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Lectures on Singular Foliations

Nguyen Tien Zung

Institute de Mathematiques de Toulouse, France

GAP XII, Sanya 10-14/March/2014

Lecture 1: Generalities on singular foliations. Tensorization.

Lecture 2: Decomposition, normal forms, and deformations.

Lecture 3: Global and dynamical aspects of singular foliations.

(Background picture: The Wave, Arizona, USA)

Lecture 1: Generalities. Tensorization

Foliations = “reasonable” partitions of manifolds into disjoint unions ofsubmanifolds called leaves . Most foliations are singular, i.e. have “specialplaces” called singular leaves.Singular foliations are geometric objects which appear naturaly in physicsand mathematics, and can also be viewed as (generalized) dynamicalsystems. They can be turned into tensor fields and studied using algebraicand analytical tools.

Examples of singular foliations

Frobenius–Clebsch–Deahna & Stefan–Sussmann

Tensorization of singular foliations

Linear and homogeneous singular foliations

(Background picture: liquid crystals)

Nguyen Tien Zung (IMT) Lectures on Singular Foliations Sanya 10-14/March/2014 2 / 75


In the nature (see background pictures)- Earth layers, landscapes, stones (slate), ...- Graphite, liquid crystals, composite materials, ...- Saturn rings, space-time, D-branes (?), parallel worlds (?), ...- Trees, flowers, bird feathers, zebra, ...- Noodles, cakes, hamburgers, sliced breads, ...- Etc.Taking clues from wood plates, we get the following proposition (not truefor regular foliations), which can be proved rigorously:

Proposition

Restriction (pull-back) of a singular foliation to a submanifold is again asingular foliation

(Background picture: Space-time)



In dynamical systems:- Integral curves of vector fields (1-dimensional foliations)- Discrete time systems generated by diffeomorphisms (turned into1-dimensional foliations via suspension)- Stable and unstable foliations (e.g. for hyperbolic systems – Anosovflows)- Foliations by level sets of first integrals- Liouville torus foliations (for proper integrable Hamiltonian andnon-Hamiltonian systems)- Foliations of underlying geometric structures (e.g. Poisson, Dirac)- Geometric control theory, with holonomic and non-holonomicconstraints: foliation by accessible sets- Etc.

(Background picture: Stable and unstable foliations of a hyperbolic geodesic flow)



In geometry:- Poisson manifolds (foliated by symplectic leaves)- Dirac manifolds (foliated by pre-symplectic leaves)- Riemannian foliations (Molino’s theorey: leaves are locally “parallel” toeach other, i.e. have locally constant distance from each other w.r.t. aRiemannian metric)- Lie group actions (foliated by orbits), Lie algebra actions- Lie groupoids and Lie algebroids- Orbit-like foliations (locally look like foliations by the orbits of proper Liegroup actions)- Foliated minimal surfaces- Webs (multi-foliations)

(Background picture: A minimal surface which can be foliated into round circles)



In geometry (cont.):- Manifolds with boundary and corners (the open manifold is one regularleaf of codimension 0, and the boundary strata are singular leaves)- Maps and singular fibrations (singular fibers/preimages are “stratified”into singular leaves), Bott-Morse foliations- Moduli problems / equivalence classes- Integrable 1-forms (give rise to codimension 1 foliations)- Representations of singular objects (orbifolds, orbispaces, Weinsteingroupoids, noncommutative spaces, etc.) as foliations.- Etc.

(Background picture: Layers of the earth – an example of orbit-like Riemannian foliations)



Definition of singular foliations (Stefan-Sussmann)

A partition M =⊔Fi (Fi are leaves) satisfies :

∀x ∈ M, Fx : the leaf contains x , ∃ local coordinates x1, . . . , xn such that

Fx = xd+1 = . . . = xn = 0each disk xd+1 = cd+1, . . . xn = cn is contained in some leaf

Regular case: all the leaves have the same dimension (each diskxd+1 = cd+1, . . . xn = cn in the above definition is an open subset ofsome other leaf)

(Background picture: Foliated structure of wood)



The Stefan–Sussmann definition may be reparaphrased as the followinginductive local splitting condition:

Local splitting condition

Near each point x of rank d , the foliation is locally isomorphic (via a localdiffeomorphism) to a direct product of Rd with a foliation on Rn−d suchthat the origin of Rn−d has rank 0.

Here, by definition, the rank of a point is the dimension of the leafthrough it.Compare with splitting/slice theorems for Poisson structures (Weinstein),Lie algebroids (Fernandes – Dufour), Dirac manifolds (Dufour – Wade),slice theorems for proper group actions (Palais), proper Lie groupoids(Weinstein – Zung)

(Background picture: A wood plate)



Definition: Smooth singular distribution

D is a field of vector subspaces of tangent spaces, which is generated by afamily Xα, α ∈ I of smooth vector fields:

Dx = Vect(Xα(x), α ∈ I ) ∀ x ∈ M

If dimDx is constant then D is a regular distribution (subbundle of TM).In the case of a singular foliation F , the associated tangent distributionDF is generated by vector fields which are tangent to (the leaves of ) F :Dx = TxS(x) ∀ x ∈ M, where S(x) denotes the leaf through x .

Problem: Given D, does there exist F such that D = DF ? If it’s the casethen we say that D is integrable, i.e. we can integrate it to obtain F .

(Background picture: Foliated structure of earth in Eifel, Germany)



Involutivity

D is called involutive if for any two vector fields X ,Y tangent to D, theirLie bracket [X ,Y ] is also tangent to D

It is clear that if D = DF then D is involutive. (Being tangent to Dmeans tangent to each leaf of F in this case, and the Lie bracket can betaken leaf by leaf). The converse is also true in the regular case:

Theorem (Frobenius 1877 – Clebsch 1860 – Deahna 1840)

If D is regular then it’s integrable if and only if it’s involutive.

Attention: The above theorem is FALSE in the smooth singular case.Example: D on R2 given byD(x ,y) = Vect(∂x) if x ≤ 0 and D(x ,y) = Vect(∂x , ∂y) if x > 0D is involutive but not integrable (1-dimension leaves at x < 0 run intothe 2-dimensional leaf x > 0 at the points with x = 0)

(Background picture: A slate stone mountain)



Solution: Impose some additional conditions to avoid pathologies such asabove, e.g. D is generated by locally finitely-generated involutive modulesof vector fields (Hermann’s theorem, 1963), or the followingStefan–Sussmann invariance condition (stronger than involutivitycondition):

Theorem (Stefan-Sussmann 1973-74)

The following conditions are equivalent:

D = DF for some singular foliation FD is generated by a family C of vector fields and is invariant withrespect to the elements of C

Example: Stefan–Sussmann condition is obviously satisfied for the familyof Hamiltonian vector fields on Poisson manifolds.

