Point-Shift Foliation of a Point Process

A point-shift $F$ maps each point of a point process $\Phi$ to some point of $\Phi$. For all translation invariant point-shifts $F$, the $F$-foliation of $\Phi$ is a partition of the support of $\Phi$ which is the discrete analogue of the stable manifold of $F$ on $\Phi$. It is first shown that foliations lead to a classification of the behavior of point-shifts on point processes. Both qualitative and quantitative properties of foliations are then established. It is shown that for all point-shifts $F$, there exists a point-shift $F_\bot$, the orbits of which are the $F$-foils of $\Phi$, and which are measure-preserving. The foils are not always stationary point processes. Nevertheless, they admit relative intensities with respect to one another.


Introduction
A point process is said to be flow-adapted if its distribution is invariant by the group of translations on R d . A point-shift is a dynamics on the support of a flow-adapted point process, which is itself flow-adapted.
The main new objects of the paper are the notion of foliation of a flowadapted point process w.r.t. a flow-adapted point-shift.
Such a foliation is a discrete version of the global stable manifold (see e.g. [6] for the general setting and below for the precise definition used here) of this dynamics, i.e., two points in the support of the point process are in the same leave or foil of this stable manifold if they have the same "long term behavior" for this dynamics. This foliation provides a flow-adapted partition of the support of the point process in connected components and foils.
The point foil of a point process w.r.t. a point-shift is defined under the Palm distribution of this point process. It is the random counting measure with atoms at the points of the foil of the origin. The distribution of the point foil under the Palm probability of the point process is left invariant by all bijective shifts preserving the foliation. A point foil is not always markable, i.e., is not always a stationary point process under its Palm distribution.
The main mathematical result of the paper is the classification of pointshifts based on the cardinalities of their foils and connected components (Theorem 21) and on whether their point foils are markable or not.
The literature on point-shifts starts with the seminal paper by J. Mecke [9]. The fundamental result of [9] is the point stationarity theorem, which states that all bijective point-shifts preserve the Palm distribution of all simple and stationary point processes. The notion of point-map was introduced by H. Thorisson (see [10] and the references therein) and further studied by M. Heveling and G. Last [5]. The dynamical system analysis of point-shifts which is pursued in the present paper was proposed in [2]. The last paper is focused on long term properties of iterates of point-shifts. It introduces the notion of point-map probability, which provides an extension of Mecke's point stationarity theorem. In contrast, the present paper is focused on the stable manifold of a point-shift, as already mentioned. It is centered on the definition of this object and on the study of both qualitative and quantitative properties of its distribution.
The paper is structured as follows. Section 2 defines the setting for discrete foliations and Section 3 that for point processes and point-shifts. Section 4 combines the two frameworks and defines the discrete foliation of a point process by a point-shift. Section 5 gives the classification. Section 6 introduces the stable group of this foliation, and shows the existence of measure preserving dynamics on the foliation. It also defines the foil point process. Finally, Section 7 gathers the quantitative properties of foliations.

Foils and Connected Components
The notion of discrete foliation can be defined for any function on any set. Since the present paper is focused on stochastic objects, only measurable functions on measurable spaces will be considered.
Assume (X, F) is a measurable space where all singletons are measurable; i.e., for all x ∈ X one has {x} ∈ F and let g be a measurable map (or dynamics) on X 1 . Let ∼ g be the binary relation on the elements of X defined by x ∼ g y ⇔ ∃n ∈ N; g n (x) = g n (y).
It is immediate that ∼ g is an equivalence relation.
Definition 1. The partition of X generated by the equivalence classes of ∼ g will be called the g-foliation of X. Denote it by L g (X) or L g X . Each equivalence class is called a foil. The equivalence class of x ∈ X is denoted by L g (x).

Remark 2.
In the terminology of geometry, foils are called leaves. But since the paper uses graphs which are mostly trees, to avoid confusion with tree leaves, the word foil will be used here.
One can also see L g (x) as the limit of the increasing sets L g n (x), where L g n (x) := {y ∈ X; g n (y) = g n (x)}.
The cardinality of L g (x) (resp. L g n (x)) will be denoted by l g (x) (resp. l g n (x)). For reasons that will be explained below, the class of g(x), namely L g (g(x)) will be denoted by L g + (x). If there exists a point y ∈ X such that g(y) ∈ L g (x), L g (y) is denoted by L g − (x). One can verify that L g + (x) is well-defined and that both L g − (x) and L g + (x) are class objects; i.e., they do not depend on the choice of the element of the equivalence class.
Remark 3. For a homeomorphism g on a metric space, the stable manifold of a point x ∈ X with respect to g is W s (g, x) = {y ∈ X; lim n→∞ d(g n (x), g n (y)) = 0}.
Hence, in the case where the space X is equipped with a discrete metric, the stable manifold foliation is the g-foliation of X as defined above. This explains the chosen terminology.
The measurability of g implies all foils are measurable subsets of X.
The g-foliation of X is the finest g-invariant partition L g X of X in the sense that for all g-invariant partitions L one has ∀L ∈ L g X , ∃L ∈ L s.t. L ⊂ L .
Definition 4. The graph G g = G g (X) = (V, E) has for set of vertices V = X and for set of edges E = {(x, g(x)), x ∈ X 2 . Note that this graph can be considered either as undirected or as directed, with each edge from x to g(x). For x ∈ X, denote by C g (x) the undirected connected component of G g which contains x; i.e., the set of all points y ∈ X for which there exist nonnegative integers m and n such that g m (x) = g n (y). The set of connected component of G g will be denoted by C g (X). If x ∼ g y then x and y are in the same connected component of C g (x). In other words, the foliation is a subdivision of C g (X).
C g (x) will be said to be g-acyclic, if the restriction of G g to C g (x) is a tree.
Lemma 5. The connected component C = C g (x) of G g is either an infinite tree or it has exactly one (directed) cycle K(C); in the latter case, for all y ∈ C, there exists n ∈ N such that g n (y) ∈ K(C).
Proof. All statements follow from the fact that all vertices of C, seen as a directed graph, have out-degree equal to one and from the fact that C is connected (as an undirected graph).
Whenever it is clear from the context, the superscript g is dropped.

