say that (X, T) is minimal and in this case each point in X is a minimal point.

It is known ([P] and [Ba]) that (X, T) is not weakly mixing if and only if there are two non-empty open sets U and V such that there is no n ∈ ℕ with that U ∩T^{−n}U ≠ ∅ and U ∩ T^{−n}V ≠ ∅. Motivated by the above property we have

Definition 2.4. For a dynamical system (X, T), (x_{1}, x_{2}) ∈ X^{(2)} \ Δ_{X} is a weak mixing pair if for any open neighborhood U_{i} of x_{i}, i = 1, 2, one has N(U_{1}, U_{1}) ∩ N(U_{1}, U_{2}) ≠ ∅, where N(U, V) = {n ∈ ℤ_{+} : U ∩ T^{−n}(V) ≠ ∅}. The set of weak mixing pairs is denoted by W M(X, T).

Sequence entropy pairs and weak mixing pairs are related by the following

Proposition 2.5. For a dynamical system (X, T), (1) SE(X, T) ⊂ W M(X, T).

(2) W M(X, T) ⊂ RP (X, T^{−1}), if T is invertible.

Proof. For the proof of 1 we follow [BHM]. Assume that x_{1} ≠ x_{2} and (x_{1}, x_{2}) ∉ W M(X, T). Then there are open neighborhoods U′_{i} of x_{i}, i = 1, 2 such that N(U′_{1}, U′_{1}) ∩N(U′_{1}, U′_{2}) = ∅. Take closed neighborhoods U_{i} of x_{i} with U_{i} ⊂ U′_{i}, i = 1, 2 and U_{1} ∩ U_{2} = ∅. Clearly, N(U_{1}, U_{1}) ∩ N(U_{1}, U_{2}) = ∅. Therefore U_{1} ∩ T^{−n}U_{1} = ∅ or U_{1} ∩ T^{−n}U_{2} = ∅ for any n ∈ ℤ_{+}. Moreover, for any n ∈ ℤ_{+} we can take W_{n} = U_{1}^{c} or U_{2}^{c} with U_{1} ⊂ T^{−n}W_{n}.

Let R = {U_{1}^{c}, U_{2}^{c}}. For n ∈ ℕ and any sequence A = {t_{1}, t_{2}, ⋯ } with 0 ≤ t_{1} < t_{2} < ⋯, consider for each x ∈ X the first i ∈ {1, 2, ⋯, n} such that T^{t_{i}}x ∈ U_{1},

when there exists one. We get that the R_{n} = ⋁_{i=1}^{n} T^{−t_{i}}R admits a subcover by the

sets

T^{−t_{1}}U_{1}^{c} ∩ ⋯ ∩ T^{−t_{i−1}}U_{1}^{c} ∩ T^{−t_{i}}W_{0} ∩ T^{−t_{i+1}}W_{t_{i+1}−t_{i}} ∩ ⋯ ∩ T^{−t_{n}}W_{t_{n}−t_{i}},

i = 1, 2, ⋯, n, n + 1. Hence for all n ∈ ℕ, H(R_{n}) ≤ n + 1 and therefore h_{A}(T, R) = 0. This implies that (x_{1}, x_{2}) ∉ SE(X, T) and we have proved that SE(X, T) ⊂ W M(X, T).

Now let (x_{1}, x_{2}) ∈ W M(X, T) and assume that T is invertible. Then for any open neighborhood U_{i} of x_{i}, i = 1, 2, one has N(U_{1}, U_{1}) ∩ N(U_{1}, U_{2}) ≠ ∅. Thus there exists n ∈ ℤ_{+} such that U_{1} ∩ T^{−n}U_{1} ≠ ∅ and U_{1} ∩ T^{−n}U_{2} ≠ ∅. This implies that there are x′_{1} ∈ U_{1} and x′_{2} ∈ U_{2} with T^{−n}x′_{1}, T^{−n}x′_{2} ∈ U_{1}. Thus, we have (x_{1}, x_{2}) ∈ RP (X, T^{−1}) \ Δ(X). □

In Section 3 and Section 4 we will give conditions under which a weak mixing pair is a sequence entropy pair. To show Theorem 2.7 we need

Proposition 2.6. Let (X, T) be a dynamical system and (x_{1}, x_{2}) ∈ X^{(2)} \ Δ_{X}. If for any neighborhood U_{i} of x_{i} (i = 1, 2), there is a sequence A = {t_{1}, t_{2}, … } of ℤ_{+} such that for any n ∈ ℕ and any s = (s(1), …, s(n)) ∈ {1, 2}^{n}, ⋂_{i=1}^{n} T^{−t_{i}}U_{s(i)} ≠ ∅, then (x_{1}, x_{2}) ∈ SE(X, T).

Proof. Let U_{i} be closed mutually disjoint neighborhood of x_{i}, i = 1, 2. By the assumption, we know that there exists a sequence 0 ≤ t_{1} < t_{2} < t_{3} < ⋯ such that

for any n > 0 and s ∈ {1, 2}^{n}, we can find x_{s} ∈ ⋂_{i=1}^{n} T^{−t_{i}}U_{s(i)}.

Let A = {t_{1}, t_{2}, t_{3}, · · · · · · } and let X_{n} = {x_{s} : s ∈ {1, 2}^{n}}. Note that for every

s ∈ {1, 2}^{n}, one has | ⋂_{i=1}^{n} T^{−t_{i}}U_{s(i)}^{c} ∩ X_{n}| = 1. Combining the fact and |X_{n}| = 2^{n},

one gets N( ⋁_{i=1}^{n} T^{−t_{i}}U) = 2^{n}, where U = {U_{1}^{c}, U_{2}^{c}}.

Hence h_{A}(T, U) = lim sup_{n→+∞} \frac{1}{n}H( ⋁_{i=1}^{n} T^{−t_{i}}U) = log 2 and thus (x_{1}, x_{2}) ∈ SE(X, T).

□

With the above preparation we can prove now

Theorem 2.7. Let (X, T) be a dynamical system. Then the following statements are equivalent:

(1) (X, T) is weakly mixing.

(2) W M(X, T) = X^{(2)} \ Δ_{X}.

(3) (X, T) is s.u.p.e..

Proof. By Proposition 2.5, (3) implies (2) and by [P] or [Ba] (2) implies (1). It remains to show (1) implies (3).

Now let (X, T) be weakly mixing. Then we have the following fact (*): for nonempty open sets U_{1}, U_{2}, V_{1} and V_{2} of X there are non-empty open sets U_{3} and V_{3} such that N(U_{1}, V_{1}) ∩ N(U_{2}, V_{2}) ⊃ N(U_{3}, V_{3}).

Let (x_{1}, x_{2}) ∈ X^{(2)} \ Δ and U_{i} be an open neighborhood of x_{i}, i = 1, 2. We are going to find 0 < t_{1} < t_{2} < … such that for each n ∈ ℕ and s ∈ {1, 2}^{n}, U_{s(0)} ∩ T^{−t_{1}}U_{s(1)} ∩ … ∩ T^{−t_{n−1}}U_{s(n−1)} ≠ ∅.

By the fact (*) there is t_{1} ∈ N(U_{1}, U_{1}) ∩ N(U_{1}, U_{2}) ∩ N(U_{2}, U_{1}) ∩ N(U_{2}, U_{2}). Assume that we have found 0 < t_{1} < … < t_{l} with the required property. By the fact (*)

⋂_{s∈{1,2}^{l+1},i=1,2} N(U_{s(0)} ∩ T^{−t_{1}}U_{s(1)} ∩ … ∩ T^{−t_{l}}U_{s(l)}, U_{i}) ≠ ∅.

Let t_{l+1} be any number in the above intersection with t_{l+1} > t_{l}. Then t_{1}, …, t_{l+1} have the required property.

By Proposition 2.6, (x_{1}, x_{2}) ∈ SE(X, T). Thus (X, T) has s.u.p.e.. □

The proof of the following theorem is similar to that of Proposition 6 [B2], and in section 4 we will strength it by showing that scattering system is disjoint from any null minimal system. We remark that a system is null if and only if SE(X, T) = 0 (see Proposition 2.3(1)).

Theorem 2.8. Let (X, T) be a weakly mixing system. Then (X, T) is disjoint from any null minimal system.