(Background picture: A roof made of slate)


Some references

- F. Deahna, Ueber die Bedingungen der Integrabilitat linearerDifferentialgleichungen erster Ordnung zwischen einer beliebigen Anzahlveranderlicher Grossen, Journ. reine angew. Math. 20 (1840), 340–349.- A. Clebcsh, Ueber die simultane Integration der linearer partiellerDifferentialgleichungen, Journ. reine angew. Math. 65 (1866), 257–268.- F.G. Frobenius, Ueber das Pfaffsche Problem, Journ. reine angew. Math.82 (1877), 267–282.- J. Milnor, Foliations and foliated vector bundles, 1970 (Lecture notes).- H. J. Sussmann, Orbits of families of vector fields and integrability ofdistributions, Trans. AMS 180 (1973), 171–188.- P. Stefan, Accessible sets, orbits, and foliations with singularities, Proc.London Math. Soc. (3) 29 (1974), 699–713.For the proofs, see e.g. Chapter 1 of the book:- Dufour–Zung, Poisson structures and their normal forms, 2005.

(Background picture: Foliations by tori and Klein bottles – a drawing by Fomenko)


Tensorization of singular foliations

Idea: to represent a singular foliation by a tensor fieldWhy do it? Because tensor fields live in vector spaces, and one can applyalgebric methods (cohomology) and analytical methods (integration,averaging, norms and estimates, etc.) to them. Leads to theories ofnormalization and deformation of singular foliations.Other approaches to singular foliations:

Heafliger’s Gamma-structures: in terms of cocycles with values inDiff (Kq) where q is the codimension. Gives holonomy and universalclassifying space. But not every singular foliation can be given by aHaefliger structure ?!

Skandalis (+ Androulidakis, Zambon): define singular foliations vialocally finitely-generated modules of tangent vector fields. OK forholonomy. But what about normalization and deformation theory ?!

Maybe one can combine tensorization and holonomy groupoids to get abetter picture ?!

(Background picture: Artifical wood made of leather)


Tensorization: Integrable differential forms

The case of 1-forms (codimension 1 foliations): If α is a differential1-form, then near a point x s.t. α(x) 6= 0, Ker(α) is a regular distributionof corank 1. The Frobenius involutivity condition on Ker(α) is equivalentto the equation α ∧ dα = 0.Proof: Take vector fields X ,Y ,Z such that X ,Y lie in the kernel of α.Then by Cartan’s formula,

α ∧ dα(X ,Y ,Z ) = α(Z )dα(X ,Y ) = −α(Z )α([X ,Y ]) = 0

for all Z iff [X ,Y ] is in the kernel of α.

Definition

A differential 1-form α is called integrable if α ∧ dα = 0

It follows that, near a non-zero point, an integrable 1-form α can bewritten locally as: α = fdg , where f , g are two functions

(Background picture: Graphite foliated by graphenes)


Tensorization: Integrable differential forms

The case of q-forms where q is arbitrary (codimension-q foliations)

Definition

A differential q-form ω is called integrable if:

ω ∧ iAω = 0 & dω ∧ iAω = 0 ∀ (q − 1)-vector A

The kernel of an integrable q-form ω near a point x where ω(x) 6= 0 is aninvolutive distribution of corank q → codimension q foliation outsidesingular points (will talk about singular leaves later)

Integrable 1-forms can be traced back to back to Thom (1960s), Cartan,Nemytskii (1940s), ... all the way to Frobenius (?). Integrable q-forms arein use since 1970-80s only (?) Malgrange (wedge product of 1-forms),Camacho, Lins-Neto, Medeiros, ...

(Background picture: D-branes)


Tensorization: Nambu structures

Fix a volume form Ω. Then for each q-form ω there is a unique p-vectorfield Λ (where p + q = n is the dimension of the manifold) such that

ω = ΛyΩ

Λ is called an integrable p-vector field, or also a Nambu structure oforder p, if its dual differential form ω = ΛyΩ is integrable. Equivalentdefinition:

Definition

A p-vector field Λ is called a Nambu structure iff near every point x suchthat Λ(x) 6= 0 there is a local coordinate system (x1, . . . , xn) such that

Λ = f ∂x1 ∧ . . . ∧ ∂xp

(One can put f = 1). One may view a Nambu structure as a singularfolitation + leafwise contravariant volume form.

(Background picture: Parallel worlds)


Tensorization: Nambu structures

Nambu (1973): generalization of Hamiltonian formalisim from a binarybracket (the Poisson bracket) to a p-ary bracket (for p = 3). Takhtajan(1994) gave a definition in the general case:

Definition

A Nambu-Poisson structure (bracket) is a p-ary antisymmetric map

., . . . , . : C∞(M)× · · · × C∞(M)→ C∞(M)

which satisfies the Leibniz rule, and satisfies the so-called (generalizedJacobi) fundamental identity for any functions f1, . . . , fp−1, g1, . . . , gp:

f1, . . . , fp−1, g1, . . . , gp =∑i

g1, . . . , f1, . . . , fp−1, gi, . . . , gp

When p = 2: it’s nothing but Poisson.

(Background picture: A hamburger)


Nambu–Poisson = Nambu if p 6= 2

Due to Leibniz condition and anti-commutativity, a Nambu-Poissonstructure is given by a p-vector field (which satisfies the fundamentalidentity).

f1, . . . , fp = 〈df1 ∧ · · · ∧ dfp,Λ〉

Hamiltonian vector fields: Xf1,...,fp−1 = (df1 ∧ · · · ∧ dfp−1)yΛFundamental identity ⇔ Hamiltonian vector fields preserve Λ

Theorem (Alekseevsky – Guha, Gautheron, Nakanishi, ...)

When p 6= 2: Λ is Nambu–Poisson iff it’s a Nambu tensor, i.e. dual to anintegrable differential form.


Tensorization: The tangency condition

Question: Given a singular foliation F , how to associate to it a Nambustructure Λ, so that one can essentially recover F from Λ?

Intuitively, Λ would be tangent to F in the sense that at every regularpoint x of Λ (i.e. Λ(x) 6= 0) we can write Λ = ∂x1 ∧ . . . ∧ ∂xp in a localcoordinate system such that ∂x1, . . . , ∂xp generate F near x . In particular,Λ vanishes at every singular point of F

However, in some special situations the above intuitive tangency conditionwould imply that the singular set of Λ is too big compared to the singularset of F .

