Dynamics of conformal foliations

The purpose of this article is to review the author’s results on the existence and structure of minimal sets and attractors of conformal foliations. Results on strong transversal equivalence of conformal foliations are also presented. Connections with works of other authors are indicated. Examples of conformal foliations with exceptional, exotic and regular minimal sets which are attractors are constructed.


Introduction
The purpose of this article is to review the author's results on the existence and structure of minimal sets and attractors conformal foliations. The results are illustrated by various examples. Connections with works of other authors are indicated.
Remind that two Riemannian metrics h and g on a manifold M are called conformally equivalent if there exists a positive smooth function f on M with h = f g. A conformal equivalence class [g] of Riemannian metrics on M is called a conformal structure on M, and the pair (M, [g]) is said to be a conformal manifold.
A group of conformal transformations of a Riemannian manifold (M, g) is called inessential if it is a group of isometries of a Riemannian manifold (M, h) with some h ∈ [g]. Otherwise, it is called essential.
Lichnerowicz conjectured that for n ≥ 3 every n-dimensional compact Riemannian manifold admitting an essential group of conformal transformations is conformally equivalent to the standard n-dimensional sphere S n . The articles by M. Obata [22], D.V. Alekseevskii [1,2], J. Ferrand [13] and others are devoted to this conjecture.
It was also established that if a group of conformal transformations of a non-compact Riemannian manifold M is essential then M is conformally equivalent to the n-dimensional Euclidean space. In 1996 J. Ferrand [13] gave a complete proof of the Lichnerowicz conjecture, including the case of non-compact manifolds.
C. Tarquini [26] and then C. Tarquini and C. Frances [14] posed the following question about conformal foliations: Is every codimension q ≥ 3 conformal foliation on a compact manifold either a Riemannian foliation or a (Con f (S q ), S q )-foliation?
C. Frances and C. Tarquini called the question a foliated analog of the Lichnerowicz conjecture. For q ≥ 3 a conformal foliation is a (Con f (S q ), S q )-foliation if and only if it is transversally conformally flat. C. Frances and C. Tarquini [14] gave a positive answer to this question under some additional assumptions. We proved that in general the answer to the C. Frances and C. Tarquini question is positive (Theorem 7).
Our main results included in this article are as follows: 1) Different interpretations for the germ holonomy group of a leaf of a conformal foliation (Theorem 2).
2) A criterion for a conformal foliation to be Riemannian (Theorem 3).
3) The existence of an attractor which is a minimal set for every non-Riemannian conformal foliation of codimension q ≥ 3 (Theorem 4). 9) The existence of a two-dimensional suspension conformal foliation in every class of transversely equivalent non-Riemannian conformal foliations (Theorem 18).
In Section 3.5 we compare our results with some results of B. Deroin and V. Kleptsyn [11]. Section 7 contains examples of conformal foliations with global attractors. In particular, we construct a conformal foliation with two global attractors, one of which is a transitive attractor but not a minimal set (Example 22).
Notations In this article we denote by M a smooth manifold of dimension n and by C ∞ (M) the space of smooth functions on M. Let X(M) be the C ∞ (M)-module of smooth vector fields on M. Also, we denote by X c (M) the C ∞ (M)-module of compactly supported smooth vector fields on M.
2 Conformal foliations and associated constructions

