Topological Dynamics of Groupoid Actions

Some basic notions and results in Topological Dynamics are extended to continuous groupoid actions in topological spaces. We focus mainly on recurrence properties. Besides results that are analogous to the classical case of group actions, but which have to be put in the right setting, there are also new phenomena. Mostly for groupoids whose source map is not open (and there are many), some properties which were equivalent for group actions become distinct in this general framework; we illustrate this with various counterexamples.


Introduction
Classically, topological dynamics is understood as the study of group and semigroup actions on topological spaces. It is an important chapter of modern mathematics originating from physics and the theory of differential equations, and its theoretical and practical outreach need not be outlined here. The point of view we adopt is that of the abstract theory, as exposed in references such as [2,7,9,12].
Basic to topological dynamics in the classical sense is the idea of global symmetry. However, many interesting systems only present local (or partial) forms of symmetry. Such systems have acquired an increasing role in modern mathematics, and their impact in applications is already largely acknowledged.
Partial symmetry is treated using concepts as groupoids, partial group actions or inverse semigroup actions. Several aspects are already very well developed. In fact our interest was raised by the applications to C * -algebras, as in [10,11,16,18,20]. However, up to our knowledge, the basic theory in the spirit of classical topological dynamics has not yet been developed systematically. In the literature, when notions and results are needed, they are briefly introduced and used in an ad hoc way. In addition, certain basic concepts barely appear in the partial symmetry setting.
The present article is dedicated to a systematic study of the most elementary dynamical notions in the framework of continuous groupoid actions on topological spaces. We emphasize recurrence phenomena. Some connections with partial group actions are also made, but the interaction with inverse semigroup dynamical systems, as well as a deeper study of more advanced dynamical notions, are deferred to a subsequent article.
As a general convention, all the topological spaces (including the groupoids) are Hausdorff. Local compactness is required only when needed. In certain cases we ask all the fibers of the groupoid to be non-compact, in order to avoid triviality.
In most cases (but not always), guessing the analog of the standard notions in the groupoid case is quite straightforward. Both in the statements and the proofs, one has to take into account the fibered nature of a groupoid action. When a groupoid Ξ with unit space X acts on a topological space Σ , the action θ is only defined on a subset of the product Ξ×Σ . The element θ ξ (σ) is defined only if the domain d(ξ) of ξ ∈ Ξ coincides with the image ρ(σ) of σ ∈ Σ by the anchor map ρ : Σ → X, which is part of the definition of the action. There are also some technical difficulties that we discuss briefly: (a) When dealing with θ ξ (σ) , sometimes one needs to approximate ξ by a net {ξ i | i ∈ I} . Usually θ ξi (σ) makes no sense, and this requires more involved arguments. For group actions this phenomenon is absent, since the action is composed of (global) homeomorphisms. Therefore, extra care will be needed in some proofs.
Let us describe the content of the article. Section 2 introduces the basic definitions. To read the article, the reader only needs to know what a (topological) groupoid is, and to be familiar with the elementary properties a classical dynamical system might have. The notations involving groupoids and continuous groupoid actions are introduced in 2.1. The canonical action on the unit space will be very important; it is a terminal action, by Lemma 7.2. Notions as orbit, orbit closure, invariant set and saturation are obvious extensions of the standard ones. However, if the groupoid Ξ is not open, the interior, the closure or the boundary of an invariant subset of Σ might not be invariant. This will have far-reaching negative consequences. The central notion of recurrence set Ξ N M appears in 2.2. For M, N ⊂ Σ , this consists in the elements ξ ∈ Ξ under which some point of M is sent inside N (taking into account domain issues).
Section 3. In certain situations, a groupoid (or a groupoid action) is canonically constructed from other related mathematical objects. We present here a couple of such situations, stressing the way recurrence sets are determined by these objects. Our selection is probably inspired by our interest in groupoid C * -algebras; geometrically oriented readers could prefer others. Equivalence relations are important in groupoid theory; for us, they will often serve as counterexamples. In fact, the most general of our counterexamples are subgroupoids of direct products between equivalence relations and groups. In 3.2, the usual group actions are encoded in two different ways to fit our setting. It is shown that the second way yields a better grasp at the level of recurrence. The Deaconu-Renault groupoid of a local homeomorphism appears in 3.3. In 3.4 and 3.5 we arrive at groupoid actions making non-invariant restrictions in larger, maybe global, dynamical systems; this includes partial group actions. In 3.6, to a groupoid action (Ξ, θ, Σ) we associate a larger transformation (crossed product) groupoid Ξ⋉ θ Σ . It only has a partial relevance for recurrence issues. But it will also be used later, in connection with factors and extensions. A pull-back construction appears in 3.7.
Section 4 mainly treats various types of topological transitivity. Consider the following properties of a continuous action θ of the groupoid Ξ in the topological space Σ : (i) Σ is not the union of two proper invariant closed subsets, (ii) non-empty open invariant subsets of Σ are dense, (iii) for every U, V ⊂ Σ open and non-void, θ ξ (U ) ∩ V = ∅ for some ξ ∈ Ξ , (iv) invariant subsets of Σ are ether dense, or nowhere dense. For group actions, these properties (suitably reformulated) are equivalent and go under the common name of "topological transitivity". This also happens for groupoid actions, if Ξ is open (i. e. the domain map is open). But if it is not, then one only has (iv) ⇒ (iii) ⇒ (ii) ⇒ (i) . After proving this and some connected results in Theorem 4.1, we indicate a couple of counterexamples showing that in general none of the implications can be reverted. It is also shown that pointwise transitivity (existence of a dense orbit) implies (iii) but not (iv). The failure of some implications can be tracked back to the fact that the closure of an invariant set could not be invariant if the groupoid is not open. We decided to call (iii) recurrent transitivity and (iv) (the strongest notion) topological transitivity; it is debatable whether the terminology is the best one. Properties (i) and (ii) did not receive a name. Proposition 4.5 makes the necessary specifications for the case of a Baire second-countable space Σ . Then we particularize to some of the examples from Section 3. In 4.2 we introduce and study a final transitivity notion, the weak pointwise transitivity. It is similar to pointwise transitivity, but the orbit closure of a point (that might not be invariant if Ξ is not open) is replaced by the smallest closed invariant set containing the point. Section 5 is concerned with limit points, recurrent points and wandering and non-wandering properties. Limit sets are mostly studied for actions of one of the groups Z or R on topological spaces, where one distinguishes between positive and negative limit points. A good reference is [7,Chapter II.2]. We are going to adapt the basic part of the theory to groupoids. But besides being groups, Z and R have two extra features: (1) they have 'two ends' and (2) they are not compact. The first feature leads to various ramifications, as distinguishing between positive and negative limit sets. Trying to imitate this for groupoid actions is possible, but would require a specialized and rather intricate setting. On the other hand, too much compactness in a groupoid would make some constructions and results trivial or noninteresting. So in 5.1, we will restrict our attention to strongly non-compact groupoids , those for which all the fibers Ξ x are non-compact. The limit points of a given point σ are defined by their asymptotic behavior under the action, where "asymptotic" is encoded, equivalently, by complements of compact sets, diverging nets (both in Ξ) or suitable recurrence sets. They are contained in the orbit closure of σ and form a closed subset having an attractor behavior. This subset is invariant if the groupoid is open, but a counterexample shows that in general this can fail. In 5.2 a point is called recurrent if it is a limit point of itself. In Proposition 5.8 this is related to other types of descriptions, as in classical Dynamical Systems, but once again full equivalences hold under the openness assumption (a counterexample is presented). This assumption is also needed to insure invariance of the family of all recurrent points. Wandering and non-wandering are similar to the group case notions, but defined in terms of groupoid recurrence sets Ξ W W and compactness, where W is a neighborhood of the point we study. The family of non-wandering points is closed; to be invariant one also requires Ξ to be open and locally compact. If Σ is compact, it is non-void and attracts the points of Σ in a suitable sense. We indicate examples involving pull-backs and action groupoids. Section 6. We start in 6.1 with the set Σ fix ⊂ Σ of fixed points of a groupoid action. It is invariant; it is also closed if Ξ happens to be open, but not in general, as counterexamples show. We inspect the origin of the fixed points for global and partial group actions, for non-invariant restrictions, for the Deaconu-Renault groupoid and for pull-backs. In 6.2 we show the inclusion (1.1) Besides fixed, recurrent and non-wandering points, already defined, we introduce three other types of points: periodic, weakly periodic and (most important) almost periodic. For the first two, terminology could be debated even in the group case. The precise meaning can be found in Definition 6.8, where an adaptation to groupoids of the standard notion of syndeticity is also included. We discuss issues such as invariance and closure for the new sets. In [3] the authors work with locally compact, second countable, open groupoids with compact unit space. A unit x is called a periodicoid if its orbit is closed (i.e. compact). Among others they show that if, in addition, the groupoid isétale, the orbit of a periodicoid point is actually finite. Obviously, the notion makes sense and is relevant also for general groupoid actions. In 6.3 we connect it with our periodicity. Proposition 6.16 proves that a periodic point σ has a compact orbit and in Proposition 6.18 we prove the converse implication, adding some extra conditions. Minimality is explored in 6.4. After stating the definition and providing the most elementary properties, we exhibit in Theorem 6.23 the connection between minimality and almost periodicity. For group actions, this is the standard result that can be found in every textbook [2,7,9,12]. The proof is similar, but slightly more involved, because of the groupoid setting. Corollary 6.28 deals with semisimplicity and point almost periodicity. In Proposition 6.30 we provide conditions under which minimality implies the non-wandering property. Section 7. Homomorphisms (equivariant maps) are an important topic in Topological Dynamics. In 7.1 we study surjective homomorphisms (epimorphisms) in the framework of groupoid actions (equivariance now also requires a compatibility of the two anchor maps). They lead to the usual concepts of extension and factor. The canonical action of a groupoid Ξ on the unit space is terminal, being a factor of any other Ξ-action. We study the fate under epimorphisms of most of the dynamical properties already introduced. Two main results are Theorem 7.6 (referring to the sets (1.1)) and Proposition 7.7 (treating various types of transitivity, including recurrent transitivity). Note that, by Example 7.9, topological transitivity do not always transfer to factors when the groupoid is not open. Proposition 7.10 refers to the behavior of minimality under epimorphisms. It also contains conditions under which a minimal subsystem of the factor admits a minimal pre-image and an almost periodic point of the factor is reached from an almost periodic element of the extension. Subsection 7.2 connects extensions of groupoid actions as in 7.1 with action (crossed product) groupoids as in 3.6. In the recent preprint [8] the authors use transformation groupoids to study group action extensions. They develop a very interesting theory, that however has little in common with the content of the present article. By a straightforward generalization of a construction from [8], given a groupoid action Θ ′ = Ξ, θ ′ , Σ ′ , we show that there is a one-to-one correspondence between extensions of Θ ′ and actions of the crossed product groupoid Ξ(Θ ′ ) := Ξ ⋉ θ ′ Σ ′ . We also exhibit an isomorphism between two different crossed products. In Propositions 7.13 and 7.14 we prove that the dynamical properties are the same under the mentioned bijective correspondence. Section 8. We dedicate this short final section to convince the reader that mixing, as presented in Definition 8.1, is not an interesting concept outside the classical group case, if the anchor map ρ : Σ → X is surjective and the unit space X is Hausdorff (standing assumptions in the present paper).