Example: F in C2 with leaves x = const 6= 0, x = 0, y 6= 0 andx = y = 0. Then Λ = f ∂y . If Λ is analytic and vanishes at the singularpoint of F then the singular set of Λ (i.e. the level set f = 0 is ofdimension (at least) 1 while the singular set of F is of dimension 0.

(Background picture: Singular foliation of S3 into tori)


Tensorization: Associated Nambu structure

Need a compromise between “being tangent everywhere” and “withoutunwanted singular points”. The following definitions work well in theanalytic case: Denote by S(Λ) (resp. S(F)) the singular set of a Nambustructure Λ of order p (resp. of a p-dimensional foliation F).

Λ a tangent Nambu structure to F if codim(S(F) \ S(Λ)) ≥ 2 andnear each point x /∈ S(Λ) ∪ S(F) there is a local coordinate system inwhich Λ = ∂x1 ∧ . . . ∧ ∂xp and F is generated by ∂x1, . . . , ∂xp

Moreover, if codim(S(F) \ S(Λ)

)≥ 2, and is without multiplicity in

the sense that Λ can’t be written as Λ = f 2Λ′, where f is a functionwhich vanishes somewhere, then we say that Λ is an associatedNambu structure to F .

(Background picture: Hopf circle fibration of S3)


Tensorization: Construction

The following simple construction of associated Nambu structures workswell locally, at least in the analytic case:

Take p local vector fields X1, . . . ,Xp which are tangent to F andwhich are linearly independent almost everywhere, and put

Π = X1 ∧ . . . ∧ Xq.

Decompose Π = hΛ, where codimS(Λ) ≥ 2.

If codimS(F) ≥ 2 then Λ is an associated Nambu structure of F .

If codimS(F) = 1, we find a reduced function s such thatS(F) = s = 0 then sΛ is an associated Nambu structure of F .

(Background picture: Hopf circle fibration)


Tensorization: Existence and uniqueness

Proposition

In the holomorphic category, there always exists a local associated Nambustructure which is unique up to multiplication by an invertible function.

Example: F on R3 or C3 with leaves x2 + y 2 + z2 = const. Take two

tangent vector fields X = y ∂∂z − z ∂

∂y , Y = z ∂∂x − x ∂

∂z , and put

Π = X ∧ Y = z

(x∂

∂y∧ ∂

∂z+ y

∂

∂z∧ ∂

∂x+ z

∂

∂x∧ ∂

∂y

)then Λ = Π

z is an associated Nambu structure of F .

Global situation: Sheaf of tangent Nambu structures = sheaf of sectionsof the anti-canonical line bundle of the foliation. Associated Nambustructure = section which doesn’t vanish anywhere.

(Background picture: Pinched torus in Lagrangian torus fibrations)


From Nambu structures to foliations

An “obvious” foliation generated by Λ consists of 2 kinds of leaves: regularleaves (through regular points of Λ) and 0-dimensional leaves (singularpoints of Λ). But this foliation is “stupid” (doesn’t have leaves of otherdimensions). Need more sophisticated constructions.

Definition

We say that a vector field X is tangent to Λ if

X ∧ Λ = 0

The set of tangent vector fields forms an integrable distribution andhence defines a singular foliation.

However, the foliation defined in this way may lose many singularitiesof Λ.

(Background picture: A Morse foliation)


Associated foliations

Example:

Λ = x∂

∂x∧ ∂

∂y.

Then codimS(Λ) = codimx = 0 = 1, but the foliation F defined by thetangent vector fields of Λ consists of just one leaf, which is the wholespace C2. Put an additional condition to avoid this situation:

Definition

A vector field X is called a conformally invariant tangent (CIT) vectorfield of a Nambu structure Λ if X is tangent to Λ and X conformallypreserves Λ, i.e.

LXΛ = f Λ

for some function f . The set of CIT vector fields of Λ will be denoted byCIT (Λ).

(Background picture: Spider web)


Associated foliations

Proposition

The set of CIT vector fields of a Nambu structure Λ of order p generatesan integrable singular distribution and hence defines a singular foliation FΛ

which will be called the associated foliation of Λ.

For previous example, Λ = x ∂∂x ∧

∂∂y

∂∂x is a tangent but not an associated vector field of Λ.

FΛ is generated by x ∂∂x ,

∂∂y and consists two leaves x = 0 and

x 6= 0.

Proposition

If Λ is a holomorphic Nambu structure and codim(Λ) ≥ 2, then everytangent vector of Λ is also CIT vector field of Λ.

(Background picture: Zebra)


From Nambu structures to singular foliations and back

Λ→ FΛ → ΛFΛ

We will say Λ conformally preserves a function f if there is Σ such that

[f ,Λ] = f Σ,

where the bracket means the Schouten bracket. Denoted by µ(Λ) the setof functions which are conformally preserved by Λ.

Proposition

If codimS(Λ) ≥ 2, then ΛFΛ= uΛ for some invertible function u.

If Λ =∏

f mii

∏gmj

j Λ1, where codimS(Λ1) ≥ 2, fi , gj are irreducible,fi ∈ µ(Λ1), gj 6∈ µ(Λ1), then ΛFΛ

= u∏

gjΛ1 for some invertiblefunction u.

(Background picture: Noodles)


From foliations to Nambu structures and back

Example: Let f , g ∈ O2 be irreducible and (f , g) = 1. Consider Λ = fgXf

where Xf = ∂f∂y ∂x − ∂f

∂x ∂y . Then f ∈ µ(Xf ), g 6∈ µ(Xf ) and ΛFΛ= gXf

From foliations to Nambu structures and back: F → ΛF → FΛF

Proposition

Let F be a holomorphic singular foliation and ΛF be its associated Nambustructure. Suppose that FΛF is an associated foliation of ΛF then FΛF isa saturation of F . Moreover, if codimS(F) ≥ 2 then codimS(FΛF ) ≥ 2.

Saturation means that each leaf of the latter foliation is saturated by theleaves of the former one.

Reference for Nambu↔foliation correspondence: Minh & Zung,”Commuting Foliations”, Regular and Chaotic Dynamics 2013.

(Background picture: The Wave again)


Morphisms, pull-backs, stratification, etc.