Conformal foliations
First, we recall some basic notions. A diffeomorphism f : N 1 → N 2 of Riemannian manifolds (N 1 , g 1 ) and (N 2 , g 2 ) is called conformal if there exists a smooth function λ on N 1 with f * g 2 = λ g 1 . A conformal diffeomorphism f from a Riemannian manifold (N, g) to itself is also called a conformal transformation. A conformal transformation f of (N, g) is said to be a similarity, if f * g 2 = λ g for a constant λ .
Conformal foliations were studied by some authors as foliations admitting a transverse conformal structure.
• a possibly disconnected Riemannian manifold (N, g); • an open cover {U i | i ∈ J} of M; j∈J , satisfying the above properties, determines a new topology on M, whose base is the set of fibers of all submersions f i . This topology is called the leaf topology and denoted by τ.
Path-connected components of the topological space It is also said that (M, F) is modeled on the conformal geometry (N, [g]).
Definition 3. If each γ i j is a local similar transformation of (N, g), then (M, F) is called a transversely similar Riemannian foliation. Definition 4. If each γ i j is an isometry, then (M, F) is called a Riemannian foliation.
Let Con f (S q ) be the Lie group of all conformal transformations of the q-dimensional sphere S q .
where N = S q and each γ i j is the restriction of a transformation f ∈ Con f (S q ), then (M, F) is referred to as a (Con f (S q ), S q )-foliation.
A conformal transformation of the complex plane is conformal in the above sense. Therefore, every transversely complex analytic foliation of complex codimension one is a conformal foliation of real codimension two in the sense of Definition 1. The converse is also true, i.e., every conformal foliation of codimension two can be considered as a transversely complex analytic foliation of complex codimension one.
Conformal foliations of codimension two have been studied by several authors, using methods of complex analysis. For example, in [6] P. Baird and M. Eastwood gave a detailed twistor description of conformal foliations of codimension two in the 3-dimensional Euclidean space. Holonomy diffeomorphisms of these foliations preserve the conformal equivalence class of Riemannian metrics on orthogonal complements to their leaves.
Since every 2-dimensional manifold is locally conformally flat, every conformal foliation of codimension two is transversely conformally flat. A transversal conformal manifold (N, [g]) is locally conformally flat if and only if, in codimension q > 3, its Weyl conformal curvature tensor vanishes or, in codimension q = 3, its Schouten tensor vanishes.
Like S. Morita [21], in contrast to the case of conformal foliations of codimension two, we study conformal foliations of codimension q, q ≥ 3, by differential geometric methods, considering them as Cartan foliations.

The foliated bundle over a conformal foliation
The construction of the foliated bundle is essentially used by us. Foliated bundles were introduced in works of P. Molino, F. Kamber and Ph. Tondeur.
Recall that a foliation admitting an effective Cartan geometry of type (G, H) [10] as a transverse structure is said to be a Cartan foliation of the same type. Remark that any conformal foliation may be considered as a Cartan foliation of type (G, H), where G = Con f (S q ) is the full group of conformal transformations of S q and H is the stabilizer of G at an arbitrary point b ∈ S q . Let g and h be the Lie algebras of the Lie groups G and H, respectively. Then we have the following statement [27]. (ii) R * a ω = Ad G (a −1 )ω for any a ∈ H, where Ad G is the adjoint representation of the Lie group G in its Lie algebra g; (iii) the Lie derivative L X ω vanishes for every vector field X tangent to the leaves of (R, F ).
The H-bundle π : R → M is said to be foliated. The foliation (R, F ) is called the lifted foliation, and (R, F ) is a transversely parallelizable foliation, i.e., an e-foliation.

Holonomy groups of a conformal foliation
We consider conformal foliations as Cartan foliations and use the construction of the principal foliated bundle. Due to this, we can apply the results of our previous work [27].
In foliation theory, the germ holonomy groups are usually used. By analogy with [27], we give the following interpretations for the germ holonomy group of a leaf of a conformal foliation.
As above, let G = Con f (S q ) be the Lie group of conformal transformations of the sphere S q and let H be the stabilizer of G at an arbitrary point in S q . Theorem 2. Let (M, F) be a conformal foliation of codimension q ≥ 3, R be the foliated H-bundle over (M, F) with the projection π : R → M and (R, F ) be the lifted foliation. Then, for every leaf L = L(x), x ∈ M, of (M, F) the restriction π| L : L → L to the leaf L = L (u), u ∈ π −1 (x), of the lifted foliation (R, F ) is a regular covering map, and the germ holonomy group Γ(L, x) of L is isomorphic to each of the following groups:  Existence of minimal sets and description of their structure are one of the central problems in the foliation theory. A. H. Aranson, V. Z. Grines [4,5] and J. Levitt [19] obtained a topological classification of nontrivial minimal sets of flows and foliations on closed surfaces of genus p ≥ 2. Minimal sets of Riemannian foliations were studied by P. Molino [20], A. Haefliger [15] and E. Salem [23] without using the term minimal set.
Every foliation on a compact manifold has a minimal set. This is wrong for non-compact manifolds. Foliations without minimal sets (on non-compact manifolds) are constructed in the works of J.-C. Beniere and G. Meigniez [7], T. Inaba [16] and M. Kulikov [18]. Moreover, J.-C. Beniere and G. Meigniez proved the existence of foliations without minimal sets on every non-compact manifold of dimension n ≥ 2, different from surfaces of finite kind [7].
A minimal set M is called regular, if it is a submanifold of M. Let {U i |i ∈ N} be an at most countable locally finite open cover of M by foliated coordinate neighborhoods. Let T i be a submanifold in U i , which intersects transversely each leaf of the induced foliation (U i , F U i ), and let T = i∈N T i be the complete transversal. A minimal set M , having no interior points, is called: In Section 7 we construct examples of conformal foliations with exotic, exceptional, and regular minimal sets.