For a dynamical system the topological Pinsker factor, i.e. the maximal factor with zero topological entropy, exists [BL]. Similarly we have (following [BL])

Theorem 2.9. For a dynamical system (X, T) the smallest closed invariant equivalence relation containing SE(X, T) induces the maximal null factor.

For a minimal system (of ℤ_{+}-action) the regionally proximal relation is a closed invariant equivalence relation which coincides with the set of complexity pairs up to the diagonal ([BHM] and [HY2]). As the notion of weak mixing pairs can be generalized to the group actions on compact Hausdorff spaces, we will discuss its relation with regionally proximal relation in the general setting. By a flow we mean (X, T, π) or (X, T), where X is a compact Hausdorff space, T is a topological group and π : T × X ⟶ X is continuous with π(e, x) = x and π(t_{1}, π(t_{2}, x)) = π(t_{1}t_{2}, x). Let W M(X, T) be the collection of points (x_{1}, x_{2})\Δ such that for each neighborhood U_{i} of x_{i}, there is t ∈ T such that U_{1} ∩ t(U_{1}) ≠ ∅ and U_{1} ∩ t(U_{2}) ≠ ∅. Let P (X, T) = ∩_{α∈U}T α and RP (X, T) = ∩_{α∈U} \overline{T α}, where U is the unique uniformty on X induced by the topology of X.

For (x_{1}, x_{2}) ∈ RP (X, T) let L(x_{1}, x_{2}) be the subset of X such that for each x_{3} in the set, each neighborhood U_{i} of x_{i},i = 1, 2, 3, there are x, y ∈ U_{3} and t ∈ T with tx ∈ U_{1} and ty ∈ U_{2}. It is easy to see that L(x_{1}, x_{2}) is non-empty, closed and invariant. We have the following simple observation

Proposition 2.10. Let (X, T) be a flow and (x_{1}, x_{2}) ∈ RP (X, T). Then (x_{1}, x_{2}) ∈ W M(X, T) if and only if x_{1} ∈ L(x_{1}, x_{2}).

Theorem 2.11. Let (X, T) be a minimal flow. Then W M(X, T) = RP (X, T)\Δ_{X}.

Proof. It is easy to see that W M(X, T) ⊂ RP (X, T) and it remains to show W M(X, T) ⊃ RP (X, T) \ Δ_{X}.

Let (x_{1}, x_{2}) ∈ RP (X, T) \ Δ_{X}. Then there is x ∈ L(x_{1}, x_{2}). As (X, T) is minimal, we have X = L(x_{1}, x_{2}). In particular x_{1} ∈ L(x_{1}, x_{2}). Thus (x_{1}, x_{2}) ∈ W M(X, T). □

We now show a simple and useful lemma.

Lemma 2.12. Let (X, T) be a flow. Then (X, T) is not equicontinuous if and only if there are x ∈ X, x_{β} ⟶ x and t_{β} ∈ T such that t_{β}x ⟶ x′, t_{β}x_{β} ⟶ x′′, x′ ≠ x′′ and (x′, x′′) ∈ RP (X, T).

Proof. (⇒) Assume that (X, T) is not equicontinuous. Then there is an index α such that for each index β there is t ∈ T with tβ ⊄ α. That is, for each β there are (z_{β}, y_{β}) ∈ β, t_{β} ∈ T with (t_{β}z_{β}, t_{β}y_{β}) ∉ α. Let α_{1} be an index with α_{1}^{2} ⊂ α. Assume lim z_{β} = x = lim y_{β} (if necessary pass to a subnet). Then we have (t_{β}x, t_{β}x_{β}) ∉ α_{1}, where x_{β} = z_{β} or y_{β}. Passing to a subnet if necessary we assume that t_{β}x ⟶ x′ and t_{β}x_{β} ⟶ x′′. Then it is clear that x′ ≠ x′′, (x′, x′′) ∈ RP (X, T).

(⇐) It is clear. □

Generally speaking a pair in regionally proximal relation can not be lifted by factor maps. But the special pair which we have in Lemma 2.12 does have lifting property. This allows us give simple proofs of some known theorems.

Corollary 2.13. 1. The factor of an equicontinuous flow is equicontinuous.

2. Let (X, T) be a flow. Then the factor (Y, T) of (X, T) induced by the smallest invariant equivalence relation containing RP (X, T) is the maximal equicontinuous factor.

Proof. 1. Let π : (X, T) ⟶ (Y, T) be a factor map with (X, T) equicontinuous. Assume that (Y, T) is not equicontinuous. By Lemma 2.12, there are y ∈ Y, y_{β} ⟶ y and t_{β} ∈ T such that t_{β}y ⟶ y′, t_{β}y_{β} ⟶ y′′ and y′ ≠ y′′. Take x_{β} ∈ X with π(x_{β}) = y_{β}. Suppose that x_{β} ⟶ x, t_{β}x ⟶ x′ and t_{β}x_{β} ⟶ x′′ (if necessary pass to a subnet). It is clear that π(x) = y, π(x′) = y′ and π(x′′) = y′′. Moreover, x′ ≠ x′′. This implies that (X, T) is not equicontinuous, a contradiction.

2. We only need to show that (Y, T) is equicontinuous. Let π : (X, T) ⟶ (Y, T) be the factor map. Assume that (Y, T) is not equicontinuous. Then following the argument in 1, we know that there are y ∈ Y, y_{β} ⟶ y and t_{β} ∈ T such that t_{β}y ⟶ y′, t_{β}y_{β} ⟶ y′′ and y′ ≠ y′′. Moreover, there are x ∈ X, x_{β} ⟶ x and t_{β} ∈ T such that t_{β}x ⟶ x′, t_{β}x_{β} ⟶ x′′ with π(x) = y, π(x′) = y′ and π(x′′) = y′′. This implies that x′ ≠ x′′.

It is clear that (x′, x′′) ∈ RP (X, T). By the definition of π we have π(x′) = π(x′′), i.e. y′ = y′′, a contradiction. □

With the help of Lemma 2.12 we can prove

Theorem 2.14. Let (X, T) be a flow.

1. W M(X, T) = ∅ if and only if (X, T) is equicontinuous.

2. The factor (Y, T) of (X, T) induced by the smallest invariant equivalence relation containing W M(X, T) is equicontinuous.

Proof. 1. We only need to show that if W M(X, T) = ∅ then (X, T) is equicontinuous.

Assume that (X, T) is not equicontinuous. By Lemma 2.12 there are x ∈ X, x_{β} ⟶ x and t_{β} ∈ T such that t_{β}x ⟶ x′, t_{β}x_{β} ⟶ x′′, x′ ≠ x′′ and (x′, x′′) ∈ RP (X, T). It is easy to see that x ∈ L(x′, x′′). As x′ is in the orbit closure of x, we have x′ ∈ L(x′, x′′). This implies that (x′, x′′) ∈ W M(X, T), a contradiction. This ends the proof of 1.

2. Let π : X ⟶ Y be the factor map. Assume that (Y, T) is not equicontinuous. Then Lemma 2.12 there are y, y_{β} ⟶ y and t_{β} such that t_{β}y ⟶ y′ and t_{β}y_{β} ⟶ y′′ with y′ ≠ y′′. Take x_{β} ∈ X with π(x_{β}) = y_{β}. Assume that lim x_{β} = x, lim t_{β}x = x′ and lim t_{β}x_{β} = x′′. Then π(x) = y, π(x′) = lim t_{β}y = y′ and π(x′′) = lim t_{β}y_{β} = y′′. It is clear that x′ ≠ x′′. By the argument in 1, we have x′ ∈ L(x′, x′′). This implies that (x′, x′′) ∈ MW (X, T). By the definition of π, we have y′ = π(x′) = π(x′′) = y′′, a contradiction. Thus, (Y, T) is equicontinuous. □

Remark 2.15. 1. In [Au] the author gave a proof of the fact that the smallest closed invariant equivalence relation containing RP (X, T) induces the maximal equicontinuous factor using a characterization of equicontinous systems, namely (X, T) is equicontinuous if and only if each element in the Ellis semi-group is a homeomorphism. Corollary 2.13 shows that this can be done directly without using Ellis semigroup theorem. Moreover, using Lemma 1 of [P] (only a few line) we get immediately that if T is Abelian and (X, T) is minimal, then (X, T) is weakly mixing if and only if W M(X, T) ∪ Δ = RP (X, T) = X^{(2)} if and only if the maximal equicontinuous factor is trivial.

