Center-unstable foliations do not have compact leaves

0.09 MB·English

by  Federico Rodriguez Hertz

CENTER-UNSTABLE FOLIATIONS DO NOT HAVE COMPACT
5 LEAVES
1
0
2 F.RODRIGUEZHERTZ,J.RODRIGUEZHERTZ,ANDR.URES

n
a
J ABSTRACT. Forapartiallyhyperbolicdiffeomorphismona3-manifold,
we show that any invariant foliation tangent to the center-unstable (or
1
3 center-stable)bundlehasnocompactleaves.

]
S
D
h. 1. INTRODUCTION
t
a This article deals with the integrability of invariant distributions arising
m
in certain dynamical systems. The integrability of tangent distributions of
[
k−planes is an important problem that has been studied for more than a

1 century. Under certain hypotheses, there have been quite satisfactory an-
v
swers to the unique integrability problem. For instance, Picard’s theorem
5
2 in the one-dimensional case, and Frobenius’ theory in higher dimensions.
1
However,bothresultsinvolveregularityofthedistributions.
0
0 The problem is that, in general, distributions arising from a dynamical
2. system are only Ho¨lder-continuous, even if the system is smooth. There-
0 fore, other elements of analysis are required in order to establishnecessary
5
orsufficient conditionsforintegrability.
1
: Consider, for instance, the case of hyperbolic systems. Namely, diffeo-
v
i morphims f : M → M for which there exists an invariant splitting of the
X
tangent bundle into two invariant distributions, called stable and unstable:
r
a TM =Es⊕Eu such that, forsomeRiemannian metrick.k and all unitvec-
torsv ∈Es and v ∈Eu wehave:
s u
kDf(x)v k<1<kDf(x)v k.
s u
Inthiscase,bothdistributions,stableandunstable,areuniquelyintegrable.
This can be achieved by applying essentially either one of the following
two methods: Hadamard’s method [8], which consists in seeing each in-
variant integral manifold as a fixed point of a contracting operator acting
in an appropriate space of functions, or Perron’s method [12], which con-
sists in applying the Implicit Function Theorem to an operator acting in an
appropriatefunctionspace.
Date:February3,2015.
1
2 F.RODRIGUEZHERTZ,J.RODRIGUEZHERTZ,ANDR.URES
Hereweshallfocusontheintegrabilityofsomedistributionsarisingfrom
a more general kind of systems, the partially hyperbolic ones. These sys-
tems involve not only a contracting and an expanding bundle as the ones
mentioned above, but also a center bundle, which has an intermediate be-
havior. Namely; a diffeomorphism f : M → M is partially hyperbolic if
the tangent bundle of M admits an invariant splitting into 3 bundles, called
respectively,stable,centerandunstablebundles,TM=Es⊕Ec⊕Eu,such
that, for some Riemannian metric k.k, and all unit vectors v ∈Es, v ∈Ec
s c
andv ∈Eu wehave
u
kDf(x)v k<kDf(x)v k<kDf(x)v k and kDf(x)v k<1<kDf(x)v k
s c u s u
Hadamard’s and Perron’s methods can be used to show that also in this
setting, the stable and unstable bundles are uniquely integrable, see for in-
stance[4,11]. However,itisamoredelicatemattertodeterminewetherEc
oreven Ecs =Es⊕Ec orEcu =Ec⊕Eu are integrable.
Wewillsaythat f iscs-dynamicallycoherentifthereexistsan f-invariant
foliation tangent to Ecs. The cu-dynamical coherence is defined analo-
gously. A diffeomorphism f is said to be dynamicallycoherent ifit is both
cs-andcu-dynamicallycoherent. Thisimpliesinparticularthatthereexists
an f-invariantfoliationtangentto Ec.
Partially hyperbolic diffeomorphisms are not dynamically coherent in
general. Indeed,asitwasobservedbyA.Wilkinsonin[17],anon-dynamically
coherent example is given by an algebraic Anosov diffeomorphim in a six
dimensional manifold, that was presented in the well-known survey by S.
Smale [16] and is attributed to A. Borel. In this example, the sub-bundles
¥
areC , thecenterbundleis4-dimensionalanditcannotbeintegrablesince
theFrobenius conditionisnot satisfied.
