Minimality, distality and equicontinuity for semigroup actions on compact Hausdorff spaces

Let $\pi\colon T\times X\rightarrow X$ with phase map $(t,x)\mapsto tx$, denoted $(\pi,T,X)$, be a \textit{semiflow} on a compact Hausdorff space $X$ with phase semigroup $T$. If each $t\in T$ is onto, $(\pi,T,X)$ is called surjective; and if each $t\in T$ is 1-1 onto $(\pi,T,X)$ is called invertible and in latter case it induces $\pi^{-1}\colon X\times T\rightarrow X$ by $(x,t)\mapsto xt:=t^{-1}x$, denoted $(\pi^{-1},X,T)$. In this paper, we show that $(\pi,T,X)$ is equicontinuous surjective iff it is uniformly distal iff $(\pi^{-1},X,T)$ is equicontinuous surjective. As applications of this theorem, we also consider the minimality, distality, and sensitivity of $(\pi^{-1},X,T)$ if $(\pi,T,X)$ is invertible with these dynamics. We also study the pointwise recurrence and Gottschalk's weak almost periodicity of $\mathbb{Z}$-flow with compact zero-dimensional phase space.


Introduction
Let T be a topological semigroup with a neutral element e; that is, T is a T 2 -space and meanwhile it is a multiplicative semigroup with te = et = t for all t ∈ T such that (s, t) → st of T × T to T is continuous.
Let X be a non-singleton compact T 2 -space, unless otherwise stated, in this paper. Given any A ⊂ X, by Int X A and cls X A we will denote respectively the interior and closure of A relative to X. We will write ∆ X = {(x, x) | x ∈ X} for the diagonal set of X × X.
We say that T ×X → X, (t, x) → tx is a semiflow with phase space X and with phase semigroup T , denoted (T, X), if (t, x) → tx is jointly continuous from T × X to X such that ex = x ∀x ∈ X and t(sx) = (ts)x ∀s, t ∈ T, x ∈ X. When T is a topological group, i.e., (s, t) → st −1 of T × T onto T is continuous, then we shall call (T, X) a flow with the phase group T (cf. [35,26,30,4,20]).
Standing notation 1.1. Given (T, X), x ∈ X and subsets A, U, V of X, by T x we denote the orbit {tx | t ∈ T } of x, write T A = t∈T tA = x∈A T x, and set In addition, t −1 x = {y ∈ X | ty = x} for all x ∈ X and t ∈ T .

(T, X)
is called a surjective semiflow if each t ∈ T is a surjective self-map of X, i.e., tX = X for all t ∈ T ; 2. (T, X) is said to be invertible if each t ∈ T is bijective. In this case, by T we will denote the smallest group of self-homeomorphisms of X containing T , and then ( T , X) is a flow.
However, it should be noted that since T is in general neither a syndetic nor a normal subsemigroup of T , (T, X) and ( T , X) do not possess the same dynamics in general. In fact, contrary to what one might hope or expect, the passage from group to the semigroup case is not straightforward in many important cases; cf., e.g., [21]. First let us see an explicit example for this as follows.
1. There exists an invertible semiflow (T, X) such that there are points of X which are almost periodic for (T, X) but not for ( T , X).

Distality, proximity and regional proximity
The concept of "distality" has been proved to be a very fruitful one for topological dynamics of flows, giving rise to a rather extensive theory; see [26,31,4]. We will discuss this for semiflows in this paper.
(f) We say that x ∈ X is proximal to y ∈ X, write (x, y) ∈ P(T, X) or P(X) or y ∈ P[x], if there exist a net {t n } in T and z ∈ X with t n (x, y) → (z, z). By definition, (g) (T, X) is called distal if for all x, y ∈ X with x y, one can find some α ∈ U X with t(x, y) α for every t ∈ T .
(h) An x ∈ X is called a distal point of (T, X) if there exists no point other than itself in cls X T x to be proximal to it under (T, X).
(i) (T, X) is called a point-distal semiflow if there exists a point x ∈ X such that x is a distal point of (T, X) and T x is dense in X (cf. Veech [53]).
(⋆) A flow (T, X) is equicontinuous iff it is uniformly distal.
Proof. Let (T, X) be an equicontinuous flow with ε-δ as in §1.1.1 (a). Then if (x, y) ε, then (t −1 x, t −1 y) δ ∀t ∈ T . Since T −1 = T for T is a group, so (T, X) is uniformly distal. Conversely, assume (T, X) is uniformly distal with ε-δ as in Definition 1.9. It is obvious that (x, y) ∈ δ implies that (tx, ty) ∈ δ for all t ∈ T by T −1 = T again.
Notice that the group structure of T plays a role in both of the "if" and "only if" parts. However, since there is no T = T −1 for a general semiflow with T not a group, hence according to Example 1.7 "Equicontinuous ⇔ Uniformly distal" is not obvious for semiflows with which we will be mainly concerned. See Theorem 1.14 below.
Let (T, X) be an arbitrary semiflow. Next we will introduce another important relation which is weaker than proximity on X.
(j) We say that x ∈ X is regionally proximal to y ∈ X, denoted (x, y) ∈ Q(T, X) or Q(X), if there are nets {x n }, {y n } in X and {t n } in T such that t n (x n , y n ) → (z, z) for some z ∈ X. Clearly, is a closed symmetric reflexive relation on X. Then: Lemma 1. 10. Let (T, X) be a semiflow with Q(X) = ∆ X . Then X × T → X, which is defined by (x, t) → xt := t −1 x, is equicontinuous.
Proof. Given ε ∈ U X , by Q(X) ⊂ ε and by the finite intersection property, there is some δ ∈ U X such that cls X×X t∈T t −1 δ ⊆ ε. Thus δt ⊆ ε for all t ∈ T . This shows that X × T → X is equicontinuous.
It should be noticed that although P(X) and Q(X) both are reflexive symmetric relations on X, yet if T is non-abelian they need not be invariant in our semigroup setting. In view of this, even if P(X) and Q(X) are closed equivalence relations on X, (T, X/P) and (T, X/Q) do not need to make sense in general semiflows.

Amenability and C-semigroup
It is known that the structure of a topological semigroup is closely related to some dynamics of its actions; see e.g. [14]. We will consider here two kinds of phase semigroups as follows.
(k) A discrete semigroup T is called amenable if every semiflow (T, Y) with the phase semigroup T permits an invariant Borel probability measure, i.e., there is a Borel probability measure µ on Y such that µ(B) = µ(t −1 B) ∀t ∈ T for all Borel subset B ⊆ Y (cf. [19,18]).
(l) Let T be a topological semigroup, not necessarily discrete; then T is called a C-semigroup if T \ sT and T \ T s are relatively compact in T for all s ∈ T (cf. [41]).
In particular each abelian semigroup is amenable by the classical Markov-Kakutani fixedpoint theorem. If T is a topological group, then sT = T s = T for all s ∈ T so it is a Csemigroup. Clearly, T = (Z + , +) is a C-semigroup. In addition, under the usual non-discrete topology, T = (R + , +) is a C-semigroup, but not under the discrete topology. 7

Ellis enveloping semigroups
Let X X be the set of all functions from X to itself, continuous or not. The topology of pointwise convergence for X X is defined as follows: A net { f n } in X X converges to f if and only if f n (x) → f (x) for each x ∈ X (cf. [39]). A subbase of this topology is the family of all subsets of the form { f | f (x) ∈ U}, where x is a point of X and U is open in X.
Then we recall several notions based on the semiflow (T, X) as follows: (m) By E(T, X) or simply E(X), we denote the Ellis semigroup of (T, X); that is, E(X) is the closure of T in X X in the sense of the pointwise topology (cf., e.g., [26,30,4]).
(n) An element u ∈ E(X) is called an idempotent in E(X), denoted u ∈ J(E(X)), if u 2 = u.
(o) I ∅ is called a minimal left ideal in E(X) if E(X)I ⊆ I and no proper non-empty subset of I has this property.
• Since E(X) is a compact right-topological semigroup (i.e., E(X) is a semigroup and a compact T 2 -space with R q : p → pq continuous, for all q ∈ E(X)), there always exists an idempotent in each minimal left ideal in E(X) (cf. [26,4]).
• Moreover, (x, x ′ ) ∈ P(X) iff ∃ p ∈ E(X) with p(x) = p(x ′ ) iff there is a minimal left ideal I in E(X) such that p(x) = p(y) ∀p ∈ I.
Clearly E(X) associated to (T, X) is independent of the topology of the phase semigroup T . For a flow (T, X) we will consider whether or not T is a topological group under the pointwise topology in §8.
The proof of the following basic lemma is taken nearly word-for-word from [26, 2 of Proposition 5.16]. We will postpone the details in §6.1 following Lemma 6.3. Lemma 1.11 (cf. [26,4] for T in groups). Given any semiflow (T, X), P(X) is an equivalence relation on X iff there is only one minimal left ideal in E(X).

Main theorems
Although its proof is very easy (cf., e.g., [26,4,29]), yet it is a very useful important fact in topological dynamics that If (T, X) is an equicontinuous flow, then it is distal (cf. [26,Proposition 4.4 and Corollary 5.4]).
In fact, an equicontinuous flow is uniformly distal by §1. 1.3(⋆). We note that the group structure of T plays a role in its various proofs available in the literature (cf. [26,4,29]). Moreover, if T is only a semigroup, the above important result need not be true. For instance, Example 1.7 is equicontinuous but not distal with P(X) = X × X ∆ X .
Let us see a more simple counterexample with abelian phase semigroup, which together with Examples 1.4 shows that some dynamics in semiflows are very different with the flow case.
Then the cascade ( f, X) with phase semigroup Z + is equicontinuous but it is not distal.
Here T is abelian and (T, X) is not minimal. Of course, if (T, X) is minimal equicontinuous with T an abelian (or more general, amenable) semigroup, then we shall show it is uniformly distal (cf. Corollary 3.3). 8 In both of Examples 1.7 and 1.12, each t ∈ T is not surjective. In view of this, recently Ethan Akin and Xiangdong Ye have independently suggested (in personal communications) the following assertion: If (T, X) is an equicontinuous surjective semiflow on a compact metric space with T an abelian semigroup, then (T, X) is distal.
In fact, by constructing an equivalent isometric metric d T on X, Akin's [1, (d) In §2, using several different approaches, we shall be able to prove the Akin-Ye assertion without the metric condition on the phase space X and with no the abelian hypothesis on the phase semigroup T .
Precisely speaking we shall prove the following theorem, which consists of Theorem 2.1, Lemma 6.7, (1) of Proposition 6.10, and Theorem 6.12.
Theorem 1.13. Let (T, X) be a semiflow with phase semigroup T . Then the following statements hold: Since T is in general much more bigger than T , (3) and (4) of Theorem 1.13 are non-trivial. They are useful in our later applications.
Recall that 'equicontinuity' asserts that if two initial points x, y are sufficiently close, then their orbits T x and T y are synchronously close. And 'uniform distality' asserts that if two initial points x, y are sufficiently far away, then their orbits T x and T y are synchronously far away.
Thus, "equicontinuity" looks very different with "uniform distality". However, as a result of Theorem 1.13 we can obtain that they are in fact equivalent to each other. Theorem 1.14. A semiflow (T, X) is uniformly distal if and only if it is equicontinuous surjective.
(2) The "only if" part: First by Theorem 1.13, (T, X) is invertible. Then by Definition 1.9, it follows that given ε ∈ U X , there is a δ ∈ U X such that if (x, y) ∈ δ then (t −1 x, t −1 y) ∈ ε for all t ∈ T and x, y ∈ X. Thus T −1 acts equicontinuously on X. Whence by (4) of Theorem 1.13, ( T −1 , X) is equicontinuous so that (T, X) is invertible equicontinuous. This proves Theorem 1.14.
Let (T, X) be a semiflow and M a closed invariant subset of X. If (T, X) is equicontinuous, then so is (T, M). However, if (T, X) is surjective (even though invertible), (T, M) need not be surjective. For instance, 1 of Examples 1.4. We can here construct a more simple example as follows. Let f : As a consequence of Theorem 1.14, we can easily obtain the following, which is evident if T is a group because tM ⊆ M ∀t ∈ T with T = T −1 implies that M ⊆ t −1 M ⊆ M ∀t ∈ T so that tM = M ∀t ∈ T . However, for a semigroup T , it becomes non-trivial. Proof. By Theorem 1.14, (T, X) is uniformly distal so that (T, M) is also uniformly distal. Then (T, M) is equicontinuous surjective by Theorem 1.14 again.
As be mentioned before, if using no (2) and (4) of Theorem 1.13, then we need a long zigzag proof for this result as in [17].
In 1963 Furstenberg proved that "If X is simply connected non-trivial, then X does not admit of a minimal distal flow for any locally compact abelian group T " (cf. [29,Theorem 11.1]). It is natural to ask if there admits of a minimal distal semiflow or not. In fact, by (3) of Theorem 1.13 there exists no minimal distal semiflow too. Theorem 1.17. If X is simply connected non-trivial, then X does not admit of a minimal distal semiflow for any locally compact abelian semigroup T .
Proof. Assume (T, X) is a minimal distal semiflow with T a locally compact abelian semigroup. Then under the discrete topology of T , ( T , X) is a minimal distal flow by (3) of Theorem 1.13. This then contradicts Furstenberg's theorem.
In particular, the n-sphere, n ≥ 2, cannot support a minimal distal semiflow for any abelian semigroup T .