Groupoids and groupoid actions 2.1 The framework
We deal with groupoids Ξ over a unit space Ξ (0) ≡ X, seen as small categories in which all the morphisms (arrows) are inverible. The source and range maps are denoted by d, r : Ξ → Ξ (0) and the family of composable pairs by Ξ (2) ⊂ Ξ × Ξ . For M, N ⊂ X one uses the standard notations is called the isotropy bundle of the groupoid. The subset ∆ of the topological groupoid Ξ is called a subgroupoid if for every (ξ, η) ∈ (∆×∆)∩Ξ (2) one has ξη ∈ ∆ and ξ −1 ∈ ∆. This subgroupoid is wide if ∆ (0) = Ξ (0) .
If the action θ is understood, we will write ξ • σ instead of ξ • θ σ. If ρ is not supposed surjective, then ρ(Σ) is an invariant subset of the unit space and only the reduction Ξ ρ(Σ) ρ(Σ) really acts on Σ , so asking ρ to be onto seems convenient.
Example 2.2. Each topological groupoid acts continuously in a canonical way on its unit space. In this case, we have Σ = X and ρ = id X , and then (note the special notation) ξ • x := ξxξ −1 as soon as d(ξ) = x. Putting this differently, ξ sends d(ξ) into r(ξ) . One could also name this the terminal action; see Lemma 7.2.