A criterion for a conformal foliation to be Riemannian
Definition 6. The holonomy group of a leaf L of a conformal foliation is said to be relatively compact or inessential if the corresponding subgroup H(L ) of the Lie group H is relatively compact.
Since H(L ) is defined up to conjugacy in H, then this definition is correct. We established the following criterion for a conformal foliation to be Riemannian [29,Theorem 3]. Remark 1. According to [31,Theorem 3], Theorem 3 holds for transversely similar Riemannian foliations of codimension q ≥ 2. Note that it is true also for q = 1.

The existence of attractors of conformal foliations
Let (M, F) be a foliation. In the case when there exists a dense leaf in M , the attractor M is referred to be transitive.
Using the criterion for conformal foliation to be Riemannian (Theorem 3) and some results from local conformal geometry, we prove the following theorem on two alternative possibilities for conformal foliations. • either it is Riemannian, • or it has an attractor M which is the closure of a leaf L with essential holonomy group, M =L, and a minimal set. Moreover, the restriction (A ttr(M ), F) of the foliation to the attraction basin is a (Con f (S q ), S q )-foliation.
Recall that a foliation (M, F) is called proper if all its leaves are embedded submanifolds of M. A leaf L is called closed if L is a closed subset of M.
Corollary 5. Every proper non-Riemannian conformal foliation of codimension q ≥ 3 has a closed leaf with essential holonomy group that is an attractor. Corollary 6. A non-Riemannian conformal foliation (M, F) of codimension q ≥ 3 has a minimal set that is an attractor of this foliation.
Remark 2. Theorem 4 is proved without assumption that the foliated manifold M is compact.

An analog of the Lichnerowicz conjecture for conformal foliations
The following theorem [28,Theorem 4] gives a positive answer to the C. Frances and C. Tarquini question stated in Introduction in the general case. • or a (Con f (S q ), S q )-foliation with finitely many minimal sets. All these sets are attractors given by the closures of leaves with essential holonomy group, and each leaf of F belongs to the basin of at least one of these attractors.
For the proof of Theorem 7, we applied the results of D.V. Alekseevskii [1] and J. Ferrand [12] concerning with local differential geometry as well as Riemannian geometry on non-Hausdorff manifolds.
Let us emphasize that transversely similar foliations form a subclass of conformal foliations. Since any Riemannian manifold of dimension 1 or 2 is locally conformally flat, every transversely similar Riemannian foliation of codimension q = 1 or 2 is a (Sim(E q ), E q )-foliation. By using similar arguments as in the proof of [28,Theorem 4], we prove the following statement. • or a (Sim(E q ), E q )-foliation with finitely many minimal sets. They are all attractors formed by the closures of leaves with essential holonomy group, and each leaf of the foliation belongs to the basin of at least one of them.