Applications
There has already been some applications of Theorem 1.13 in the recent work [17] 1 and in the proofs of Theorems 1.14 and 1.17 and Corollary 1.16. Next we shall give some other applications here.
Here (X, T ) will be called the reflection of (T, X), which is also thought of as the 'history' of (T, X).
It should be noted that the phase semigroup of the reflection (X, T ) is T , not T −1 , with the discrete topology in general.
1. Let G be a non-discrete topological group. If (G, X) is a flow and if T is a subsemigroup of G, then (X, T ) is of course a semiflow where T with the non-discrete topology inherited from G.
then (x, t) → xt is jointly continuous by Corollary 8.17 in Appendix so (X, T ) is a semiflow where T with the non-discrete topology.
If (T, X) is a minimal cascade corresponding to a Z + -action, then for every minimal set X 0 of its reflection (X, T ) we have X 0 T = X 0 and furthermore X 0 = T X 0 ; so X 0 = X. This indicates that (X, T ) is also minimal. However, for an invertible semiflow (T, X) with T Z + , 'X 0 T = X 0 ' need not imply X 0 = T X 0 . Moreover, (T, X) and (X, T ) do not share the same dynamics in general. For example, a recurrent/transitive point of a cascade (T, X) need not be a recurrent/transitive point of its reflection (X, T ). In addition, in 1 of Examples 1.4, every point x of Λ is minimal for (T, X) but not minimal for its reflection (X, T ); for otherwise, x is minimal for ( T , X).
Nevertheless, as applications of Theorem 1.13, we will consider in §6 the minimality, distality, and equicontinuity dynamics of (X, T ) as (T, X) itself possesses these dynamics. We will mainly show the following three reflection principles.
Reflection principle I (cf. Propositions 6.1 and 6.10). Let T × X → X be an invertible semiflow with the reflection X × T → X. Then: Reflection principle I may be utilized for proving the "only if" part of Theorem 1.14 above. Moreover, it will be useful for us to show Furstenberg's structure theorem of distal minimal semiflows (Theorem 6.14) in §6.2.
Here we are going to give another application. Let (T, X) and (T, Y) be two semiflows and π : X → Y a continuous surjective map. If π(tx) = tπ(x) for all t ∈ T and x ∈ X, then (T, Y) is called a factor of (T, X) and π an epimorphism of (T, X) and (T, Y), denoted π : (T, X) → (T, Y). Definition 1.20. We will say that (T, X) is a relatively equicontinuous (or an almost periodic) extension of (T, Y) in case there is an epimorphism π : (T, X) → (T, Y) such that for every ε ∈ U X there exists δ ∈ U X satisfying that if (x, x ′ ) ∈ δ with π(x) = π(x ′ ) then (tx, tx ′ ) ∈ ε ∀t ∈ T (cf. [4, p. 95]). If Y is one-pointed, then this reduces to Definition (a) in §1.1.1. Theorem 1.21 below is actually a relativized version of (1) of Theorem 1.13 before. Its special case that (T, X) is a skew-product semiflow driven by (T, Y) with T = R + has been proven in [46,47] and [48] by using Ellis' enveloping semigroup. Proof. Let π : (T, X) → (T, Y) be a relatively equicontinuous epimorphism. First by Theorem 1.13, (T, Y) is invertible and then (Y, T ) is distal by Reflection principle I. Note that π : X → Y is also a homomorphism from (X, T ) onto (Y, T ). We will show that (X, T ) is distal.
For that, let x, x ′ ∈ X with (x, x ′ ) ∈ P(X, T ). Then by distality of (Y, T ), π(x) = π(x ′ ). Taking a net {t n } in T with t −1 n (x, x ′ ) → (z, z) for some z ∈ X, by the relative equicontinuity of (T, X) and π(t −1 n x) = π(t −1 n x ′ ), it follows that x = x ′ and so (X, T ) is distal. Again using Reflection principle I, (T, X) is distal. This proves Theorem 1.21.
The above proof implies that if π : (T, X) → (T, Y) is a relatively equicontinuous epimorphism of flows π is of distal type (cf., e.g., [4, p. 95]). It would be interested to know if this holds for any invertible semiflow or not.
Next for invertible semiflows with amenable semigroups we can obtain the following Reflection principle II.
Reflection principle II (cf. Theorem 6.19 and Proposition 6.20). Let (T, X) be invertible with T an amenable semigroup and x ∈ X. Then: It should be mentioned that in light of Examples 1.4 the amenability of the phase semigroup T is very important for the statement of Reflection principle II. As a result of our Reflection principle II, we can generalize Theorem 1.21 in amenable semigroups as follows: Proof. Let π : X → Y be a relatively equicontinuous extension. Let y ∈ Y, with π(x) = y for some x ∈ X, be a distal point of (T, Y), then y is also a distal point of (Y, T ) by Reflection principle II so that if (x, x ′ ) ∈ P(X, T ) then π(x ′ ) = y, i.e., x ′ ∈ π −1 (y). Hence as in the proof of Theorem 1.21, we can easily show that x ′ = x. Thus x is a distal point of (X, T ) and then of (X, T ).
In addition, using Reflection principle II in the classical case that T = R + and T = R we can easily obtain the following: In particular, if R + × X → X is a distal semiflow, then R × X → X is a distal flow (cf. Sacker-Sell [47]).
The following is another consequence of Reflection principle II. Theorem 1.23 (cf. [49] for T = R + ). Let (T, X) be invertible with T an amenable semigroup such that P( T , X) is an equivalence relation on X. Then P( T , X) = P(T, X) = P(X, T ).
Proof. We only prove P( T , X) = P(T, X). Clearly, P(T, X) ⊆ P( T , X). To show the converse inclusion, let (x, x ′ ) ∈ P( T , X). Let I be the unique minimal left ideal in E( T , X) by Lemma 1.11. Then there exists some u ∈ I with u(x) = u(x ′ ). We will show that u ∈ E(T, X).
For that, we need to consider the natural flow π * : T × E( T , X) → E( T , X), which has only one minimal subset I. Clearly E(T, X) ⊆ E( T , X) and π * : T × E(T, X) → E(T, X) is compatible with ( T , E( T , X)). We can choose a minimal subset K ⊆ E(T, X) with respect to (T, E(T, X)). Then by 1 of Reflection principle II, K is also T −1 -invariant so that K is Tinvariant and then T -minimal. Thus K = I. This implies u ∈ E(T, X) and thus (x, x ′ ) ∈ P(T, X).
The proof of Theorem 1.23 is therefore complete.
Notice that in general, T ∪ T −1 T for an invertible semiflow (T, X). 2 of Reflection principle II says that an x ∈ X is a distal point of (T, X) iff x is a distal point of (X, T ). However,

Is x a distal point of the induced flow ( T , X)?
It is exactly the localization problem of (3) of Theorem 1.13. As a consequence of Theorem 1.23, the answer to this question is YES in the setting of Theorem 1.23.
In the same situation of Theorem 1. 23, an x ∈ X is a distal point of (T, X) iff x is a distal point of ( T , X).
In fact, this still holds without the condition that P( T , X) is an equivalence relation on X; see 1 of Theorem 5.4 by purely topological methods.
We note that if T = R + in the proof of Theorem 1.23 and so T = R, then I is contained in the ω-limit set ω(e) of e ∈ T under ( T , E( T , X)). Since e = id X ∈ E(T, X) and ω(e) ⊆ E(T, X) by T = R, then I ⊆ E(T, X). This is actually the idea of Yi's proof in the R-action case in [49, p. 7]. Clearly Yi's idea does not work for our Theorem 1.23 here.
In addition, when T is abelian and if E( T , X) ⊂ C(X, X), then by Theorem 6.6 it follows that P( T , X) is an equivalence relation on X. Moreover, if P( T , X) is closed in X × X, then it is an equivalence relation (cf. [4,Corollary 6.11]).
Next we consider invertible semiflows with C-semigroups as our phase semigroups instead of amenable semigroups.
Reflection principle III (cf. Theorem 6.31). Let (T, X) be invertible with T a C-semigroup not necessarily discrete. Then (T, X) is minimal iff so is (X, T ).
Comparing with Reflection principle II, we pose the following Question 1.24. Let (T, X) be invertible with T a C-semigroup and x ∈ X. Then, does it hold that x is minimal for (T, X) iff so is x for (X, T )?
As a simple application of our reflection principles, we will study the sensitivity of semiflows and their reflections in §7.

Distality of equicontinuous surjective semiflows
Recall that a semiflow (T, X) is distal if and only if no diagonal pair is proximal (cf. §1.1.3 (g)). This section will be mainly devoted to proving (1) of Theorem 1.13 using three different approaches, which asserts that every equicontinuous surjective semiflow is distal. That is the following Theorem 2.1. 13 Theorem 2.1. If (T, X) is an equicontinuous surjective semiflow, then (T, X) is distal.
Our new approaches (Proofs (I), (II) and (III) below) introduced in proving Theorem 2.1 are all certainly valid for flows with phase groups.

Using pointwise recurrence of transition maps
For Proof (I) of Theorem 2.1, in preparation we first recall a classical notion. Let f be a continuous self-map of X.
. Then the following is easily seen by definition.
Thus if ( f, X) is pointwise recurrent, then f is surjective (cf. [2, Lemma 3.1]). The following simple observation is very useful for Theorem 2.1.
Let ε ∈ U X and let δ correspond to ε in the definition of equicontinuity. Let n > 0 and s > 0 be integers such that (x n , x n+s ) ∈ δ, so ( f n+s (x n ), f n+s (x n+s )) ∈ ε. Then (x, f s (x)) ∈ ε and thus x is (forwardly) recurrent for ( f, X). Now we can prove Theorem 2.1 by using the pointwise recurrence as follows: Proof (I) of Theorem 2.1. For t ∈ T , (t, X×X) is equicontinuous surjective and thus by Lemma 2.3 it is pointwise recurrent. Suppose (y, y ′ ) ∈ P(T, X) with y y ′ . Let ε ∈ U X such that (y, y ′ ) ε. Let δ ∈ U X correspond to ε/3 as in §1.1.1 (a). Since y is proximal to y ′ , we now can take τ ∈ T such that (τy, τy ′ ) ∈ δ, so (τ n y, τ n y ′ ) ∈ ε/3 for all n > 0. Then there cannot be n i with τ n i (y, y ′ ) → (y, y ′ ). But this contradicts the pointwise recurrence. The proof of Theorem 2.1 is thus completed.

Using almost periodicity
Let (T, X) be a semiflow with T a topological semigroup not necessarily discrete here. We will first recall the concept of "almost periodicity".
If every point of X is almost periodic, then (T, X) is called pointwise almost periodic.
(iv) (T, X) is called uniformly almost periodic (u.a.p.) if given ε ∈ U X , there exists a syndetic set A in T such that Ax ⊆ ε[x] for all x ∈ X.
Here "thick set" is weaker than the notion-replete set [35,Definition 3.37] that requires containing some bilateral translate K s ∪ sK of each compact subset K of T .
Given k ∈ T , let L k : T → T, t → kt be the left translation mapping of T . Then for subsets K, A of T , we simply write Since here T is only a semigroup, K −1 A is possibly empty. If e ∈ K then A ⊆ K −1 A. Thus a subset A of T being syndetic in T can be equivalently described as follows:

A is syndetic in T if and only if there exists a compact subset K of T with T
Note that in some literature, an "almost periodic" point is defined as N T (x, U) is "syndetic" in the sense that there is a compact subset K of T with T = KN T (x, U).
It should be mentioned that " is not permitted to be replaced by "T = KN T (x, U)" in semiflows here; see [9, Proposition 4.8] for a counterexample which says that there is a semiflow on a compact metric space such that it has an almost periodic point in the sense of (iii) of Definition 2.4 but has no "almost periodic" points in the latter sense.
The following two equivalent conditions will be very useful for our later arguments involving almost periodicity.