Recurrence sets
Definition 2.9. For every M, N ⊂ Σ one defines the recurrence set as The set Ξ N M is increasing in M and N . In the group case, one also uses the term "dwelling set".
The recurrence set can be also described in terms of the function If we denote by q the projection on the first variable Ξ×Σ → Ξ and by q its restriction to Ξ ⋊ ⋉ Σ , then It follows immediately that Ξ N M ⊂ Ξ ρ(N ) ρ(M) . Actually, when ρ is also injective, one has Ξ N M = Ξ ρ(N ) ρ(M) . This applies, in particular, to Example 2.2. The stabilizer Ξ σ σ = {ξ ∈ Ξ | ξ • σ = σ} is a closed subgroup of the isotropy group Ξ ρ(σ) ρ(σ) . They coincide whenever ρ is injective. Example 2.11. In the setting of Example 2.4 one has Example 2.12. Given a topological space Σ , the fundamental groupoid Ξ , typically denoted by Π 1 (Σ) , is just the set of homotopy classes of paths between pairs of points. The space of all paths is given the compact-open topology and this induces in Ξ the quotient topology. We observe that X is basically Σ and the action ξ • σ moves the starting point σ through the path ξ . The recurrence sets are expressed in terms of the path-connectedness of Σ : The stabilizer of σ ∈ X is the fundamental group Ξ σ σ = π(Σ, σ) of Σ rooted at σ. The next straightforward results will be useful in the next sections. Proof. We only show the first equality: Lemma 2.14. Let M, N ⊂ Σ . Then For the converse: Remark 2.15. Let (Ξ, ρ, θ, Σ) be a continuous groupoid action and ∆ a wide subgroupoid of Ξ . In terms of the restricted action from Example 2.3, if M, N ⊂ Σ , the contention ∆ N M ⊂ Ξ N M between the corresponding recurrence sets is obvious. It is also clear that the ∆-orbit of any point of Σ is contained in the Ξ-orbit of this point and that the invariant sets under Ξ are also invariant under ∆ . From this one deduces many simple connections between dynamical properties of the two actions, that we will not write down.

Some examples 3.1 Equivalence relations
If Π ⊂ X ×X is an equivalence relation on the Hausdorff topological space X, one can make Π into a topological groupoid by using the product topology in X ×X and the operations The unit space is Diag(X) and we identify it with X, via the homeomorphism (x, x) → x . As a particular case of Example 2.2, the groupoid Ξ := Π acts in a canonical way on X by There are two extreme particular cases: (i) Π = Diag(X) (the trivial groupoid), for which Ξ N M may be identified with N ∩ M , and (ii) Π = X × X (the pair groupoid), when Ξ N M = N × M . Actually, an equivalence relation on X is a wide subgroupoid of the pair groupoid.
Non-open equivalence relations will be used repeatedly as counterexamples. It is true, however, that in some situations one considers on equivalence relations topologies which are different from the one induced from the Cartesian product.

Group actions
We indicate two ways to encode group actions by groupoids. The second one will be convenient for our purposes. We thought it would be interesting to also mention the first one, since it seems natural. Example 3.2. As a particular case of Example 2.2, the transformation groupoid Ξ ≡ G ⋉ γ X associated to the continuous action γ of the topological group G on the topological space X naturally acts on X by (a, x) • x := γ a (x) . We recall that, as a topological space, it is just The first projection p : G ×X → G restricts to a surjection where Rec γ (M, N ) is the usual recurrence set for dynamical systems [2,7,9]. Injectivity fails in general: for instance, Ξ X X = G ×X while Rec γ (X, X) = G . On the other hand, if M or N are singletons, injectivity holds. Using slightly simplified notations, one has Ξ N x0 Although the relation (3.2) is quite concrete, it will not be precise enough to make suitable connections between dynamical properties in the group and in the groupoid framework. Example 3.3. So we implement differently the classical dynamical system (G, γ, Σ) (for a better correspondence of notations, we set Σ for the space of the group action). The group is an open groupoid in the obvious way; so we have Ξ := G and the unit space X = {e} is only composed of the unit of the group. The source and the domain maps are constant, the same being true for ρ : Σ → {e} ; it follows that G ⋊ ⋉ Σ = G ×Σ . One sets a • σ := γ a (σ) for every a ∈ G, σ ∈ Σ (thus θ = γ). Note that this is not covered by Examples 2.2 or 2.4. A simple inspection of the definitions shows that and this will be very convenient below.
An important tool is the canonical cocycle (a groupoid morphism) which is, of course, the restriction to Ξ(ν) of the middle projection. For every M, N ⊂ X set where notationally we identify singletons to points. The canonical cocycle restricts to a surjection In contrast with the global case, if M = {y} , (3.4) could still fail to be injective. But clearly c restricts to a one to one map allowing to identify Ξ(ν) x y with Z ν (x, y) for every x, y ∈ X .
Let Σ an open subset of Σ ; we do not suppose it invariant. One defines X := ρ(Σ) , so ρ := ρ| Σ : Σ → X is a continuous surjection. We also have the (maybe non-invariant) groupoid restriction that acts naturally by the restriction of • on Σ . Obviously One gets equality whenever Σ is ρ-saturated, i.e. when ρ −1 ρ(Σ) = Σ , in particular when ρ is a bijection. On the other hand, if M, N ⊂ Σ , one always gets equality in (3.6), and in this case To get the last term, one notices that