- The pul-back of a singular foliation by a map is again a singular foliation(via the pull-back of local associated integrable differential forms)- Morphisms? Many kinds: isomorphisms (no problems); sending Nambuto Nambu (preserve the dimension), sending distribution to distribution, ....- Foliation vs stratification of singular fibers of maps/fibrations: singularleaves of the associated singlar foliations are often strata of the (Whitney– Thom – Mather) stratification. It’s true, for example, for nondegeneratesingularities of integrable Hamiltonian systems.- Counter-example (suggested by Mattei): Level sets of the functionf (x , y , z) = x(x − y)(x − 2y)(x − zy) in C3. An 1-dimensional stratum ofthe singular level set is not a leaf, but the leaves in it are just points (dueto changing biratios). Nothing wrong with singular foliations or Nambustructures, it’s just that sometimes the stratification of a singular fibercan’t be made foliated)

(Background picture: Dragonfish)


What are linear foliations?

There are two different non-equivalent notions of linear singular foliations:

Lie-linear: foliations generated by linear vector fields = generatedbylinear representations of Lie algebras = generated by linear actions ofLie groups. (Big theory of linear representations)

Nambu-linear: associated to a linear Nambu structure, i.e. whosecoefficients in a coordinate systems are linear.

Nambu-linear are also Lie-linear though the converse is not true (there arefew Nambu-linear foliations): contraction of a linear Nambu p-vector fieldwith constant (p − 1)-forms give rise to generating linear vector fields.

Classification of Nambu-linear: Dufour–Z (Compositio Math. 1999). Someother people (Grabowski, ...) arrived at similar results.

(Background picture: A wood slice)


Classification of linear Nambu structures: 2 types

Type I (piles of cabbages):

Λ = ωy(∂x1 ∧ · · · ∧ ∂xn)

whereω = dx1 ∧ · · · ∧ dxp−1 ∧ dQ

where Q is a quadratic functionx1, . . . , xp−1 are regular linear first integralsQ is a quadratic first integralIn the nodegenerate case, Q depends only on xp, . . . , xnIf Q is positive definite then the leaves are p-dimensional spheresThe foliation looks like a pile of cabbages or parallel worlds

(Background picture: A pile of cabbages)


Classification of linear Nambu structures: 2 types

Type II (books):Λ = ∂x1 ∧ · · · ∧ ∂xp−1 ∧ X

whereX =

∑i ,j≥p

aijxi∂xj

is a linear vector field in the variables xp, . . . , xnThis foliation can be splitted into direct product of a linear vector fieldwith Rp−1.Looks like an open book (especially if X is hyperbolic).

In a sense, the two types are dual to each other: in Type 2 the Nambustructure is decomposabe, in Type 1 the integrable differential form isdecomposable. Very often, singular foliations are locally of these twotypes, because of the linearization theorems (see Lecture 2).

(Background picture: An open book)


(Quasi-)homogeneous foliations

What is a homogeneous singular foliation of degree k? Generated byhomogeneous vector fields of degree k?The problem is that, the Lie bracket of two vector fields of degree k is avector field of degree 2k − 1 6= k unless k = 1. So a family of vector fieldsof degree k can’t be involutive in general, and if one generates aninvolutive family from some vector fields of degree k by taking Liebrackets, we will get vector fields of different degrees.To avoid this problem: replace vecto fields by (quasi-)homogeneousNambu structures or integrable differential forms.Example: ω = dF 1 ∧ · · · ∧ dFq, where F1, . . . ,Fq are homogeneouspolynomial functions.Example: Any Lie-linear foliation is a (Nambu-)homogeneous foliation.Example: Diract product of homogeneous foliations is again homogeneous.

(Background picture: Saturn’s rings look like S1-orbits)


H-degree

H-degree = degree of the associated homogeneous Nambu structure. (Adiscrete invariant for linear representations)

Example: Let g be a semisimple Lie algebra, and consider the associatedcoadjoint foliation on g∗. Leaves = coadjoint orbits. The singular set ofcodimension 3. First integrals = Casimir functions. The associated Nambustructure is

Λ = ∧mΠ

where m is half the dimension of coadjoint orbits. The H-degree is also m,because Π is linear. Up to a multiplicative constant, the dual integrabledifferential form is

dF1 ∧ · · · ∧ dFd

where F1, . . . ,Fd are generators of the algebra of Casimir functions. (d isthe dimension of the Cartan subalgebra).

(Background picture: A rose)


Example: g = gl(n,K)

The coordinates are xij , 1 ≤ i , j ≤ n. The tangent vector fields are (fori 6= j)

Xij = xij(∂xjj − ∂xii ) + (xii − xjj)∂xji +∑k 6=i ,j

(xik∂xjk − xkj∂xki )

The tangent Nambu structure ∧i 6=jXij is divisible by∏

i<j(xii − xjj) and

Λ = ∧i 6=jXij/∏i<j

(xii − xjj)

is the associated Nambu structure of order n(n − 1) (equal to thedimension of the generic orbits), which is homogeneous of degreen(n − 1)/2 (half of the order). It is proportional to ∧n(n−1)/2Π where Π isthe associated linear Poisson structure on the dual of the Lie algebra.

(Background picture: Feathers of a gull)


Lecture 2: Decomposition, normal forms, and deformations

Main topics of this lecture:

Decomposition (Kupka’s phenomenon)

Normal forms (mainly results on linearization)

Deformation theory (mainly deformation cohomology)


Kupka’s phenomenon for codimension 1 foliations

Recall that a codimension 1 foliation is given locally by an integrable1-form α: α ∧ dα = 0. If α(x) 6= 0 then x is a regular point of theassociated foliation: the leaf through x has dimension n − 1 andcodimension 1. Intuitively, if a singular point x (where α(x) = 0) is“generic” then dα(x) 6= 0.Question: What is the local structure of α (or the dual Nambu structure)and of the associated foliation near such a “generic” singular point? Inparticular, what is the dimension of the leaf through it? The answer isgiven by Kupka’s theorem (1964) and is known as Kupka’s phenomenon.

Theorem (Kupka 1964)

If α is integrable 1-form in (Kn, 0) such that α(0) = 0 but dα(0) 6= 0,then locally it’s the pull-back of an 1-from (K2, 0) by a local submersionfrom (Kn, 0) to (K2, 0).


