Dynamics of foliations
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Contents |
1 Introduction
Foliations of Riemannian manifolds can be considered as multidimensional dynamical systems: Riemannian structure (more precisely, Riemannian structure along the leaves) plays the role of time, the leaves play the role of orbits [Walczak2004]. The dynamics of a foliation can be described in terms of its holonomy psudogroup (see Foliations#Holonomy).
2 Pseudogroups
The notion of a pseudogroup generalizes that of a group of transformations. Given a space , any group of transformations of consists of maps defined globally on , mapping bijectively onto itself and such that the composition of any two of them as well as the inverse of any of them belongs to the group. The same holds for a pseudogroup with this difference that the maps are not defined globally but on open subsets, so the domain of the composition is usually smaller than those of the maps being composed.
To make the above precise, let us take a topological space and denote by Homeo the family of all homeomorphisms between open subsets of . If Homeo, then is its domain and .
Definition 2.1. A subfamily of Homeo is said to be a pseudogroup if it is closed under composition, inversion, restriction to open subdomains and unions. More precisely, should satisfy the following conditions:
(i) whenever and ,
(ii) whenever ,
(iii) whenever and is open,
(iv) if Homeo, is an open cover of and for any , then .
Moreover, we shall always assume that
(v] id (or, equivalently, ).
As examples, one may have (1) all the homeomorphisms between open sets of a topological space, (2) all the diffeomorhisms of a given class (say, , , between open sets of a manifold, (3) all the restrictions to open domains of elements of a given group of homeomorphisms of a given topological space. (In the last case, we say that the pseudogroup is generated by .) Any set of homeomorphisms bewteen open sets (with domains covering a space ) generates ma pseudogroup which is the smallest pseudogroup containing ; precisely a homeomorphism belongs to if and only if for any point there exist elements , exponents and a neighbourhood of such that on . If is finite, is said to be finitely generated.
3 Holonomy pesudogroups
Definition 3.1. A foliated atlas on a foliated manifold is said to be nice (also, nice is the covering of by the domains of the charts ) if
(i) the covering is locally finite,
(ii) for any , is an open cube,
(iii) if and , and , then there exists a foliated chart chart and such that is an open cube, contains the closure of and .
Since manifolds are supposed to be paracompact here, they are separable and hence nice coverings are denumerable. Nice coverings on compact manifolds are finite. On arbitrary foliated manfiolds, nice coverings do exist.
Given a nice covering of a foliated manifold . For any , let be the space of the plaques (i.e., connected components of intersections . being a leaf of ) of contained in . Equip with the quotient topology: two points of are equivalent iff they belong to the same plaque. is homeomorphic (-diffeomorphic when is -differentiable and ) to an open cube () via the map , where is a foliated chart on . The disjoint union
is called a complete transversal for . Transversality refers to the fact that, if is differentiable of class , , each of the spaces can be mapped homeomorphically onto a -submanifold transverse to : if and is the leaf of passing through , then
Completeness of means that every leaf of intersects at least one of the submanifolds .
Definition 3.2. Given a nice covering of a foliated manifold and two sets and such that the holonomy map , being the open subset of which consists of all the plaques of for which , is defined in the following way:
All the maps () generate a pseudogroup on . is called the holonomy pseudogroup of .
Certainly, holonomy pseudogroups of arbitrary compact foliated manifolds are finitely generated. For example, gluing together two 2-dimensional Reeb foliations (see Foliations#Reeb Foliations) on one gets a foliation of the 3-dimensional sphere for which any arc intersecting the unique toral leaf is a complete transversal; can be identified with a segment (), the point of intersection with the number , while the holonomy group with the one on our segment generated by two maps such that , , snd contract their domains towards .
4 Growth
5 Geometric entropy
6 Invariant measures
7 Results on entropy
8 References
- [Walczak2004] P. Walczak, Dynamics of foliations, groups and pseudogroups, Birkhäuser Verlag, 2004. MR2056374 (2005d:57042) Zbl 1084.37022
To make the above precise, let us take a topological space and denote by Homeo the family of all homeomorphisms between open subsets of . If Homeo, then is its domain and .
Definition 2.1. A subfamily of Homeo is said to be a pseudogroup if it is closed under composition, inversion, restriction to open subdomains and unions. More precisely, should satisfy the following conditions:
(i) whenever and ,
(ii) whenever ,
(iii) whenever and is open,
(iv) if Homeo, is an open cover of and for any , then .
Moreover, we shall always assume that
(v] id (or, equivalently, ).
As examples, one may have (1) all the homeomorphisms between open sets of a topological space, (2) all the diffeomorhisms of a given class (say, , , between open sets of a manifold, (3) all the restrictions to open domains of elements of a given group of homeomorphisms of a given topological space. (In the last case, we say that the pseudogroup is generated by .) Any set of homeomorphisms bewteen open sets (with domains covering a space ) generates ma pseudogroup which is the smallest pseudogroup containing ; precisely a homeomorphism belongs to if and only if for any point there exist elements , exponents and a neighbourhood of such that on . If is finite, is said to be finitely generated.