Partial group actions
Let G be a group assumed, for simplicity, to be discrete. The unit is denoted by e .
Definition 3.4. A partial action [11] of G on the topological space Y is a family of homeomorphisms The domain of the composition above is It is easily shown that β −1 a = β a −1 for any element a . To such a partial action, one associates on which one considers the induced product topology. The structure maps are If the action is global, meaning that Y a = Y for every a , then we recover the situation of Example 3.2.
The subset T ⊂ Y is said to be β-invariant if β a (y) ∈ T whenever a ∈ G and y ∈ T ∩ Y a −1 . Of course, this happens exactly when T is invariant, seen as a set of units of Ξ[β] .
For S, T ⊂ Y it seems natural to define the recurrence set (3.8) Very much as in Example 3.2, one gets The first projection (a, y) → p(a, y) := a restricts to a surjection A very direct way to get a partial group action is to make a non-invariant restriction in a group (global) action. In a certain sense this is the most general situation, since any partial action can be extended to a global one [10].
So let (G, γ, X) be a continuous group action and Y an open, maybe non-invariant, subset of X. For every a ∈ G, set Y a := Y ∩ γ a (Y ) ⊂ Y . It is straightforward to check that becomes a partial action. The groupoid Ξ[β] ≡ G ⋉ [β] Y may be seen as the non-invariant restriction to Y of the transformation groupoid G ⋉ γ X from Example 3.2, as described in Subsection 3.4. For this notice that, if (a, x) ∈ G × X, the conditions d(a, x), r(a, x) ∈ Y mean exactly that x ∈ Y a −1 . The relationship between the two types of groupoid recurrence sets may be read off from (3.6), or from (3.7) in the most interesting case, when the two sets are already contained in Y . In this last case, one gets which can also be checked directly. In terms of the actions themselves, using a notation from (3.2), the relation (3.8) converts into . It is also easy to verify that (3.11) follows from (3.9), (3.10) and (3.2).

The action groupoid of a groupoid action
A continuous groupoid action (Ξ, ρ, •, Σ) being given, one constructs the action (or transformation, or crossed product) groupoid which, as a set, is the closed subspace Ξ ⋊ ⋉ Σ of Ξ×Σ introduced in (2.1) and the structure maps are To stress the origin of the construction, we are going to denote by Ξ⋉ θ Σ this groupoid, in analogy with the group case which is a particular example. The space of units may also be written (ξ, σ) • σ = ξ • σ, and thus reproduces in some way the initial action. The invariant subsets, in particular the orbits, are the same. This has a series of obvious consequences on the way dynamical properties are preserved when passing from (Ξ, ρ, •, Σ) to Ξ⋉ θ Σ . The connection between the relevant recurrent sets is similar with (and in fact generalizes) that of where p is the restriction of the first projection of the product Ξ × Σ . This restriction is not always injective, being constant on any set of the type . It follows that for recurrence phenomena it is not always a good idea to replace the initial action by the action groupoid.

Groupoid pull-backs
There is a powerful method to construct new more sophisticated groupoids from simpler ones, which, however, does not loose control over the orbit structure or the recurrence sets. Let d, r : Ξ → X be the domain and the range maps of a topological groupoid, Ω a topological space and h : Ω → X an open continuous surjection. Let be the associated pull-back groupoid [5,14]. We recall its structural maps: Note the relations between orbits and orbit closures in the two groupoids (closures commute with open continuous surjections): For M, N ⊂ Ω , by inspection one gets  (iv) Each invariant subset of Σ is ether dense, or nowhere dense (topological transitivity).
Then the following implications hold: Proof. 1. The first implication is trivial (and obviously it is not an equivalence). We verify now the second one. Assume that Σ has a dense orbit O σ and let ∅ = V 1 , V 2 be open sets. One has The fact that all the other implications fail without extra assumptions will be showed in a series of counterexamples below.
3. Provided that d is open, it is enough to prove that (i ′ ) implies (iv) . So let us assume (i ′ ) , but let A ⊂ Σ be invariant, neither dense, nor nowhere dense.  It is easy to see that X = X 1 ∪X 2 is the single (non-void) open invariant set, since any invariant set is a union of orbits, and X 1 , X 2 are not open. Thus the condition (ii) from Theorem 4.1 holds. On the other hand the condition (iv) definitely fails, since the invariant sets X 1 , X 2 are neither dense, nor nowhere dense. Of course recurrent transitivity also fails. To see this directly, take for instance U = (2, ∞) and V = (−∞, −2) , for which by (3.1) or by a simple computation.
Example 4.3. The previous example can be easily modified to show that (i ′ ) ⇒ (ii). In R x , define the equivalence relation Υ associated to the partition and consider the canonical action of Ξ = Υ on R x . As invariant sets are union of orbits, the only non-void invariant open sets are R x and which have non-empty intersection, so (i ′ ) holds. But the set Y 1 ∪ X 2 is not dense, hence (ii) fails.
Example 4.4. Define on Σ = X = R the equivalence relation This provides a non-open groupoid acting on R , and this action is pointwise transitive (Q is a dense orbit), hence recurrently transitive. The invariant set (0, 1) \ Q is neither nowhere dense nor dense, so this action is not topologically transitive. This example shows that pointwise transitivity ⇒ (iv) and (consequently) (iii) ⇒ (iv). Since Σ is second-countable, its topology has a countable basis {V n = ∅} n∈N . By defining U n = Ξ • V n we get countably many dense open subsets of a Baire space, so U = ∩ n U n is also a dense (invariant) set. Let W = ∅ be an open subset of Σ . By the definition of a basis, there exists some V n ⊂ W . Hence we have U ⊂ U n = Ξ • V n ⊂ Ξ • W . Therefore, if σ ∈ U then ξ −1 • σ ∈ W for some ξ ∈ Ξ . Hence σ has a dense orbit and the action is pointwise transitive.
We recall that Hausdorff locally compact spaces and complete metric spaces are Baire.
Remark 4.6. What we used in the proof of Proposition 4.5 is the fact that the intersection U is non-void. Actually, one could improve: under the given requirements, the set of points with dense orbit is a dense G δ -set.
Example 4.7. In the classical dynamical system case of Example 3.3 we recover known results [2,7,12].
Since the source map G → {e} is clearly open, Theorem 4.1 simplifies a lot. Even in this particular case, without second countability the full equivalence from Proposition 4.5 fails.    In [11,Prop. 5.5] it is shown that any partial action (G, β, Y ) may be obtained from a global one, with the extra condition that the total space X is the saturation of the initial one, in an essentially unique way. In general X could fail to be Hausdorff, cf. [11,Prop. 5.6]. The results are attributed to F. Abadie [1]. Example 4.14. Although the pullback groupoid of subsection 3.7 might be much more complicated than the initial one, h ↓↓ (Ξ) and Ξ are simultaneously recurrently transitive. This follows from (3.14). This is, the smallest closed and invariant set including A . One very special case (deserving its own notation) is C σ = C({σ}) , the invariant orbit closure of σ.   17. An action (Ξ, ρ, θ, Σ) is called weakly pointwise transitive (wpt) if exists a point σ ∈ Σ such that C σ = Σ . In this case, we say that σ is a weakly transitive point. • In example 4.2, one has C σ = R x for all σ ∈ Σ = R x . So every point is weakly transitive. We recall that this example is not topologically transitive or recurrently transitive.