Lemma 2.5 (cf. [30] for T = Z + ). A subset S of T is syndetic in T if and only if S ∩ R ∅ for each thick set R in T .
Proof. Let S be syndetic in T and let K a compact subset of T defined by syndeticity of S . Then for each thick set R in T , there is some t 0 ∈ T with Kt 0 ⊆ R. Since (Kt 0 )∩S ∅, hence R∩S ∅. Conversely, let S ∩ R ∅ for all thick set R in T . If S is not syndetic, then for each compact subset K of T there is t K ∈ T such that Kt K ∩ S = ∅. Set R = K∈K Kt K where K is the set of all non-empty compact subsets of T . Clearly R is thick in T , but S ∩ R = ∅, a contradiction. This proves Lemma 2.5.
Since our phase space X is a compact T 2 -space, every orbit closure contains a minimal set by Zorn's lemma. So it contains an almost periodic point by the following basic result (cf. [33,30,9]). We will present a proof here for self-closeness. Lemma 2.6 (cf. [35] and [26,Proposition 2.5] for T in groups). Let (T, X) be a semiflow where T not necessarily discrete; then a point x of X is almost periodic iff it is a minimal point.

Notes.
1. Instead of 'compact X' 'regular X' is enough for the necessity. 2. Moreover, this lemma shows that the almost periodicity of any point of X is independent of the topology of the phase semigroup.
Proof. Let x be almost periodic for (T, X); and if Λ is a minimal subset of cls Conversely, let x be a minimal point and let U be an open neighborhood of x. Since T y is dense in cls X T x for all y ∈ cls X T x, hence {t −1 U | t ∈ T } is an open cover of the compact subspace cls X T x. Thus one can find a finite subset Proof. By the joint continuity of tx, if (x, y) ∈ P(X), then for every ε ∈ U X , {t ∈ T | t(x, y) ∈ ε} is a thick subset of T . Thus we can obtain the conclusion.
The following lemma is a generalization as well as strengthening of Lemma 2.3. See [4, Lemma 2.3] for T in groups.

Lemma 2.8. If (T, X) is equicontinuous surjective, then every point of X is an almost periodic point of (T, X).
Proof. Let x ∈ X and let M be a minimal subset of cls X T x. If x M, then there is an ε ∈ U X with x ε[M]. Let tx be arbitrarily close to some y ∈ M. Since x is a recurrent point for (t, X) by Lemma 2.3, there is a net {n k } in N with n k → ∞ and t n k x → x. Then by equicontinuity, it follows that t n k x is arbitrarily close to t n k −1 y ∈ M and so x is arbitrarily close to M, contradicting x ε[M]. Thus x ∈ M and so every point of X is almost periodic by Lemma 2.6.
Note that in view of Example 1.12 the 'surjective' condition is important for Lemma 2.8. However, it is not a necessary condition for almost periodicity; for instance, Example 1.7. Now, based on Lemma 2.7 and Lemma 2.8, we can present another concise proof of Theorem 2.1 as follows.
It is interesting to notice that "equiconinuous + surjective ⇒ distal" (Theorem 2.1) and "equiconinuous + surjective ⇒ almost periodic" (Lemma 2.8) can not be localized. In fact we can easily construct a counterexample on the interval I = [0, 1] with the usual topology. However, "distality ⇒ almost periodicity" may be localized (cf. Theorem 5.1 in §5). Example 2.9. Let f : I → I be defined by x → x 2 . Then 0 and 1 are the only recurrent (fixed) points of ( f, I). Moreover, ( f, I), as a flow with phase group Z, is equicontinuous at each x ∈ (0, 1) but x ∈ (0, 1) is neither an almost periodic point and nor a distal point of ( f, I).
Using our Theorem 2.1 as an important tool, Dai and Xiao in [17] have proved the following equivalence of 'uniformly almost periodic' and 'equicontinuous surjective' (see 1 and 2 of [26,Proposition 4.4] for T in groups).

Theorem 2.10 ([17]). Let (T, X) be a semiflow with phase semigroup T not necessarily discrete. Then (T, X) is u.a.p. if and only if it is equicontinuous surjective.
It should be noted that the compactness of X is essential for Theorem 2.10. For example, the Proof. Let (T, Y) be a factor of an equicontinuous surjective semiflow (T, X) via an epimorphism Although the notion 'syndetic' is closely related to the topology of the phase semigroup T , yet Theorem 2.10 shows that 'u.a.p.' does not depend on it, since distality and equicontinuity do not depend on it.
Let C(X) be the algebra of real-valued continuous functions on X; then based on (T, X), T can act from the right on Similar to [26,Proposition 4.15] for T in groups, with the aid of Theorem 2.1, we can characterize u.a.p. and equicontinuity of (T, X) in terms of C(X).
Proposition 2.12. Let (T, X) be a semiflow with phase semigroup T . Then:

(T, X) is u.a.p. if and only if (T, X) is surjective and for every f ∈ C(X) the uniform closure of f T is a compact subset of C(X). 2. (T, X) is equicontinuous if and only if for every f ∈ C(X) the uniform closure of f T is a compact subset of C(X).
Proof. Let (T, X) be u.a.p. and f ∈ C(X). Then by Theorem 2.10, f T and its uniform closure are equicontinuous and (T, X) is surjective. Hence f T is relatively compact in C(X) under the supremum norm by Ascoli's theorem (cf. [39, Theorem 7.18 in p. 234]). Now assume f T is relatively compact in C(X) under the supremum norm, for all f ∈ C(X); and let α ∈ U X be any given. Then there exist ε > 0 and F a finite subset of Hence T δ ⊆ α and (T, X) is equicontinuous and so u.a.p. by Theorem 2.10.
The same argument implies the second assertion of Proposition 2.12 and we thus omit the details here.

Using Ellis' enveloping semigroup
Based on Ellis' semigroup (cf. §1.1.5), the following another short proof of Theorem 2.1 without using the pointwise recurrence of an equicontinuous surjection is the other important idea of this paper.
X is jointly continuous and hence the topology of pointwise convergence coincides with the compact-open topology for E(X) (cf. [39,Theorem 7.15]). It follows easily from equicontinuity and surjectivity of each t ∈ T that all p ∈ E(X) are surjective. Indeed, let T ∋ t n → p ∈ E(X) and p(X) X; then there is an ε ∈ U X so small that U = ε[p(X)] X. Since p(X) ⊂ U and t n → p in the sense of compact-open topology, t n (X) ⊆ U as n sufficiently large. This contradicts } is a non-empty closed subsemigroup of E(X) and so there is an idempotent u in E(X) with u(x) = u(y) and so x = y. This proves Theorem 2.1.

When is a semiflow surjective?
In light of Examples 1.7 and 1.12, the "surjective" condition is essential for our assertion of Theorem 2.1. In this section we will now introduce some sufficient conditions for that 'each t ∈ T is surjective' for a semiflow (T, X) with some special phase semigroups T .

Homogeneity condition
Let (T, X) be a semiflow. Since X is compact by our convention, each (t, X) must have almost periodic points by Lemma 2.6 and so it has (forwardly) recurrent points. This point is very useful for us to justify the surjectiveness of a semiflow by the so-called "homogeneity" condition as follows.
Proof. Let t ∈ T . Since (T, X) is homogeneous, then the (forwardly) recurrent points are dense in X for (t, X). Because if x is recurrent for (t, X) it is such that x ∈ cls X X {t n x | n ≥ 1} ⊆ tX and tX is closed, it follows that t is a self-surjection of X for each t ∈ T . Thus (T, X) surjective, and then it is distal by Theorem 2.1 if it is equicontinuous.
Particularly, if (T, X) is minimal with T abelian, then it is homogeneous and thus (T, X) is surjective by Proposition 3.2. Here we will present a more simple independent proof for this as follows.

Corollary 3.3. Let (T, X) be a minimal semiflow with T abelian. Then (T, X) is surjective and hence if (T, X) is in addition equicontinuous it is (uniformly) distal.
Proof. Let Z = tX for all t ∈ T . Then Z is closed and since T abelian Z is T -invariant. Thus Z = X. This completes the proof by Theorem 2.1 (and Theorem 1.14).
It should be noticed here that in view of Example 1.7 the abelian condition in Corollary 3.3, which guarantees the homogeneity, is essential. This result will be generalized to amenable semigroups by Proposition 3.7 in §3.2, using ergodic theory.
Given any integer d ≥ 1, as a consequence of Proposition 3.2 and Theorem 2.1, the following corollary seems to be non-trivial because it is beyond Ellis' joint continuity theorem.
which is an additive abelian semigroup. First, under the discrete topology of T , (T, X) is a minimal semiflow. Then by Corollary 3.3, it follows that for each t ∈ T , x → tx is a continuous surjection of X. Therefore, (R d + , X) is distal by Theorem 2.1. 18

Amenable semigroup condition
More general than the case of abelian phase semigroup, now we will consider amenable one (cf. §1.1.4 (k)). Lemma 3.6. Let µ be an invariant quasi-regular Borel probability measure of (T, X); then Proof. Set S = supp (µ). By definition, it easily follows that S is closed; and moreover, S is of µ-measure 1. Otherwise, by the quasi-regularity of µ there exists a compact subset K of X with K ∩ S = ∅ such that µ(K) > 0; then K contains at least one point of S . For, if not, then there is an open neighborhood V x of any x ∈ K with µ(V x ) = 0 and so by the compactness of K, µ(K) = 0 contradicting µ(K) > 0. Now given t ∈ T , since tS is a Borel set and µ(tS ) = 1, we can easily obtain that tS = S . Indeed, tS ⊇ S is obvious. (If S \ tS ∅, then X \ tS is an open set containing points of S so that µ(X \ tS ) > 0, a contradiction to µ(tS ) = 1.) Next assume tS S and then we can choose some y ∈ tS − S and x ∈ tS such that tx = y. Now we can pick an open set U with y ∈ U ⊂ X − S .
This thus completes the proof of Lemma 3.6.
Now we can easily conclude the following by Theorem 2.1 together with Lemma 3.6, which generalizes Corollary 3.3.

Proposition 3.7. Let (T, X) be a semiflow with T an amenable semigroup and with a dense set of almost periodic points. Then (T, X) is surjective; and hence if (T, X) is in addition equicontinuous it is distal.
Note. In view of Lemma 3.6, the statement of Proposition 3.7 is still true if (T, X) is only a general minimal semiflow admitting an invariant Borel probability measure.
Proof. Let x ∈ X be an almost periodic of (T, X) and write X x = cls X T x. Then (T, X x ) is a minimal subsemiflow of (T, X). Since T is amenable by hypothesis, hence by amenability and Riesz's theorem there exists an invariant quasi-regular Borel probability measure µ for (T, X x ). Moreover, since (T, X x ) is minimal and supp (µ) ⊆ X x is T -invariant, thus supp (µ) = X x . Then by Lemma 3.6, it follows that each t ∈ T restricted to X x is a surjection of X x . Thus x ∈ tX for all t ∈ T . This shows that tX = X for all t ∈ T , because almost periodic points are dense in X and tX is closed. Finally by Theorem 2.1, it follows that (T, X) is distal, if it is equicontinuous. This therefore proves Proposition 3.7.