2. Generally W M(X, T) = RP (X, T) \ Δ is not true.

Let X = {x_{∞}} ∪ {x_{i} : i ∈ ℤ} with x_{i} ≠ x_{j}, i ≠ j. Define T : X ⟶ X such that x_{∞} is a fixed point and T (x_{i}) = x_{i+1}, i ∈ ℤ. It is easy to see that RP (X, T) = X^{(2)} but (x_{i}, x_{j}) ∉ W M(X, T) if i ≠ j.

3. There are examples where SE(X, T) is a proper subset of W M(X, T) for minimal systems.

In [G] Goodman obtained a minimal system whose sequence entropy is zero for any sequence. Thus SE(X, T) = ∅. At the same time as the system is a subshift, there is an asymptotic pair (x, y) with x ≠ y. Thus W M(X, T) = RP (X, T) ≠ ∅.

4. W M(X, T) is not symmetric generally.

Let (X, T) be a transitive system, x_{1} be a transitive point and p be a fixed point. For each neighborhood U_{1} of p, we have N(U_{1}, U_{1}) = ℤ_{+}, and hence (p, x_{1}) ∈ W M(X, T). If in addition that T is almost equicontinuous [ABB], then (x_{1}, p) ∉ W M(X, T). That is, W M(X, T) is not symmetric even for a transitive system.

5. Similarly one can define sequence complexity pairs. It is easy to see that a pair is a sequence complexity pair if and only if it is a complexity pair.

§3 Transitive non-minimal cases

Generally speaking, it is difficult to check whether a pair (x, y) is in SE(X, T), but it is relatively easy to check whether it is in W M(X, T). In this section and section follows we will give conditions under which a weak mixing pair is a sequence entropy pair.

Proposition 3.1. Let (X, T) be a transitive dynamical system, x_{1} a transitive point of T and p a fixed point of T. Then (x_{1}, p) ∈ SE(X, T) if and only if (x_{1}, p) ∈ W M(X, T).

Proof. By Theorem 2.5 SE(X, T) ⊂ W M(X, T). It remains to show that if (x_{1}, p) ∈ W M(X, T) then (x_{1}, p) ∈ SE(X, T). Let N(U′_{1}, U′_{1}) ∩ N(U′_{1}, U′_{2}) ≠ ∅ for any open neighborhood U′_{1} of x_{1} and open neighborhood U′_{2} of p.

Fixing an open neighborhood U_{1} of x_{1} and an open neighborhood U_{2} of p, we have Claim: There exists a sequence 0 ≤ t_{1} < t_{2} < t_{3} < ⋯ such that for any l > 0 and

s ∈ {1, 2}^{l}, we can find M_{s} ∈ ℤ_{+} with T^{M_{s}}x_{1} ∈ ⋂_{i=1}^{l} T^{−t_{i}}U_{s(i)} and T^{M_{s}}x_{1} ∈ U_{1}.

Proof of claim. We choose t_{1}, t_{2}, ⋯ by induction. As N(U_{1}, U_{1}) ∩ N(U_{1}, U_{2}) ≠ ∅, there exists t_{1} ≥ 0 such that U_{1} ∩ T^{−t_{1}}U_{1} ≠ ∅ and U_{1} ∩ T^{−t_{1}}U_{2} ≠ ∅. Furthermore, there exist M_{1}, M_{2} ∈ ℤ_{+} such that T^{M_{i}}x_{1} ∈ U_{1}∩T^{−t_{1}}U_{i}, i = 1, 2, as x_{1} is a transitive point. Hence, T^{M_{i}}x_{1} ∈ T^{−t_{1}}U_{i} and T^{M_{i}}x_{1} ∈ U_{1}, i = 1, 2.

Suppose 0 ≤ t_{1} < t_{2} < ⋯ < t_{l}, l ≥ 1 satisfy that for any s ∈ {1, 2}^{l} there exists

M_{s} ∈ ℤ_{+} such that T^{M_{s}}x_{1} ∈ ⋂_{i=1}^{l} T^{−t_{i}}U_{s(i)} and T^{M_{s}}x_{1} ∈ U_{1}.

Now, we are going to find t_{l+1}. Take δ > 0 such that for any z_{1}, z_{2} ∈ X with

d(z_{1}, x_{1}) < δ and d(z_{2}, p) < δ, one has T^{M_{s}}z_{1} ∈ ⋂_{i=1}^{l} T^{−t_{i}}U_{s(i)}, T^{M_{s}}z_{1} ∈ U_{1} and

T^{M_{s}}z_{2} ∈ U_{2} for any s ∈ {1, 2}^{l}. Let U_{1}^{δ} = {z ∈ X : d(x_{1}, z) < δ} and U_{2}^{δ} = {z ∈ X : d(p, z) < δ}.

Without loss of generality (if need, let δ smaller), we assume that

N(U_{1}^{δ}, U_{1}^{δ}) ∩ N(U_{1}^{δ}, U_{2}^{δ}) ⊂ {t_{l} + 1, t_{l} + 2, ⋯ }.

As ∅ ≠ N(U_{1}^{δ}, U_{1}^{δ}) ∩ N(U_{1}^{δ}, U_{2}^{δ}) ⊂ {t_{l} + 1, t_{l} + 2, ⋯ } there exists t_{l+1} > t_{l} such that U_{1}^{δ} ∩ T^{−t_{l+1}}U_{1}^{δ} ≠ ∅ and U_{1}^{δ} ∩ T^{−t_{l+1}}U_{2}^{δ} ≠ ∅. Moreover, there exist P_{1}, P_{2} ∈ ℤ_{+} such that T^{P_{i}}x_{1} ∈ U_{1}^{δ} ∩ T^{−t_{l+1}}U_{i}^{δ} for i = 1, 2.

For any r ∈ {1, 2}^{l+1}, let s ∈ {1, 2}^{l} with s(i) = r(i), i = 1, 2, ⋯, l. Let M_{r} = M_{s} + P_{r(l+1)}. As T^{P_{r(l+1)}}x_{1} ∈ U_{1}^{δ}, we have T^{M_{s}}(T^{P_{r(l+1)}}x_{1}) ∈ U_{1} and T^{M_{s}+t_{i}}(T^{P_{r(l+1)}}x_{1}) ∈ U_{s(i)} = U_{r(i)} for i = 1, 2, ⋯, l. That is, T^{M_{r}}x_{1} ∈ U_{1} and T^{M_{r}+t_{i}}x_{1} ∈ U_{r(i)} for i = 1, 2, ⋯, l.

As T^{P_{r(l+1)}}x_{1} ∈ U_{1}^{δ} ∩ T^{−t_{l+1}}U_{r(l+1)}^{δ}, one has T^{P_{r(l+1)}+t_{l+1}}x_{1} ∈ U_{r(l+1)}^{δ}. Therefore, T^{M_{r}+t_{l+1}}x_{1} = T^{M_{s}}(T^{P_{r(l+1)}+t_{l+1}}x_{1}) ∈ U_{r(l+1)}. This proves the claim.

By the claim and Proposition 2.6 we get that (x_{1}, p) ∈ SE(X, T). □

Let S be a subset of ℤ_{+}. The upper Banach density of S is

BD^{⋆}(S) = lim sup_{|I|⟶+∞} \frac{|S ∩ I|}{|I|},

where I ranges over intervals of ℤ_{+}. We say the lower Banach density of S is one if for each a < 1 there is N such that for any subinterval I of ℤ_{+} with |I| ≥ N we have |S ∩I| ≥ a|I|. Let (X, T) be a dynamical system. We shall say that (X, T) is Banachtransitive if for any non-empty open subsets U, V of X, N(U, V) has positive upper Banach density. For x ∈ X and U ⊂ X let N(x, U) = {n ∈ ℤ_{+} : T^{n}(x) ∈ U}. Note that a dynamical system is an E−system if it is transitive and there is an invariant measure with full support. It is known [GW1] that if (X, T) is an E−system, then N(U, V) is syndetic for any non-empty open subsets U, V of X, and thus is Banachtransitive. Now we can prove

Theorem 3.2. Let (X, T) be a dynamical system. If (X, T) is a non-minimal and Banach-transitive system, then SE(X, T) ≠ ∅.