Foil Order
The g-foliation of each connected component of X can be equipped with some form of order. Consider g(x) as the father of x. Then L g (x) denotes the g-generation of x i.e., the set of its g-cousins of all orders; L g n (x) denotes the set of its g-cousins with common n-th g-ancestor. In addition, L g + (x) is the g-generation senior to x's, i.e., that of its father, whereas L g − (x) (if it exists) is the g-generation junior to x's, i.e., that of its sons (if any) or that of the sons of its cousins (again if any).
Definition 7. Note that if C(x) is acyclic, this definition of generations gives a linear order on the foils of C(x) which is that of seniority: by definition L g (y) < L g + (y) for all y ∈ C(x). This order is then similar to the order of either Z or N (total order with either no minimal element or with a minimal element).
Note that g n (X) is a sequence of decreasing sets in n. Its limit (which may be the empty set) is denoted by g ∞ (X) and, consistently with the seniority order, the set g ∞ (X) will be called the set of g-primeval elements of X.
Definition 8. Let n be a positive integer. For all x ∈ X, let D n (x) = D n (g, x) be the set of all descendants of x which belong to the n-th generation w.r.t. x; i.e., D n (x) := {y ∈ X; g n (y) = x}.
The cardinality of D n (x) (which may be zero, finite or infinite) is denoted by d n (x). Also, let D(x) = D(g, x) denote the set of all descendants of x; i.e., Finally the cardinality of D(x) is denoted by d(x).

Point Processes and Point-Shifts
Whenever (R d , +) acts on a space, the action of t ∈ R d on that space is denoted by θ t . It is assumed that (R d , +) acts on the reference probability space (Ω, F).

Counting Measures and Point Processes
Let N be the space of all locally finite and simple counting measures on R d . It contains all measures φ on R d such that for all bounded (relatively compact) Borel subsets B of R d , φ(B) ∈ N (counting measure condition) and for all x ∈ R d , φ({x}) ≤ 1 (simplicity condition). Let N be the cylindrical σ-field on N generated by the functionals φ → φ(B), where B ranges over the elements of B, the Borel σ-field of R d . The flow θ t acts on counting measures as and therefore on R d as θ t x = x − t. Let N 0 be the subspace of N of counting measures with an atom at the origin.
A (random) point process is a couple (Φ, P) where P is a probability measure on a measurable space (Ω, F) and Φ is a measurable mapping from (Ω, F) to (N, N ). Note that the point process (Φ, P) is a.s. simple by construction.
The stationarity of a point process translates into the assumptions that for all t ∈ R d , θ t P = P and and that Φ(θ t ω) = θ t Φ(ω).
When the point process (Φ, P) has a finite and positive intensity, its Palm probability [3] is denoted by P Φ . Expectation w.r.t. P Φ is denoted by E Φ .

Flow-Adapted Point-Shifts
A point-shift on N is a measurable function F which is defined for all pairs (φ, x), where φ ∈ N and x ∈ φ, and satisfies the relation F φ (x) ∈ φ.
In order to define flow-adapted point-shifts, it is convenient to use the notion of point-map. A measurable function f from the set N 0 to R d is called a point-map if for all φ in N 0 , f (φ) belongs to φ.
If f is a point-map, the associated flow-adapted point-shift, F = F f , is a function which is defined for all pairs (φ, x), where φ ∈ N and x ∈ φ, by In the rest of this article, point-shift always means flow-adapted point-shift. Point-shifts will be denoted by capital letters and the point-map of a given point-shift will be denoted by the associated small letter (F 's point-map is hence denoted by f ). The n-th image of φ under F is inductively defined as

Examples
This subsection introduces a few basic examples which will be used to illustrate the results below. These examples will be based on two types of point processes: Poisson point processes and Bernoulli grids. The latter are defined as follows: it is well known that the d dimensional lattice Z d can be transformed into a stationary point process in R d by a uniform random shift of the origin in the d unit cube. The Bernoulli grid of R d is obtained in the same way when keeping or discarding each of the lattice points independently with probability p. The result is again a stationary point process whose distribution will be denoted by P p .

Strip Point-Shift
The Strip Point-Shift was introduced by Ferrari, Landim and Thorisson [4].
It is easy to verify that S is flow-adapted. Denote its point-map by s. The strip point-shift is not well-defined when there are more than one left most point in St(x), nor when the point process has no point (other than x) in St(x). Note that such ambiguities can always be removed, and some refined version of the strip point-shift can always be defined by fixing, in some flow-adapted manner, the choice of the image and by choosing f φ (x) = x in the case of non-existence. By doing so one gets a refined point-shift defined for all (φ, x).

Mutual Nearest Neighbor Point-Shift
Two points x and y in φ are mutual nearest neighbors if φ(B o (x, ||x − y||)) = φ(B o (y, ||x − y||)) = 1 and φ(B(x, ||x − y||)) = φ(B(y, ||x − y||)) = 2, where B o (z, r) (resp. B(z, r)) denotes the open (resp. closed) ball of center z and radius r. The Mutual Nearest Neighbor Point-Shift N is the involution which maps x to y when these two points are mutual nearest neighbors and maps z to itself if z has no mutual nearest neighbor. This point-shift is bijective.