Dimension of the singular leaf

In other words, α can be “reduced” to just 2 dimensions: there is acoordinate system (x1, . . . , xn) in which

α = f1(x1, x2)dx1 + f2(x1, x2)dx2

In terms of the dual Nambu structures of order n − 1:

Λ = g∂x3 ∧ · · · ∧ ∂xn ∧ (a(x1, x2)∂x1 + b(x1, x2)∂x2)

near a “generic” singular point, where g is some multiplicative factor(which can be eliminated by further normalization).Geometrically, it means that the singular leaf is of dimension n − 2 and isgenerated locally by (∂x3, . . . , ∂xn). The codimension 1 foliation can besplitted locally at such a singlar point into the direct product of Kn−2 andan 1-dimensional foliation in dimension 2.


Splitting degree of Λ

We want to extend Kupka’s phenomenon (reduction/splitting result) tothe cases of higher codimensions and more general singularities.

Definition

A p-vector field Λ is said to be of Type IIr , or r-splittable, a point z , if itcan be written in a local coordinate system near z as

Λ = ∂x1 ∧ · · · ∧ ∂xr ∧ Π

where Π is a (p − r)-vector field which is independent of x1, . . . , xr .

Remark: The apparently weaker splitting condition

Λ = f ∂x1 ∧ · · · ∧ ∂xr ∧ Π,

where f is a function such that f (0) 6= 0, is equivalent to the abovecondition. (One can eliminate f by changing the coordinates x1, . . . , xrwhile keeping xr+1, . . . , xn intact). If Λ is Nambu then Π is also Nambu.


Splitting vs projection

Example: At a regular point, a Nambu structure of order q is q-splittable.A linear Nambu structure of Type II is (q − 1)-splittable. (Hence the nameType IIr in the book Dufour–Z 2005).

An integrable q-form ω (dual to Λ) is r-conformally projectable at apoint z if locally, up to multiplication by an invertible function, it can bewritten as the pull-back of another form β by a local submersion from Kn

to Kn−r :ω = f .proj∗β

ω is r-projectable if ω = proj∗β. (Example: generic fdg is 1-conformallyprojectable and 2-projectable)

Theorem

Λ is r -splittable ⇔ ω is r -conformally projectable ⇔ the foliation isr -splittable ⇔ the leaf of FΛ through the point is of dimension at least r


The curl operator (modular operator)

Fix a volume form Ω, get 2 duality operators (isomorphisms): Ω[: frommulti-vector fields to differential forms, by contraction with Ω, andΩ] = (Ω[)−1 is the inverse. The operator

DΩ = Ω] d Ω[

is called the curl operator. It’s use in Poisson geometry: if Π is Poissonthen DΩΠ is its modular vector field.

If Λ is Nambu of order q then DΩΛ is Nambu of order q − 1. If Λ ishomogeneous of degree d then DΩΛ is homogeneous of degree d − 1.

The idea is to study DΩΛ, which is “simpler” than Λ, then inferinformation about Λ.


Generalized Kupka’s phenomenon

Using the curl operator, we arrive at the following results (first for Nambustructures, then translated to results for integrable differential forms):

Theorem (Kupka–Medeiros–Doufour–Z)

If ω is integrable q-form, dω 6= 0 a.e., and dω is s-projectable at z then ωalso is s-projectable z .

Kupka (1966) for the case q = 1, s = 2, Medeiros (1977) for the cases = q + 1, Dufour–Z (Poisson Book 2005) for the general case.

Theorem (Dufour–Z 2005)

If Λ is Nambu of order p such that DΩΛ is nonzero a.e. and is r -splittableat z then Λ is also r -splittable at z .


Singular set and decomposability

General idea: if the singular set is “small” (i.e. have high codimension)then the tensor field is “highly decomposable”. Works for both integrabledifferential forms and Nambu structures

Theorem (Medeiros 2000)

Let ω be a holomorphic integrable q-form on (Cn, 0) (2 ≤ q ≤ n − 1).i) If codimΠyω = 0 ≥ 3 where Π = ∂x1 ∧ · · · ∧ xq−1 then

ω = fdg ∧ dy1 ∧ · · · ∧ dyq−1

in some local coordinate system, f (0) 6= 0.ii) If the singular set of the pull-back of ω to a (q + 1)-dimensionalsubmanifold through 0 is of codimension at least 3, then the dual Nambutensor Λ of ω is (p − 1)-splittable, where p = n − q is the order of Λ.

Proof: use division theorems (De Rham, Saito), Malgrange’s “Frobeniuswith singularities”


(Quasi-)homogenization problem

Let Λ be a (formal, analytic, or smooth) Nambu structure on (Kn, 0) suchthat Λ(0) = 0. Then

Λ = Λ(hom) + h.o.t

where Λ(hom) is the (quasi)homogeneous part of Λ. Similarly to thePoisson case, Λ(hom) is automatically a Nambu structure.

Problem: Is Λ locally isomorphic to Λ(hom)? (If not, can we develop a kindof Poincare-Birkhoff normal forms for Λ, where there will be higher degree“resonant” terms?)

In particular, when Λ(hom) is of degree 1, we have the linearization problem.


Linearization of Type II

Λ = Λ1 + h.o.t is of Type II, i.e.

Λ1 = ∂x1 ∧ · · · ∧ ∂xp−1 ∧∑i ,j≥p

aijxi∂xj

Lemma (corollary of generalized Kupka’s phenomenon)

If∑

i aii 6= 0 then Λ is (p − 1)-splittable.

After the splitting, the linearization problem for Λ becomes thelinearization problem for a vector field, so we immediately get:

Theorem (Dufour–Z 1999)

If∑

i ,j≥p aijxi∂xj is non-resonant then Λ is smoothly linearizable. Ifmoreover it satisfies Bruno’s condition (or a Diophantine condition) and Λis analytic then Λ is analytically linearizable.


Linearization of Type I

Λ = Λ1 + h.o.t is of Type II, i.e.

Λ1y(dx1 ∧ · · · ∧ dxn) = dQ ∧ dxp+2 ∧ · · · ∧ dxn

where Q is a quadratic function. We will consider only the nondegeneratecase, i.e. Q is nondegenerate quadratic in variables x1, . . . , xp+1

Theorem (Dufour–Z 1999, Z 2013)

a) If Λ is formal then it’s formally linearizableb) If Λ is analytic then it’s analytically linearizablec) If Λ is smooth and the signature of Q is different from (2, ∗) then Λ issmootly linearizable. If the signature is (2, ∗) then there arecounter-examples.


