Dynamics of foliations
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Revision as of 20:06, 26 November 2010
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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 spaceTex syntax error, any group of transformations of
Tex syntax errorconsists of maps defined globally on
Tex syntax error, mapping
Tex syntax errorbijectively 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 spaceTex syntax errorand denote
by Homeo the family of all
homeomorphisms between open subsets ofTex syntax error. 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 setTex syntax errorof homeomorphisms bewteen open sets (with domains covering a space
Tex syntax error) generates ma pseudogroup which is the smallest pseudogroup containing
Tex syntax error; precisely a homeomorphism belongs to if and only if for any point there exist elements , exponents and a neighbourhood
Tex syntax errorof such that on
Tex syntax error. If
Tex syntax erroris 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 .Tex syntax errorbeing 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 andTex syntax erroris 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 .
Tex syntax erroris 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
Tex syntax errorconsists of maps defined globally on
Tex syntax error, mapping
Tex syntax errorbijectively 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 spaceTex syntax errorand denote
by Homeo the family of all
homeomorphisms between open subsets ofTex syntax error. 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 setTex syntax errorof homeomorphisms bewteen open sets (with domains covering a space
Tex syntax error) generates ma pseudogroup which is the smallest pseudogroup containing
Tex syntax error; precisely a homeomorphism belongs to if and only if for any point there exist elements , exponents and a neighbourhood
Tex syntax errorof such that on
Tex syntax error. If
Tex syntax erroris 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 .Tex syntax errorbeing 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 andTex syntax erroris 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 .
Tex syntax erroris 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
Tex syntax errorconsists of maps defined globally on
Tex syntax error, mapping
Tex syntax errorbijectively 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 spaceTex syntax errorand denote
by Homeo the family of all
homeomorphisms between open subsets ofTex syntax error. 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 setTex syntax errorof homeomorphisms bewteen open sets (with domains covering a space
Tex syntax error) generates ma pseudogroup which is the smallest pseudogroup containing
Tex syntax error; precisely a homeomorphism belongs to if and only if for any point there exist elements , exponents and a neighbourhood
Tex syntax errorof such that on
Tex syntax error. If
Tex syntax erroris 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 .Tex syntax errorbeing 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 andTex syntax erroris 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 .
Tex syntax erroris 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