Next Row Point-Shift on the Bernoulli Grid
The Next Row point-shift, which will be denoted by R, is defined on the d-dimensional Bernoulli grid as follows: It is easy to verify that if d ≥ 2 and p > 0, R is a.s. well-defined.

Condenser Point-Shifts
Assume each point x ∈ φ is marked with Note that m c (x) is always positive. The condenser point-shift acts on marked point process as follows: it goes from each point x ∈ φ to the closest point y with a larger first coordinate such that m c (y) = m c (x)+1. It is easy to verify that the condenser point-shift is flow-adapted and almost surely well-defined on the homogeneous Poisson point process.

On Finite Subsets of Point Process Supports
This subsection contains some of the key technical results to be used in the proofs below. Below, a counting measure will often be identified with its support, namely with a discrete subset of R d .
Lemma 9. Let X ⊂ N(R d ) be a family of discrete subsets of R d such that, for all t ∈ R d and X ∈ X , one has θ t X ∈ X . Assume π : X → N(R d ) is a measurable finite and non-empty flow-adapted inclusion, i.e., for all t ∈ R d and all X ∈ X , 0 < |π(X)| < ∞ (finite and non-empty), π(θ t X) = θ t π(X) (flow-adapted), π(X) ⊂ X (inclusion).
Then, there exists a flow-adapted numbering of the points of the elements of X .
Proof. For all X ∈ X one can choose a point y 1 of π(X) in a flow-adapted manner; e.g. the least point in lexicographic order of R d . Then considering y 1 as the first point, one can number the other points of X according to their distance to y 1 in an increasing order; if there are several points equidistant to y, one can sort them in increasing lexicographic order. Note that the fact that X is discrete implies there are at most finitely many points in B r (y) for all given y and hence the above numbering is well-defined.
Theorem 10. Let (Φ, P) be a stationary point process and n = n(Φ) be a measurable flow-adapted random variable taking its values in N = N ∪ {∞}.
is a flow-adapted collection of subsets of Φ such that for all i, Ξ i is a finite subset of Ψ i then, almost surely, all Ξ i -s are empty.
In words, Theorem 10 states that no stationary point process (and no collection of its infinite disjoint subsets) possesses a finite non-empty flowadapted inclusion.
Proof. If, for some i, Ξ i = ∅, one has a flow-adapted numbering of the points of Ψ i from Lemma 9. Let A(m, Ξ i ), m ≥ 1, be the m-th point of Ψ i in this numbering. Now define the following point-shift: The compatibility assumptions imply that F Φ is indeed a point-shift. It is clear from the definition that F Φ is injective from the support of Φ to itself, and according to Corollary 27 below, F Φ is almost surely bijective. But, for all i for which Ξ i = ∅, A(1, Ξ i ) is not the image of any point. Therefore almost surely there is no non-empty Ξ i .
Remark 11. In Theorem 10, the condition that the Ξ i -s are disjoint is not necessary. But for the sake of space, the proof of this more general case is skipped.
Corollary 12. Letting Ψ = Φ in Theorem 10 gives that one cannot choose a finite non-empty subset of a point process in a flow-adapted manner.

Partitions of the Support of a Point Process
A partition of counting measures, T , is a map that associates to each φ ∈ N a partition T (φ) = {T n (φ); n ∈ N} of the support of φ into a countable collection of non-empty sets.
This partition is flow-adapted if for all φ ∈ N and all t ∈ R d , One of the simplest cases of flow-adapted partitions is the singleton partition; i.e., For a partition T , the element of T (φ) that contains t ∈ φ is denoted by T t (φ). Using this notation, it is easy to see that each flow-adapted partition T of counting measures is fully characterized by a measurable map T 0 : An enumeration of the elements of a set is an injective function ν from from this set to N (or equivalently to Z). There are several enumerations of the elements of the partition T ; e.g. based on the distance to the origin.
Any element T of the partition is a countable collection of points of φ. Since φ has no accumulation points, one can define the distance of T to the origin as the minimum of the distances from the points of T to the origin. If the set of distances to the sets of the partition are all different, one defines T 0 as the element of the partition with the smallest distance to the origin, T 1 as the one with the second smallest distance to the origin, and so on. Ties are treated in the usual way, e.g. by using lexicographic ordering. Note that this enumeration is not flow-adapted.
A natural question is about the existence of translation invariant enumerations. This is not always granted. For example, it is well known, and can be seen from Corollary 12, that the singleton partition of a stationary point process (Φ, P) cannot be enumerated in a measurable and flow-adapted manner.
Definition 13. A flow-adapted partition of a stationary point process will be said markable if there exists an enumeration of the elements of the partition which is invariant by translations.
Proposition 14. The flow-adapted partition T of the stationary point process (Φ, P) is markable if and only if (Φ, P) can be partitioned as where (Φ i , P) is a sub-stationary point process such that its support is an element of T .
Proof. Assume T is markable with the enumeration function ν. Then, The injectivity of ν implies ν −1 (i) is either an element of T or the empty set with positive probability. In addition, the translation invariance of ν gives ν −1 (i) is a sub-stationary point process.
On the other hand if Φ possesses a markable decomposition (3), then the function ν which assigns to each element T of T , the index i for which the support of Φ i is T , is an enumeration function of for T and hence T is markable.
The reason for this terminology is that if the partition is defined by a selection of the points of Φ based on marks (see e.g. [3] for the definition of marks of a point process), then such an enumeration exists 3 . For instance, the singleton partition of a stationary point process is flow-adapted but is not a markable partition.
Definition 15. Let T be a flow-adapted partition of the support of Φ. Let H be a point-shift. One says H preserves T if for all T ∈ T , H −1 (T ) = T . If H is bijective, this is equivalent to the property that for all T ∈ T , H(T ) = T .
Definition 16. Let Γ T := Γ T (Φ) be the set of all bijective and T -preserving point-shifts. The set Γ T can be equipped with a group structure by composition of point-shifts. This group, which is as subgroup of the symmetric group on the support of Φ, is called the T -stable group. An element H of this T -stable group is said T -dense if P-almost surely, for all x ∈ φ, the orbit of x under H spans the whole set of the partition that contains x; i.e.,