Tools used in the proof

Division theorems (De Rham, Saito)

Decomposition of the dual integrable differential form

Godbillon–Vey algorithm (to formally linearize the foliation)

Malgrange’s “Frobenius with singularity” theorem (for the existenceof analytic first integrals)

Blowing up (in the compact case, when Q is positive definite)

Levi decomposition (for the existence of SO(p + 1) symmetry group)

Slicing method (for dealing with the smooth concompact case)

Equivariant smooth linearization of vector fields(Sternberg–Belitskii–Kopanskii)


Levi decomposition

- Classical Levi-Malcev theorem for finite-dim Lie algebras: l = gn r,where r is the solvable radical, g is semisimple (called Levi factor).- For geometric structures: l is infinite-dim (vector fields, sections of Liealgebroids, etc.) → want an extention of Levi-Malcev to infinite-dim case.- Important observation (going back to Moshe Flato ?): Levidecomposition implies linearization when the linear part is semi-simple.- Cerveau’s theorem for analytic singular foliations (1979)- Formal Levi decomposition for Poisson structures: Wade (1997),Chloup-Arnold (1997?)- Abstract formal Levi decompostion: Dufour-Z (book, 2005)- Even if the linear part is not semi-simple, Levi decomposition still gives anice normal form (partial linearization), which leads to a full linearizationin some problems. Examples: nondegeneracy of aff(n) = gl(n) nKn

(Dufour-Z 2002); smooth (non) degeneracy of semi-simple Lie algebras ofreal rank 1 (Monnier-Z, in progress)


Slicing method

Turn a non-compact situation to a compact situation by slicing!Used to prove linearizability of Type 1 structures in the non-compact case(of signature different from (2,*)), in combination with other tools (Levidecomposition, formal linearization, SO(k) actions, Belitskii-Kopanskiiequivariant linearization of vector fields, ...)Ref: NT Zung, New results on the linearization of Nambu structures, J.Math. Pures Appliquees, 2013.


Deformation cohomology

Linearization theorems show that Nambu-linear foliations are rigid (do notadmit nontrivial deformations). Mor generally, we want to develop adefrmation theory for singular foliations.A fundamental tool at the infinitesimal level: deformation cohomologyGeneral idea: Given a certain structure S- Infinitesimal deformations: D such that the formal deformation S + εDof S also satisfies structural equations modulo ε2

- Trivial deformations: terms of the type (1 + εX )∗S − S modulo ε2

Hdef (S) =infinitesimal deformationstrivial deformations

Hdef (S) = formal tangent space to the moduli space of deformations (i.e.nearby structures). If Hdef (S) = 0 then S is called formally (orinfinitesimally) rigid. In many situations, formal rigidity implies rigidity(Richardson–Nijenhuis, Mather, etc.).


The case of Nambu structures and singular foliations

Let Λ be a Nambu structure of order p.

A multi-vector field Π of order p is called an infinitesimal deformationof Λ if Λ + εΠ is a Nambu structure modulo ε2. The condition“Λ + εΠ is Nambu modulo ε2” is a linear system of equations on Π(linear first order PDEs + linear algebraic equations), so the set ofinfinitesimal deformations is a vector space.

If Π = LXΛ for some vector field X , then Π is called a trivialdeformation of Λ. The set of all trivial deformations is also a vectorspace.

If Π = LXΛ + f Λ for some vector field X and some function f , thenΠ is called a trivial deformation of the associated foliation FΛ. Theset of all trivial deformations of the foliation is also a vector space.


Dedinition of deformation cohomologies

Definition: Deformation cohomologies of Λ and FΛ

Hdef (Λ) =Infinitesimal deformations of Λ

LXΛ,

Hdef (FΛ) =Infinitesimal deformations of Λ

LXΛ + f Λ.

Remark: If a local associated Nambu structure doesn’t exist globally(because the anti-canonical line bundle is non-trivial, use a ”twistedassociated Nambu structure” (twisted by the canonical bundle) in thedefinition of deformation cohomology of the foliation.

Problems: Computations of deformation cohomologies, relations toproblems of rigidity and (true) deformations, comparison with othercohomology theories, characteristic classes and indices, etc.


13/March/2014: real 3rd talk starts here

Recall:- Foliations arise naturally in mathematics and physics, and most of themare singular. When people in mechanics/physics talk about regularfoliations, it’s because singularities are more difficult to deal with than theregular part, so they hide them.- We want to study things like normal forms, stability, and deformations ofsingular foliations- In order to do it, we need to represent singular foliations as”algebro-analytical” objects, like tensor fields, so that we can apply usualnormalization and deformation methods- The right way to do it is to represent foliations by Nambu structure /integrable differential forms. There is an almost 1-1 correspondencebetween Nambu structures and singular foliations.


Recall of some main ideas

- In general, associated Nambu structures do not lose singularities offoliations. Nor do they create new artificial singular points.- The sheaf of local tangent Nambu strucures = the sheaf of local sectionsof a line bundle (the anti-canonical bundle of the foliations). If this linebundle is not globally trivial then a global Nambu structure doesn’t exist,but it’s no big deal: one can talk about twisted associated Nambustructures by taking tensor product with the dual (canonical) line bundle,so the theory still works.- Nambu are good not only for foliations, but also for analysis (singularitytheory). Example: manifold with boundary and corners can also berepresented by Nambu. Near a corner, Λ is monomial:

Λ = x1 . . . xk∂x1 ∧ · · · ∧ ∂xk ∧ · · · ∧ ∂xn

One recovers the boundary strata as singular leaves of the associatedfoliation. (One can often views tratifications as singular foliations)


Recall: Deformation cohomology

Definition: Deformation cohomologies of Λ and FΛ

Hdef (Λ) =Infinitesimal deformations of Λ

LXΛ,

Hdef (FΛ) =Infinitesimal deformations of Λ

LXΛ + f Λ.

Rigidity of the structure often corresponds to the fact that thedeformation cohomology is trivial. And if the deformation cohomology isfinite-dimensional, one expects that the moduli space ofdeformations/normal forms of perturbations is also locallyfinite-dimensional.


Computation of deformation cohomology

Joint work in progress with Philippe Monnier and Truong Hong Minh.New people welcome! (A lot of things to be computed)

The regular case

The case of top order (singular volume forms)

The case of zero order (functions)

The case of order n − 1 (codimension 1 foliations)

The case of order 1 (vector fields)

The case of linear representations of Lie algebras

Etc.