Proof. Without loss of generality, we assume that (X, T) has a fixed point p (using Proposition 2.3). Let x_{1} be a transitive point and U_{1} and U_{2} be open neighborhoods of x_{1} and p respectively. There are two cases.

Case 1: (X, T) is an E-system. As (X, T) is an E-system, N(U_{1}, U_{1}) is syndetic [GW1]. As p is a fixed point, N(x_{1}, U_{2}) is thick. Thus N(U_{1}, U_{2}) = N(x_{1}, U_{2}) − N(x_{1}, U_{1}) is a thick set. Combining the above facts, one has N(U_{1}, U_{1})∩N(U_{1}, U_{2}) ≠ ∅. By Proposition 3.1, we know that (x_{1}, p) ∈ SE(X, T).

Case 2: (X, T) is not an E-system. Without loss of generality, we assume that (X, T) is unique ergodic and the unique ergodic measure is δ_{p} (collapsing the closure of the union of the supports of all invariant measures and using Proposition 2.3). We have Claim: The lower Banach density of N(x_{1}, U_{2}) is 1.

Proof of claim. If this is not the case, then N(x_{1}, U_{2}^{c}) has positive upper Banach density. Thus, there exist k ∈ ℕ and a_{k} < b_{k} such that

lim_{k→+∞} \frac{1}{b_{k} − a_{k}} |N(x_{1}, U_{2}^{c}) ∩ \{a_{k}, a_{k} + 1, ⋯, b_{k} − 1\}| > 0 and lim_{k→+∞}(b_{k} − a_{k}) = +∞.

Now, set μ_{k} = \frac{1}{b_{k}−a_{k}} ∑_{i=a_{k}}^{b_{k}−1} δ_{T^{i}x_{1}}. Let μ = lim_{i→+∞} μ_{k_{i}} be a limit point of \{μ_{k}\}_{k=1}^{+∞} in

the weak^{⋆}-topology. Clearly, μ is an invariant measure of (X, T) and

μ(U_{2}^{c}) ≥ lim_{i→+∞} μ_{k_{i}}(U_{2}^{c})

= lim_{i→+∞} \frac{1}{b_{k_{i}} − a_{k_{i}}} ∑_{i=a_{k_{i}}}^{b_{k_{i}} −1} δ_{T^{i}x_{1}}(U_{2}^{c})

= lim_{i→+∞} \frac{|N(x_{1}, U_{2}^{c}) ∩ \{a_{k_{i}}, a_{k_{i}} + 1, ⋯, b_{k_{i}} − 1\}|}{b_{k_{i}} − a_{k_{i}}}

> 0.

Thus μ ≠ δ_{p} which contradicts the unique ergodicity of (X, T). This finishes the proof of the Claim.

As (X, T) is Banach-transitive, N(U_{1}, U_{1}) has positive upper Banach density. By the claim, the lower Banach density of N(x_{1}, U_{2}) is 1. This implies that the lower Banach density of N(U_{1}, U_{2}) = N(x_{1}, U_{2}) − N(x_{1}, U_{1}) is 1. Thus, N(U_{1}, U_{1}) ∩ N(U_{1}, U_{2}) ≠ ∅. By Proposition 3.1, we know that (x_{1}, p) ∈ SE(X, T). □

A dynamical system (X, T) is topologically ergodic if for each pair of non-empty open sets U and V, N(U, V) is syndetic and a dynamical system is extremely scattering, if its product with any topologically ergodic system is transitive. It is proved in [HY3] that an extremely scattering system is Banach-transitive. Thus we have

Corollary 3.3. If a dynamical system is non-trivial and extremely scattering, then SE(X, T) ≠ ∅, and thus the maximal null factor is trivial.

Proof. If (X, T) is minimal then it is weakly mixing and SE(X, T) ≠ ∅ by Theorem 2.7. If it is not minimal then SE(X, T) ≠ ∅ by Theorem 3.2. □

Remark 3.4. Though we could not supply a transitive non-minimal null system, we conjecture that such a system exists.

4 Minimal cases

In this section we consider homeomorphisms (but we still use the definitions of W M(X, T) and RP (X, T) for ℤ_{+} action). To get the results for surjective systems (X, T) we need to consider the natural extension (X̃, T̃) of (X, T). As the minimality of (X, T) implies the minimality of (X̃, T̃), for any sequence A, h_{A}(T) = h_{A}(T̃) ([G]) and the projection to the first coordinate is almost one-to-one ([KST]), Theorem 4.9 (using [AG, p.140, Lemma 4]), 4.12 and Corollary 4.11 are also valid for surjective systems.

First let’s give some basic definitions of topological dynamical systems. Let (X, d) be a compact metric space and T : X → X a homeomorphism. Two points x_{1}, x_{2} of X are called proximal if there exists a sequence n_{i} → ∞ such that d(T^{n_{i}}(x_{1}), T^{n_{i}}(x_{2})) → 0. x_{1}, x_{2} are called distal if they are not proximal.

Given two dynamical systems (X, T), (Y, S), a continuous map π : X → Y is called a homomorphism of systems (X, T) and (Y, S) if it is onto and πT = Sπ. We say (X, T) is an entension of (Y, S). If π is also injective then it is called an isomorphism. An extension π : X → Y is called proximal (resp. distal) if π(x_{1}) = π(x_{2}) implies that x_{1} and x_{2} are proximal (resp. distal). It is called equicontinuous if for any ϵ > 0, there exists δ > 0 such that π(x_{1}) = π(x_{2}) and d(x_{1}, x_{2}) < δ imply d(T^{n}(x_{1}), T^{n}(x_{2})) < ϵ for any n ∈ ℤ. An equicontinuous extension is also called isometric extension. The extension is called highly proximal if for every nonempty open set U of X and every point y of Y, there exists n ∈ ℤ such that T^{n}(π^{−1}(y)) ⊂ U. If π : X → Y is highly proximal then X, Y are necessarily minimal and ϕ is proximal. It is well known that when X and Y are metric space π is highly proximal if and only if it is almost one to one, i.e., there exists a dense G_{δ} set X_{0} ⊂ X such that π^{−1}π(x) = {x} for any x ∈ X_{0}. Finally the extension is called weak mixing if the subsystem R_{π} = {(x_{1}, x_{2}) : π(x_{1}) = π(x_{2})} is toplogically transitive under T × T.

The following theorem is the structure theorem for minimal systems. ([Au, Theorem 14.30])

Theorem 4.1. Given a compact metric minimal system (X, T), there exists a countable ordinal η and a canonically defined commutative diagram of minimal systems (PI-tower):

X = X_{0} ⟵^{id} X_{0} ⟵^{ϕ_{1}} X_{1} ⟵ ⋯ X_{ν} ⟵^{id} X_{ν} ⟵^{ϕ_{ν+1}} X_{ν+1} ⟵ ⋯ X_{η} = X_{∞}
π_{0}↓  ^{σ_{1}}↓  ^{π_{1}}↓  ^{π_{ν}} ↓  ^{σ_{ν+1}}↓  ^{π_{ν+1}}↓  ^{π_{∞}}↓
pt = Y_{0} ⟵_{ρ_{1}} Z_{1} ⟵_{ψ_{1}} Y_{1} ⟵ ⋯ Y_{ν} ⟵_{ρ_{ν}} Z_{ν+1} ⟵_{ψ_{ν+1}} Y_{ν+1} ⟵ ⋯ Y_{η} = Y_{∞}

where for each ν ≤ η, ρ_{ν} is equicontinuous, ϕ_{ν} and ψ_{ν} are proximal and π_{∞} is weakly mixing. For a limit ordinal ν, X_{ν}, Y_{ν}, π_{ν} etc. are the inverse limits of X_{λ}, Y_{λ}, π_{λ} etc. for λ < ν.

We say that an extension ϕ : X → Y of minimal systems is a strictly PI-extension or SPI-extension if there is a countable ordinal η and a family of minimal systems X_{λ} (0 ≤ λ ≤ η) such that (i) X_{0} = Y and X_{η} = X. (ii) for every λ < η, there exists an extension ϕ_{λ} : X_{λ+1} → X_{λ} which is either proximal or isometric. (iii) for a limit ordinal ν ≤ η the system X_{ν} is the inverse limit of the systems X_{λ} (λ < ν) and (iv) ϕ is the transfinite composition of extensions ϕ_{λ}(λ < η). An extension ϕ : X → Y of minimal systems is called PI-extension if there exists a proximal extension ψ : Z → X of minimal systems such that ϕ ∘ ψ : Z → Y is a strictly PI-extension. If Y is the trival system and ϕ : X → Y is a (strictly) PI-extension then X is called a (strictly) PI-system.