Point-Map Foliations
This section introduces two dynamics associated with a flow-adapted pointshift F = F φ (or equivalently to its associated point-map f ) and discuss the associated foliations.
1. For all fixed φ ∈ N, consider the map g = F φ , from the discrete space support(φ) to itself. The F φ -foliation of φ is a partition of the set support(φ). It will be denoted by L F φ φ . The set of connected components will be denoted by C F φ φ . Whenever the context allows it, the subscript φ is dropped, so that the latter set is denoted by C F and the former by L F .
definition of the θ f -foliation is nevertheless that of Definition 1 4 . The reason for this choice of definition is given in Corollary 17 below. The associated foliation (resp. set of connected components) is denoted by L Note that this partition of N 0 is very different in nature from that discussed for dynamics 1 above: each connected component of L θ f (and hence each foil or each component of the graph G θ f ) is still discrete, whereas N 0 is a noncountable set. So the number of connected components of this foliation must be non-countable.
Although L F and L θ f are defined on different spaces, they are closely related because of the following statement, which follows from the compatibility of the point-shift F .
Example 18. Consider the Next Row point-shift, R, on d-dimensional Bernoulli grid defined in Subsection 3.3.3. If p ∈ (0, 1), one can show that Thus each foil looks like a 1-dimensional Bernoulli grid and each connected component looks like a 2-dimensional Bernoulli grid ( Figure 1). If d = 2, the graph G R has a singe connected component.
Consider now a stationary point process (Φ, P), with Palm version denoted by (Φ, P Φ ). The expectation with respect to P (resp. P Φ ) is denoted by E (resp. E Φ ). The above dynamics lead to the following stochastic objects: 1. F Φ is a random map from the discrete random set support(Φ) to itself.
Both the component partition C F = C F Φ Φ and the foil partition L F = L F Φ Φ are flow-adapted partitions of this random set, with the latter being a refinement of the former. 2. L θ f = L θ f N 0 is a deterministic partition of the whole set N 0 (in contrast to the random partition of a random set described above). Note however that it is sufficient that θ f be defined P Φ -almost surely and hence it may be undefined for some elements of N 0 of null measure for P Φ .
Here are some observations on the flow-adapted partitions C F and L F . These two partitions do not depend on P at all (since they are defined on realizations). In particular, they do not depend on whether the point process is considered under P or P Φ .
The elements of each of these two partitions can be enumerated in a natural way following the method discussed just before Definition 13.
The dichotomy of Section 3.5 applies: there are cases where L F (resp. C F ) is a markable partition and cases where it is not 5 . A simple instance of the latter case is obtained when F is bijective; then the foil partition coincides with the singleton partition, which is not a markable partition.
The following solidarity properties hold: Proposition 19. If the foil partition L F is markable, so is the component partition C F . Conversely, if C is a component which is the support of a flow-adapted point process Ξ, then either the foil partition of Ξ, L F Ξ Ξ is markable or there is no flow-adapted point process with a positive intensity having for support a foil of L F Ξ Ξ . Proof. The first assertion is immediate. The proof of the converse leverages the foil order introduced in Subsection 2.2. First observe that Ξ has a single component. Assume that, for some foil L of Ξ, the point process Ψ(L) with support L is a flow-adapted point process. Then Ψ(L + ) = Ψ(F (L)) is a flow-adapted point process with non empty support (as L is non empty) and hence with positive intensity. Hence all foils that are senior to L are flow-adapted point processes with a positive intensity. Similarly, either L − is empty, and then there is no foil junior to L, or Ψ(L − ) = Ψ(F −1 (L)) is also a flow-adapted point process with a positive intensity. It then follows that the foil partition of Ξ is markable.
Remark 20. Under P Φ , the foil order leads to a natural enumeration of the foils of the component of the origin. The foil of the origin is numbered 0 and will be denoted by L F (0) = L 0 , the foil senior (resp. junior) to it will be numbered 1 and will be denoted by L 1 (resp. L −1 if non empty), and so on. Note that this enumeration is not flow-adapted.

Point-Map Cardinality Classification
In the rest of this work (Φ, P) is a stationary point process with Palm version (Φ, P Φ ).
The foliation L F partitions the support of the point process Φ into a discrete set of connected components; each component is in turn decomposed in a discrete set of F -foils, and each foil in a set of points. The present subsection proposes a classification of point-maps based on the cardinality of these sets.

Connected Components
The cardinality classification of connected components of the two dynamics differ.
The partition C F Φ Φ is countable. Its cardinality is a random variable with support on the positive integers and possibly infinite. If (Φ, P) is ergodic, this is a positive constant or ∞ almost surely.
As already mentioned, in contrast, the partition C θ f is deterministic and non-countable in general.

