Zero sets of Abelian Lie algebras of vector fields

0.06 MB·

by  Morris W. Hirsch

Zero sets of Abelian Lie algebras of vector
fields
6 Morris W. Hirsch
1
0 Mathematics Department
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University of Wisconsin at Madison

n
a University of California at Berkeley
J

2
January 13, 2016
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]
S
D Abstract
.
h Assume M is a 3-dimensional real manifold without boundary, A is an
t
a abelian Lie algebra of analytic vector fields on M, and X ∈ A. The fol-
m
lowingresultisproved:
[ If K is a locally maximal compact set of zeroes of X and the Poincare´-

1 Hopf index of X at K is nonzero, there is a point in K at which all the
v elementsofAvanish.
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2 Contents
0
.
1
1 Introduction 1
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6 Applicationtoattractors . . . . . . . . . . . . . . . . . . . . . . 3
1
:
v 2 Background material 4
i
X
3 ProofofTheorem 1 5
r
a
1 Introduction
Throughoutthispaper M denotesa real analytic,metrizablemanifoldthatis con-
nected and hasfinitedimensionn, fixedat n = 3 inthemainresults.
Thespaceof(continuous)vectorfieldson M endowedwiththecompactopen
topology is V(M), and VrM is the subspace of Cr vector fields. Here r denotes
a positive integer, ∞, or ω (meaning analytic); this convention is abbreviated by
1 ≤ r ≤ ω.
The zero set of X ∈ V(M) is Z(X) := {p ∈ M: X = 0}. If Z(X) = ∅ (the
p
empty set), X is nonsingular. The zero set of a subset S ⊂ V(M) is Z(S) :=
Z(S).
TX∈S
1
A compact set K ⊂ Z(X) is a block of zeros for X— called an X-block for
short— if it lies in a precompact open set U ⊂ M whose closure U contains no
otherzeros of X; such anopen set isisolatingfor X, and for(X,K).
WhenU isisolatingforX thereisauniquemaximalopenneighborhoodN ⊂
U
V(M) of X withthefollowingproperty(Hirsch[9]):
IfY ∈ N hasonlyfinitelymanyzerosinU, thePoincare´-Hopfindex
U
ofY|U dependsonlyon X and K.
This index is an integer denoted by i (X), and also by i(X,U), with the latter
K
notationimplyingthat U is isolatingfor X.1
ThecelebratedPoincare´-HopfTheorem[14,23]connectstheindextotheEu-
lercharacteristicχ(M). Amodernformulation(seeMilnor[19])runsas follows:
Theorem (Poincare´-Hopf). Assume M is a compact n-manifold,X ∈ V(M), and
Z(X)∩∂M = ∅. If X is tangent to ∂M at all boundary points, or points outward
at all boundary points then i(X,M) = χ(M). If X points inward at all boundary
points,i(X,M) = (−1)n−1χ(M).
ForcalculationsoftheindexinmoregeneralsettingsseeGottlieb[6],Jubin[15],
Morse[21], Pugh [24].
The X-block K is essential if i (X) , 0. When this holds every Y ∈ N (X)
K U
has an essential block of zeros in U (Theorem 6). If M is a closed manifold
(compact, no boundary)and χ(M) , 0, thePoincare´-HopfTheorem impliesZ(X)
isan essential X-block.
C. Bonatti’s proved a remarkable extension of the Poincare´-Hopf Theorem to
certain pairs of commuting analytic vector fields on manifolds that need not be
compact:
Theorem (Bonatti [2]). Assume dimM ≤ 4 and ∂M = ∅. If X,Y ∈ Vω(M) and
[X,Y] = 0, thenZ(Y)meets everyessential X-block.2
Related resultsare inthearticles [3,9, 10,13, 16, 17, 22, 26].
Ourmainresult isan extensionofBonatti’sTheorem:
Theorem 1. Let M be a connected 3-manifold and A ⊂ Vω(M) an abelian Lie
algebra of analytic vector fields on M. Assume X ∈ A is nontrivial and Z(X) ∩