If in the above definitions proximal is replaced by highly proximal then we get the notions of strictly HPI-extension (system) and HPI-extension (system)

The following lemma can be found in [Au, Theorem 14.31].

Lemma 4.2. Let (X, T) be a minimal dynamical system. Then X is PI if and only if it satisfies the following property: whenever W is a closed invariant subset of X × X which is topologically transitive and has a dense subset of minimal points, then W is minimal.

Theorem 4.3. Let (X, T) be a minimal dynamical system. If SE(X, T) = ∅, then (X, T) is PI.

Proof. If (X, T) is not PI, then exists a closed invariant subset W of X × X which is a non-minimal transitive system and has a dense subset of minimal points by Lemma 4.2. Clearly, (W, T ×T) is Banach-transitive. By Theorem 3.2, we know that SE(W, T × T) ≠ ∅; say ((x_{1}, y_{1}), (x_{2}, y_{2})) ∈ SE(W, T × T). As (x_{1}, y_{1}) ≠ (x_{2}, y_{2}), we have that (x_{1}, x_{2}) ∈ SE(X, T) or (y_{1}, y_{2}) ∈ SE(X, T) (Proposition 2.3(3)(b)). This contradicts the assumption SE(X, T) = ∅.

□

In order to get furthur properties of null minimal systems, we need to consider the relationships between sequence entropy and isometric extension. First we give a lemma concerning the conditions under which a weak mixing pair is a sequence entropy pair. Call a continuous map f : X → Y between toplogical space semi-open if the image of every open set of X has nonempty interior.

Lemma 4.4. Let T : X → X be a homeomorphism of compact metric space. Let x_{1}, x_{2} ∈ X and x_{1} ≠ x_{2}. A = \overline{orb(x_{1}, x_{2})}. If π_{1} : A → X is semi-open, then (x_{1}, x_{2}) ∈ SE(X, T) if and only if (x_{1}, x_{2}) ∈ W M(X, T).

Proof. Suppose (x_{1}, x_{2}) ∈ W M(X, T) and π_{1} : A → X is semi-open. By proposition 2.6 we only need to prove the following claim. Claim: For any neighborhood U_{1}, U_{2} of x_{1}, x_{2}, there exists a sequence 0 ≤ t_{1} < t_{2} < t_{3} < ⋯ such that for any l > 0 and s ∈ {1, 2}^{l}, we can find M_{s} ∈ ℤ with T^{M_{s}}(x_{1}) ∈ ⋂_{i=1}^{l} T^{−t_{i}}U_{s(i)} and T^{M_{s}}(x_{j}) ∈ U_{j}, j = 1, 2. Proof of Claim: Fix open neighborhood U_{1}, U_{2} of x_{1}, x_{2}. Since π_{1} : A → X is semi-open, W_{1} = int(π_{1}((U_{1} × U_{2}) ∩ A)) is a nonempty open set and W_{1} ⊂ U_{1}. Let W_{2} = U_{2}.

We are going to prove x_{1} ∈ W_{1}. Since A is the orbit closure of (x_{1}, x_{2}), there exists n ∈ ℕ such that T^{n}(x_{1}) ∈ W_{1}. T^{−n}(W_{1}) is an open neighborhood of x_{1} and for any x ∈ T^{−n}(W_{1}), T^{n}(x) ∈ W_{1} and therefore there exists y ∈ W_{2} such that (T^{n}(x), y) ∈ A. As A is invariant we have (x, T^{−n}(y)) ∈ A and therefore x_{1} ∈ T^{−n}(W_{1}) ⊂ W_{1}.

Since (x_{1}, x_{2}) ∈ W_{1} × W_{2} and (x_{1}, x_{2}) ∈ W M(X, T), we have N(W_{1}, W_{1}) ∩ N(W_{1}, W_{2}) ≠ ∅. So there exists t_{1} ≥ 0 such that W_{1} ∩ T^{−t_{1}}W_{1} ≠ ∅ and W_{1} ∩ T^{−t_{1}}W_{2} ≠ ∅. Since the orbit of (x_{1}, x_{2}) is dense in A, there exist M_{1}, M_{2} ∈ ℤ such that (T^{M_{i}}(x_{1}), T^{M_{i}}(x_{2})) ∈ ((W_{1} ∩ T^{−t_{1}}W_{i}) × U_{2}) ∩ A, i = 1, 2. Thus T^{M_{i}}(x_{1}) ∈ U_{1} ∩ T^{−t_{1}}U_{2} and T^{M_{i}}(x_{j}) ∈ U_{j} for i, j = 1, 2.

Now suppose 0 ≤ t_{1} < t_{2} < ⋯ < t_{l}, l ≥ 1 have been defined and satisfy that for any s ∈ {1, 2}^{l} there exists M_{s} ∈ ℤ such that T^{M_{s}}(x_{1}) ∈ ⋂_{i=1}^{l} T^{−t_{i}}U_{s(i)} and T^{M_{s}}(x_{j}) ∈ U_{j}, j = 1, 2.

We are going to define t_{l+1}. take δ > 0 such that for any z_{1}, z_{2} ∈ X with d(z_{1}, x_{1}) < δ and d(z_{2}, x_{2}) < δ, one has T^{M_{s}}(z_{1}) ∈ ⋂_{i=1}^{l} T^{−t_{i}}U_{s(i)} and T^{M_{s}}(z_{j}) ∈ U_{j}, j = 1, 2 for any s ∈ {1, 2}^{l}. Let U_{1}^{δ} = {z ∈ X : d(z, x_{1}) < δ}, U_{2}^{δ} = {z ∈ X : d(z, x_{2}) < δ}.

Since π_{1} : A → X is semi-open, W_{1}^{δ} = int(π_{1}((U_{1}^{δ} × U_{2}^{δ}) ∩ A)) is nonempty open set and W_{1}^{δ} ⊂ U_{1}^{δ}. Let W_{2}^{δ} = U_{2}^{δ}. Using the same argument as above we have x_{1} ∈ W_{1}^{δ} and therefore (x_{1}, x_{2}) ∈ W_{1}^{δ} × W_{2}^{δ}. As (x_{1}, x_{2}) ∈ W M(X, T), N(W_{1}^{δ}, W_{1}^{δ}) ∩ N(W_{1}^{δ}, W_{2}^{δ}) ≠ ∅. Without lose of generality (if needed, let δ small) we may assume that N(W_{1}^{δ}, W_{1}^{δ}) ∩ N(W_{1}^{δ}, W_{2}^{δ}) ⊂ {t_{1} + 1, t_{1} + 2, ⋯ }. So there exist t_{l+1} > t_{l} such that W_{1}^{δ} ∩ T^{−t_{l+1}}W_{i}^{δ} ≠ ∅, i = 1, 2. We can choose P_{1}, P_{2} ∈ ℤ such that T^{P_{i}}(x_{1}) ∈ W_{1}^{δ} ∩ T^{−t_{l+1}}W_{i}^{δ} and T^{P_{i}}(x_{j}) ∈ U_{j}^{δ} for i, j = 1, 2.