Is the lack of Frobenius condition the only reason for non-integrability
of the center bundle? What about the case of one-dimensional Ec, where
Frobenius condition is always trivially satisfied? The question of whether
a partially hyperbolic diffeomorphism existed for a one-dimensional non-
integrable center bundle remained open since the 70’s. In [15] the au-
thorsansweredthequestionnegatively. Theyconstructedexamplesofnon-
dynamicallycoherent partially hyperbolicdiffeomorphismson T3. In [15],
it is also given (using the same methods) an example of a dynamically co-
herent diffeomorphism with non-locally uniquely integrable center folia-
tion. The existence of such examples contrasts with the result obtained by
M. Brin, D. Burago and S. Ivanov [2] proving that diffeomorphisms on T3
satisfyingamorerestrictivedefinitionofpartialhyperbolicity(theabsolute
partialhyperbolicity)aredynamicallycoherent.
CENTER-UNSTABLEFOLIATIONSDONOTHAVECOMPACTLEAVES 3
Anaturalquestionisthen: aretherenecessaryorsufficientconditionsfor
dynamicalcoherence? Inthispaperweprovidesomenecessaryconditions:
Theorem 1.1. Let M be a closed 3-dimensional manifold and f :M →M
be a cu-dynamically coherent partially hyperbolic diffeomorphism. Then,
thecenter unstablefoliationhas nocompact leaves.
Let us note that any compact leaf tangent to Ecu must be a torus, by
Poincare´-Hopf, due to the fact that is is foliated by lines. We shall call
cu-torusany2-torustangent toEcu.
As a matter of fact, the non-dynamically coherent example [15] was in-
spiredinthisresult. Sinceanycenter-unstablefoliationcannothaveatorus
leaf,weconstructedaplanefieldthatwas uniquelyintegrableoutsideacu-
torus in such a way that if a foliation existed, it should contain the torus
as a leaf. The fact that this is not possible provided the first example of
a non dynamically coherent partially hyperbolicdiffeomorphismwith one-
dimensionalcenterbundle.
Observe that Theorem 1.1 does not prevent the existence of tori, even
invariant,tangent to Ecu. Theorem 1.1 asserts theimpossibilityof theexis-
tenceofsuchtoriaspartofaninvariantfoliationtangenttoEcu. Indeed,in
[15]wealsoprovideanexampleofadynamicallycoherentpartiallyhyper-
bolicdiffeomorphismofT3 withan invariantcu-torus.
Theorem 1.1 then states that if there is a center-unstable invariant folia-
tion,no leafcan beatorus. Weconjecturetheconverseisalsotrue:
Conjecture 1.2. If a partially hyperbolic diffeomorphism f : M3 → M3 is
notdynamicallycoherent,then itadmitseithera cu-oran sc-torus.
In Section 2 we prove that, in fact, not every manifold can contain a cu-
oran sc-torus:
Theorem 1.3. Let f :M →M be a partially hyperbolic diffeomorphism of
an orientable 3-manifold. If there exists a torus tangent to either Es⊕Eu,
Ec⊕Eu orEs⊕Ec, thenthemanifoldM iseither:
(1) the3-torusT3
(2) themappingtorusof−id :T2 →T2
(3) themappingtorusofa hyperbolicautomorphismA:T2 →T2
This follows essentially from Proposition 2.3, where we prove that the
dynamics on any invariant torus tangent to Ecu, or Esc is isotopic to the
one generated by a hyperbolic linear automorphism. When a manifold M
admitsan embedded 2-torus and a global diffeomorphismg:M →M such
that g(T) = T and g| is isotopic to an Anosov diffeomorphism, then we
T
saythat M admitsan Anosovtorus.
4 F.RODRIGUEZHERTZ,J.RODRIGUEZHERTZ,ANDR.URES
In[13],weprovedthattheonlyirreduciblemanifoldsadmittinganAnosov
torusaretheoneslistedinTheorem1.3. Butontheotherhand,ifM admits
a partially hyperbolic diffeomorphism, then M is an irreducible manifold.
SeeSection 2 formoredetails.
As a consequence of Theorem 1.3, if Conjecture 1.2 were true, then
the only manifolds supporting non-dynamically coherent diffeomorphisms
wouldbetheoneslistedinTheorem1.3. InProposition2.2weshowthatthe
existenceofacu-torusimpliestheexistenceofaperiodiccu-torus,whichis
attracting. An analogous statement holds for sc-tori. Therefore, were Con-
jecture1.2true, allpartiallyhyperbolicdiffeomorphisms f withW (f)=M
would be dynamically coherent. Hammerlindl and Potrie have proven that
Conjecture 1.2 is true for manifoldswith solvablefundamental group [10];
namely,forthemanifoldsthatarefinitelycoveredbytheoneslistedinThe-
orem 1.3. This is the greatest advance in Conjecture 1.2 so far. Theorem
1.1isused intheirproof.