C-semigroup condition and ℓ-recurrence
It was already known that if x is a recurrent point of a continuous self-map f of X then x ∈ f (X) (by Lemma 2.2). However, even for a minimal semiflow (T, X), X tX in general; see Example 1.7. Now we will generalize Lemma 2.2 to semiflows with a kind of special phase semigroups.
Definition 3.8 ( [41]). Let T be a topological semigroup, which is not necessarily discrete. Then: When T is right and left C-semigroup, it is called a C-semigroup as in Definition (l) in §1.1.4. For example, let T = [1, ∞) with e = 1; then T is a multiplicative C-semigroup under the usual topology.
Next we need the notion-recurrent point-for a semiflow with general phase semigroup beyond T = Z + . Definition 3.9. Let (T, X) be a semiflow, where T is a non-compact topological semigroup, not necessarily discrete. By K e we will denote the family of all compact neighborhoods of e. Then: Of course, even if T = Z, an ℓ-recurrent point need not be an almost periodic point for a general semiflow. For instance, every point of X is ℓ-recurrent for ( T , X) in 1 of Examples 1.4, but it is not almost periodic except the two ends −1 and 2 of X.
for every K ∈ K e , there is some n K with t n ∈ K c ∀n ≥ n K .
Note. If a net {t n } in T satisfies condition (2), then we shall say t n → ∞.
Proof. The sufficiency is obvious; so we only need to prove the necessity. For this, assume x is an ℓ-recurrent point of (T, X).
Let U x be the neighborhoods filter of x. Define a binary relation ≥ on U x × K e as follows: is a directed set. Now for every (U, K) ∈ U x × K e , we can take some t U,K ∈ T such that t U,K x ∈ U and t U,K ∈ K c . It is easy to see that {t U,K } is a net in T satisfies conditions (1) and (2). This proves Lemma 3.10. 20 For example, let X = R ∪ {∞} be the one-point compactification of the 1-dimensional Euclidean space R (so X is homeomorphic with the unit circle) and let T = (R, +) with the usual topology. Define a flow on X with the phase group T as follows: When T is a group, an almost periodic point is always an ℓ-recurrent point.
Proof. If A is a syndetic subset of T , then A is never contained in any K ∈ K e for T is non-compact.
(c) More generally than the above (b), let T be such that each syndetic set is not relatively compact in T . Then every almost periodic point is ℓ-recurrent for (T, X).
Remarks 3.12. The almost periodicity is a strong form of recurrence, yet (b) of Remark 3.11 is false in general if T is not a group, even for semiflows on compact metric spaces with no isolated points. Let's construct such an example as follows.
(1) Let Y be a locally compact, non-compact, Polish space with no isolated points like Y = R d ; Then T is a locally compact, non-compact, σ-compact (in fact separable), and non-abelian topological subsemigroup of C(Y, Y) under the topology defined by the way: for every net {t n } in T , In this case, e is an isolated point of T and T \ {e} is homeomorphic with Y, i.e., t n → t in T iff t n → t in Y.
(2) We now consider the naturally induced semiflow on Y with the phase semigroup T as follows: Given But (T, Y) shows that these statements need not be true in general semiflows.) (3) Further based on (1) and (2), define X = Y ∪ {∞} to be the one-point compactification of Y. We now consider the naturally induced semiflow on X For every x ∈ X and all neighborhood U of x, U) is syndetic, which is left-thick but not right-thick.) Clearly, T x = Y dense in X for all x ∈ Y and T ∞ = X. This shows that • (T, X) is minimal, pointwise almost periodic and equicontinuous, but not distal. Nevertheless, Proof. For every x ∈ X, y ∈ Y, and taking a compact neighborhood K of y in Y, (Note here that T is neither an amenable semigroup nor a C-semigroup.) Remarks 3.13. Let T be a locally compact, σ-compact, and non-compact topological semigroup with an increasing sequence {K n } of compact neighborhoods of e such that T = n K n and let (T, X) be a semiflow. Then: (1) ℓ T (x) = n cls X K c n x for all x ∈ X. Thus, if X is a metric space, then y ∈ ℓ T (x) if and only if ∃ t n ∈ K c n with t n x → y as n → ∞. (2) If s −1 K is relatively compact in T for all s ∈ T and K ∈ K e , then ℓ T (x) is invariant for (T, X) with X a metric space. Thus ℓ T (x), for x ∈ X, is an invariant closed non-empty set if X is a compact metric space.
Proof. Indeed, for all y ∈ ℓ T (x) and s ∈ T , let t n ∈ K c n with t n x → y. Then st n x → sy. For every compact subset K of T , there is some n 0 > 0 such that st n K as n > n 0 . This shows that we can select out a subsequence {τ n } from {st n } with τ n ∈ K c n such that τ n x → sy. Thus ℓ T (x) is invariant for all x ∈ X.
We notice that the classical topological semigroups T = R d + and Z d + both are locally compact non-compact σ-compact.
Remarks 3.14. Let (T, X) be a semiflow on a uniform T 2 -space (X, U X ) not necessarily compact with phase semigroup T . Then: Whenever T is a topological group and X is a locally compact metric space instead of a compact metric space, this statement still holds (cf. [9,Corollary 4.2]). In view of this, the following question is natural: Does the statement of [9,Theorem 1.3] still hold if (T, X) is a semiflow on a locally compact metric space X? (cf. [9, Question 4.9]) (c) Now in the same situation of (2) of Remark 3.12, Y is a locally compact, non-compact, Polish space. If y ∈ Y were Birkhoff recurrent for (T, Y), then cls Y T y = Y would be compact by [9,Lemma 3.4]. Therefore, every point of Y is almost periodic but not Birkhoff recurrent. This thus gives us a negative solution to [9,Question 4.9].
Remark 3.15. Let (T, X) be a semiflow with T a locally compact non-compact semigroup and Proof. Given y ∈ X, let U be an arbitrary neighborhood of y and K ∈ K e . Then U K x; otherwise, Int X T x ∅. Then tx ∈ U for some t ∈ K c . Thus y ∈ ℓ T (x). Now we can generalize Lemma 2.2 from the special case T = Z + to every left C-semigroup (cf. 2 of Definition 3.8) as follows: Proposition 3.16. Let (T, X) be a semiflow with T a locally compact, non-compact, left Csemigroup and x ∈ X. If y ∈ ℓ T (x), then y ∈ tcls X T x for every t ∈ T . Hence ℓ T (x) ⊆ tX for all t ∈ T .
Proof. Let t ∈ T . Since T \ tT is relatively compact in T and y is a limit point of x (cf. 2 of Definition 3.9), there is a net {t n } in T with tt n x → y. Take t n x → z ∈ cls X T x (passing to a subnet if necessary). Thus tz = y so y ∈ tcls X T x. This proves Proposition 3.16.
The following is a simple consequence of Proposition 3.16, which generalizes [2, Lemma 3.1] from T = Z + to a general left C-semigroup.
Note that an ℓ-recurrent point need not be a minimal point. So Corollary 3.17 is comparable with Proposition 3.7. Moreover, (3) of Remark 3.12 shows that the left C-semigroup condition is essential for Corollary 3.17, since ∞ tX for all t ∈ T, t e. Corollary 3.18. Let (T, X) be a semiflow with T a locally compact, non-compact, left C-semigroup and x ∈ X. Then: Then the assertion (1) follows at once from Proposition 3.16.
(2) Based on (1) it follows that (T, X) is surjective. Then by Theorem 2.10, (T, X) is u.a.p. and so it is minimal. This proves Corollary 3.18.
In view of Example 1.12, the condition "Int X T x = ∅" is important for the assertions of Corollary 3.18.

Inheritance theorems
It is a well-known fact that for every flow (T, X) and for all syndetic subgroup S of T , (T, X) is distal if and only if (S , X) is distal (cf. [26,Proposition 5.14]). In fact, this kind of inheritance theorem also holds for semiflows with phase semigroups as follows: Note. When T is a topological group, see [26,Lemma 5.13] for (1)  Proof. (1) Evidently P(S , X) ⊆ P(T, X). On the other hand, let (x, y) ∈ P(T, X) and let α ∈ U X , then A α := {t ∈ T | t(x, y) ∈ α} is a thick set of T . Since S is syndetic in T , thus S ∩ A α ∅. This shows (x, y) ∈ P(S , X). Thus P(S , X) = P(T, X).
Conversely, if (S , X) be u.a.p., then (T, X) is u.a.p. because every syndetic subset of S is also syndetic in T by (ii) of Definition 2.4.
(4) This follows evidently from Theorem 2.1, (3) and Theorem 2.10. (5) Let (S , X) be invertible. Since S is syndetic in T , there is a compact subset K of T such that for any t ∈ T , there are k ∈ K and s ∈ S with kt = s. Let K ′ = {k ∈ K | ∃t ∈ T s.t. kt ∈ S }; then for any t ∈ T , there is some k ′ ∈ K ′ with k ′ t = s ∈ S . This implies that every t ∈ T is an injection of X and each k ′ ∈ K ′ is a surjection of X. Thus each k ∈ K ′ is a homeomorphism of X so that each t ∈ T is a homeomorphism of X.
(6) Clearly Q(T, X) ⊇ Q(S , X). Let K be a compact subset of T such that for any t ∈ T , there are k t ∈ K and s t ∈ S such that k t t = s t . Given any α ∈ U X , there is some β ∈ U X with Kβ ⊆ α.
Note that in (4) of Proposition 4.1, since the syndetic subsemigroup S need not be dense in T , this statement is thus non-trivial. Moreover according to Theorem 1.14, it can be equivalently illustrated as follows: (4) (T, X) is uniformly distal if and only if so is (S , X).
Next we can obtain a simple consequence of Proposition 4.1. The following is, more or less, motivated by Clay's [12,Theorem 9]. Proof. We first show that T x × T x ⊂ P(T, X). In fact, for all t, s ∈ T and α ∈ U X , we can find some τ ∈ T such that {t, s}τx ⊂ α[p]. Then τ(tx, sx) ⊂ α. This implies that (tx, sx) ∈ P(T, X).
Thus T x × T x ⊆ P(T, X) = P(S , X) by (1)   Moreover, if starting from Theorem 2.10 as Proof (II) of Theorem 2.1, we can easily obtain Theorem 2.1. But the proof of Theorem 2.10 is itself based on Theorem 2.1 in [17].
Finally, we note that the compactness of the phase space X is important for Theorem 2.1. Otherwise, the statement is false; see [17, Example 3.7].

Distality of points by product almost periodicity
It is well known that (T, X) is distal iff (T, X × X) is pointwise almost periodic (cf. [17, Proposition 2.5]; also see 1 and 3 of [26, Proposition 5.9] for flows). In fact, by a purely topological proof, we can obtain the following characterization of distal points, which implies that every distal point is an almost periodic point. Proof. (1) Necessity: Let y ∈ X be any almost periodic point of (T, X). By Zorn's lemma, there exists a maximal subset A of X with y ∈ A such that for all a 1 , . . . , a k in A, (a 1 , . . . , a k ) is almost periodic for (T, X k ), for all k ≥ 1. Now for z = (z a ) a∈A ∈ X A with z a = a ∀a ∈ A, we can take an almost periodic point (z ′ , x ′ ) in cls X A ×X T (z, x) for (T, X A × X). Since z is almost periodic for (T, X A ), then there is a net {t n } in T with t n (z ′ , x ′ ) → (z, x * ) and (z, x * ) is also almost periodic for (T, X A × X). So x * ∈ A by maximality of A. Further we can select a net {s n } in T such that s n (z, x) → (z, x * ) and then s n (x * , x) → (x * , x * ) with x * ∈ cls X T x. Thus x = x * ∈ A by distality of (T, X) at x. Then x, y ∈ A. Therefore by definition of A, (x, y) is almost periodic for (T, X × X).
(2) Sufficiency: Since X is compact, by Zorn's lemma we can choose a y 0 ∈ X which is almost periodic for (T, X). So x is almost periodic for (T, X) and further every y ∈ cls X T x is almost periodic. Thus, for all y ∈ cls X T x, (x, y) is almost periodic. This implies that x must be distal (by Lemma 2.7).
The proof of Theorem 5.1 is thus completed.
It should be noticed that by using IP * -recurrence of a distal point and his central sets of Z + , Furstenberg's [30, (i) ⇔ (iv) in Theorem 9.11] says that x ∈ X is distal for (T, X) iff (x, z) is almost periodic for (T, X × Z) for all almost periodic point z ∈ Z, for the special case T = Z + with X a compact metric space (cf. [15,Theorem 4] for general semiflows on compact T 2 -spaces).
Definition 5.2. We say that (T, X) satisfies the Bronstein condition if the set of almost periodic points of (T, X × X) is dense in X × X.
The Bronstein condition is a very important one in topological dynamics; see, e.g., [54]. Then as a consequence of Theorem 5.1 and Lemma 1.8, we can easily obtain the following result, which says that the point-distal (cf. §1.1.3 (i)) implies the Bronstein condition. Proof. Since (T, X) is surjective point-distal, then by Lemma 1.8 it follows that the distal points are dense in X. Then by Theorem 5.1, for all distal point x ∈ X and every y ∈ X, (x, y) is almost periodic for (T, X × X). This proves Corollary 5.3.
If (T, X) is invertible point-distal with T an amenable semigroup, then we shall show later on that its reflection (X, T ) is point-distal (cf. Proposition 6.20). Here, based on Theorem 5.1, we can first prove that ( T , X) is point-distal.