The regular case

The foliation F is regular. J. L. Heitsch (1973-75) defined a differentialcomplex whose first cohomology class is the deformation cohomologyspace of F . In fact, Heitsch cohomology is nothing else but the algebroidcohomology of the natural linear action of the tangent Lie algebroid TFon the normal bundle NF = TM/TF of F .Assume that there is a global Nambu structure Λ associated to F (i.e.regular leafwise volume contravariant form). Then F = FΛ and ourdeformation cohomology for the foliation coincides with Heitsch’s:

Theorem

Hdef (F) = H1(TF ,NF )

The differential complex here is Ω∗(TF ,NF ) of leafwise differential formswith values in the normal bundle: it’s very similar to the usual De Rhamcomplex of differential forms with values in R


Why H1(TF ,NF) ?

η ∈ Ωk(TF ,NF ) means it’s a field of anti-symmetric k-linear maps

η(x) : ∧kTxF → NxF

The normal bundle NF is locally trivial by a unique canonical trivialization(flat connection: parallel transport along the leaves), and forms withvalues in NF are locally like forms with values in Rq. The differential dηof η is defined by Cartan’s formula:

dη(X0, . . . ,Xk) =∑i

(−1)iXi .η(X0, . . . Xi . . . ,Xk) +∑i<j

η([Xi ,Xj ],X0, . . . Xi . . . Xj . . .Xk)

Each infinitesimal deformation of the tangent distribution of F is given bya 1-form α ∈ Ω1(TF ,NF ), and it’s easy to verify that this infinitesimaldeformation is infinitesimally involutive iff α is closed, i.e. is a 1-cocycle.Similarly, each trivial infinitesimal deformations is 1-coboundary. (Draw apicture!)


The regular case

The theorem is still valid when a global assocated Nambu structuredoesn’t exist: replace Nambu by a twisted Nambu in this case.Hdef (Λ) can be much larger than Hdef (FΛ), also in the regular case.

For example, let M = P × Q compact, where Q is simply-connected, andthe foliation F is given by the projection to P, i.e. the leaves arept × Q. Then the deformation cohomology of the foliation is trivial (itfollows from the above theorem, and agrees with Reeb stability theorem).On the other hand, Hdef (Λ) becomes the deformation cohomology of afunction f on P (the value of f at a point x in P equals the volume ofx × Q with respect to the contravariant volume form Λ.

Theorem

Let f : P → R be a smooth simple Morse function on a compact manifoldP. Then dim Hdef (f ) is the number of singular points of f .


Case: Top order without singularities

If Λ is regular of top order, then only 1 leaf (the manifold itself), thefoliation is trival (no deformation), but Λ can be deformed (by changingthe total volume)

Theorem

If Λ is a regular Nambu structure of top degree on a compact manifoldthen Hdef (Λ) = R and Hdef (FΛ) = 0

Consistent with Theorem of Moser [Famous paper on the path method,1964]: Two volume forms ω1 and ω2 on a compact manifold M arediffeomorphic if and only if they have the same global volume:∫

Mω1 =

∫Mω2


Case: Top order, with nondegenerate singularities

Locally Λ = x1∂x1 ∧ ∂x2 ∧ · · · ∧ ∂xn near singular points (Type II). Twokinds of leaves: regular n-dimensional domains, and singular(n − 1)-submanifolds.

Classification of these structures is done by Olga Radko (2002, for n = 2:Poisson structures on surfaces) and David Martinez Torres (2004, for narbitrary). [Draw a picture for the case n = 1!]. Numerical invariants ofthe classifiation (besides topological and orientation invariants):

Regularized Liouville volume of the manifold

(n − 1)-dimensional volume of each singular leaf (induced from Λ)

Theorem

Let Λ be a Nambu structure of top order with nondegenerate singularitieson a compact manifold M. Then dim Hdef (FΛ) = 0 anddim Hdef (Λ) = k + 1, where k is the number of (n− 1)-dimensional leaves.


Case: Top order, local cohomology

Assume that

Λ = f∂

∂x1∧ . . . ∧ ∂

∂xn

where f (0) = 0 and moreover 0 is a singular point of f , i.e. df (0) = 0.We will work here in the local holomorphic category, with germs offunctions. In this case we have:

Hdef (FΛ) ∼=On⟨

f , ∂f∂x1, . . . , ∂f∂xn

⟩and dim Hdef (FΛ) = τ(f ) is the Tjurina number of f at 0.

Hdef (Λ) ∼=On

X (f )− (divX )f |X ∈ Xand dim Hdef (Λ) = some number of f (?! Don’t know the name)


Case: Order 0 (functions)

Λ = f is a function (0-vector field). Locally:

Hdef (f ) =On

X (f )|X ∈ X=

On⟨∂f∂x1, . . . , ∂f∂xn

⟩ ,Hdef (Ff ) =

On

X (f ) + cf |X ∈ X, c ∈ On=

On⟨f , ∂f∂x1

, . . . , ∂f∂xn

⟩ .So, dimHdef (f ) = µ(f ), dimHdef (Ff ) = τ(f ) (Milnor and Tjurinanumber).


Case of vector fields & linear structures

If Λ = X is a vector field, the leaves of the foliation are integral curves ofX .

Normalization of the foliation = Orbital normalization of X .

Local deformation cohomology (for germs of vector fields) is given byresonant terms, and can be infinite-dimensional. If the vector field has anon-resonant linear part, then the local deformation cohomology is trivial.Global deformation = complicated dynamical problem.

Dufour-Z linearization theorems and proof imply in particular that thedeformation cohomology of nondegenerate linear Nambu structuresof TypeI is trivial. For Type II, it depends on resonances.


Decomposable Nambu structures with small singularities

Λ is a Nambu structure and ω = iΛΩ, Ω is a volume form.

If ω is decomposable (i.e. ω = ω1 ∧ . . . ∧ ωp) and codim(ω) ≥ 3 thenby Malgrange (1977):

ω = udf1 ∧ . . . ∧ dfp

Proposition

Let ω = udf1 ∧ . . . ∧ dfp be an integrable p-form and η is an infinitesimaldeformation ω. If codimS(ω) ≥ p + 2 then

η = a0df1 ∧ . . . ∧ dfp + u

p∑i=1

df1 ∧ . . . ∧ dfi−1 ∧ dai ∧ dfi+1 ∧ . . . ∧ dfp.

It means that ω + εη s also decomposable and admits first integralsmodulo ε2.


Decomposable Nambu structures with small singularities

Corollary

If ω = df (Nambu structure of order n − 1) and codimS(df ) ≥ 3 then

Hdef (Fdf ) =On

ai∂f∂x1

+ . . .+ an∂f∂xn

+ h f |ai ∈ On, h ∈ O

In particular, µ(f ) ≥ dimHdef (Fdf ) ≥ τ(f )− 1.