∂M = ∅. If K isan essential X-block, thenZ(A)∩K , ∅.
The proof, in Section 3, relies heavily on Bonatti’s Theorem. An analog for sur-
faces isprovedin Hirsch[9, Thm.1.3].
1TheindexcanbeequivalentlydefinedastheintersectionnumberofX(U)withthezerosection
of the tangentbundle of U ([2]); and as the the fixed-pointindex of the time-t map of the local
flowofX|U forsufficientlysmallt>0. ([5,11,12].)
2“Thedemonstrationofthisresultinvolvesabeautifulandquitedifficultlocalstudyoftheset
ofzerosofX,asananalyticY-invariantset.” —P.Molino[20]
2
Applicationto attractors
The interior Int(L) of a subset L ⊂ M is the union of all open subsets of M
containedin L.
Fix a metric on M. If Q ⊂ M is closed, the minimumdistance from z ∈ M to
pointsof Q isdenotedby dist(z,Q).
Let X ∈ V1(M) have local flow Φ. An attractor for X (see [1, 4, 7, 25])
is a nonempty compact set P ⊂ M that is invariant under Φ and has a compact
neighborhood N ⊂ M suchthat
Φ(N) ⊂ (N)
t
and
limdist(Φ(x,P),P) = 0 uniformlyinx ∈ N. (1)
t
t→∞
Such an N can bechosen sothat
t > s ≥ 0 =⇒ Φ(N) ⊂ Int(Φ (N)), (2)
t s
which is assumed from now on. In addition, using a result of F. W. Wilson [27,
Thm.2.2]wechoose N so that:
N is acompactC1 submanifoldand X is inwardlytransverseto∂N.3 (3)
Theorem2. Let M,AandX beasinTheorem1. IfP ⊂ M isacompactattractor
for X andχ(P) , 0, thenZ(A)∩P , ∅.
Proof. P is a proper subset of M— otherwise M is a closed 3-manifold having
nonzero Euler characteristic, an impossibility(e.g., Hirsch [8, Thm. 5.2.5]). Fix
N as aboveandnotethatχ(N) , 0.
By (3)andPoincare´-HopfTheoremthereisanessential X-block K ⊂ N\∂N,
andK ⊂ Pby(1). Standardhomologytheoryand(2)implythattheinclusionmap
P ֒→ N inducesan isomorphismsonsingularhomology,henceχ(N) = χ(P) , 0.
Theconclusionfollowsfrom Theorem1 appliedto thedata M′,A′,X′:
M′ := N, A′ := Y|N: Y ∈ A , X′ := X|N.
(cid:8) (cid:9)
Example 3. Denote the inner product of x,y ∈ R3 by hx,yi and the norm of x by
kxk. Let B ⊂ R3 denotetheopen ballabouttheoriginofradiusr > 0.
r
• AssumeAisanabelianLiealgebraofanalyticvectorfieldsonanopenset
M ⊂ R3 thatcontains B . Let X ∈ A andr > 0 besuchthat
r
kxk = r =⇒ hX ,pi < 0.
p
Then Z(A)∩B , ∅.
r
Proof. ThisisaconsequenceofTheorem2: B containsanattractorforX because
r
X inwardlytransverseto∂B and χ(B ) = 1.
r r
3ThismeansX isnottangentto∂N if p∈∂N.
p
3
2 Background material
Lemma 4 (Invariance). If T,S ∈ Athen Z(S)is invariantunderT.
Proof. Let Φ := {Φ} and Ψ := {Ψ } denotethe local flows of T and S, respec-
t t∈R s
tively. If t,s ∈ Rare sufficientlycloseto0, because[T,S] = 0wehave
Φ ◦Ψ = Ψ ◦Φ,
t s s t
and
Z(S) = Fix(Ψ) := Fix(Ψ ),
\ s
s
whereFixdenotes thefixed pointset. Suppose p ∈ Z(S). Then p ∈ Fix(Ψ), and
Ψ ◦Φ(p) = Φ ◦Ψ (p) = Φ(p).
s t t s t
Consequently Φ(p) ∈ Fix(Ψ ) for sufficiently small |t|,|s|, implying the conclu-
t s
sion.
AclosedsetQ ⊂ MisananalyticsubspaceofM,oranalyticin M,providedQ
hasalocallyfinitecoveringbyzero setsofanalyticmapsdefinedonopensubsets
of M. Thisisabbreviatedtoanalyticspacewhentheambientmanifold M isclear
from the context. The connected components of analytic spaces are also analytic
spaces.
Analytic spaces have very simple local topology, owing to the theorem of
Łojasiewicz [18]:
Theorem 5 (Triangulation). If T is a locally finite collection of analytic spaces
in M, there is a triangulation of M such that each element of T is covered by a