In particular
• In example 4.3, • In example 4.4, All of these are weakly pointwise transitive systems. This examples also show that the sets {C σ | σ ∈ Σ } doesn't need to be disjoint: some could intersect or even (strictly) contain others. Of course, this often happens for orbit closures, but for invariant closures the overlaps tend to be larger. Proof. Suppose that Σ = N 1 ∪ N 2 , being N 1 , N 2 closed and invariant sets. Without loss of generality, there exists σ ∈ N 1 such that C σ = Σ . As N 1 is closed, invariant and contains σ, the relation Σ = C σ ⊂ N 1 follows. We conclude that (i) holds.
Remark 4.21. Example 4.3 shows that weakly pointwise transitivity does not imply the condition (ii) of Theorem 4.1, in general. We recall that pointwise transitivity does imply (iii) , which is stronger than (ii), so none of these properties is implied by weak topological transitivity. We summarize most of the implications in the following diagram, in which the arrows indicate implications:

Limit sets
A continuous action (Ξ, ρ, θ, Σ) will be fixed, with Ξ strongly non-compact. We recall from the Introduction that this means that none of the d-fibers of the groupoid is compact. Of course, the topological spaces Σ and X = Ξ (0) are allowed (but not required) to be compact. Note that closed equivalence relations on compact spaces X, with the induced topology, are excluded. Let K(T ) denote the family of compact subsets of the topological space T . Note that . It follows that one can also write The limit set would be void if Ξ ρ(σ) was allowed to be compact (which is not), but it can also be void in other situations.
For any unit x we say that the net The existence of divergent nets relies on our strongly non-compact assumption.
Lemma 5.2. The following statements for σ, τ ∈ Σ are equivalent: (i) τ belongs to the limit set L σ .
(ii) For every neighborhood V of τ there exists a divergent net (ξ i ) i∈I in Ξ ρ(σ) such that ξ i • σ ∈ V for any i ∈ I.
(ii') For every neighborhood V of τ , the recurrence set Ξ V σ is not relatively compact.
(ii) ⇒ (iii) Consider the set N τ of neighborhoods of τ , and order it by reversing the inclusions. For each neighborhood V of τ , select some divergent net (ξ i,V ) i∈I such that ξ i,V • σ ∈ V (it can be built over the same labels). Observe that, for each k ∈ K(Ξ ρ(σ) ) , there exists i V k such that ξ i,V ∈ k for every i ≥ i V k . Define η k,V = ξ i V k ,V , which forms a net when N τ × K(Ξ ρ(σ) ) is given the product order. By construction, we get a divergent net and τ = lim It follows that τ ∈ L σ .
The next easy lemma is sometimes useful to compute limit sets.

Lemma 5.3. Suppose that there exists a family of compact sets {k λ } λ∈Λ that exhausts Ξ ρ(σ) . That is,
Λ is a directed set, Ξ ρ(σ) = λ∈Λ k λ and k λ1 ⊂ k • λ2 whenever λ 1 ≤ λ 2 . Then the limit set L σ can be computed as Proof. Obviously, L σ ⊂ λ∈Λ (Ξ ρ(σ) \ k λ ) • σ . For the opposite inclusion, it is enough to find for every compact subset k of Ξ ρ an index µ ∈ Λ such that k ⊂ k µ . Indeed, k is covered by the family of interiors of the sets k λ so, by compactness, it is also covered by a finite subfamily k • λ1 , . . . k • λm . Since Λ is directed, that index exists. So we have k ⊂ k µ implying that The conclusion follows. (i) All the points in the orbit of σ have the same limit set L σ . One has If Ξ is open and locally compact, the closed set L σ is invariant.
(iii) If the orbit of σ is relatively compact, L σ is non-empty and it attracts the points of the orbit O σ : for every neighborhood W of L σ , there is a compact subset k of Ξ ρ(σ) such that Proof. (i) We observe that, for k ⊂ Ξ ρ(ξ•σ) , with rigorous computations based on the definitions implying that L ξ•σ ⊂ L σ (because kξ is compact), from which the first statement follows. The ⊃ inclusion in (5.1) is obvious. For each K ∈ K(Ξ) one can write from which ⊂ follows.
(iii) It is enough to show that, for a fixed open neighborhood W of L σ , there is a compact subset k of Ξ ρ(σ) such that (5.2) holds: if L σ were void, the empty set would be a neighborhood, which contradicts the inclusion (Ξ is strongly non-compact). For any k ∈ K(Ξ ρ(σ) ) , we set Σ σ (k) := Ξ ρ(σ) \ k • σ ; complements will refer to O σ . Since L σ ⊂ W ∩ O σ , the family Σ σ (k) c k ∈ K(Ξ ρ(σ) ) is an open cover of the complement of W ∩ O σ in the compact space O σ . We extract a finite subcover Σ σ (k i ) c i = 1, . . . , n . Then and the proof is finished, since a finite union of compact sets is compact.
Example 5.6. We indicate now an example that is relevant for the problem of invariance of the limit sets. Let X be a topological space, G a non-compact group and Π ⊂ X 2 an equivalence relation. The obvious product groupoid Ξ = Π × G has unit space Σ ≡ We prove now that L x = O x . Let y ∈ O x and let (x i ) i∈I ⊂ O x be a net converging to y . As G is not compact, for every k ∈ K(G) there exists some g k ∈ G \ k . The net (x i , x, g k ) (i,k)∈I×K(G) belongs to Ξ x , is divergent and fulfills lim Therefore L x = O x , because of Lemma 5.2 and equation (5.1).
In particular, one can choose X = R , G = Z . If Π is given by , which is not invariant! Note that the equivalence relation is not open.