A foliation is taut if there is an embedded circle that transversely inter-
sects each one of its leaves. Taut foliations play an important role in the
descriptionof3-dimensionalmanifolds. In Section 3 weshow:
Theorem 1.4. Let Es be the strong stable bundle of a partially hyperbolic
diffeomorphism of a closed orientable 3-dimensional manifold M. If F is
a foliationtransverse to Es that is not taut, then there exists a periodic cu-
torus. In particular,M admitsan Anosovtorus.
A foliation F like the one mentioned in Theorem 1.4 always exists, due
to Burago-Ivanov [3], see more details in Theorem 3.1. As a consequence
of Theorem 1.4, all manifolds supporting partially hyperbolic diffeomor-
phismsarefinitelycoveredbymanifoldssupportingtautfoliations. Perhaps
thetheoryoftautfoliationscouldgivesomeenlighteningtothedescription
ofpartiallyhyperbolicsystems(see, forinstance,[5]).
2. DYNAMICS ON cu- AND sc-TORI
In this section, we shall proveTheorem 1.3, which will follow from cer-
taindynamicalpropertiesofcu-and sc-tori.
As we said in the Introduction, a manifold M admits an Anosov torus if
thereexistsadiffeomorphismg:M→M andanembeddedg-invarianttorus
T such that (g| ) :p (T2)→p (T2) is hyperbolic. Admitting an Anosov
T ∗ 1 1
torusisaglobalproperty. In [13], weprovethatveryfew3-manifoldshave
suchaproperty.
Theorem2.1. [13]LetMbeanirreducibleorientable3-manifoldadmitting
anAnosovtorus,thenM is either:
CENTER-UNSTABLEFOLIATIONSDONOTHAVECOMPACTLEAVES 5
(1) the3-torusT3
(2) themappingtorusof−id :T2 →T2
(3) themappingtorusofa hyperbolicautomorphismA:T2 →T2
A 3-manifold admitting a partially hyperbolicdiffeomorphismis always
irreducible,see[6,Lemma6.3]. Theorem1.3thenfollowsfromthefollow-
ingpropositions:
Proposition 2.2. The existence of a cu-torus implies the existence of a pe-
riodiccu-torus.
Proposition 2.3. The dynamics on an invariant cu-torus is isotopic to hy-
perbolic.
Proposition2.4. Amanifoldadmittingansu-torus,admitsanAnosovtorus.
Proposition2.3isadirectcorollaryofLemma2.5below. SeealsoPropo-
sition2.1of[1]fora similarresult andproof.
Lemma 2.5. Let W be a foliation of T2 with continuous tangent bun-
dle TW and invariant by a diffeomorphism g. Suppose, in addition, that
||dg| ||>1. Then, g :p (T2)→p (T2) ishyperbolic.
TW ∗ 1 1
Proof. By taking g2 if necessary we can suppose that g preserves the ori-
entation of TW. Since g preserves a foliation without compact leaves,
g : Z2 → Z2 (we identify p (T2) with Z2) has an eingenspace of irra-
∗ 1
tional slope. This implies that either g is hyperbolic or g = Id. In the
∗ ∗
second case g has a lift g˜ : R2 → R2 such that g˜ = Id+a where a is a
periodic, and in particular bounded, function. As a consequence we ob-
tain that there exists a constant K >0 such that given any subset of R2, X,
diam(gn(X))≤ diam(X)+nK. Let g be an arc contained in a leaf of W.
Then, the length of g grows exponentiallywhileits diameter grows at most
linearly. Thisimpliesthatgivenasmalle >0thereexistsaniterateofg that
contains a curve of length arbitrarily large and with end points at distance
less than e . Using Poincare´-Bendixon we obtain a compact leaf. This is a
contradictionand then,g ishyperbolic. (cid:3)
∗
ProofofProposition2.2. Let T be a cu-torus, and consider the sequence
f−n(T). Since the family of all compact subsets of M, considered with the
Hausdorff metric dH, is compact, there is a subsequence f−nk(T) converg-
ing to a compact set K ⊂M. Therefore, for each e >0 there are arbitrarily
largeN >>L>0such thatd (f−N(T), f−L(T))<e .