Theorem 5.4. Let (T, X) be invertible with T an amenable semigroup and x ∈ X. Then:
1. x is a distal point of (T, X) iff x is a distal point of ( T , X).

(T, X) is point-distal iff ( T , X) is a point-distal flow.
Proof. (1). Clearly if x is a distal point of ( T , X), then it is a distal point of (T, X). Conversely, let x be a distal point of (T, X); we will show x is also a distal point of ( T , X). Given y ∈ cls X T x, by Theorem 5.1, (y, x) is an almost periodic point of (T, X × X). Then W = cls X×X T (y, x) is a minimal subset of (T, X × X) by Lemma 2.6. Since T is amenable, by Proposition 3.7 or by 1 of Reflection principle II, it follows that ( T , W) is a minimal subflow of ( T , X × X) and so cls X T x = cls X T x. Thus by Lemma 2.6 again, (y, x) is an almost periodic point of ( T , X × X). Using Theorem 5.1 again, x is a distal point of ( T , X).
(2). In view of 1 of Theorem 5.4, we only need to show that if ( T , X) is minimal, then (T, X) is minimal. In fact, when Λ is a minimal subset of (T, X), by 1 of Reflection principle II we can see Λ = X. This proves Theorem 5.4.

Dynamics of reflections of invertible semiflows
This section will be mainly devoted to proving (2), (3) and (4) of Theorem 1.13 and our Reflection principles I, II and III using Theorem 2.1. As applications of our reflection principles, we will prove Furstenberg's structure theorem of minimal distal semiflows in §6.2 and we shall consider minimal non-sensitive invertible semiflows in §7.
Recall that a semiflow (T, X) is invertible iff each t ∈ T is bijective; and in this case, T denotes the group generated by T . Then ( T , X) is a flow on X. However since T is neither a syndetic nor a normal subsemigroup of T in general, the dynamics properties of ( T , X) can not be naturally inherited to (T, X) in many cases.
When (T, X) is invertible, (X, T ) denotes its reflection or 'history' defined as in Definition 1.18. If (T, X) had certain dynamical property P in the past, i.e., (X, T ) has P, then does (T, X) have P? This kind of dynamics is called satisfying "reflection principle" here.

Distality and equicontinuity of reflections
Theorem 2.1 implies the following, for which we will present a direct proof with no uses of Ellis' joint continuity theorem (Theorem 8.8 in §8) and Ellis's semigroup.

Proposition 6.1. Let (T, X) be an invertible semiflow; then (T, X) is equicontinuous if and only if so is (X, T ).
Proof. By symmetry we only prove the "only if" part and so assume (T, X) is equicontinuous. To be contrary, suppose that (X, T ) is not equicontinuous at some point x ∈ X. Then there are where z z ′ . This shows that (z, z ′ ) ∈ Q(T, X); i.e., z is regionally proximal to z ′ for (T, X) (cf. Definition (j) in §1.1.3). Then it follows easily from the equicontinuity of (T, X × X) that (z, z ′ ) is a proximal pair of (T, X), contradicting (T, X) distal by Theorem 2.1. Thus (X, T ) must be equicontinuous. This proves Proposition 6.1. This is more general than §1.1.5 (o); yet we will be mainly interested to the special case E = E(X) associated to a semiflow (T, X). Since E is a compact T 2 right-topological semigroup in this case, hence J(I) ∅ for all minimal left ideal in E (by [4, Lemma 6.6]).
Proof of Lemma 1.11. Let I be the only minimal left ideal in E(X) and (x, y), (y, z) ∈ P(X). Then p(x) = p(y) and p(y) = p(z) for all p ∈ I. So (x, z) ∈ P(X). For the "only if" part, let P(X) be transitive, I, K minimal left ideals in E(X), and u ∈ J(I) and v ∈ J(K) with uv = v, vu = u by (10) of Lemma 6.3. Let x ∈ X. Then (x, ux) ∈ P(X) and (x, vx) ∈ P(X) implies (ux, vx) ∈ P(X). But v(ux, vx) = (ux, vx) implies that (ux, vx) is an almost periodic point of (T, X × X). Hence ux = vx by Lemma 2.7 and u = v so that I = K.
Following §1.1.3 (h), an x ∈ X is a distal point of (T, X) if and only if it is proximal only to itself in cls X T x. Lemma 6.4 (cf. Veech [53] for T in groups). Let (T, X) be a semiflow and x ∈ X. Then x is a distal point of (T, X) iff x = u(x) for all u ∈ J(E(X)).
Proof. Let x be a distal point of (T, X) and u ∈ J(E(X)). Since (x, u(x)) ∈ P(X) is almost periodic (by Theorem 5.1), hence x = u(x).
Conversely, assume x = u(x) for all u ∈ J(E(X)) and let y ∈ cls X T x such that (x, y) ∈ P(X). There is a minimal left ideal I such that p(x) = p(y) ∀p ∈ I. Since x ∈ I x and so y ∈ I x, it follows that Iy = I x so y ∈ Iy. Then there is u ∈ J(I) with y = u(y) = u(x) = x. This shows that x is a distal point of (T, X).
As a consequence of Lemmas 6.3 and 6.4, the following (2) of Theorem 6.5 is more or less motivated by [53, Proposition 2.1], which is useful for proving the "if" part of Lemma 1.8. Theorem 6.5. Let (T, X) be a semiflow with Ellis' semigroup E(X) and x ∈ X. Then:

(1) For every minimal left idea I in E(X), pI ∩ J(I) ∅ for all p ∈ E(X).
(2) x is a distal point of (T, X) iff x ∈ p(X) for all p ∈ E(X) iff x ∈ u(X) for all u ∈ J(E(X)).
Proof. (1) Let p ∈ E(X) and I a minimal left ideal in E(X). Then pI ⊆ I and further by (7) and (4) of Lemma 6.3 there are q ∈ I, δ ∈ I, and v ∈ J(I) such that pqδ = v. Since qδ ∈ I, hence pI ∩ J(I) ∅.
(2) Assume x is a distal point; then x = v(x) for every v ∈ J(E(X)) by Lemma 6.4. Thus x ∈ p(X) for all p ∈ E(X) by (1). Conversely, suppose that x ∈ v(X) for all v ∈ J(E(X)). Let u ∈ J(E(X)) be arbitrary. Then there exists y ∈ X such that u(y) = x. So u(x) = u 2 (y) = u(y) = x. Thus by Lemma 6.4, x is a distal point of (T, X).
The proof of Theorem 6.5 is thus complete. 28 Proof of the "if" part of Lemma 1.8. Let the set of distal points of (T, X) be dense in X and t ∈ T . Since every distal point belongs to tX by Theorem 6.5 and tX is a closed set, hence tX = X. The proof of Lemma 1.8 is thus complete.
Recall that if E(T, X) ⊂ C(X, X) then (T, X) is called "weakly almost periodic" in some literature, for example, [27]. So the following says that the proximal relation is an equivalence relation for every weakly almost periodic semiflow with abelian phase semigroup. Theorem 6.6. Let (T, X) be a semiflow with T an abelian semigroup and J(E(X)) ⊂ C(X, X). Then the following two statements hold:

exists a unique minimal left ideal I in E(X) and moreover I contains a unique idempotent u. Hence P(X) is an equivalence relation on X. (2) If x ∈ X is an almost periodic point, then it is a distal point. Hence if there is a dense set of almost periodic points, then (T, X) is distal.
Note. In fact, if E(X) ⊂ C(X, X) and there is a dense set of almost periodic points, then (T, X) is not only distal but also equicontinuous by (c) of Lemma 6.7 and Ellis's joint continuity theorem (cf. Theorem 8.8 in §8).
Proof. (1) Let I 1 and I 2 be two minimal left ideals in E(X). Then by (10) of Lemma 6.3, it follows that there are idempotents u ∈ I 1 and v ∈ I 2 such that uv = v. Thus there is a net {t n } in T with t n → v in E(X) such that I 1 ∋ lim t n u = lim ut n = uv = v. Then I 1 ∩ I 2 ∅ and thus I 1 = I 2 . This shows that there is only one minimal left ideal I in E(X). Thus P(X) is an equivalence relation on X by Lemma 1.11. Let u, v ∈ J(I). Then by (2) of Lemma 6.3, uv = u = uu. By the above argument, we can see vu = uv = uu and so by (9) of Lemma 6.3 u = v.
(2) Let x be an almost periodic point of (T, X). Then by x ∈ I x where I is as in (1), x = ux. Thus by Lemma 6.4, x is a distal point of (T, X). Because u ∈ C(X, X) by weak almost periodicity, if the set of almost periodic points of (T, X) is dense in X then u = id X . Thus (T, X) is pointwise distal by Lemma 6.4, and so it is distal.
Ellis' classical characterization of distality states that (T, X) is a distal flow if and only if E(T, X) is a group (cf. [24,Theorem 1], [26,Proposition 5.3] and [4,Theorem 5.6]). Another important consequence of Lemma 6.3 is the following semiflow version of Ellis' characterization, which has already played an important role in [17]. Lemma 6.7. Let (T, X) be a semiflow, where T is a discrete semigroup (but not necessarily e ∈ T ). Then the following statements are pairwise equivalent:

is a minimal left ideal in itself with id X ∈ E(X). (c) E(X) is a group with the neutral element id X .
Notes.
1. Condition (b) implies that (T, X) is pointwise minimal, because E(X)x is a minimal set of (T, X) and x ∈ E(X)x for id X ∈ E(X).

In particular, if
Proof. Condition (a) ⇒ (b): Let I be a minimal left ideal in E(X) and u ∈ J(I). Then by Lemma 6.4, x = u(x) for all x ∈ X. Thus u = id X and further E(X) is a minimal left ideal with the unique idempotent id X ∈ E(X).
is a group by (4) of Lemma 6.3 with u = id X ∈ E(X). Condition (c) ⇒ (a): Suppose (x, y) ∈ P(X). Then p(x) = p(y) for some p ∈ E(X) so x = y by p −1 p = id X , since E(X) is a group with e = id X ∈ E(X). Thus (a) holds.
The proof of Lemma 6.7 is thus completed.
The most important part of Lemma 6.7 is "(a) ⇒ (c)" which implies (2) of Theorem 1.13. Now we will present an independent direct proof for it without using Lemma 6.3.
Another proof of "(a) ⇒ (c)" of Lemma 6.7. Note that 'distal' implies 'pointwise almost periodic' (by Theorem 5.1). Since (T, X X ) is distal, E := E(T, X) which is the orbit closure of id X is minimal. Now for every p ∈ E, since E p is closed T -invariant, E p = E. This easily follows that every p ∈ E has an inverse.
This algebraic characterization of distality is very useful. Notice that if e T and (T, X) is distal, then either id X is a pointwise limit point of T in E(X) or tx = x ∀x ∈ X for some t ∈ T by (b) of Lemma 6.7. Now by Lemma 6.7 or by the fact that every distal map is pointwise recurrent, we can obtain the following, (2) of which is just (2) of Theorem 1.13. Corollary 6.8. Let (T, X) be a semiflow with phase semigroup T . Then:

then it is invertible and admits an invariant Borel probability measure. (2) If (T, X) is equicontinuous, then it is distal if and only if it is surjective. (3) If (T, X) is point-distal surjective with E(X) ⊂ C(X, X), then (T, X) is an u.a.p. semiflow.
Proof. (1) Let (T, X) be distal; then by Lemma 6.7, E(X) is a group with e = id X . Let Homeo (X) be the group of all self-homeomorphisms of X. Then T ⊂ Homeo (X) and E(X) = cls X X T . Thus by Furstenberg's structure theorem of distal minimal flows [29] (cf. Theorem 6.14 below), it follows that ( T , X) and so (T, X) admit invariant Borel probability measures.
(3) Let x be a distal point with cls X T x = X. By Lemma 1.8, each point of T x is distal for (T, X). Given any u ∈ J(E(X)), sx = usx for all s ∈ T . Since u ∈ C(X, X) and T x is dense, so u = id X . Thus E(X) is a group by (4) of Lemma 6.3 and so (T, X) is minimal distal by Lemma 6.7. This implies that (E(X), X) is equicontinuous. Thus (T, X) is u.a.p.
The proof of Corollary 6.8 is thus completed.
It is interesting that a distal map is always surjective, while an equicontinuous map is not by Examples 1.7 and 1.12 in §1. Also this indicates that distal is the more natural concept. However, under locally (weakly) almost periodic condition, the equicontinuous is equivalent to the distal in flows (cf. [26,6]). Lemma 6.9. If (T, X) is minimal invertible such that for each t ∈ T , (t −1 , X) is rigid, that is, id X ∈ cls X X {t −n | n = 1, 2, . . . }, then (X, T ) is minimal.
Proof. Let X 0 be a minimal set of (X, T ) by Zorn's lemma, and let t ∈ T, t e be any given. Then there exists a net {n k } in N with t −n k → id X in X X under the pointwise topology. Thus for every point x 0 ∈ X 0 , t −n k x 0 → x 0 and so t(t −n k x 0 ) = t −n k +1 x 0 → tx 0 . Since −n k + 1 ≤ 0, then t −n k +1 x 0 ∈ X 0 and so tx 0 ∈ X 0 . Hence T X 0 ⊆ X 0 and then X 0 = X for (T, X) is minimal. 30 As another result of Lemma 6.7, we can then obtain using algebraic approaches the following simple observation for distal semiflows, which are (3) of Theorem 1.13 and 2 and 3 of Reflection principle I. Proposition 6.10. Let (T, X) be a semiflow with phase semigroup T . Then: then so is ( T , X).
Proof. (1) Since E(T, X) is a group with e = id X by Lemma 6.7, then (T, X) is invertible and If p(x) = p(y) for some p ∈ E(X, T ) then by distality of (T, X) we see x = y. Therefore, (X, T ) is distal. Moreover, since T ⊆ E(X), thus ( T , X) is distal by Lemma 6.7 again.
(2) By (1), we only need prove the minimality of (X, T ). To this end, let t ∈ T . Since the distal cascade (t −1 , X) induces a distal semiflow f : (n, x) → t −n x of N × X to X where N is discrete additive, then by Lemma 6.7 the Ellis semigroup of ( f, N, X) contains id X ; i.e., (t −1 , X) is rigid. Then (2) follows from Lemma 6.9.
The proof of Proposition 6.10 is therefore completed.
We note that using Ellis' semigroup (Lemma 6.7) we have easily concluded Proposition 6.10. However, if we make no use of this and the β-compactification of T , based on Theorem 5.1 in §5 and using only topological approaches we can prove it as follows.
Proof II of Proposition 6.10. Let (T, X) be distal. We will divide our non-enveloping semigroup proof into relatively independent four steps.
Step 1. Every point of X is almost periodic for (T, X). Moreover, (T, X) is invertible.
Proof. The first part of Step 1 follows at once from Theorem 5.1. Now given t ∈ T , since (t, X) is pointwise almost periodic, then tX = X. This shows that (T, X) is invertible.
Although (T, X) is pointwise almost periodic by Step 1, yet because T need not be syndetic in T and (T, X) need not be minimal the following Step 2 is non-trivial.
Proof. Let x ∈ X be any given and write Y x = cls X T x. Clearly by Step 1, (T, Y x ) is minimal distal so that cls X T y = Y x for all y ∈ Y x . Given y ∈ Y x and t ∈ T , since y is a (forwardly) minimal point for (π t , Y x ) by Step 1, there is a net {n k } in N with t n k y → y. So t n k −1 y → t −1 y ∈ Y x , for t n k −1 y ∈ Y x and Y x is closed. This shows Y x T ⊆ Y x . Thus Y x = cls X T y ⊆ cls X T y ⊆ Y x for all y ∈ Y x . This shows that each y ∈ Y x and so x are almost periodic for ( T , X).
Step 3. (T, X × X) is distal and so ( T , X × X) is pointwise almost periodic.
Proof. It follows easily from definition that (T, X × X) is distal. Then ( T , X × X) is pointwise almost periodic by Steps 1 and 2.
Step 4. Let (T, Z) be a semiflow with any phase semigroup T . If (T, Z × Z) is pointwise almost periodic, then (T, Z) is distal. Now, since ( T , X × X) is pointwise almost periodic by Step 3, ( T , X), which is minimal if so is (T, X), is distal by Step 4. Thus (X, T ) is distal.
Next, assume (T, X) is minimal distal. Then (t −1 , X) is pointwise almost periodic (forwardly) and so every negatively-invariant closed subset of X is also π-invariant. This implies the minimality of (X, T ). The proof II of Proposition 6.10 is therefore complete.
We will continue to consider the minimality of the reflection (X, T ) under much more weaker conditions in §6.3. Moreover, for an amenable phase semigroup, we will show in §6.3 that (T, X) is distal at some point x ∈ X if and only if so is (X, T ) at the same point x (see Corollary 6.27).
The following result is originally due to Ellis [23,Theorem 3] (also see [4,Theorem 3.3]) in the case that (T, X) is a flow. Proof. First from equicontinuity, all p in E(X) are continuous. Then the necessity follows at once from Theorem 2.1 and Lemma 6.7. Conversely, if E(X) is a group of homeomorphisms of X, then by Ellis' joint continuity theorem (cf. [4,Theorem 4.3] and also Theorem 8.8 in Appendix), it follows that E(X) and so T acts equicontinuously on X. This proves Corollary 6.11.
In Corollary 6.11, it is essential that T consists of surjections, and not merely a semigroup of continuous maps. Corollary 6.11 may follows from (3) of Corollary 6.8.
Given any semigroup T of bijections of X, T ∪ T −1 is not necessarily equal to the group T . In addition, if T acts equicontinuously on X, then so does T ∪ T −1 by Proposition 6.1. However, since T need not be abelian, the equicontinuity of T cannot be trivially obtained.
Nevertheless Theorem 2.1 together with Lemma 6.7 implies the following important fact, which is just (4) of Theorem 1.13.

Theorem 6.12. Let G be a semigroup of self-homeomorphisms of X. Then G is equicontinuous on X if and only if so is G .
Proof. It suffices to show the "only if" part. Let G is equicontinuous on X. By Corollary 6.11, E(G, X) is a group consist of self-homeomorphisms of X. Further E(G, X) acts equicontinuously on X. Since G ⊆ E(G, X), thus G is equicontonuous on X.
Motivated by Proof (III) of Theorem 2.1, we can present another self-contained topological proof of Theorem 6.12 without using Lemmas 6.3 and 6.7.
Proof II of Theorem 6.12. We only need to show the "only if" part; and then assume G is equicontinuous on X. By C cpt-op (X, X) we denote the space C(X, X) of all continuous self-maps of X equipped with the compact-open topology, and let E be the closure of G in C cpt-op (X, X). Then by Ascoli's theorem E is compact and moreover, each p ∈ E is a surjection of X. We will show that E is a group.
First we note that ( f, g) → f g := f • g of C cpt-op (X, X) × C cpt-op (X, X) to C cpt-op (X, X) is separately continuous. This implies that EE ⊆ E and thus E is a compact semi-topological semigroup. Since each p ∈ E is surjective, E has the unique idempotent id X .
Given any p ∈ E, since E p is a closed subsemigroup of E so that it contains an idempotent, hence id X ∈ E p and so there is some q ∈ E such that qp = id X . This shows that E is a group of self-homeomorphisms of X. Finally, since G and then E acts equicontinuously on X, so does G because of G ⊆ E. 32 Finally we notice that whereas Proposition 6.1 may be a consequence of Theorem 6.12, its direct proof is of independent interest.
Let (T, X) be an invertible semiflow and let G = T be the discrete group of self homeomorphisms of X generated by T associated to (T, X).
Of course if T is amenable and X is metric, then the statement of Corollary 6.15 can follow from Corollary 5.5.

Minimality of reflections
If (T, X) is a flow, x ∈ X, and U a neighborhood of x, then (N T (x, U)) −1 = N (X,T ) (x, U) is right-syndetic in T by T = T −1 . So if x is almost periodic for (T, X), then it is also almost periodic for the reflection (X, T ). But if (T, X) is only an invertible semiflow, then (N T (x, U)) −1 need not be a right-syndetic subset of T so that x need not be almost periodic for (X, T ); see 1 of Examples 1.4. However, we will be concerned with a question or reflection principle as follows: In this subsection, we shall show that this question is in the affirmative if T is an amenable semigroup (cf. §1.1.4 (k)) or if T is a right C-semigroup (cf. Definition 3.8).

Abelian phase semigroup
First of all, whereas the following observation is simple, it might be useful for our later proof of Proposition 6.17. Motivated by Proposition 6.10 stated in §6.1, we can easily obtain the following result using Lemma 6.16. Proposition 6.17. If (T, X) is minimal invertible with T an abelian semigroup, then (X, T ) is minimal.
Proof. Let X 0 be a minimal set of (X, T ) and t ∈ T . Let x 0 ∈ X 0 be a minimal point for (t −1 , X 0 ) with phase semigroup Z + . As x 0 is recurrent for t −1 , it follows from Lemma 6.16 that tx 0 ∈ X 0 . Then by commutativity of T , tT −1 x 0 = T −1 tx 0 ⊆ X 0 so tX 0 ⊆ X 0 . Whence X 0 is invariant for (T, X). This proves Proposition 6.17.
In fact, Proposition 6.17 can be differently proved as follows: Proof II of Proposition 6.17. Let X 0 be a minimal set of (X, T ). Then if t ∈ T , then X 0 ∩ tX 0 ∅ (since tt −1 x = x by Corollary 3.3). But since T is abelian, then tX 0 is minimal for (X, T ) so tX 0 = X 0 . This shows that X 0 = X.
The lighting point of Proposition 6.17 is that T is not necessarily a syndetic subsemigroup of the group T of homeomorphisms of X generated by T .
Let N be a non-empty closed invariant set of (X, T ) and t ∈ T ; then for every x ∈ N, its the α-limit points set α t (x) under (t, X) is such that t n α t (x) ⊆ N for all n ∈ Z + . More generally, we can obtain the following. Proof. Let S be an abelian subsemigroup of T . Since N − is invariant for (X, T ), it is invariant for (X, S ). Then there is a minimal set N 0 for (X, S ) with N 0 ⊆ N − . By Proposition 6.17, N 0 is a minimal set for (S , X), so S N 0 = N 0 ⊆ N − . This proves Corollary 6.18.

Amenable phase semigroup
Recall that as in §1.1.4 (k) a semigroup T is said to be amenable iff every semiflow on a compact T 2 -space with the phase semigroup T admits an invariant Borel probability measure.
Since each abelian semigroup is an amenable semigroup, then the following theorem covers Proposition 6.17 by different ergodic approaches.

Theorem 6.19. Let (T, X) be an invertible semiflow with T an amenable semigroup and x ∈ X. Then x is an almost periodic point of (T, X) if and only if x is an almost periodic point of (X, T ).
Moreover, if x is an almost periodic point of (T, X), then cls X T x = cls X xT .
Note. If "with T an amenable semigroup" is replaced by "admitting an invariant Borel probability measure", then the statement still holds.
Proof. Let X 0 be a minimal subset of (T, X). Since T is amenable, there is an invariant quasiregular Borel probability measure µ for (T, X 0 ) such that supp (µ) = X 0 . Then by Lemma 3.6, it follows that for each t ∈ T is a surjection of X 0 and so is t −1 and then all t restricted to X 0 are self-homeomorphisms of X 0 . This shows that X 0 is also a closed invariant subset of (X, T ). We will show that X 0 is also minimal for (X, T ).
To be contrary assume that X 0 is not minimal for (X, T ); then by Zorn's lemma, there exists a proper non-empty closed subset Y of X 0 such that (Y, T ) is a minimal semiflow. Since T is amenable, there is an invariant quasi-regular Borel probability measure ν for (Y, T ) such that supp (ν) = Y. Then by Lemma 3.6 again, it follows that for each t ∈ T , t −1 : Y → Y is surjective and so is t −1 and then t restricted to Y is a self-homeomorphism of Y. This shows that Y is also a closed invariant subset of (T, X 0 ). But this contradicts that (T, X 0 ) is minimal.
By symmetry, we can show that every minimal set of (X, T ) is a minimal set of (T, X). The proof of Theorem 6.19 is therefore complete.
In view of 1 of Examples 1.4, the condition that T is amenable is essential for the above proof of Theorem 6.19. In fact, the key idea is that each t ∈ T is surjective restricted to every minimal subset. Amenability just guarantees this condition.
Recall that Proposition 6.10 claims that if (T, X) is distal, then so is (X, T ). However, from Theorem 6.19 we can obtain the following "reflection principle of distality" which asserts that if x ∈ X is a distal point of (T, X) and if the phase semigroup T is amenable, then x is also a distal point for (X, T ). So if f : X → X is a homeomorphism such that it is forwardly distal at a point x, then it is backwardly distal at x. 35 Proposition 6.20. Let (T, X) be invertible with T an amenable semigroup and x ∈ X. If x is a distal point of (T, X), then x is a distal point of (X, T ).