Recurrent points and wandering
We keep the framework of the preceding subsection.
Definition 5.7. When σ ∈ L σ holds, we say that σ is a recurrent point. We denote by Σ θ rec ≡ Σ rec the family of all the recurrent points of the groupoid action.
In (5.1) the union could be disjoint or not.
Proposition 5.8. For a point σ ∈ Σ , consider the following five conditions: To finish the proof, assume now that Ξ is open and locally compact and we will show (a) ⇒ (b). We know from Proposition 5.5 (ii) that L σ is (closed and) invariant, so if (a) holds it contains the closure of the orbit O σ . But it cannot be strictly bigger, by (5.1).

Corollary 5.9. If Ξ is open and locally compact, Σ rec is invariant.
Proof. Since Ξ is open and locally compact, we can describe its fellowship to L σ by condition (b) . Using Proposition 5.5, (ii), for a recurrent point σ and for ξ ∈ Ξ ρ(σ) we can write so ξ • σ is also recurrent.
Definition 5.10. The point σ ∈ Σ is wandering with respect to the action (Ξ, ρ, θ, Σ) with strongly non-compact groupoid if σ has a neighborhood W such that Ξ W W is relatively compact. In the opposite case, we say that σ is non-wandering. We denote by Σ θ nw ≡ Σ nw the family of all the non-wandering points. If Σ θ nw = Σ one says that the action is non-wandering.
Example 5.11. If (G, γ, Σ) is a topological group dynamical system, one says that σ ∈ Σ is wandering if Rec γ (W, W ) is relatively compact for some neighborhood W of σ (actually, this is mainly used for G = Z, R) . The discussion in Example 3.3 shows that by Definition 5.10 one gets the same concept.  [15]. One says that the action is proper if the map Example 5.14. The invariance may fail without openness. Let Ξ be the groupoid constructed in final part of Example 5.6, the one associated to the partition R = (−∞, 0] ∪ (0, ∞) and to the group G = Z . Then consider the wide (locally compact) subgroupoid ∆ := {(x, y, n) ∈ Ξ | x, y > 0 or n = 0}.
If we denote by θ 1 and θ 2 the canonical actions of Ξ and ∆ in Σ = R , respectively, we have To check the second equality, note that ∆ W W = W ×W ×{0} if W ⊂ (−∞, 0) , but ∆ W W = W ×W × Z when W ⊂ (0, ∞) . The second set in (5.3) is not invariant, since 0 is orbit-equivalent with any negative number. In addition, it follows easily that Proof. The first inclusion follows from the definition of recurrent points. So we only need to prove that L σ ⊂ Σ nw . Pick τ ∈ L σ and let U be a neighborhood of τ . By Lemma 5.2, we already know that Ξ U σ is not relatively compact. If ξ ∈ Ξ U σ , then ξ • σ ∈ U ; using Lemma 2.13 we get showing that the latter set is not relatively compact. (Notice that d Ξ U σ = {r(ξ −1 )}, hence Ξ U σ ξ −1 is relatively compact if and only if Ξ U σ is relatively compact)

Corollary 5.16. If at least one of the orbits is relatively compact (in particular if
Proof. This follows from (5.4) and Proposition 5.5 (iii).
Proposition 5.17. If Σ is compact, Σ nw attracts the points of Σ : for every σ ∈ Σ and for every neighborhood V of Σ nw , one has ξ • σ ∈ V for every ξ ∈ Ξ ρ(σ) outside some compact set.
Proof. One has to show that for every neighborhood V of Σ nw , the complement in Ξ ρ(σ) of the set Ξ V σ is relatively compact. If V is a neighborhood of Σ nw , by (5.4), it is also a neighborhood of the limit set L σ . One applies Proposition 5.5 (iii) to infer that there exists a compact subset k of Ξ ρ(σ) such that Then the complement of Ξ V σ in Ξ ρ(σ) is contained in k and the proof is finished.
Example 5.18. If Ξ⋉ θ Σ is the action groupoid of the groupoid action (Ξ, θ, ρ, Σ) , for every σ ∈ Σ the two limit sets that make sense are equal. The recurrent sets are also equal. These are easy consequences of the definitions. Formula (3.13) shows that if σ is non-wandering for the action θ, then it is also non-wandering for the action groupoid Ξ⋉ θ Σ . If Σ is compact, the non-wandering sets coincide.
Example 5.19. We refer now to the pull-back construction of 3.7, assuming that Ω is locally compact. We leave it to the reader to check the formula where the limit set in the l. h. s. is computed in h ↓↓ (Ξ) , being a subset of Ω . It follows easily that ω ∈ Ω is h ↓↓ (Ξ)-recurrent if and only if h(ω) ∈ X is Ξ-recurrent. Using (3.14), one checks easily that ω is h ↓↓ (Ξ)-wandering if and only if h(ω) is Ξ-wandering.
Definition 6.1. A fixed point is a point σ ∈ Σ such that ξ • σ = σ for every ξ ∈ Ξ ρ(σ) . This is equivalent to Ξ σ σ = Ξ ρ(σ) . We write σ ∈ Σ θ fix ≡ Σ fix .   Example 6.6. An element ω ∈ Ω is a fixed point of the pull-back groupoid h ↓↓ (Ξ) introduced in subsection 3.7 if and only if h(ω) is a fixed point of the canonical action of Ξ on its unit space.
If Ξ is open and σ ∈ Σ fix , then exists a net (σ i ) i∈I of fixed points converging to σ. Let ξ ∈ Ξ ρ(σ) . By Fell's criterion [20,Prop. 1.1], applied to the open map d and the net ρ(σ i ) ∈ X, there exists a net (ξ j ) and a subnet (σ ij ) such that ξ j → ξ and ρ(σ ij ) = d(ξ j ) . By continuity, we have So σ is a fixed point for the action. Examples 6.9 and 7.9 will illustrate that openness of Ξ is important.