Comparison with the results B. Deroin and V. Kleptsyn [11]
Theorem 7 strengthens (in an appropriate smoothness class and for q ≥ 3) the first part of the main theorem of B. Deroin and V. Kleptsyn [11], which claims that every conformal foliation of arbitrary codimension q ≥ 1 on a compact manifold either admits a transversal invariant measure or has finitely many minimal sets satisfying some properties. As emphasized in the introduction of [11], E. Ghys conjectured that in the absence of attractors of the conformal foliation, the invariant transverse measure is given by a Riemannian metric. It follows from our Theorem 7 that this conjecture of E. Ghys holds for conformal foliations of codimension q ≥ 3.
It is known (see, for example [1]) that there are nontrivial second order conformal diffeomorphisms f of the sphere S q , q ≥ 2, such that the differential f * x at a fixed point x is an orthogonal transformation of the tangent Euclidean vector space T x S q at x. These conformal transformations are essential. The 1-dimensional conformal foliation (M, F) formed by the suspension of such a conformal diffeomorphism f does not admit a transversely invariant measure and does not have hyperbolic holonomy; therefore, (M, F) does not satisfy [11,Cor. 1.3]. Thus, unlike us, in [11] the authors do not consider conformal foliations with such holonomies.
Note that in [28,29] we use the methods of local and global differential geometry, including foliated principal bundles, geometries on non-Hausdorff manifolds as well as Ehresmann connections for foliations. While B. Deroin and V. Kleptsyn apply methods of random dynamical systems, including Lyapunov exponents of harmonic measures. In [11], a Laplace operator along leaves is considered and the existence of a transverse invariant measure is investigated.
It is said that the vector fields ξ 1 , ..., ξ k generate the restriction M| U of M to U.  Let (M, F) be a conformal foliation of codimension q ≥ 3 on a smooth n-dimensional manifold M. We keep notation introduced in Section 2.2. Denote by R the foliated bundle over (M, F) with the projection π : R → M and by ω the g-valued 1-form on R. Let (R, F ) be the lifted e-foliation. Consider a q-dimensional distribution M on M. Denote by π * M the distribution on R generated by vector fields Y ∈ X(R) for which π * Y ∈ M.
Definition 9. A conformal foliation (M, F) of codimension q ≥ 3 is referred to as complete, if there exists a smooth transverse q-dimensional distribution M on M such that every vector field Y ∈ π * M, for which ω(Y ) = const ∈ g, is complete.

Ehresmann connection for foliations
The notion of an Ehresmann connection for foliations was introduced by R.A. Blumenthal and J.J. Hebda in [8]. R.A. Blumenthal and J.J. Hebda [8] considered a smooth distribution on a smooth manifold M as a smooth subbundle of the tangent bundle T M. We clarify the notion of Ehresmann connection for foliations due to the new definition of a distribution in sense I. Androulidakis and G. Skandalis which we have accepted.
Consider a smooth foliation (M, F) of codimension q, q ≥ 1. Let M be a smooth q-dimansional distribution in sense I. Androulidakis and G. Skandalis, and M is transverse to (M, F).
All maps considered here are assumed to be piecewise smooth. The integral curves of the foliation (M, F) are called vertical; the integral curves of the distribution M are called horizontal. Emphasize that a vertical curve lies in some leaf of (M, F).
A map H : I 1 × I 2 → M, where I 1 = I 2 = [0, 1], is called a vertical-horizontal homotopy if for each fixed t ∈ I 2 , the curve H |I 1 ×{t} is horizontal, and for each fixed s ∈ I 1 , the curve H |{s}×I 2 is vertical. The pair of curves (H |I 1 ×{0} , H |{0}×I 2 ) is called the base of H.

Minimal sets of Riemannian foliations
The following theorem was proved by us in [29, Theorem 1], see also [32]. For compact manifolds this theorem was earlier proved by P. Molino [20]. In the case when a bundle-like Riemannian metric on M with respect to (M, F) is complete, it was proved by E. Salem [23] without using the term "minimal set". In particular, if (M, F) is a proper foliation, then all its leaves are closed, and the leaf space is a smooth q-dimensional orbifold.