H
Since T is transverse to the stable foliation, the union of all local stable
leaves of T forms a small tubular neighborhood of T, U(T). Since stable
leavesgrowexponentiallyunder f−1,ifN >>Lasabovearelargeenough,
then f−L(U(T))⊂ f−N(U(T)). Thisimpliesthat fN−L(U(T))⊂U(T).
6 F.RODRIGUEZHERTZ,J.RODRIGUEZHERTZ,ANDR.URES
ThesetT =∩¥ fk(N−L)(U(T))isaperiodiccu-torus. Indeed,itiseasy
0 k=0
to see that T is periodic and homeomorphicto a torus. On the other hand,
0
for each point x of T , its tangent space is limit of the tangent spaces of
0
pointsx in fk(N−L)(T),which arecu-tori. HenceT T =Ec⊕Eu. (cid:3)
k x 0 x x
ProofofProposition2.4. Assume f admits an su-torus, and consider the
laminationL ofallsu-toriof f. Thisisacompactlamination[9]. Therefore,
thereisarecurrent leaf.,thatis,thereisatorusT andaniteraten,suchthat
d (fn(T),T) < e for small e . There exists a dipheotopy i on M, taking
C1 t
fn(T) into T. Then f = fn◦i fixes T and f |T is isotopic to an Anosov
1
diffeomorphism. (cid:3)
3. WEAK FOLIATIONS OF PARTIALLY HYPERBOLIC DIFFEOMORPHISMS
In this section we prove Theorem 1.4. For any partially hyperbolic dif-
feomorphismofa3-manifoldsuchthattheinvariantbundlesareorientable,
BuragoandIvanov[3]haveprovedthatthereare(notnecessarilyinvariant)
foliationsthat“almost”integrateEcs =Ec⊕Es , s =s,u.
Theorem 3.1(KeyLemma2.2,[3]). Let f beapartiallyhyperbolicdiffeo-
morphism of a closed 3-manifold and let E∗ be orientable for ∗ = s,c,u.
Then, for every e >0 there is a foliation Fecs such that TFecs is a contin-
uous bundle and the angles between TFcs and Ecs are no greater than e ,
e
s =s,u.
In this section we prove that these foliations, if the manifold is different
fromtheoneslistedinTheorem1.3,aretaut. Recallthatacodimensionone
foliation is taut if there exists an embedded S1 that intersects transversely,
andnontrivially,everyleaf ofthefoliation(see[5]).
LetF beacodimension-onefoliation. Adead-endcomponentisanopen
submanifold N M which is a union of leaves of F, such that there is no
properly immersed line transverse to F. That is, there is no a :[a,b]→M
transversetoF suchthat a (a,b)⊂N, a (a),a (b)∈¶ N.
A Reeb component is a solid torus whose interior is foliated by planes
transverse to the core of the solid torus, such that each leaf limits on the
boundary torus, which is also a leaf. Observe that the interior of a Reeb
component is a dead-end component. Other examples of dead-end compo-
nents are obtained by taking a Reeb foliation of the annulus, multiplying
by the interval and gluing the two boundary annulus using a rotation that
preservestheReebfoliation. Observethatinbothcasestheboundaryofthe
dead-end component consists of tori. This is a general fact that is stated in
thefollowinglemma.
CENTER-UNSTABLEFOLIATIONSDONOTHAVECOMPACTLEAVES 7
Lemma3.2(Lemma4.28,[5]). LetMbea3-dimensionalorientableclosed
manifold. A foliation F of M is taut if and only if it contains no dead-end
components. If N is a dead-end component, then the restriction of F to N
is transversely orientable and N\N consists of a union of tori leaves of F.
Moreover,boundaryleavesofN cannotbejoinedbyanarcinN transverse
toF.
Observe that the last assertion implies that the boundary leaves of any
dead-end component of a foliation have half-neighborhoods in N with size
uniformlyboundedby below.
It is an obvious corollary of Lemma 3.2 that foliations without compact
leaves are taut. For instance, the weak stable and weak unstable foliations
ofAnosovflowsof3-dimensionalmanifoldsare taut.
ProofofTheorem 1.4. Partialhyperbolicityimpliesthattheforwarditerates
ofTF convergeto Ecu. Let N bea dead-end ofF and let T bea boundary
component of N. Hence T is a torus, transverse to Es. All iterates f−n(T)
are tori transverse to Es. The proof follows now exactly as in Proposition
2.2.