Inside Connected Components
In view of Corollary 17, the cardinality classification of the foils belonging to a given connected component is the same for L F Φ Φ and for L θ f . It is easy to see that the cardinality of the set of foils of a component is a random variable with support in N = N ∪ {∞}. The same holds true for the set of points of a non-empty foil. The following theorem shows that only a few combinations are however possible: Theorem 21 (Cardinality classification of a connected component). Let (Φ, P) be a stationary point process. Then P almost surely, each connected component C of G F (Φ) is in one of the three following classes: Class F/F: C is finite, and hence so is each of its F -foils. In this case, when denoting by 1 ≤ n = n(C) < ∞ the number of its foils: • C has a unique cycle of length n; • F ∞ (C) is the set of vertices of this cycle.
Class I/F: C is infinite and each of its F -foils is finite. In this case: • C is acyclic; • Each foil has a junior foil, i.e., a predecessor for the order of Definition 7; • F ∞ (C) is a unique bi-infinite path, i.e., a sequence of points (x n ) n∈Z of φ such that F φ (x n ) = x n+1 for all n.
Class I/I: C is infinite and all its F -foils are infinite. In this case: • C is acyclic; The following definitions will be used: In particular, for all n, d n (0) is P Φ -almost surely finite. If, in addition, G F (Φ) is P Φ -almost surely acyclic, then Proof. The map w(φ, x, y) := 1{F n φ (x) = y} is a flow-adapted mass transport (see [8]). The first statement is hence an immediate consequence of Proposition 48. For the second part, when G F is acyclic, the D n -s form a partition of D and hence Remark 25. Note that G F (Φ) is P Φ -almost surely acyclic if and only if G F (Φ) is P-almost surely acyclic. Proof. If F φ is surjective (resp. injective), then almost surely d 1 (0) ≥ 1 (resp. d 1 (0) ≤ 1). Since E Φ [d 1 (0)] = 1], almost surely d 1 (0) = 1, and hence the point-shift is bijective. Proof. According to Lemma 5 each connected component of G F (φ) has at most one cycle. If the latter is finite, it possesses exactly one cycle. This proves the first statement.
Let n = n(Φ) be the number of connected components of G F (Φ) which are infinite and possess a cycle. Let Ψ = {Ψ i } n i=1 denote the collection of such components. Note that n may be infinite. According to Lemma 5, each Ψ i has exactly one cycle. This cycle is a finite non-empty flow-adapted subset of Ψ i , which contradicts Theorem 10. Therefore almost surely, there is no such component.

The next corollary follows from Lemma 5.
Corollary 29. If G F (Φ) is almost surely connected, it is almost surely a tree.
Proof of Theorem 21. The result for connected components of Class F/F is an immediate consequence of Lemma 5.
Assume C is an infinite component. According to Proposition 28, C is acyclic. Consider the collection of all connected components with both finite and infinite foils. Denote this collection by Ψ = {Ψ i } n i=1 , where n may be infinity. Corollary 26 implies that each Ψ i should have a largest finite foil, say Ξ i , where the order is that based on seniority (Definition 7). Therefore, Ψ = {Ψ i } n i=1 and Ξ = {Ξ i } n i=1 satisfy the assumptions of Theorem 10 and this is hence a contradiction. So, almost surely, there is no connected component with both finite and infinite foils, which proves that each acyclic component is either of Class I/F or I/I. Let C be a connected component of Class I/F. Almost surely, C cannot have a smallest foil. Otherwise the latter would again be a finite flow-adapted subset of the infinite connected component C, which contradicts Theorem 10. This proves the second assertion on the foils of C in this case. Now let L 0 be an arbitrary foil of C and, for all integers i, let L i be the foil F i φ (L 0 ). Since L 0 is finite, there exists a least non-negative integer n such that F n (L 0 ) is a single point. Let C 0 := {F m φ (L 0 ), −∞ < m < n}, The graph G F (C 0 ) is infinite, connected and all its vertices are of finite degree. It hence follows from König's infinity lemma [7] that G F (C 0 ) has an infinite path {x i } i≤0 . For i > 0, define . Since all edges of G F (C) are from a foil L to the foil L + , (x i ) i∈Z has exactly one vertex in each foil. Finally for each point y in an arbitrary foil L(x i ), there exists m > 0 such that F m φ (y) = F m φ (x i ) = x i+m and hence F ∞ (C) ⊂ (x i ) i∈Z , which completes the proof of the properties of Class I/F. Consider now C of Class I/I and assume that F ∞ φ (C) is not empty. If x is a primeval element of C, then G F (D(x)) is an infinite connected graph with vertices of finite degree and hence it possesses an infinite path, which in turn gives a bi-infinite path using the same construction as what was described above for the Class I/F. Hence, in order to prove that F ∞ φ (C) is empty, it is sufficient to show that C has no bi-infinite path. If C has finitely many bi-infinite paths, then the intersections of these bi-infinite paths with each foil of C give a collection of finite subsets of infinite sets, which contradicts Theorem 10. Consider now the case where C has infinitely-many bi-infinite paths. Since C is connected, each two bi-infinite paths should intersect at some point. Let J = J(C) be the set of all points x ∈ C such that at least two bi-infinite paths join at x. It is now shown that, almost surely, the intersection of a bi-infinite path and J has neither a first nor a last point for the order induced by F . If it has a first (resp. last) point, then the part of the path before the first (resp. after the last) point is an infinite flow-adapted set with a finite flow-adapted subset, which contradicts Theorem 10. Therefore, for each point x ∈ J, there is a smallest positive integer n(J, x) such that F n(J,x) φ ∈ J. Now define a point-shift h on the whole point process as follows Since the intersection of any bi-infinite path with J does not have a first point, h is almost surely surjective. But from the very definition of J, all points of this set have at least two pre-images, which contradicts Corollary 27. Hence the situation with infinitely-many bi-infinite paths is not possible either, which concludes the proof.
In graph theoretic terms, one can summarize the results discussed in the last proof as follows:

Comments and Examples
Here are a few observations on the cardinality classification.
A point process (Φ, P) can have a mix of components of all three classes. If (Φ, P) only has F/F components, then it should have an infinite number of connected components. An example of this situation is provided by the Mutual Nearest Neighbor Point-Shift N on Poisson point process on R 2 (see below).
If (Φ, P) only has I/F components, then the cardinality of C F Φ Φ may be finite or infinite. An example of the first situation is provided by the Royal Line of Succession Point-Shift on Poisson point processes on R 2 (see below). An example of the latter is provided by the Strip Point-Shift S on the Bernoulli grid of dimension 2.
If (Φ, P) only has I/I components, then the cardinality of C F Φ Φ may again be finite or infinite. An example of the first situation is provided by the Strip Point-Shift S on Poisson point processes on R 2 . An example with infinite cardinality is provided in Subsection 8.3 of the appendix.
The end of this section gathers examples of the three classes.