Periodic and almost periodic points
Although it is not always necessary, in this subsection we prefer to assume that in the continuous action (Ξ, ρ, θ, Σ) the groupoid Ξ is strongly non-compact (all the d-fibers are non-compact).
(c) The point σ is called weakly periodic (we write σ ∈ Σ wper ) if the subgroup Ξ σ σ is not compact.
(d) The point σ ∈ Σ is said to be almost periodic if Ξ U σ is syndetic in Ξ ρ(σ) for every neighborhood U of σ in Σ . We denote by Σ alper the set of all the almost periodic points. If Σ alper = Σ , the action is pointwise almost periodic. Example 6.9. Consider the equivalence relation Π introduced in Remark 3.1, and form the groupoid product Ξ = Π × Z . By considering the action of Ξ in Σ = R , we see that which also illustrates that, in general, the set Σ fix is not closed. Example 6.10. We fix a unit y ∈ X of the Deaconu-Renault groupoid of the local homeomorphism ν : X → X. We remarked that (3.4) becomes a bijection for singletons. Then y is periodic if and only if it is weakly periodic, and this happens exactly when Z ν (x, x) (a subgroup of Z) does not coincide with {0} (then it will be both infinite and syndetic). This means that for some positive integers k = l one has ν k (x) = ν l (x) . Example 6.11. Let Ξ[β] be the groupoid associated to the partial action β of the discrete group G on the topological space Y . After (3.9), we established an identification between Ξ[β] N y and Rec [β] (y, N ) for every point y ∈ Y and subset N ⊂ Y . This allows rephrasing the periodicity properties in this case. For example, y is periodic in the groupoid Ξ[β] if and only if the subgroup of G composed of all the elements "defined at y and leaving it invariant" is syndetic (which is equivalent to having finite index in G). Proposition 6.12. Let Ξ be a strongly non-compact groupoid. Then one has Proof. The inclusions (1) and (3) are obvious. We proved (5) and (6) previously, in Proposition 5.15.
To deduce (2) from the definitions, note that Ξ σ σ ⊂ Ξ U σ if σ ⊂ U and that a syndetic set is not compact (since the d-fibers are not compact).
The inclusion (4) also follows easily from the definitions, by the same type of arguments: use Proposition 5.8 (e) to describe recurrent points.
Example 6.13. The situation is particularly simple for equivalence relations, outlined in subsection 3.1, especially because of equation (3.1). In particular, for y ∈ X ≡ Σ one gets To insure that the associated groupoid is strongly non-compact, we require that for every y ∈ X the set {x ∈ X | x Π y} is non-compact. Then Σ wper = ∅ (so there are no fixed points or periodic points). With some abuse of notation and interpretation, y will be almost periodic if and only if {x ∈ U | x Π y} is syndetic in {x ∈ X | x Π y} for every neighborhood U of y . If X is locally compact, one could choose a relatively compact neighborhood and syndeticity contradicts the fact that the fiber in y is non-compact. Therefore, in the locally compact strongly non-compact case, equivalent relations do not exhibit almost periodic points.
If the equivalence relation is not forced to lead to a strongly non-compact groupoid, the situation might be very different. In particular, some of the inclusions in (6.1) no longer hold. We leave this to the reader.
Example 6.14. Denoting by β any of the properties "periodic", "weakly periodic" and "almost periodic", it is easy to check that ω ∈ Ω has β in the pull-back h ↓↓ (Ξ) if and only if h(ω) ∈ X has β in Ξ . This happens mostly because of equality 3.14.

Compact orbits
We connect now periodicity with the notion of a periodicoid point, introduced in [3, Section 3.1] in a more restricted context. Proof. If σ ∈ Σ per , then K Ξ σ σ = Ξ ρ(σ) for some compact set K ⊂ Ξ . Any net (σ i ) ⊂ O σ can be written Proof. Let us indicate a second proof, using a construction that will also be useful for Proposition 6.18. For σ ∈ Σ , let us define the continuous surjective function If σ ∈ Σ per , then K Ξ σ σ = Ξ ρ(σ) for some compact set K ⊂ Ξ . By using the definition of Ξ σ σ , one gets which is compact, as a direct continuous image of a compact set.
For deriving a converse proposition, we will use a well known lemma (with proof, for the convenience of the reader): Lemma 6.17. Let Ξ = G be a locally compact, second countable group acting on a topological space Σ. If σ ∈ Σ has a compact orbit, then G σ σ ≡ Rec(σ, σ) (see subsection 3.2) is syndetic in G .
Proof. Let N g ⊂ G be a relatively compact, open neighborhood of g , for every g ∈ G . Observe that Let us, for the sake of the argument, use (and prove it later) that the set N g • σ is a neighborhood of σ. By compactness we can extract a finite index set F = {g 1 , . . . , g n } such that Define K = n i=1 N gi and notice that for every g • σ ∈ O σ , there exists h ∈ K such that meaning that h −1 g ∈ G σ σ and g ∈ K G σ σ . As K is compact, we conclude that G σ σ is syndetic in G. Now, we fill the remaining gap: Pick W ⊂ G, another relatively compact, open neighborhood of g but satisfying W ⊂ N g . As G is second countable, it has the Lindelöf property, so exists a countable subset C ⊂ G making G = CW true. If N g • σ had void interior, so would c • W • σ = cW • σ. But then we could decompose O σ as a countable union of closed nowhere dense sets: contradicting Baire's category theorem. Proposition 6.18. Assume that Ξ is locally compact, second countable and open. If σ ∈ Σ has a compact orbit, then it is a periodic point.
Proof. Let us first treat the case of the canonical action of Ξ in its unit space X (and notice that X, being closed in Ξ , is a locally compact, second countable space by its own). We will use a small amount of information from [20,Sect. 22], treating the Mackey-Glimm-Ramsay Dichotomy for groupoids; see also [17]. For x ∈ X, let us define the continuous surjective function x . This leads to a continuous bijection The quotient map p :  [20] (see the non-trivial implication (2) ⇒ (e) on pages 41-42), this happens if the orbit O x is Baire. In our case this is insured, since it is (Hausdorff and) compact. So we conclude that (6.2) is a homeomorphism and thus Ξ x / Ξ x x is compact. To finish this part of the proof, we show now that the compactness of Ξ Then p(ξ) = p(η) for some η ∈ K . By the definition of p , this means ξ ∈ η Ξ x x and we are done.
Now, we will derive the full result: Suppose that Ξ acts on a very general space Σ and O σ ⊂ Σ is compact, for some σ ∈ Σ . As the anchor map is continuous, compact and by the previous discussion, the decomposition Ξ ρ(σ) = K 1 Ξ ρ(σ) ρ(σ) holds for some compact set By applying Lemma 6.17 (with a change of notations), we obtain another compact set K 2 ⊂ Ξ such that finishing the proof.
There is a shorter proof, in only one step and avoiding the use of Lemma 6.16, but it requires Σ to be locally compact and second countable, which we succeeded to avoid.