Notes.
1. If "with T an amenable semigroup" is replaced by "admitting an invariant Borel probability measure", then the statement still holds. 2. Proposition 6.20 is in fact a corollary of Theorem 5.4. But we will present an independent proof here.
Proof. Let x ∈ X be distal for (T, X). Then by Theorem 5.1, x is minimal for (T, X). By Theorem 6.19, x is a minimal point for (X, T ). Let Z = cls X xT corresponding to (X, T ). Clearly Z = cls X T x by Theorem 6.19 again. We will show that x is not proximal to any x ′ x in Z in the sense of (X, T ). In fact, if x ′ is in Z, then x ′ is a minimal point of (X, T ). Whence x ′ is also a minimal point of (T, X) by Theorem 6.19 once more. Then by Theorem 5.1, (x, x ′ ) is a minimal point for (T, X × X). This implies by Theorem 6.19 that (x, x ′ ) is a minimal point of (X × X, T ). Thus, if x is proximal to x ′ for (X, T ), then cls X×X (x, x ′ )T is contained in the diagonal of X × X by minimality of (x, x ′ ) under (X × X, T ). Thus x = x ′ . The proof of Proposition 6.20 is thus complete.
In preparation for our next equicontinuity consequence of Proposition 3.7, we need to recall a notion for our convenience. If E(X) is a topological group with the pointwise topology and if (T, X) is minimal, then (T, X) is equicontinuous (cf. [17,Proposition 5.5]). However, if E(X) is only a topological semigroup but E(X) ⊂ C(X, X), then (T, X) is still equicontinuous by the following. Proof. The "only if" parts are obvious. Next we show the "if" parts of Theorem 6.22. In fact, we only need prove that if E(X) ⊂ C(X, X) is a topological semigroup, then (T, X) is equicontinuous and surjective by Theorem 2.1 and Theorem 2.10. For this, we now assume that E(X) ⊂ C(X, X) is a topological semigroup in the sense of the pointwise topology p.
Let I be a minimal left ideal in E(X). Then we can first show that (i) Given any p ∈ I, pI = I.
Proof. Indeed, applying Proposition 3.7 with T × I → I, (t, p) → tp, it follows that for every t ∈ T , tI = I. Then if T ∋ t n p − → p and q ∈ I, there are q n ∈ I with t n q n = q and q n p − → r for some r ∈ I so that pr = q by the joint continuity of ( f, g) → f • g. Thus, pI = I.
Then by (i) there follows that (ii) up = p, for u ∈ J(I) and p ∈ I.
Next, if u, v are idempotents in I, then by (ii), it follows that (u, v)u = (u, u). (9) of Lemma 6.3 implies that Therefore, I has a unique idempotent u in I. Of course u ∈ C(X, X). Since (T, X) has a dense set of almost periodic points, hence ux = x for each x ∈ X. This implies by (4) of Lemma 6.3 that I = E(X) ⊂ C(X, X) is a group. Thus (T, X) is equicontinuous by Corollary 6.11.
If T is a topological group, then we can improve the statement of Theorem 6.22 by a completely different proof as follows: It should be noticed that the 'topological semigroup' condition is essential for the above theorem (cf. Theorem 8.14 in §8) as shown by the following example.
Example 6.23. Let X be the one-point compactification of the reals and define a homeomorphism f : X → X by x → x + 1 for all x ∈ X. Then ∞ is the unique almost periodic point and ( f, X) with phase group Z is not equicontinuous; but E( f, X) ⊂ C(X, X) consists of the powers of f together with the constant map c : x → ∞. By Theorem 6.6, I = {c} is the unique minimal left ideal in E( f, X). Moreover, it is easy to see that E( f, X) is a semi-topological semigroup but not a topological semigroup. Indeed, let t n = f n and s n = f 1−n for any n ≥ 1. Clearly, t n → c and s n → c but f = lim t n s n (lim t n )(lim s n ) = c.
On the other hand, let us consider ( f, X) in Example 6.23 from the viewpoint of semiflow. It shows that the condition 'with a dense set of almost periodic points' is essential in Theorem 6.22.
Example 6.24. Let f : X → X be same as in Example 6.23. But here we now consider ( f, X) with phase semigroup Z + . Clearly ∞ is also the unique almost periodic point and ( f, X) is not equicontinuous; but E( f, X) ⊂ C(X, X) consists of the powers f n , n ≥ 0, together with the constant map c : x → ∞ of X into itself. By Theorem 6.6, I = {c} is the unique minimal left ideal in E( f, X). Moreover, it is easy to see that E( f, X) is a topological semigroup but not a topological group.
Recall that any subset A of X is called non-trivial if A ∅ and moreover A X. Then it is easy to verify that However, in our semigroup situation, this becomes a non-trivial case. First of all, we can easily get the following simple fact for an invertible semiflow (π, T, X).
Lemma 6.25. Let (T, X) be an invertible semiflow; then the following two statements hold: (1) W ⊂ X is an invariant open set of (T, X) iff X \ W is an invariant closed set of (X, T ).
(2) (X, T ) is minimal iff T U = X for every non-empty open set U.
The following seems to be helpful for considering the minimality of the reflection (X, T ) with T a non-abelian semigroup. See  Proof. Let (X, T ) be minimal and assume U is a non-trivial open invariant subset of (T, X). Then X \ U is invariant non-empty closed for (X, T ) by Lemma 6.25 and so X \ U = X contradicting U non-trivial. Thus X does not contain a non-trivial open invariant subset for (T, X).
Conversely, let X have no non-trivial open invariant subset for (T, X) and assume (X, T ) is not minimal. Then we can find a non-trivial closed invariant subset Θ of (X, T ). Then X \ Θ is a non-trivial open invariant subset of (T, X) by Lemma 6.25 again. Thus this concludes that (X, T ) is a minimal semiflow.
It is clear that every minimal flow admits no non-trivial open invariant set. Now, by Theorem 6.19 and Theorem 6.26, we can easily obtain the following semigroup-action result. Proof. If this were false, then (X, T ) would not be minimal by Theorem 6.26. But this contradicts Theorem 6.19. This completes the proof of Corollary 6.27.
Another result of Theorem 6.19 is the following theorem, which is a generalization of a classical theorem of Tumarkin [45,Theorem V7.13] from the important case of T = (R, +) to the case of general amenable semigroups.
Theorem 6.28. Let T be an amenable semigroup. If Λ is a minimal subset of (T, X) such that Int X Λ ∅, then Λ is clopen in X.
Proof. Let y ∈ Λ be an interior point of X. Then we can pick some index ε ∈ U X such that ε[y] ⊆ Λ. Then U := t∈T tε [y] is an open, invariant, and non-empty subset of X such that U ⊆ Λ. Thus by Theorem 6.19 (more precisely by Corollary 6.27), it follows that U = Λ. This proves Theorem 6.28.
Let n be a positive integer. From Urysohn's theorem the dimension of a compact subset of an n-dimensional manifold which has no interior points does not exceed n−1 (cf. [ Let (T, M n ) be an invertible semiflow on an n-dimensional manifold M n , n ≥ 1, such that T is amenable. If A is a compact minimal subset with A M n , then Int M n A = ∅ and dim A ≤ n − 1.
Proof. If Int M n A = ∅, then by Urysohn's theorem dim A ≤ n − 1. Now assume Int M n A ∅; then by Theorem 6.28, it follows that A is clopen non-trivial in M n . This is a contradiction.
Let (G, X) be a flow with phase group G and T a normal syndetic subgroup of G. Then it is a well-known fact that • An x ∈ X is an almost periodic point of (G, X) if and only if x is an almost periodic point of (T, X) (cf., e.g., [ By Theorem 6.28 we can obtain the following same flavor result using amenability instead of the normality of T . Corollary 6.30. Let (G, X) be an invertible semiflow on a compact connected T 2 -space X and T a discrete syndetic amenable subsemigroup of G. Then (G, X) is minimal iff so is (T, X).
Proof. Let x 0 ∈ Tran − (T, X) and x ∈ Equi ε (T, X) both be any given points. We then need to verify x 0 ∈ Equi ε (T, X). For this, let η ∈ U X such that if y, z ∈ η[x] then t(y, z) ∈ ε for all t ∈ T . Since x 0 is negatively transitive, there is an s ∈ T such that s −1 x 0 ∩ η[x] ∅. Then by the openness of s, there exists a δ ∈ U X such that δ[x 0 ] ⊆ s (η[x]). Now for y, z ∈ δ[x 0 ] and t ∈ T , t(y, z) = ts(y ′ , z ′ ) ∈ ε for some y ′ , z ′ ∈ η[x] with sy ′ = y, sz ′ = z. This shows that x 0 ∈ Equi ε (T, X) for all ε ∈ U X . Proposition 7.3. Let (T, X) be a minimal non-sensitive semiflow with T not necessarily discrete. Then the following two statements hold: (1) If (T, X) is invertible with T an amenable semigroup, then (T, X) is equicontinuous.
(2) If T is a right C-semigroup, then (T, X) is equicontinuous.
(2) Given ε ∈ U X , there are x 0 ∈ X and δ ′ ∈ U X such that T (δ ′ [x 0 ], x 0 ) ⊂ ε 3 by non-sensitivity. Now since (T, X) is minimal, for every x ∈ X there are s ∈ T and δ ⊆ δ ′ such that s(δ[x]) ⊆ δ ′ [x 0 ]. In addition, since T \ T s is relatively compact in T , we can take an η ∈ U X with η ⊆ δ so small Since ε is arbitrary, Equi (T, X) = X and so (T, X) is equicontinuous by Lemma 1.6.

Corollary 7.4. Let (T, X) be a minimal semiflow with T not necessarily discrete. Suppose that (1) T is a right C-semigroup or (2) (T, X) is invertible with T amenable. Then (T, X) is either sensitive or equicontinuous.
Proof. If (T, X) is sensitive, then it evidently not equicontinuous. Now if (T, X) is non-sensitive, then it is equicontinuous by Proposition 7.3.
Since Z + is a right C-semigroup, hence the case (1) of Corollary 7.4 is a generalization of the Auslander-Yorke dichotomy theorem [7]. Proof. If Equi (T, X) ∅, then (T, X) is non-sensitive and so it is equicontinuous by Proposition 7.3. This proves Corollary 7.5.
The following corollary is a reflection principle on sensitivity of invertible semiflows in amenable semigroups or C-semigroups. Corollary 7.6. Let (T, X) be a minimal invertible semiflow with T not necessarily discrete such that T is either a C-semigroup or an amenable semigroup. Then (T, X) is sensitive if and only if so is (X, T ).
Proof. Assume (T, X) is sensitive. If (X, T ) were not sensitive, then it would be non-sensitive minimal by Theorems 6.19 and 6.31 and so equicontinuous by Proposition 7.3. Moreover, by Reflection principle I (Proposition 6.1), it follows that (T, X) would be equicontinuous. This is a contradiction. Conversely, if (X, T ) is sensitive, then we could similarly prove that (T, X) is sensitive.
Then by Theorem 2.1 and Corollary 7.4, we can easily obtain the following.
Corollary 7.7. Let (T, X) be minimal surjective with T a right C-semigroup not necessarily discrete. If there is some t ∈ T non-invertible, then (T, X) is sensitive.
Proof. If (T, X) were non-sensitive, then by Corollary 7.4 (T, X) would be equicontinuous and so distal by Theorem 2.1. This is a contradiction. Now by using distality instead of amenability and C-semigroup, we can obtain the following dichotomy. This shows that the dynamics of any non-equicontinuous minimal distal semiflow could not be predictable.