Class F/F
For the Mutual Nearest Neighbor Point-Shift N on a Poisson point process, there is no evaporation and there is an infinite number of connected components, all of class F/F. The order of the foils is that of Z mod 2 or Z mod 1. The foil partition is the singleton partition, hence not markable. The connected component partition is not markable either.

Class I/F
An example of this class is provided by the strip point-shift on 2-dimensional Bernoulli grids. It is easy to see that the connected components of F s are the horizontal sub-processes which look like 1-dimensional Bernoulli grid. The point-shift F s is a bijection which leaves each of the connected components invariant. It follows that there is a countable collection of connected components. Each of them is of Class I/F, and hence not markable. The foil of a point is the singleton containing this point. The set of descendants of a point x consists of those points on the same horizontal line which are located on left side of x. Each connected component has two ends. Its foliation has the same order as Z but is not markable. Another example of this class is leverages the Strip point-shift on a stationary Poisson point process (Φ, P) of R 2 and R 3 . This point-shift has a single connected component [4]. The RLS ordering (see the proof of Proposition 35) hence defines a total order on (Φ, P), which is equivalent to that of Z. This allows one to define the RLS Point-Shift F rls which associates to x ∈ Φ the unique point y ∈ Φ such that x comes next to y in this total order. This point-shift is clearly bijective. Hence the foil of x is {x}. The unique connected component of this point-shift is thus of class I/F. The unique connected component has two ends. Its foliation has the same order as Z. It is not markable.
Note that these two examples are bijective point-shifts. But there are cases of type I/I which are not bijective.

Class I/I
Here are three examples illustrating that this class of point-shifts can have either markable or non markable foliations. grid with 0 < p < 1, has a single connected component of type I/I. Foils of this connected component are not markable.
Proof. There is a single connected component and each foil consists of all points of the point process on a vertical line (Figure 1). Therefore all foils are infinite and the connected component is of type I/I. If foils of this connected component were markable, the partition of the point process into horizontal lines would form a collection of infinite disjoint subsets of the Bernoulli grid. The intersection of these subsets with a some fixed foil is a finite non-empty inclusion, which contradicts Theorem 10.

Partition Preserving Point-Maps
It is well-known that all bijective point-shifts preserve the Palm probability of stationary point processes and that this property characterizes the Palm probabilities of stationary point processes [9].
This section features a fixed point-shift F and considers the class of bijective point-shifts which preserve the two partitions of a stationary point process Φ.

Bijective Point-Shifts Preserving Components
Let Γ F C := Γ C F (Φ) be the C F -stable group as defined in Definition 16. As mentioned in Definition 16, Γ F C is a subgroup of the symmetric group on the support of Φ.
Proposition 34. For each point-shift F and each stationary point process (Φ, P), there exists a C F -dense element (see Definition 16) of the C F -stable group; i.e., there exists H ∈ Γ F C such that for all x ∈ Φ, There is no uniqueness in general.
Proof. The construction of H is different for each of the three classes of components identified in Theorem 21. In each case, the first step is the construction of a total order on the points of C which is flow-adapted and the second one is the definition of a dense and bijective point-shift preserving C.
is of F/F class, then it is easy to create a total order which is translation invariant on the points of C as it is a finite set (e.g. using lexicographic order) with points that can be numbered 0, 1, . . . , n − 1 for some integer n = n(x) ≥ 1. A flow-adapted bijection H preserving C is then easy to build by taking H = M n with M n (k) = k + 1 mod n.
If C = C F Φ (x) is of I/F class, then, the existence of a single bi-infinite path in G F (C) (see Theorem 21) and the finiteness of the foils can be used to construct a total order. Let {x n } n∈Z be the bi-infinite path in question and let L n denote the foil of x n . Since L n is finite for all n, one can use the lexicographic order to create a total order between its points. The total order is then obtained by saying that all points of L n have precedence over those of L n−1 . This total order, which is that of Z, is flow-adapted. The bijective point-shift is that associating to a point x of C its direct successor for this order. This point-shift will be referred to as the Bi-Infinite Path Point-Shift B, with associated point-map b. On such a component, one takes H = B.
If C = C F Φ (x) is of I/I class, then the construction uses a total order on the nodes of G F (C) known as RLS (Royal Line of Succession). The latter order is based on two ingredients: 1. A local (total) order among the sons of a given node in G F (C). This can be done as follows: for a given point x of C let B F (x) = B F φ (x) be the set of its brothers; i.e., The elements of B F (x) can then be ordered in a flow-adapted manner using the lexicographic order of the Euclidean space.
2. The Depth First Search (DFS -see Appendix 8.2) pre-order on rooted trees.
The RLS order on a rooted tree is a total order on a finite tree obtained by combining (1) and (2): DFS is used throughout and the sons of any given node are visited in the order prescribed by (1), with priority given to the older son.
It is now explained how this also creates a total order on the nodes of C. For x, y ∈ C, there exist positive integers m and n such that F m φ (x) = F n φ (y). One says that x ≥ r y if x has RLS priority over y in the rooted tree of descendants of F m φ (x). This tree is a.s. finite because in the I/I case, there is evaporation of C by the point-shift, which this in turn implies that for all points z ∈ φ, the total number of descendents of z is a.s. finite. The DFS preordering on descendants of a node in a tree forms an interval of this preordering. This implies that ≥ r is a well-defined order on C and also that it orders elements of C in the same linear order as that of an interval in Z. Furthermore, since C is infinite, this order on C cannot have a greatest element or a least element. Otherwise the greatest and the least elements would be a finite flow-adapted subset of C or the foil, which contradicts Theorem 10. Therefore the order on C as well as its restriction to a foil is a linear order, similar to that of Z.
On such a components, one defined H = R where R denotes the RLS point-shift, namely the point-shift that associates to each point its successor in the RLS order, which is bijective and translation invariant.
Let h denote the point-map of the point-shift H defined in the last theorem. Notice that since H is bijective, the dynamical system (N 0 , θ h ) preserves P Φ .