Minimal sets
Minimality is a very important property in classical topological dynamics; it extends straightforwardly to groupoid actions, denoted below by (Ξ, ρ, θ, Σ) . During this subsection, both Ξ and Σ are assumed to be locally compact. Remark 6.21. A function ϕ : Σ → R is called invariant with respect to the groupoid action if ϕ(σ) = ϕ(τ ) whenever σ θ ∼ τ . By an obvious proof, one shows that if the action is minimal and ϕ is continuous at least at one point, it has to be constant.
We proceed now to characterize minimality. Proof. Suppose that σ is almost periodic; we show first that its orbit closure O σ is compact. Let U 0 be a compact neighborhood of σ. Using the assumptions, for some compact set K one has If O σ is not minimal, it strictly contains a minimal (and compact) set M . The point σ does not belong to M , so there are disjoint open sets U, V ⊂ Σ such that σ ∈ U and M ⊂ V . For an arbitrary compact set K ⊂ Ξ we will now show that K Ξ U σ = Ξ ρ(σ) , implying that in fact σ is not almost periodic. The set M being invariant, By compactness of O σ applied to the open cover above, for a finite set F = {ξ 1 , . . . , ξ k } ⊂ Ξ we get If η ∈ Ξ ρ(σ) then η • σ ∈ N ξj • U for some j, and then one has η • σ ∈ η j • U for some η j ∈ N ξj . It follows immediately that r(η) = r(η j ) = d η −1 This means that η −1 j η ∈ Ξ U σ or, equivalently, that Since η is arbitrary one gets Ξ ρ(σ) ⊂ K Ξ U σ , where K := k i=1 N ξi is compact. We checked that Ξ U σ is syndetic in Ξ ρ(σ) , so σ is almost periodic.
Example 6.24. In the case of the transformation groupoid associated to a topological dynamical system (G, γ, X) , one recovers the classical result ([2, pag.11] and [9, pag. 28, 38, 39]). For this, we use Example 3.3. First of all, it is clear that minimality of the group action coincides with minimality in the sense of groupoids, since the orbits are the same. The relevant recurrence sets also coincide: set S = {σ} and T = U in (3.3). Finally, syndeticity has the same meaning in the two cases.
Example 6.25. Consider the case of the pair groupoid Ξ := X ×X acting on its unit space X, taken to be compact. The action is transitive (only one orbit), so every x ∈ X should be almost periodic. And it is, since Ξ x = X ×{x} and Ξ U x = U ×{x} . The set K := X ×{x} itself is compact, and This example also shows another difference between the group and the groupoid case. For groups, the compact set K in the definition of syndeticity can always be taken finite (see for example [7, pag.271], where this property is called 'discrete syndeticity'). In this groupoid no finite set K ⊂ X × {x} makes the equality K Ξ U x = Ξ x true if X itself is infinite. If Σ is a compact space, Zorn's Lemma implies that it has a minimal subset M ⊂ Σ . If in addition Ξ is open, every x ∈ M is almost periodic, by Theorem 6.23. Thus, in this setting, almost periodic points always exist.
meaning that Ξ V σ is also syndetic.
Proof. By Proposition 5.2, τ ′ ∈ f L θ σ if and only if exists a divergent net Then the statement about recurrence follows from the definitions.
Let us see what happens with recurrence sets under epimorphisms.
Proof. One verifies easily that the next sequence of equivalences and implications is rigorous: The second equivalence is true because f is onto. In general one has f (A ∩ B) ⊂ f (A) ∩ f (B) and the inclusion could be strict; this shows why (and when) there is no equality in the statement.
To see the usefulness of this Lemma, we hurry to apply it. On many occasions we are going to use the equality f f −1 (B ′ ) = B ′ for B ′ ⊂ Σ ′ , valid by surjectivity.
Proof. Using Lemma 7.2 the last statement follows from the first, that we now prove.
For α = rec this is already known from Proposition 7.4. The statement for α = fix follows immediately from (7.1).
and the proof is finished.
Corollary 7.8. We say that the groupoid Ξ has the property P if its canonical action on its unit space has this property. Suppose that the topological groupoid Ξ admits a continuous action (ρ, θ, Σ) having one of the properties P mentioned in Proposition 7.7. Then Ξ itself has this property.
Proof. This follows from Proposition 7.7 and Lemma 7.2.
If we require Ξ to be open, there is a direct proof that the property (iv) from Theorem 4.1 (called topological transitivity) also transfer to factors; it uses Lemma 2.7. This also follows joining Theorem 4.1 and Proposition 7.7. But see the next example for a non-topologically transitive groupoid, which however possesses a topologically transitive action. It follows that topological transitivity is not preserved by factors (which is rather surprising). Example 7.9. Form the product groupoid Ξ = Π × R , with the relation x Π y ⇔ x, y ∈ Q or x = y over X = R , and consider the wide subgroupoid ∆ = {(x, y, g) ∈ Ξ | x, y ∈ Q ⇒ g = 0} = Q×Q×R ∪ Diag(R×R)×{0} .
By analogy with Example 4.4, its easy to see that the canonical action of ∆ on X = R is not topologically transitive. Actually, the orbits of rational points all coincide with Q , but each irrational point is a fixed point, so (s, t) \ Q is invariant for every s < t , without being dense or nowhere dense. Now we will exhibit a topologically transitive action of Ξ : Let Σ = {(y, h) ∈ R 2 | h = 0 or y ∈ Q} = (R×{0}) ∪ (Q×R) with the topology inherited from R 2 and define the continuous action ρ(y, h) = y and (x, y, g) • (y, h) = (x, g + h) .
Notice that the orbits of • are Q×R and (the fixed points) {(y, 0) | y ∈ R \ Q} , implying that all of the invariant sets are either dense or nowhere dense, since Σ\(Q×R) = (R\Q)×{0} is already nowhere dense.
We finish this subsection with a result on the behavior of minimality under epimorphisms, in both directions.

The action associated to an extension
The next result is a straightforward generalization of a recent construction from [8], in which the authors showed how to encode extensions of classical group actions by groupoids. Then we are going to study the resulting correspondence with respect to recurrence and other dynamical properties.
Here r is the range map of Ξ⋉ θ ′ Σ ′ . The two procedures are inverse to each other.