3 Holonomy pesudogroups
Definition 3.1. A foliated atlas on a foliated manifold is said to be nice (also, nice is the covering of by the domains of the charts ) if
(i) the covering is locally finite,
(ii) for any , is an open cube,
(iii) if and , and , then there exists a foliated chart chart and such that is an open cube, contains the closure of and .
Since manifolds are supposed to be paracompact here, they are separable and hence nice coverings are denumerable. Nice coverings on compact manifolds are finite. On arbitrary foliated manfiolds, nice coverings do exist.
Given a nice covering of a foliated manifold . For any , let be the space of the plaques (i.e., connected components of intersections . being a leaf of ) of contained in . Equip with the quotient topology: two points of are equivalent iff they belong to the same plaque. is homeomorphic (-diffeomorphic when is -differentiable and ) to an open cube () via the map , where is a foliated chart on . The disjoint union
is called a complete transversal for . Transversality refers to the fact that, if is differentiable of class , , each of the spaces can be mapped homeomorphically onto a -submanifold transverse to : if and is the leaf of passing through , then
Completeness of means that every leaf of intersects at least one of the submanifolds .
Definition 3.2. Given a nice covering of a foliated manifold and two sets and such that the holonomy map , being the open subset of which consists of all the plaques of for which , is defined in the following way:
All the maps () generate a pseudogroup on . is called the holonomy pseudogroup of .
Certainly, holonomy pseudogroups of arbitrary compact foliated manifolds are finitely generated. For example, gluing together two 2-dimensional Reeb foliations (see Foliations#Reeb Foliations) on one gets a foliation of the 3-dimensional sphere for which any arc intersecting the unique toral leaf is a complete transversal; can be identified with a segment (), the point of intersection with the number , while the holonomy group with the one on our segment generated by two maps such that , , snd contract their domains towards .
4 Growth
5 Geometric entropy
6 Invariant measures
7 Results on entropy
8 References
- [Walczak2004] P. Walczak, Dynamics of foliations, groups and pseudogroups, Birkhäuser Verlag, 2004. MR2056374 (2005d:57042) Zbl 1084.37022
To make the above precise, let us take a topological space and denote by Homeo the family of all homeomorphisms between open subsets of . If Homeo, then is its domain and .
Definition 2.1. A subfamily of Homeo is said to be a pseudogroup if it is closed under composition, inversion, restriction to open subdomains and unions. More precisely, should satisfy the following conditions:
(i) whenever and ,
(ii) whenever ,
(iii) whenever and is open,
(iv) if Homeo, is an open cover of and for any , then .
Moreover, we shall always assume that
(v] id (or, equivalently, ).
As examples, one may have (1) all the homeomorphisms between open sets of a topological space, (2) all the diffeomorhisms of a given class (say, , , between open sets of a manifold, (3) all the restrictions to open domains of elements of a given group of homeomorphisms of a given topological space. (In the last case, we say that the pseudogroup is generated by .) Any set of homeomorphisms bewteen open sets (with domains covering a space ) generates ma pseudogroup which is the smallest pseudogroup containing ; precisely a homeomorphism belongs to if and only if for any point there exist elements , exponents and a neighbourhood of such that on . If is finite, is said to be finitely generated.
3 Holonomy pesudogroups
Definition 3.1. A foliated atlas on a foliated manifold is said to be nice (also, nice is the covering of by the domains of the charts ) if
(i) the covering is locally finite,
(ii) for any , is an open cube,
(iii) if and , and , then there exists a foliated chart chart and such that is an open cube, contains the closure of and .
Since manifolds are supposed to be paracompact here, they are separable and hence nice coverings are denumerable. Nice coverings on compact manifolds are finite. On arbitrary foliated manfiolds, nice coverings do exist.
Given a nice covering of a foliated manifold . For any , let be the space of the plaques (i.e., connected components of intersections . being a leaf of ) of contained in . Equip with the quotient topology: two points of are equivalent iff they belong to the same plaque. is homeomorphic (-diffeomorphic when is -differentiable and ) to an open cube () via the map , where is a foliated chart on . The disjoint union
is called a complete transversal for . Transversality refers to the fact that, if is differentiable of class , , each of the spaces can be mapped homeomorphically onto a -submanifold transverse to : if and is the leaf of passing through , then
Completeness of means that every leaf of intersects at least one of the submanifolds .
Definition 3.2. Given a nice covering of a foliated manifold and two sets and such that the holonomy map , being the open subset of which consists of all the plaques of for which , is defined in the following way:
All the maps () generate a pseudogroup on . is called the holonomy pseudogroup of .
Certainly, holonomy pseudogroups of arbitrary compact foliated manifolds are finitely generated. For example, gluing together two 2-dimensional Reeb foliations (see Foliations#Reeb Foliations) on one gets a foliation of the 3-dimensional sphere for which any arc intersecting the unique toral leaf is a complete transversal; can be identified with a segment (), the point of intersection with the number , while the holonomy group with the one on our segment generated by two maps such that , , snd contract their domains towards .
4 Growth
5 Geometric entropy
6 Invariant measures
7 Results on entropy
8 References
- [Walczak2004] P. Walczak, Dynamics of foliations, groups and pseudogroups, Birkhäuser Verlag, 2004. MR2056374 (2005d:57042) Zbl 1084.37022