Bijective Point-Shifts Preserving Foils
The results of this subsection parallel those of the last subsection, with an important refinement which is that of order preservation.
Let Γ F L := Γ F L (Φ) denote the set of all bijective and L F Φ -preserving pointshifts. This group, which is called the L F -stable group, is a subgroup of the C F -stable group.
As above, an element H of the L F -stable group is said to be Each L F Φ -dense element H of the L F -stable group induces a total order H on the elements of each infinite foil of C by This total order is flow-adapted. It is said to be preserved by F if The following proposition uses the fact (proved in Theorem 21) that in a connected component C of G F (Φ), either all foils of C have finite cardinality or all foils have infinite cardinality.
Proposition 35. For each stationary point process (Φ, P), and each pointshift F , there exists a L F Φ -dense element F ⊥ of the L F -stable group. In addition F ⊥ can be chosen such that the F ⊥ order is preserved by F on components with all its foils with infinite cardinality. There is no uniqueness in general.
Proof. If the connected component C of a realization φ has finite foils, the following construction can be used: F ⊥ (x) is the element coming next to x in the lexicographic order. This rule is applied to all elements of a foil except the greatest element for this order, whereas F ⊥ of the greatest element is the least element.
For a connected component C with all its foils with infinite cardinality, the construction uses the RLS total order on the nodes of G F (C).
One defines F ⊥ (x) as the next element in L(x), i.e., the greatest element of L(x) which is less than x, makes F ⊥ a bijection, and the orbit of each point x of L(x) is the foil L(x).
The property that F ⊥ is preserved by F follows from the fact that if x has priority over y for DFS, then the father of x also has priority over the father of y for DFS.
Let f ⊥ denote the point-map of the point-shift F ⊥ defined in the last theorem. For the same reasons as above, the dynamical system (N 0 , θ f ⊥ ) preserves P Φ .
In the next definition and below, in order to simplify notation, F ⊥ (defined in (6) It is easy to verify that for all x and y in the same foil,

Point Foils and Components
This subsection discusses some properties of the foil and the component of the origin, seen as point processes. For all countable sets S of points of R d without accumulation, let Ψ(S) denote the counting measure with support S.
Let L 0 (resp. C 0 ) denote the foil (resp. component) of the origin under P Φ . The counting measure Ψ(L 0 ) under P Φ (resp. Ψ(C 0 ) under P Φ ) will be called the point foil (resp. the point component) of Φ w.r.t. the point-shift F .
The terms point foil and point component are used to stress that these random counting measures are not always Palm versions of flow-adapted point processes. More precisely, let Q 0 denote the distribution of the point foil Ψ(L 0 ). If the foliation of C 0 is not markable, then Q 0 is not the Palm distribution of a flow-adapted point process (see Subsection 3.5). Similarly, if C 0 is not markable, then the distribution R 0 of Ψ(C 0 ) is not the Palm distribution of a stationary point process.
It follows from the above considerations that both in the markable and the non-markable cases, Q 0 (resp. R 0 ) is preserved by θ f ⊥ (resp. θ h ). This invariance property is of course classical in the markable case.
The fact that it holds in general can be phrased as follows: for all (nonnecessarily measure preserving) dynamics f on a stationary point process, there exists a dynamics f ⊥ on the typical leaf of the stable manifold of f , which is bijective, dense (has the whole leaf as orbit), and which preserves the law of the leaf.
For all x and y in φ, w + (x) = h(l n (x)) l n (x) , w − (y) = 1{d n (y) = 0}h(d n (y)), and therefore using Lemma 48, The announced quantitative results are given in the following corollaries of Proposition 37.
If in (8) h(x) is replaced by xh(x), one get: Corollary 38. For all n ≥ 0, Corollary 39. For all n ≥ 0, In addition Proof. The first result is obtained by putting h ≡ 1 in (8). Equation (11) is obtained when letting n → ∞ in Equation (10) and when using monotone convergence.