For any r ∈ {1, 2}^{l+1}, let s ∈ {1, 2}^{l} with s(i) = r(i), i = 1, 2, ⋯, l and let r(l + 1) = k. Put M_{r} = M_{s} + P_{k}. Then T^{M_{r}}(x_{1}) = T^{M_{s}}(T^{P_{k}}(x_{1})) ∈ ⋂_{i=1}^{l} T^{−t_{i}}U_{s(i)} . As T^{t_{l+1}}T^{P_{k}}(x_{1}) ∈ W_{k}^{δ}, T^{t_{l+1}}T^{M_{r}}(x_{1}) = T^{M_{s}}T^{t_{l+1}}T^{P_{k}}(x_{1}) ∈ U_{k}. So T^{M_{r}}(x_{1}) ∈ ⋂_{i=1}^{l} T^{−t_{i}}U_{s(i)} ⋂ T^{−t_{l+1}}U_{k} = ⋂_{i=1}^{l+1} T^{−t_{i}}U_{r(i)}. Also, T^{M_{r}}(x_{j}) = T^{M_{s}}T^{P_{k}}(x_{j}) ∈ U_{j}. This finished the proof of the claim.. □

Corollary 4.5. Let π : (X, T) → (Y, T) be a distal extension of minimal dynamical systems and let x_{1}, x_{2} ∈ X with x_{1} ≠ x_{2} and π(x_{1}) = π(x_{2}). Then (x_{1}, x_{2}) ∈ SE(X, T) if and only if (x_{1}, x_{2}) ∈ W M(X, T).

proof. Since π is distal, (x_{1}, x_{2}) is an almost periodic point of (X × X, T × T) ([V, Propostion V.2.7]). Therefore A = \overline{orb(x_{1}, x_{2})} is a minimal set. It is well known that a homomorphism between minimal systems is semi-open ([Au, Theorem 1.15]), therefore π_{1} : A → X is semi-open and the corollary follows from Lemma 4.4. □

Corollary 4.6. If (X, T) is a minimal distal system then SE(X, T) = RP (X, T)\Δ.

Proof. As (X, T) is distal, π : X ⟶ Y is a distal extension, where (Y, S) is the trivial system. Thus SE(X, T) = W M(X, T) = RP (X, T) \ Δ, following Proposition 4.2 and 2.11 (it is easy to check that the definitions W M(X, T) and RP (X, T) for ℤ_{+} action and for ℤ action are the same in our case). □

To prove our main theorem we need some knowledge on the algebraic theory of minimal dynamical systems. Let M be a universal minimal system for ℤ-action (M can be taken as a minimal left ideal of the Stone-Čech compactification βℤ of the discrete group ℤ). M has a semi-group structure inherited from βℤ and it has an action (may not continuous) on every minimal system. Let u be an idempotent of M and G = uM. Then G is a group. For any minimal dynamical system (X, T), choose x ∈ X such that ux = x. The pointed system (X, x) is called a minimal ambit. The Ellis group of an ambit (X, x) is defined to be G(X, x) = {g ∈ G : gx = x}. The following lemma can be found at [Au, Theorem 10.1].

Lemma 4.7. Let π : (X, x) ⟶ (Y, y) be a homomorphism of minimal ambits. Then G(X, x) ⊂ G(Y, y) and G(X, x) = G(Y, y) if and only if π is a proximal extension.

Now we are ready to prove our main theorem.

Theorem 4.8. Let (X, T) be a minimal system with SE(X, T) = ∅. Then (X, T) is an almost one to one extension of an equicontinuous system.

Proof. We consider the PI-tower of (X, T) as constructed in Theorem 4.1. As (X, T) is a null minimal system, Theorem 4.3 implies (X, T) is a PI system and therefore π_{∞} in the PI-tower is an isomorphism. ([Au,??]).

We divide the proof into the following steps.

Step 1. Z_{1} is the maximal equicontinuous factor of X_{∞}.

From the construction of the PI-tower we know Z_{1} is the maximal equicontinous factor of X. As the composition of proximal extensions is proximal, by Lemma 4.7 X_{∞} and X have same Ellis group. Suppose Z is the maximal equicontinuous factor of X_{∞}. By Lemma 4.7 again Z is also a factor of X and therefore Z = Z_{1}.

Step 2. The isometric extensions ρ_{ν}(ν ≥ 2) are isomorphisms.

Suppose ρ_{ν+1} is not isomorphism for some ν ≥ 1. Then there exist z_{1} ≠ z_{2} ∈ Z_{ν+1} such that ρ_{ν+1}(z_{1}) = ρ_{ν+1}(z_{2}). The maximal equicontinuous factor of Z_{ν+1} is also Z_{1} as Z_{ν+1} is factor of X_{∞}. It is well known that the regionally proximal relation of a minimal system is a closed invariant equivalent relation. So (z_{1}, z_{2}) ∈ RP (Z_{ν+1}) as z_{1} and z_{2} have the same image in Z_{1}. Therefore (z_{1}, z_{2}) ∈ W M(Z_{ν+1}) as W M(Z_{ν+1}) = RP (Z_{ν+1}) \ Δ. Then by Corollary 4.6 and the fact that an isometric extension is always distal, we have (z_{1}, z_{2}) ∈ SE(Z_{ν+1}). Let (x′_{1}, x′_{2}) be a preimage of (z_{1}, z_{2}) in X_{∞} × X_{∞} such that (z_{1}, z_{2}) ∈ SE(X_{∞}) (Proposition 2.3). Clearly x′_{1}, x′_{2} are distal as z_{1}, z_{2} are distal. Therefore the images x_{1}, x_{2} of x′_{1}, x′_{2} in X are different and (x_{1}, x_{2}) ∈ SE(X). This contradicts the assumptin SE(X) = ∅.

Step 3. X is a proximal extension of Z_{1}.

As all ρ_{ν}(ν ≥ 2) are isomorphism, X_{∞} is a proximal extension of both Z_{1} and X. Therefore the systems X, X_{∞} and Z_{1} have the same Ellis group. By Lemma 4.7, σ_{1} : X → Z_{1} is a proximal extension.

Step 4. σ_{1} : X → Z_{1} is a HPI-extension.

If σ_{1} : X → Z_{1} is not a HPI-extension then by the theorem of Woude ([W]) there exists a closed invariant subset A ⊂ R_{σ_{1}} = {(x, y) ∈ X^{(}2) : σ_{1}(x) = σ_{1}(y)} such that (A, T × T) is a topological transitive non-minimal system and the projections π_{i} : A → X are semi-open for i = 1, 2. Thus there exists (x_{1}, x_{2}) ∈ A, x_{1} ≠ x_{2} and (x_{1}, x_{2}) is a transitive point. Since σ_{1} is a proximal extension of a equicontinuous flow we have

R_{σ_{1}} ⊂ P rox(X, T) ⊂ RP (X, T) = W M(X, T) ∪ Δ

Therefore (x_{1}, x_{2}) is a weak mixing pair and by Lemma 4.4 it is a sequence entropy pair. This contradicts the assumption that X is a null system.

Step 5. σ_{1} : X → Z_{1} is an almost one to one extension.

As σ_{1} : X → Z_{1} is a HPI-extension and Z_{1} is equicontinuous, we know X is a HPI-system. Similar to PI-tower, we can construct canonical HPI-tower for every minimal system. As X is a HPI-system, the canonical HPI-tower of X is the same as its PI-tower and all the extensions ϕ_{ν}, ψ_{ν} (1 ≤ ν ≤ η) are highly proximal. (See [V, Remark VI.4.21(3)]). In particular π_{∞} is an isomorphism. Thus step 1 and 2 can be applied in this case and since the highly proximal extension is kept under transfinite composition, we have that X_{∞} is highly proximal extension of both X and Z_{1}. As all the spaces are assumed to be metric , these two extensions are almost one to one and therefore σ_{1} : X → Z_{1} is almost one to one. This finishes the proof. □

In the following we show that in some other cases a proximal extension is almost one-to-one. To do this we need

Lemma 4.10. Let π : X ⟶ Y be a proximal extension of minimal systems, W_{n} = {(x_{1}, …, x_{n}) : π(x_{1}) = … = π(x_{n}), x_{i} ∈ X} for n ∈ ℕ. Then we have

(1) W_{n} has only one minimal set Δ_{n} = {(x, …, x) : x ∈ X}.

(2) If there is y ∈ Y such that π^{−1}(y) is finite, then π is almost one-to-one.

Proof. (1) It is easy to see that W_{2} contains only one minimal set Δ_{2}.

Assume W_{i} contains only one minimal set Δ_{i} for 1 ≤ i ≤ n. Let (x_{1}, …, x_{n+1}) ∈ W_{n+1} be a minimal point. Then (x_{i}, x_{j}) are proximal and thus (x_{i}, x_{j}) ∈ W_{2}. As (x_{i}, x_{j}) is a minimal point we have x_{i} = x_{j}. Thus W_{n+1} contains only one minimal set Δ_{n+1}.