Global attractors of a countable subgroup of Con f (S q )
Let Ψ be a homeomorphism group of a topological space B. A nonempty closed invariant subset K ⊂ B is called an attractor of Ψ if there exists an open invariant subset W ⊂ B such that the orbit closure Cl(Ψ.z) of any point z ∈ W \ K contains K . An attractor K of Ψ is global if W = B.
We recall that a minimal set of a homeomorphism group Ψ of a topological space B is a nonempty closed invariant subset K with respect to Ψ containing no proper subsets with this property. A finite minimal set is called trivial.
Let K be a minimal set of a diffeomorphism group Ψ of a manifold B. The minimal set K with empty interior is called exceptional if K is a Cantor set, and exotic if K is not a totally disconnected topological subspace of B.
It is well known that the conformal group Con f (S q ), q ≥ 2, is isomorphic to the Mobius group Mob q (R) [24]. The limit set Λ(Ψ) of an arbitrary subgroup Ψ of Con f (S q ) coincides with the intersection of the closures of all non-one-point orbits of this group, that is, Λ(Ψ) = Cl(Ψ.z), where z ∈ S q and Ψ.z = z. If the limit set Λ(Ψ) is finite, then either it is empty or it consists of one or two points. In this case, the group Ψ is called elementary.
Using some facts about subgroups of Con f (S q ) [17], we get the following statement.
Proposition 11. Let Ψ be a countable essential subgroup of Con f (S q ). Then Ψ has a global attractor which coincides with the limit set Λ(Ψ) of Ψ.

Global attractors of complete conformal foliations
The following theorem is the main result of our works [28] and [29]. Emphasize that the Ehresmann connection for (M, F) is one of the main technical tools used in the proof.

The global holonomy group
The following theorem [29,Theorem 5] describes the global structure of the foliated manifold and establishes a relationship between the attractors of conformal and transversely similar foliations and the attractors of their global holonomy groups. The following statement is proved in a constructive way for q ≥ 3 [29,Theorem 7]. For q = 2, the proof is similar.
Theorem 14. Every countable subgroup Ψ of Con f (S q ), where q ≥ 2, can be realized as the global holonomy group of some two-dimensional conformal foliation (M, F) of codimension q.
If Ψ is finitely generated, then such a foliation (M, F) exists on a closed manifold M of dimension (q + 2).
6 Strong transverse equivalence of foliations

Invariants with respect to strong transverse equivalence of conformal foliations
The notion of strong transverse equivalence of foliations was introduced and investigated by us in [30] under the name "transverse equivalence".
A continuous map p : X → Y has the covering homotopy property with respect to a topological space K if, for any continuous map G 0 : K → X and any homotopy H t : K → Y , t ∈ [0, 1], such that p • G 0 = H 0 , there exists an extension of G 0 to a homotopy G t : K → X satisfying the equality p • G t = H t , t ∈ [0, 1].
We recall that a Serre fibration is a continuous surjective map having the covering homotopy property with respect to any finite polyhedron. It is known that for Serre fibrations it is possible to construct the exact homotopy sequence for a fibration. It is also well known that any locally trivial fibration is a Serre fibration. If a Serre fibration is a submersion, then it is called a smooth Serre fibration. Using this additional requirement, we proved that the strong transverse equivalence of foliations covered by fibrations can be realized by a foliation covered by a fibration. This fact was used in the proofs of the following theorems in [30]. As we have shown, in general, this is not true for transverse equivalent foliations in the sense of P. Molino.
Theorem 15. Two complete conformal, but not transversally similar foliations (M 1 , F 1 ) and (M 2 , F 2 ) of codimension q ≥ 3 are strong transversely equivalent if and only if their global holonomy groups Ψ 1 and Ψ 2 coincide (up to conjugacy in Con f (S q )).
Theorem 16. Two complete non-Riemannian transversally similar foliations (M 1 , F 1 ) and (M 2 , F 2 ) of codimension q ≥ 1 are strong transversely equivalent if and only if their global holonomy groups Ψ 1 and Ψ 2 coincide (up to conjugacy in Sim(E q )).
The structure Lie algebra g 0 of a complete non-Riemannian conformal foliation (M, F) of codimension q ≥ 3 is isomorphic to the Lie algebra of the Lie group Ψ, where Ψ is the closure of the global holonomy group Ψ of this foliation in the Lie group Con f (S q ). If (M, F) is a transversally similar Riemannian foliation of codimension q ≥ 1, then g 0 is the Lie algebra of the closure Ψ of Ψ in the Lie group Sim(E q ). In particular, the structure Lie algebra g 0 is equal to zero if and only if Ψ is a discrete subgroup.
From Theorems 15 and 16 we obtain the following assertion.
Proposition 17. The Lie group Ψ, the structure Lie algebra g 0 and the limit set Λ(Ψ) of the global holonomy group Ψ of a complete non-Riemannian conformal foliation (M, F) are invariants with respect to the strong transverse equivalence.