Equation (11) immediately proves:
Corollary 40. F evaporates (Φ, P) if and only if the F -foil of 0 is P Φ a.s. infinite 6 . This is consistent with the result of Corollary 23 since the property that the foil of 0 is infinite a.s. is equivalent to having all connected components of Class I/I, or equivalently to having (Φ F /F , P) and (Φ I/F , P) almost surely empty.
Corollary 42. If f does not evaporate (Φ, P), then and lim sup Proof. The first assertion follows from Equation (12). The second follows from Equation (12) and simple monotonicity arguments.
This subsection is concluded with a few observations. 2. If Φ I/I is empty, then each F -foil of Φ has an a.s. finite number of points (Theorem 21); the typical point has descendants of all orders with a positive probability 7 (Corollary 39), and hence an infinite number of descendants. However, the expected number of descendants of order n does not diverge in mean (Corollary 42) as n tends to infinity. If in addition Φ I/F is empty, then the set of descendants of the typical point looks like a "finite star with a loop at the center". See Subsection 3.3.2 for an example. If in place Φ F /F is empty, then the set of descendants of the typical point is either finite or looks like an "infinite path with finite trees attached to it". The points in this infinite path constitute a sub-stationary point process. This point process always has a positive intensity. (For instance, for the RLS point-shift on the Poisson point process in R 2 , this is the whole point process. There exist cases where F is not bijective and the connected components are of type I/F and hence such that this sub-point process is not the whole point process.

Foil Intensities
This subsection is focused on the intensity of the F -foils. From Proposition 19, either all foils of a markable component are flow-adapted point processes, or none of them are. The notion of intensity only makes sense in the former case. The notion of relative intensity defined in the next subsection allows one to discuss the "density" of foils in whole generality, namely regardless of the above dichotomy.

Relative Intensities
Below, when considering a component of Class I/I, it is assumed that F ⊥ is an L F Φ -dense element of the F -stable group and that F ⊥ is F -compatible. Let f ⊥ denote the point-map of F ⊥ and let θ f ⊥ denote its related shift on N 0 . Equations (4) and (7) give that for P Φ -almost all φ ∈ N 0 , Hence if φ ∼ θ f ψ, with abuse of notation, one can define ∆(φ, ψ) as the unique integer n such that θ n f ⊥ φ = ψ. Consider the dynamical system (N 0 , θ f ⊥ ). The fact that F ⊥ is bijective implies that θ f ⊥ preserves P Φ .
Theorem 43. Let (Φ, P) be a stationary point process, F be an arbitrary point-shift and F ⊥ and ∆ be as in Definition 36. Then, for P 0 almost-all realizations φ, the limit exists, is positive and in L 1 (P 0 ). In addition, λ + (φ) is a function of the foil of 0 only; i.e., if 0 ∼ F x, λ + (θ x φ) = λ + (φ) and λ + is independent of the choice F ⊥ as far as it satisfies the properties in Proposition 35.
Remark 44. The existence of the non-degenerate limit in (16) can be seen as a proof of the fact that all foils of a connected components have the "same dimension". This fact justifies the use of the term "foliation" within this context (see e.g. [1]).
Proof. If x is in a connected component of G F (φ) which is F/F or I/F, all statements follow from finiteness of the foils. Hence assume C(0) is I/I. Hence it is sufficient to show that, for P 0 almost all φ, the limit exists and is positive, finite and constant on the foil L θ f (φ) provided the latter is infinite. Let ∆ + (φ) := ∆(θ f φ, θ f • θ f ⊥ φ). Now consider the following mass transport: w(φ, x, y) = 1 y ∈ L F + (x) and 0≤∆(F φ (x), y)<∆(F φ (x), F φ • F ⊥ (x)) 0 otherwise.
One has, for all points x, y ∈ φ, w − (y) ≤ 1 and w + (x) = ∆ + (θ x φ). Therefore Since the denominator in (17) is equal to n, one has Since F ⊥ is P-almost surely bijective, P Φ is θ f ⊥ -invariant. Therefore if one denotes by I the invariant σ-field of θ f ⊥ , the finiteness of E Φ [∆ + ] implies that the last limit exists for P Φ -almost all φ and it is equal to E Φ [∆ + |I], which is finite and invariant under the action of θ f ⊥ ; i.e., it is a function of L θ f (φ). To prove that λ + (φ) is a.s. positive, note that, if Y is the event of being the youngest son of the family, then∆ ≥ 1 Y . Hence if, with positive probability, λ + (φ) = 0, this means that, with positive probability, E Φ [1 Y |I] is zero. But since E Φ is θ f ⊥ -invariant, this means that, with positive probability, there is no youngest son on the foil of L θ f (φ), which means that all points of L θ f (φ) are brothers. Since C θ f (φ) is infinite, this contradicts the a.s. finiteness of d 1 (F φ (0)).
Finally to prove that λ + (φ) is independent of the choice of F ⊥ , it is sufficient to show that E Φ [∆ + |I] depends only on f . To do so, it is enough to prove that for all A ∈ I, E Φ [∆ + 1 A ] depends only on f . Since A ∈ I, with abuse of notation, one has 1 A (φ) = 1 A (L θ f (φ)). Let It is easy to see that L which is a flow-adapted transport kernel. If A + denotes {θ f φ; φ ∈ A}, by the mass transport principle, where ∆ ⊥ (φ) is the smallest i > 0 such that F i ⊥ (x) has a child and zero otherwise. Note that all elements of A + have at least one child and therefore 1 A + (φ) is zero whenever φ has no child. Let v φ (x, y) = 1 θ x (φ) ∈ A + and y = f i ⊥ (x) for some 0 ≤ i < ∆ ⊥ (x), 0 otherwise.
Since A ∈ I, (18) and the mass transport principle give Clearly the latter depends only on f and not on the choice of F ⊥ which completes the proof. Definition 46. The quantity λ + (Φ), defined P Φ a.s., counts the average number of different points in the foil L F + (0) per point in the foil of 0, L F (0), and is hence called the relative intensity of L F + (0) with respect to L F (0) in Φ. This notion extends to the relative intensity Λ + (x, Φ) = λ + (θ x Φ) of L F + (x) with respect to L F (x) for all x ∈ Φ.

Multi Type Strip Point-Shift
Consider