(2) Suppose that π^{−1}(y) = {x_{1}, x_{2}, …, x_{n}}. By (1) W_{n} contains only one minimal set. We have that for each ϵ > 0 there are m = m(ϵ) and x = x(ϵ) such that d(T^{m}(x_{i}), x) < ϵ for each 1 ≤ i ≤ n. Thus π is a highly proximal extension, and it is almost one-to-one. □

We say a dynamical system is semi-distal if every recurrent point in X^{(2)} is minimal. It turns out that (X, T) is semi-distal if and only if each idempotent is minimal [AA]. Now we have

Corollary 4.11. Let (X, T) be a minimal system and (Y, S) be equicontinuous and π : X ⟶ Y be a proximal extension. If (X, T) is semi-distal (in particular π is asymptotic) or there is y ∈ Y such that π^{−1}(y) is finite then π is almost one-to-one.

Proof. First we assume that X is semi-distal. By [W], a system is HPI if and only if each transitive sub-system W ⊂ X × X with each projection to X is semi-open is minimal. As X is semi-distal, X is HPI. The argument of Theorem 4.7 can be applied here and we get that π is almost one-to-one.

Now we assume that π^{−1}(y) is finite for some y ∈ Y. By Lemma 4.10, π is almost one-to-one. □

In [G] the author showed that for a finite dimensional system the restricted variational principal holds for sequence entropy. Though his result can not be applied in our situation we still can prove

Theorem 4.12. Let (X, T) be a null minimal system. Then (X, T) is uniquely ergodic.

Proof. By Theorem 4.7 we assume that Y is equicontinuous and π : X ⟶ Y is proximal. Let ν be the unique measure on Y.

Let μ be an invariant measure on X and consider the disintegration of μ over ν. That is, for a.e. y ∈ Y we have a measure μ_{y} on X such that supp(μ_{y}) ⊂ π^{−1}(y) (see [R] or [F]) and μ = ∫_{y∈Y} μ_{y}dν. We claim for a.e. y ∈ Y, μ_{y} is concentrated on one point.

Proof of the claim: Let W = {(x, y) ∈ X^{2} : π(x) = π(y)}. As supp(μ_{y}) ⊂ π^{−1}(y), a.e y ∈ Y we have supp(μ_{y} × μ_{y}) ⊂ π^{−1}(y) × π^{−1}(y) ⊂ W, a.e y ∈ Y. Thus

μ ×_{Y} μ(W) = ∫_{y∈Y} μ_{y} × μ_{y}(W)dν = 1.

If μ ×_{Y} μ(Δ) ≠ 1, then μ ×_{Y} μ(W \ Δ) > 0. Thus there is an ergodic measure w on (W, T × T) such that w(W \ Δ) > 0. As Δ is the only minimal set of (W, T × T), supp(w) is an E−system which is not minimal. By Theorem 3.2, SE(supp(w), T × T) ≠ ∅, a contradiction. Thus we have μ ×_{Y} μ(Δ) = 1. Since

μ ×_{Y} μ(Δ) = ∫_{y∈Y} μ_{y} × μ_{y}(Δ)dν = 1

we have that μ_{y} × μ_{y}(Δ) = 1, a.e. y ∈ Y. A simple calculation shows that μ_{y} is concentrated on one point, say c(y). This ends the proof of the claim.

We now show (X, T) is uniquely ergodic. Assume that μ^{1} and μ^{2} are two invariant measures on X. Let μ = \frac{1}{2}(μ^{1} + μ^{2}). By the claim μ_{y}^{i} is concentrated on c^{i}(y), a.e. y ∈ Y. Thus μ_{y} is concentrated on c^{1}(y) and c^{2}(y), a.e. y ∈ Y. By the claim we have c^{1}(y) = c^{2}(y), a.e. y ∈ Y. Thus μ^{1} = μ^{2}. This proves that (X, T) is unique ergodic. □

A dynamical system is scattering if it is weakly disjoint from all minimal systems [BHM]. It is known [AG], [HY3] that weak mixing is strictly stronger than scattering. Now we show

Theorem 4.12. A scattering system is disjoint from any minimal null system.

Proof. It is a direct consequence of Theorem 4.9 and [AG, Theorem 3.7] which states that if (X_{1}, T_{1}) is an almost one to one extension of a distal system, then a transitive system (X, T) is disjoint from (X_{1}, T_{1}) if and only if they are weakly disjoint. □

§5 Some properties of null systems

In this section we give some properties of null systems. Note that the notion of nullness of a system is valid for any continuous map of a compact Hausdorff space and Prposition 2.3 remains true in this generalized situation.

Let (X, T) and (X′, T′) be two dynamical systems. It is known [L] that h_{A}(X × X′, T × T′) = h_{A}(X, T) + h_{A}(X′, T′) does not hold in general. In contract to this fact we have

Proposition 5.1. The property of nullness of a dynamical system is stable under factor maps, arbitrary product and inverse limit.

Proof. This is a simple application of Proposition 2.3. □

In the sequel we analyze the null minimal systems obtained by Goodman in [G] and to see how far they are from equicontinuous systems. We begin by briefly reviewing the theory of the substitution minimal system on two symbols. For details, see [CK].

Let Σ_{2} be the set of all bisequences x = … x_{−1}x_{0}x_{1} … with x_{i} ∈ {0, 1}, i ∈ ℤ. A metric compatible is given by d(x, y) = \frac{1}{1+k}, where k = min\{|n| : x_{n} ≠ y_{n}\} , x, y ∈ Σ_{2}. The shift map σ : Σ_{2} ⟶ Σ_{2} is defined by (σx)_{n} = x_{n+1} for all n ∈ ℤ. The pair (Σ_{2}, σ) is called the shift dynamical system.

If n is a positive integer, a n−block C will be a finite sequence C = c_{1}c_{2} … c_{n}, with c_{i} ∈ {0, 1}. A substitution of length n(n ≥ 2) is a map θ from {0, 1} into the n−blocks, that is, θ is a pair of n−blocks θ(0) = a = a_{0}a_{1} … a_{n−1}, θ(1) = b = b_{0}b_{1} … b_{n−1}. Set H = {00, 01, 10, 11}, and λ_{θ} : H ⟶ H is given by 00 ⟶ a_{n−1}a_{0}, 01 ⟶ a_{n−1}b_{0}, 10 ⟶ b_{n−1}a_{0}, 11 ⟶ b_{n−1}b_{0}. It is easy to know the set of the fixed points of λ_{θ}^{2} is not empty. Let pq be a fixed point of λ_{θ}^{2}, then there is a unique sequence w^{pq} ∈ Σ_{2} such that θ^{2}w^{pq} = w^{pq} and w_{−1}^{pq}w_{0}^{pq} = pq. Let O_{θ} = \overline{Orb(w^{pq}, σ)} be the orbit closure of w^{pq} in Σ_{2}. We call (O_{θ}, σ) a substitution minimal system.

θ is called finite if any of the following conditions holds: (1)a_{i} = b_{i} for all i; (2)a_{i} = 0 for all i or b_{i} = 1 for all i; (3)a_{i} = 1 for all i and b_{i} = 0 for all i; (4)n is odd and a = b̄ = 0101 … 010 or 1010 … 101, where b̄ is the mirror of b, i.e., b̄ is obtained from b by replacing all the zeros in b by ones and all the ones by zeros. θ is called continuous if it is not finite and if a = b̄. θ is called discrete if it is neither finite nor continuous. If θ is finite, then the only minimal sets in O_{θ} are periodic orbits which is uninteresting as minimal sets. So in the paper we always assume θ is discrete.

By Z(n), (n ≥ 2) we denote the additive group of n−adic integers: Z(n) = {z = Σ_{i=0}^{∞}z_{i}n^{i}|z_{i} = 0, 1, …, n−1\}. A metric in Z(n) is given by d′(z, z′) = \frac{1}{min\{i:z_{i}≠z′_{i}\}+1}, where z = Σ_{i=0}^{∞}z_{i}n^{i}, z′ = Σ_{i=0}^{∞}z′_{i}n^{i} ∈ Z(n). (Z(n), d′) is a compact metric group. A map τ : Z(n) ⟶ Z(n) is defined by τ(z) = z + 1. Clearly, τ is an isometric homemorphism.