Suspension foliations
A. Haefliger introduced the construction of a suspension foliation. Let B and T be smooth connected manifolds, and ρ : π 1 (B, b) → Di f f (T ) be a group homomorphism. Let G := π 1 (B, b) and Ψ := ρ(G). Consider a universal covering map p : B → B. Define a right action of the group G on the product B × T as where B → B : x → xg is the deck transformation of the covering p induced by an element g ∈ G, which acts on B on the right. The map p : M := ( B × T )/G → B = B/G is a locally trivial bundle over B with standard fiber T . It is associated with the principal bundle p : B → B with the structural group G. Let Θ g := Θ| B×t×g . Since Using suspension foliations, we proved the following theorem. Note that the same method is used in the construction of Example 21.
Theorem 18. Every strong transverse equivalence class of a non-Riemannian conformal foliation contains a two-dimensional conformal foliation, which is a suspension foliation.

Examples
Let B k be a smooth closed 3-manifold, which is homeomorphic to the connected sum k i=1 S 1 × S 2 of k copies of S 1 × S 2 . Its fundamental group π 1 (B k , b) = < g 1 , . . . , g k > is the free group of rank k.
Bourdon proved [9] that if sin π/k < 1/ √ m, then there is an exact representation α km : Γ km → Con f (S 2m−2 ) such that the group Ψ km = α km (Γ km ) is a Fuchsian group (i.e. a finitely generated discrete subgroup of Con f (S 2m−2 )), and the limit set Λ(Ψ km ) of Ψ km is homeomorphic to the Menger curve. Thus ψ i := α(s i ) are generators of Ψ km .
. . , B − k be a finite set of disjoint closed topological balls with smooth boundary in the sphere S q , and q = 4. Let ψ i ∈ Con f (S q ) be a conformal transformation such that . The group Ψ generated by ψ 1 , . . . , ψ k is called the Schottky group. As well-known (see, for example, [17]), the Schottky group Ψ is the free group of rank k, i.e. Ψ =< ψ 1 , . . . , ψ k >, and Ψ has a minimal set Λ(Ψ) homeomorphic to a Cantor subset of the segment [0, 1]. Therefore, the topological dimension Λ(Ψ) is equal to zero.
Let B k := k i=1 S 1 × S 2 be as above. Define a group isomorphism ρ k : π 1 (B k , b) → Ψ, setting ρ k (g i ) = ψ i , i = 1, k. The suspension foliation (M k , F k ) := Sus(S q , B k , ρ k ) is a complete conformal foliation and, according to Theorems 13, (M k , F k ) has a global attractor, representing an exceptional minimal set.
If f k : M k → M k is the universal covering map, then the induced foliation F k = f * k F k is given by the fibers of a locally trivial bundle r k : M k → S q . As (M k , F k ) is a suspended foliation, (M k , F k ) admits an Ehresmann connection M k .
Let M 0 k := M k \ M . Therefore, M 0 k = f k (r −1 k (S q \Λ(Ψ))). Since the group Ψ acts freely and properly dis- Note that the closure L α in M of a leaf L α ⊂ M 0 k is equal to L α ∪ M , what agrees with Theorem 12.
Example 21. Let E q be the q-dimensional Euclidean space and E k , 2 ≤ k < q, be its k-dimensional subspace. Let e i , i = 1, k, be a basis of vector space R k corresponding to E k . Consider a subgroup Ψ of the group of similarities Sim(E q ) generated by transformations ψ j , j = 1, k + 1, where ψ j (z) := z + e j for j = 1, k and ψ k+1 (z) := λ · z for some number λ ∈ (0, 1) and all z ∈ E q . Note that Ψ is essential group of similar transformations, and the limit set Λ(Ψ) of Ψ is equal to E k . Moreover, Λ(Ψ) is