subcomplex.
TheproofofTheorem1 usesthefollowingfolktheorem:
Theorem 6 (Stability). Assume X ∈ V(M)and U ⊂ M isisolatingfor X.
(a) If i(X,U) , 0 thenZ(X)∩U , ∅.
(b) If Y ∈ V(M) is sufficiently close to X, then U is isolatingfor Y and i(Y,U) =
i(X,U).
Proof. See Hirsch[9, Thm.3.9].
Let Z(S)denotetheset ofcommonzeros ofasubsetS ⊂ Vω(M).
Proposition7. ThefollowingconditionsholdforeveryS ⊂ A:
(a) Z(S)isanalyticin M.
(b) EveryzerodimensionalA-invariantsetlies inZ(A).
Proof. Left tothereader.
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3 Proof of Theorem 1
Recall thehypothesesoftheMainTheorem:
• M isa 3-dimensionalmanifold,
• A ⊂ Vω(M)isan abelianLiealgebra,
• X ∈ Aisnontrivial, Z(X)∩∂M = ∅, and K isanessentialblockofzeroes
for X.
Theconclusiontobeprovedis: Z(A)∩K , ∅. ItsufficestoshowthatZ(A)meets
everyneighborhoodof K, becauseZ(A)isclosedand K iscompact.
Case I: dimA = d < ∞. The special case d ≤ 2 is covered by Bonatti’s
Theorem. We proceed byinductionond:
Induction Hypothesis
• dimA = d +1, d ≥ 2.
• Thezeroset ofeveryd-dimensionalsubalgebraofA meets K.
Arguingby contradiction,weassumepercontra:
(PC) Z(A)∩K = ∅.
An importantconsequenceis:
(A) dimK ≤ 2.
For otherwise dimK = 3, which entails the contradiction that X is trivial: X is
analyticand vanishesona3-simplexin theconnected 3-manifold M.
TheStabilityTheorem(6)impliesX hasaneighboroodN ⊂ Vω(X)withthe
U
followingproperty:
(B) Y ∈ N =⇒ U isisolatingforY and i(Y,U) = i(X,U) , 0.
U
LetG (A)denoted-dimensionalGrassmannmanifoldofd-dimensionallinear
d
subspacesB ofA;theseareabelian subalgebras.
Thenonemptyset
G (N ) := {B ∈ G (A): B∩N , ∅}
d U d U
isopen inG (A), hence itisad-dimensionalanalyticmanifold.
d
Bonatti’sTheorem and(B) imply:
(C) Z(B)∩K , ∅ forallB ∈ G (N ).
d U
A keytopologicalconsequenceof(C) is:
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(D) IfBandB′ aredistinctelementsofG (N ),thenZ(B)∩K andZ(B′)∩K are
d U
disjoint.
ThisholdsbecauseB∪B′ spansA,hence (PC) implies
Z(B)∩K Z(B′)∩K = Z(A)∩K = ∅.
(cid:0) (cid:1)T(cid:0) (cid:1)
EachsetZ(B)∩K isinvariant(Lemma4)andthereforehaspositivedimension
by (PC). Moreover:
(E) Theset Γ := B ∈ G (N ): dim Z(B)∩K = 2 isfinite.
U (cid:8) d U (cid:9)
For otherwise (D) implies K contains an infinite sequence of pairwise disjoint
compact subsets that are 2-dimensional and analytic in M. But this is impossible
by (A)andtheTriangulationTheorem5.
(E)showsthatΓ = ∅providedU issmallenough. Thereforewecanassume:
U
(F) dim Z(B)∩K = 1 forallB ∈ G (N ).
d U
Theset
Q := (B,p) ∈ G (N )× M: p ∈ Z(B)∩K .
(cid:8) d U (cid:9)
isanalyticinG (N )× M (Proposition7). Thenatural projections
d U
π : Q → G (N ), π : Q → K
1 d U 2
are analytic,π is surjective,π is injectiveby(D).
1 2
ThesetsZ(B)∩K arethereforepairwisedisjoint,andeachisa1-dimensional
analyticsubspaces of Q by(F). Therefore
dimQ = dimG (A)+dim(Z(B)∩K) ≤ dimK,
d
whence
dimQ = d +1 ≤ 2.
Butthisisimpossiblebecaused ≥ 2bytheInductionHypothesis. Thiscompletes
theinductiveproofoftheMainTheoremin CaseI.
CaseII: dimA isarbitrary.
ConsiderthefamilyFofcompactsubsetsof K:
F := Z(A′)∩K: A′ ⊂ Ais afinite-dimensionalsubalgebra .
(cid:8) (cid:9)
Evidently
S = Z(A)∩K.
\
S∈F
CaseIshowseveryfinitesubsetofFhasnonemptyintersection. AsK iscompact,
all theelementsofFhavenonmptyintersection,provingZ(A)∩K , ∅.
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