A basic fact is that there is a homomorphism π : (O_{θ}, σ) ⟶ (Z(n), τ) mapping w^{pq} to 0. Set J_{k} = {m : 0 ≤ m ≤ n^{k} − 1,θ^{k}(0) and θ^{k}(1) disagree at place m}, J_{∞} = {Σ_{i=0}^{∞}z_{i}n^{i} : z_{i} ∈ J_{1} for all i}, E = ⋃{τ^{k}J_{∞} : k ∈ Z}, Z^{⋆} = Z(n) − E. Then we have Z^{⋆} = {z ∈ Z(n)|π^{−1}(z) is a single point}. If z ∈ E, then π^{−1}(z) consists of exactly two points unless z ∈ Z and a and b disagree at both endpoints. In this case, π^{−1}(z) consists of four points (see [CK]).

Lemma 5.2. Let (O_{θ}, σ) be a substitution minimal flow, where θ is a substitution of constant length n(n ≥ 3) on the symbols 0 and 1. Let θ(0) = a_{0}a_{1} … a_{n−2}a_{n−1} and θ(1) = a_{0}a_{1} … a_{n−2}a′_{n−1} with a_{n−1} ≠ a′_{n−1}, then (O_{θ}, σ) is null and π : (O_{θ}, σ) ⟶ (Z(n), τ) is an asymptotic extension. Moreover, RP (O_{θ}, σ) \ Δ is consisting of one orbit under σ × σ.

Proof. By Proposition 6.2 of [G], (O_{θ}, σ) is null. Since J_{1} = {n − 1}, as above J_{∞} = {Σ_{i=0}^{∞}(n − 1)n^{i}}, E = {Σ_{i=0}^{∞}(n − 1)n^{i} + t : t ∈ Z} = Z, Z^{⋆} = Z(n) − E = Z(n) − Z. For z ∈ Z(n) − Z, π^{−1}(z) is a single point. For z ∈ Z, π^{−1}(z) consists of exactly two points. If let π^{−1}(0) = {w, w′}, then π^{−1}(k) = {σ^{k}(w), σ^{k}(w′)} for k ∈ Z. Since (O_{θ}, σ) is a subshift, there exists x ≠ x′ such that (x, x′) is an asymptotic pair. By the above discussion, we have {x, x′} = {σ^{j}(w), σ^{j}(w′)} for some j ∈ Z. This shows that {σ^{k}(w), σ^{k}(w′)} is an asymptotic pair for all k ∈ Z.

Therefore, π : (O_{θ}, σ) ⟶ (Z(n), τ) is an asymptotic extension and RP (O_{θ}, σ) \ Δ is consisting of one orbit under σ × σ. □

Corollary 5.3. There is a null minimal system which is an asymptotic extension of an equicontinuous system such that P R(X, T) \ Δ is uncountable.

Proof. Let p_{1}, p_{2}, … be all odd prime numbers and let (X_{i}, T_{i}) be the system in Lemma 5.2 with n = p_{i}. It is easy to see that Π_{i}Z(p_{i}) is minimal and corollary follows. □

A dynamical system (X, T) is LY-chaotic, if there is an uncountable set S such that for each x ≠ y ∈ S, (x, y) is proximal and not asymptotic. For the recent progress along the topic for general compact spaces, see [HY1-2], [BGKM] and [AA] and the references there. The above examples are minimal and are not LY-chaotic (even the scrambled set is finite). Thus it seems that the question to characterize minimal or transitive non-minimal systems which are not LY-chaotic is meaningful.

Finally we have some open problems:

1. Is there a null transitive system which is 2-scattering?

Remark: A dynamical system is 2-scattering if Com(X, T) ∪ Δ = X^{(2)}. Note that E(X, T) ⊂ SE(X, T) ⊂ Com(X, T) and any weakly mixing system with zero topological entropy will be an example such that E(X, T) = ∅ and SE(X, T) ∪ Δ = X^{(2)}. It will be interesting to get an example such that SE(X, T) = ∅ and Com(X, T) ∪ Δ = X^{(2)}. If such a system exists, it should be non-minimal (minimal 2-scattering implies weak mixing) and non Banach-transitive. Thus we may assume that it has only one minimal set which is a fixed point.

2. Is it true that a null minimal system is an asymptotic extension of an equicontinuous system?

3. Does there exist a transitive non-minimal null system?

References

[AA] E. Akin and J. Auslander, Distality concepts for Ellis actions, preprint, 2002.

[AAB] E. Akin, J. Auslander and K. Berg, When is a transitive map chaotic? Convergence in Ergodic Theory and Probability (Columbus, OH, 1993) (Ohio University Math. Res. Inst. Pub., 5). de Gruyter, Berlin, 1996, pp. 25-40.

[AG] E. Akin and E. Glasner, Residual properties and almost equicontinuity, J. d’Anal. Math., 84(2001), 243-286.

[Au] J. Auslander, Minimal flows and their extensions (North-Holland Mathematics Studies, 153). North-Holand, Amsterdam, 1988.

[Ba] J. Banks, Topological mapping properties defined by digraphs, Discrete Continuous Dyn. Sys. 5(1999), 83-92.

[B1] F. Blanchard, Fully positive topological entropy and topological mixing, Symbolic Dynamics and its Applications (in honour of R. L. Adler) (Amer. Math. Soc. Contemporary Mathematics). Ed. P. Walters, 1992.

[B2] F. Blanchard, A disjointness theorem involving topological entropy, Bull. Soc. Math. France,

121(1993), 465-478.

[BGKM] F. Blanchard, E. Glasner, S. Kolyada and A. Maass, On Li-Yorke pairs, preprint, 2000, J. fur

die reine und angewandte Mathematik, to appear.

[BHM] F. Blanchard, B. Host and A. Maass, Topological complexity, Ergod. Th. and Dynam. Sys.,

20(2000), 641-662.

[BL] F. Blanchard and Y. Lacroix, Zero-entropy factors of topological flows, Proc. Amer. Math. Soc., 119(1993), 985-992.

[Br] I.U. Bronstein, Extensions of minimal transformation groups, Martinus Nijhoff Publications, The Hague, 1979.

[CK] E.M. Coven and M.S. Keane, The structure of substitution minimal sets, Trans. Amer. Math.

Soc. 162(1971), 89-102.

[F] H. Furstenberg, Recurence in ergodic theory and combinatrial number theory, Princeton Univ. Press, Princeton, 1981.

[G] T.N.T. Goodman, Topological sequence entropy, Proc. London Math. Soc.(3) 29(1974), 331-350.

[GW1] E. Glasner and B. Weiss, Locally equicontinuous dynamical systems, Colloq. Math., 84/85(2000), Part 2, 345-361.

[GW2] E. Glasner and B. Weiss, On the construction of minimal skew products, Israel J. Math., 34(1979), 321-336.

[HY1] W. Huang and X. Ye, Homeomorphisms with the whole compacta being scrambled sets, Ergod. Th. and Dynam. Sys., 21(2001), 77-91.

[HY2] W. Huang and X. Ye, Devaney’s chaos and 2-scattering imply Li-Yorke’s chaos, Topology Appli., 117(2002), 259-272.

[HY3] W. Huang and X. Ye, An explicit scattering, non-weakly mixing example and weak disjointness,

Nonlinearity, to appear.

[K] A.G. Kushnirenko, On metric invariants of entropy type, Russion Math. Surveys 22(1967), 53-

61.

[KST] S. Kolyada, L. Snoha and S. Trofimchuk, Noninvertible minimal maps, Fund. Math., 168(2001),

141-163.

[L] M. Lemanczyk, The sequence entropy for Morse shift and some counterexamples, Studia Math., 82(1985), 221-241.

[P] K. Petersen, Disjointness and weak mixing of minimal sets, Proc. Amer. Math. Soc., 24(1970), 278-280.

[R] V. A. Rokhlin, On the fundamental ideas of measure theory, Transl. Amer. Math. Soc., Series 1, 10(1962), 1-52.

[W] Jaap van der Woude, Characterizations of (H)PI extensions, Pacific J. of Math., 120(1985), 453-467.

[V] J. de Vries, Elements of Topological Dynamics, Kluwer Academic Publishers, Dordrecht, 1993.

1Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, P.R. China

E-mail address: wenh@mail.ustc.edu.cn

2current address: Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153 Japan

E-mail address: lisimin@ms.u-tokyo.ac.jp

permanent address: Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, P.R. China

E-mail address: simin@ustc.edu.cn

3Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, P.R. China

E-mail address: songshao@ustc.edu.cn

4Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, P.R. China

E-mail address: yexd@ustc.edu.cn