a minimal set and a global attractor of Ψ. = Id E q where Id E q is the identical map of E q . Then (M, F):=Sus(E q , S 2 m , ρ) is a suspension two-dimensional non-Riemannian transversely similar foliation. Let f : R 2 × E q → M be the universal covering map. Then (M, F) is covered by the trivial bundle r : R 2 × E q → E q and has the global holonomy group Ψ. By properties of suspension foliations, the (q + 2)-dimensional manifold M is the space of a locally trivial bundle p : M → S 2 m with a standard fibre E q over the base S 2 m . Therefore M is not compact. Thus, we get a transversely similar foliation (M, F) of codimension q, q ≥ 3, with a regular minimal set M := f (r −1 (E k )), and M is a global attractor and a minimal set of (M, F). According to [27,Theorem 9], M and M are homotopy equivalent.
Let M 0 := M \ M and L α be an arbitrary leaf in M 0 . Emphasize that the induced foliation (M 0 , F M 0 ) is an improper Riemannian foliation without holonomy, admitting an Ehresmann connection. According to Theorem 12, the closure L α is equal to L α ∪ M , where L α is the closure of L α in M 0 . Note that if k < q − 1, then M 0 is connected and dense in M. In the case when k = q − 1, the submanifold M 0 is dense and has two connected components denoted by M + 0 and M − 0 , hence L α belongs to M + 0 or M − 0 , respectively. We emphasize that all L α are pairwise diffeomorphic, that is, L α does not depend on α. Therefore, the closure L α of any leaf L α ⊂ M 0 is also independent of α.
Example 22. Fix a point a ∈ S q . Let f : S q → R q ∪ {∞} be the homeomorphism defined via the stereographic projection S q \ {a} → R q , and f (a) = {∞}. Identify S q with R q ∪ {∞} through f . The map f induces a group isomorphism f : Con f a (S q ) → Sim(E q ) of the stationary subgroup Con f a (S q ) of the group Con f (S q ) at point a onto the similarity group Sim(E q ).
Let B := S 2 m be as above, T := S q and define the group homomorphism ρ : π 1 (B, b) → Con f (S q ) by setting it on generators: ρ(a i ) := ψ i , i = 1, k + 1, ρ(b i ) := Id S q . Then we get the suspension foliation (M, F):=Sus(S q , S 2 m , ρ) which is non-Riemannian conformal foliation of codimension q, and Ψ is its global holonomy group.
The universal covering manifold for M is the product R 2 × S q . Denote by h : R 2 × S q → M the universal covering map. Let r : R 2 × S q → S q be the canonical projection.
Since the global holonomy group Ψ of (M, F) has a fixed point a, the leaf L := h(r −1 (a)) is a unique closed leaf and a global regular attractor of (M, F). Emphasize that L is diffeomorphic to the base B := S 2 m . According to the definition of a limit set, Λ( Ψ) = f −1 (R k ∪ {∞}). Therefore, Λ( Ψ) is the canonically embedded k-dimensional sphere in S q containing the point a. As it was shown in Example 21, Λ(Ψ) is the global attractor of the respective transversely similar foliation, hence Λ( Ψ) is a global attractor of the conformal foliation (M, F). Since Λ( Ψ) contains a fixed point a, Λ( Ψ) is not a minimal set. We emphasize that for every point z ∈ Λ( Ψ) \ {a} the orbit Ψ.z is dense in Λ( Ψ). This implies that is a global transitive regular attractor of the conformal foliation (M, F).
Thus we construct a conformal foliation having two global regular attractors L and M , L ⊂ M , such that L is a closed leaf and M is transitive attractor which is not a minimal set.