Appendix: revisit to Robert Ellis' joint continuity theorems
Let T be a multiplicative topological group or semigroup and X a compact T 2 -space, and let T × X → X, (t, x) → tx be a separately continuous flow or semiflow. Then the problem to find conditions on T so that (t, x) → tx jointly continuous goes back, at least, to Baire (1899); see, e.g., [8,28,23,52,44,4]. In this Appendix, we will revisit Robert Ellis' joint continuity theorems and generalize some of them to locally compact Hausdorff semi-topological semigroups based on Isaac Namioka's theorem.
Standing notation. In this appendix "locally compact Hausdorff space" will be abbreviated as "l.c.T 2 -space".
Every l.c.T 2 -space is of the second category and every closed or open subset of an l.c.T 2 -space as a subspace itself is an l.c.T 2 -space.
Differently with Ellis' original proof [23] in which the group structure of T plays an essential role, we will employ mainly the following basic joint continuity theorem due to Isaac Namioka 1974 [44, Theorem 1.2] as our tool.
Lemma 8.1 (Namioka [44]). Let G be an l.c.T 2 -space or a separable Baire space and X a compact T 2 -space, and let (Z, d) be a metric space. If a map f : G × X → Z is unilaterally continuous, then there exists a dense G δ -set R in G, such that f is jointly continuous at each point of R × X.
Since f : G × X → Z is unilaterally continuous, then E f : G → C p (X, Z) given by g → f (g, ) is continuous, where C p (X, Z) is the space of all continuous functions from X to Z with the pointwise topology p. Thus Lemma 8.1 is a corollary of [4, Lemma 4.2] due to Troallic [50].
Under the setup of Lemma 8.1, C u (X, Z) denotes the uniform space C(X, Z) where the topology of uniform convergence on C(X, Z) is induced by the standard supremum norm: It is very convenient to reformulate Lemma 8.1 in terms of functions spaces as follows. In our later application of this lemma, Z will be the unit interval I = [0, 1] with the usual Euclidean metric.
Lemma 8.2. Let G be an l.c.T 2 -space and X a compact T 2 -space, and let (Z, d) be a metric space. If f : G × X → Z is unilaterally continuous, then there exists a dense G δ -set R in G, such that the induced map F : Proof. This is just a consequence of [44, Theorem 2.2]; however we present its proof here for reader's convenience. Let R be a dense G δ -set in G given by Lemma 8.1. Then for any τ ∈ R, f : G × X → Z is continuous at each point of {τ} × X. Since X is compact, we see that F is continuous at τ in the sense of the topology of uniform convergence on C(X, Z). Indeed, given any ε > 0, for any x ∈ X, there are open neighborhoods U x of τ in G and V x of x in X such that This concludes the desired.
A topological space Y is called completely regular iff for each member y of Y and each neighborhood U of y there is a continuous function α on Y to the closed unit interval I such that α(y) = 0 and α is identically 1 on Y \ U. It is clear that the family C(Y, I) of all continuous functions on a completely regular space Y to the unit interval I distinguishes points and closed sets in the sense that for closed subset A of Y and each point y ∈ Y \ A there is an α ∈ C(Y, I) such that α(y) does not belong to the closure of α(A).
If X is a completely regular T 1 -space, then by the classical Embedding Lemma (cf. [39,Chapter 4]) X is homeomorphic to a subspace of the cube Q = I C(X,I) . Therefore we can easily obtain the following Lemma 8.3. Let X be a completely regular T 1 -space and W a topological space. Then a map f : W → X is continuous at a point w 0 ∈ W if and only if α • f : W → I is continuous at the point w 0 for each α ∈ C(X, I).
With this lemma at hands, we do not need here to strengthen Lemma 8.2 by uniform space instead of a metric space Z as [50,42] there.
Recall that (T, X) is weakly almost periodic iff E(X) ⊆ C(X, X). Comparing with [3, Corollary 6], an interesting point of the following is that X is not necessarily to be a metric space. Proof. Let α ∈ C(X, I) where I = [0, 1] and π : X × E(X) → X by (x, p) → p(x) which is unilaterally continuous. Then by Lemma 8.1, there is an x 0 ∈ X such that α • π : X × E(X) → I is jointly continuous at each point of {x 0 } × E(X). Thus by compactness of E(X), for ε > 0 there is δ 0 ∈ U X such that |α • p(y) − α • p(x 0 )| < ε/3 for all y ∈ δ 0 [x 0 ] and p ∈ E(X). Now let x ∈ Tran (T, X); there are t ∈ T and δ ∈ U X such that t(δ[x]) ⊆ δ 0 [x 0 ]. Using E(X)t = E(X), we can conclude that |α • p(y) − α • p(x)| < ε for all y ∈ δ[x] and p ∈ E(X). This shows by Lemma 8.3 that π : X × E(X) → X is jointly continuous at each point of Tran (T, X) × E(X). Thus (T, X) is equicontinuous at each point of Tran (T, X). Definition 8.5. Given any semigroup T and non-empty set X, T × X → X, (t, x) → tx is called an algebraic semiflow if ex = x and (st)x = s(tx) for all x ∈ X and s, t ∈ T .
Following the classical work of Ellis [23] we now introduce a notion we will need in our later arguments.
Definition 8.6. Let T be a semigroup with a topology S and X a compact T 2 -space. Then an algebraic semiflow (T, X) is called an Ellis semiflow if (a) T is an l.c.T 2 -space and a right-topological semigroup under S; (b) (t, x) → tx of T × X to X is unilaterally continuous.
In view of Definition 8.6 we now introduce "admissible" time.
Definition 8.7 ([42]). We say that an Ellis semiflow (T, X) is admissible at τ ∈ T if (c) Int T cls T τ{t ∈ T | t is a surjection of X} ∅.
We shall say (T, X) is admissible if it is admissible at each element of T .
Note. In Ellis' setup that T is a group, every Ellis flow is admissible.
Clearly each T 2 non-compact C-semigroup T is such that T \ tT relatively compact and so Int T cls T tT ∅ for each t ∈ T like the additive semigroups N and R + (cf. [41]).
Proof. For simplicity, write M = {t ∈ T | t is a surjection of X}. In view of Lemma 8.3 with W = T × X, to prove Theorem 8.8 it is sufficient to show that for any θ ∈ C(X, I), the induced function ϑ : (t, x) → θ(tx) of T × X to I is jointly continuous at each point of {τ} × X. For that, by Lemma 8.2 with G = T and Z = I, it follows that there exists a residual subset R of T such that at each point of R, the map Θ induced by ϑ, Θ : T → C(X, I); t → ϑ(t, ) ∀t ∈ T, is continuous under the topology of uniform convergence on C(X, I). Next, we will prove that Θ is continuous at τ under the topology of uniform convergence on C(X, I).
Indeed, let τ be an arbitrary admissible element of T and let {t γ | γ ∈ Γ} be a net in T with t γ → τ under the topology of S. We need to show that Θ(t γ ) − Θ(τ) → 0.
By condition (c), R∩cls T τM ∅; and so it follows that we can choose an a ∈ R with τa j → a for some net {a j | j ∈ J} in M. Then t γ a j → τa j for any j ∈ J by condition (b). Now given any ε > 0, there exists a neighborhood U of a in T such that Θ(a) − Θ(t) < ε for each t ∈ U, because Θ is continuous at the point a ∈ R.
This, of course, implies that ϑ : (t, x) → θ(tx) of T × X to I is jointly continuous at each point of {τ} × X. The proof of Theorem 8.8 is thus completed.
Note that the group structure of T plays a role in Namioka's proof of Ellis' joint continuity theorem ([23, Theorem 1] and [44, Theorem 3.1]). From Theorem 8.8, we can easily obtain the following four corollaries and Ellis' joint continuity theorem.
As the first simple application of Theorem 8.8, we can obtain an affirmative answer to the following open question: Let S be a compact T 2 semi-topological semigroup with a dense algebraic subgroup G. Suppose a net g α → g in G. Does g −1 α converges to g −1 in G? (See [42, Question 10.3].) Corollary 8.9. Let S be a compact T 2 semi-topological semigroup with a dense algebraic subgroup G. Then G is a topological subgroup of S .
Proof. Let T = S , X = S and define T × X → X by (t, x) → tx and X × T → X by (x, t) → xt. Since G is a subgroup and dense in S , it follows that cls T gG = T = cls T Gg for all g ∈ G. Thus g : x → gx and g : x → xg are surjections of X for each g ∈ G and further T is admissible at each element g ∈ G. Then by Theorem 8.8, (t, x) → tx is continuous on G × X and (x, t) → xt is continuous on X × G. Now let g α → x in G and let g −1 α → y in S ; then by the continuity, xy = e = yx. Whence y = x −1 . This concludes the proof of Corollary 8.9.
The interesting point of Corollary 8.9 is that G as a subspace of S is not necessarily locally compact so Ellis' theorem (cf. Theorem 8.15 below) plays no role here. Corollary 8.10. Let T be a semigroup of continuous self-surjections of a compact T 2 -space X; and let S be a topology on T such that (T, X) is admissible. Then (t, x) → tx of T × X into X is jointly continuous.
Given any integer d ≥ 1, the following corollary seems to be non-trivial because it is beyond Ellis' joint continuity theorem. 45 Corollary 8.11. Let R d + × X → X, (t, x) → tx be a separately continuous semiflow, where (R d + , +) is under the usual Euclidean topology. If X is minimal, then (t, x) → tx is jointly continuous on R d + × X.
Proof. Write T = R d + , which is an additive abelian semigroup. First, under the discrete topology of T , (T, X) becomes a minimal semiflow. Then by Corollary 3.3, it follows that for each t ∈ T , x → tx is a continuous surjection of X. Therefore, under the Euclidean topology of R d + , the following conditions are satisfied: (a) T is a locally compact T 2 -space; and (t, x) → tx is separately continuous of T × X to X.
(b) The right translation R s : t → t + s of T to itself is continuous, for each s ∈ T .
Then by Lawson's theorem (cf. [42,Theorem 5.2] and also see Theorem 8.8), (t, x) → tx is jointly continuous on T × X. This completes the proof of Corollary 8.11. This corollary may be applied to two interesting cases. First, let R d + × X → X be a semiflow; then it is well known that the induced Ellis semiflow R d + × E(X) → E(X) is only separately continuous, not necessarily jointly continuous. However, for any minimal left ideal I of E(X), R d + × I → I is a jointly continuous semiflow by Corollary 8.11. Particularly, if (π, R d + , X) is distal, then E(X) itself is a minimal left ideal in E(X) (by Lemma 6.7) so that (π * , R d + , E(X)) is a semiflow with the phase semigroup R d + under the usual topology. Secondly, let βR d + be the Stone-Čech compactification of R d + . Then βR d + is a compact Hausdorff right-topological semigroup in a natural manner and there is a natural separately continuous semiflow R d + × βR d + → βR d + . Therefore, for any minimal left ideal I of βR d + , R d + × I → I is a jointly continuous semiflow by Corollary 8.11.
Let C p (X, X) denote the Hausdorff space C(X, X) equipped with the topology p of pointwise convergence. Clearly, C p (X, X) is a semi-topological semigroup, since the maps R g : f → f • g and L g : f → g • f of C p (X, X) to itself are continuous for each g ∈ C p (X, X). Then for any semigroup G of homeomorphisms on X, by an argument similar to the proof of [30, Proposition 8.3], we can see that the closure cls C p (X,X) G of G in C p (X, X) is a subsemigroup of C p (X, X).
The following corollary is a generalization of [23, Lemma 3] using different approach. There Ellis is for compact metric phase space X. Corollary 8.12. Let G be a group of self-homeomorphisms of a compact T 2 -space X; and let T = cls C p (X,X) G. If T is an l.c. subset of C p (X, X), then (g, x) → gx of G × X to X is jointly continuous, where G is regarded as a subspace of C p (X, X).
Proof. We consider T × X → X defined by the evaluation map (t, x) → tx. As cls T gG = T for each g ∈ G, T is admissible at each element of G. Thus Corollary 8.12 follows at once from Theorem 8.8.
We shall say that for a group G, an action G × X → X is effective if whenever g e for g ∈ G then gx x for some x ∈ X. This is only a minor technical condition. If the action is not effective, let F = {t ∈ G | tx = x ∀x ∈ X}. Then F is a closed (since X is T 2 ) normal subgroup of T . The