Corollary

If 0 is an isolated singularity of ω = udf1 ∧ . . . ∧ dfp thendimHdef (Fω) <∞

Some open problems concerning deformation

Interprete deformation cohomology as part of a bigger cohomologytheory (with a differential complex) in the singular case ?

Computation of deformation cohomology for linear actions of Liealgebras, and for other situations?

Rigidity of singular foliations given by semisimple compact groupactions: The group actions are rgid, but if we forget the group, is thefoliation still rigid ?

Singular Reeb stability (for singular points and leaves) ? Linearizedmodels along a leaf ?

Local and global (Infinitesimal) rigidity of orbit-like foliations withsimply-connected leaves ?

Etc.


Lecture 3: Dynamical aspects of singular foliations

Some global aspects

Foliations as dynamical systems

Foliations and Nambu structures associated to integrable(non)Hamiltonian systems

Integrability of foliations: first integrals, commuting flows, Liouville’stheorem,

Dynamical and structural stability

Entropy


Some global aspects of singular foliations

Questions (which are largely open):

Characteristic classes for singular foliations?

How does deformation cohomology sit in the space of characteristicclasses?

Topological and geometrical properties of foliations with “nice”singularities? For example, Riemannian foliations, orbit-like foliations(coming from proper Lie groupoids)? Foliations associated tointegrable (non)Hamiltonian systems? Foliation with (generalized)Bott-Morse singularities?

Topological obstructions to the existence of such or such foliations /Nambu structures ?


Foliations with Bott-Morse singularities

Transversally nondegenerate singularities (in particular, they aretransversally Nambu-linearizable). (Draw a picture!)If the singular leaves are simply-connected + the transversal index isdifferent from (2,*) then existence of first integrals. Even if the index is(2,*) but the regular leaves are simply-connected the there still exist firstintegrals. So in fact, the foliation is given by a map with transversallynondegenerate singularities.Codimension 1 case studied by: Camacho, Seade, Scardua in 2000s.


Singular foliations as dynamical systems

Dynamical systems are nothing but singular foliations of dimension 1. Sosingular foliations (of any dimension) may be viewed as generalizeddynamical systems. A lot of different notions for dynamical systems can beextended to the world of singular foliations, e.g. (a lot of open questionshere):

Symmetries (foliation-preserving group actions and vector fields).Inner symmetries (each leaf is preserved). How to find intrinsicsymmetries ? (Vector fields admit intrinsic formal torus actionsrelated to Poincare – Dulac normalization.Redution theory w.r.t. local/global symmetries? (Splitting is a localreduction w.r.t. local free tangent invariant action of Rr .Reconstruction problem (from reduced foliation/Nambu structure tothe original foliation? Tudor Ratiu et al. did it for dynamical systems).Invariant submanifolds (saturated by leaves). How to find them? (Fordynamical systems Chaperon has a way to find a lot of peculiarinvariant submanifolds)

(Background picture: Graphite foliated by graphenes)Nguyen Tien Zung (IMT) Lectures on Singular Foliations Sanya 10-14/March/2014 70 / 75


“Complete Integrability”: first integrals (level sets are saturated byleaves), commuting foliations (via commuting Nambu structures),extension of Liouville’s theorem.

Holonomy (extension of Poincare’s return map). See IakovosAndroulidakis’s talk (holonomy pseudogroups and groupoids).Relations between holonomy and existence of 1st integrals (extentionof the results of Mattei–Moussu, Ziglin, et al.)?

Differential Galois theory (Malgrange differential Galois groupoid) forsingular foliations. if completely integrable then the differential Galoisgroupoid is Abelian ?! (Extension of Morales–Ramis–Simo theorem tothe case of foliations). Solvable groupoids and Liouvillian solvabiity?!

(Background picture: D-branes)



Normal forms and perturbation/deformation theory

Structural stability (including structural stability of “hyperbolic,Anosov” foliations?)

Pertistence of a singular leaf. Conjecture: Dufour–Wade theoremabout the stability of singular points (of H-degree k) also holds in thecase of singular foliations.

Dynamical stability of a leaf (leads to things like Riemannianfoliations, singular Reeb stability ?)

Stable/unstable submanifolds which go through leaves?!

Entropy, as a measure of “chaos”.

Random singular foliations ? Foliation-wise diffusions ?

(Background picture: Parallel worlds)


Commuting foliations

The notion of commuting foliations (which generalizes in a natural waycommuting vector fields) can be defined using Nambu structures.For transversal Nambu structures Λ1,Λ2 (whose associated distributionshave trivial intersection), the condition is that their Schouten bracketvanishes:

Λ1,Λ2 = 0

Theorem (Simultaneous local normal form for a pair of transversalcommuting Nambu structures)

Λ1 = ∂x1 ∧ · · · ∧ ∂xq1

Λ2 = ∂xq1+1 ∧ · · · ∧ ∂xq1+q2

For Nambu structures with q1 + q2 > n : need to change the defintionotherwise it won’t mean anything.


Integrability and Liouville’s theorem

Integrability: a family of commuting foliations whose total dimension isp = p1 + · · ·+ pk ≤ n, together with q = n − p joint first integrals

Theorem (Decomposition)

Under some additional hypotheses, a regular compact level set of anintegrable foliation is locally isomorphic to a semi-direct product

(Pp11 × · · · × Ppk

k )/Γ

where the action of the discrete group Γ is proper, free, diagonal. Remark:Nondegenerate singular fibers of integrable Hamiltonian systems alsoadmit such a decomposition!


Entropy of singular foliations

Extension of the notions of metric/topological entropies to the case of(singular) foliations, using the same ideas of exponential growth in time ofthe number of points which are two-by-two epsilon-separated at somemoment of time.

Ghys–Langevin–Walczak (Acta Math. 1988) for regular foliations. Zung(Bull. Brazilian Math Soc. 2011) for general geometric structures in thesense of a normed vectoe bundle on a manifold plus an anchor map fromthat bundle to the tangent bundle of the manifold

In the case of vector fields this entropy coincides with the topologicalentropy. For symplectic foliation of linear Poisson structures (on the dualsof Lie algebras) this entropy is trivial. If the foliation is integrable then theentropy is trival (at least outside degenerate singularities). For the directprduct of two foliations, the entropy will be the sum of the two. ...


lectures on singular foliationszung.zetamu.net/maths/talks/foliationssanya2014.pdf · - moduli...

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