(cid:3)
4. PROOF OF THEOREM 1.1
Letussupposethatthereexistsan invariantfoliationFcu tangenttoEcu,
and that Fcu has a compact leaf, which must be a torus. (Then M must be
oneofthemanifoldslistedinTheorem1.3). ByProposition2.2,thereexists
aperiodiccu-torusT. Bytakinganiterate,wecanassumethatisfixed. The
dynamics on T is isotopic to hyperbolic, due to Proposition 2.3. In [13] it
is shown that cutting M along T we obtain a manifold with boundary, that
isdiffeomorphictoT2×[0,1]. Moreover, f induces adiffeomorphismgof
T2×[0,1] isotopic to A×id where A is a hyperbolic automorphism of T2
and id is the identity map of the interval [0,1]. Then, [7] implies that there
existsasemiconjugacyh:T2×[0,1]→T2,betweengandA,homotopicto
theprojection p:T2×[0,1]→T2. Observealsothat h(T2×{0})=T2.
The torus T2×{0} is foliated by a foliation Su by lines that are integral
curves ofthestrong unstablefoliation. Call S˜u thelift ofSu to R2, theuni-
versalcoverofT2×{0}. It isnotdifficultto seethatifx, yarein thesame
leaf Su of S˜u the fact that dist (x,y) goes to infinity implies that dist(x,y)
u
goes to infinity (dist (x,y) is the length of the arc of Su joining x and y).
u
Let h¯ be a lift of h| to R2. Since the diameters of the sets h¯−1(y)
T2×{0}
are uniformly bounded and f(h¯−1(y)) = h¯−1(Ay), we obtain that the map
h¯ is injective when restricted to strong unstable manifolds. Recall that the
8 F.RODRIGUEZHERTZ,J.RODRIGUEZHERTZ,ANDR.URES
imageofan unstablemanifoldby h isan unstablemanifoldofA.
Let usshowthattheimageofacentercurvebyh iscontained inastable
manifold of A. For this, it is enough to show that the length of the forward
iteratesofthecurvearebounded. Letg bea(small)centercurveandletd be
suchthatWu(x)∩Wu(y)=0/ forallx6=y∈g . LetWu(g )=S{Wu(x);x∈g }.
d d d d
Since f expandstheunstablebundlewehavethatWu(fn(g ))⊂ fn(Wu(g )).
d d
But since the angle between the center bundle and the unstable bundle is
bounded by below, we have that the area of Wu(fn(g )) goes to infinity as
d
thelengthof fn(g ) goesto infinity whichis a contradiction. Moreover,this
impliesthatEc isuniquelyintegrablefor f| . Ifthiswerenotthecasethere
T
wouldexist,intheuniversalcover,twocentercurvesg , g beginningatthe
1 2
samepointxand cuttinga nearby unstablemanifoldat two different points
y,z. By forward-iterating this “triangle” we would obtain that the distance
between fn(y) and fn(z) goes to infinity. Then, either the length of g or
1
the length of g goes to infinity, which contradicts what we have proved
2
before. Summarizing, f| has two invariant foliation by lines, one tangent
T
to the unstable bundle and the other one tangent to the center bundle; the
semiconjugacysendstheunstableleavestounstablelinesofAand thecen-
terleavesto stablelines ofA. Againthedistancebetween twopointsin the
same center leaf in the universal cover of T goes to infinity as the length
ofthe center curve goes to infinity. It is not difficult to see that this implies
that,foranyy∈T2,h−1(y)∩T isaconnectedarccontainedinacenterleaf.
Let p∈T be a periodic point. We may assume that J =h−1(h(p))∩T
p
is a very small arc. This is easy to obtain since there are infinitely many
periodicpointsin differentcenter curves. LetU ⊂T2×I bea smallneigh-
borhood of J , and let y∈h−1(h(p))∩U. We will prove that y∈Wcs(p).
p loc
On one hand, if y∈/ Wcs(p) then z=Ws (y)∩T is not in the local center
loc loc
manifold of p. On the other hand, h(z)∈h(Ws (y))⊂Ws (h(p)) is in the
loc loc
local stable manifold of h(p) for A, which contradicts the fact that it is not
inthelocalcenter manifold(in T)of p.
Let us focus onWcs(p). Theintersectionofthecenter unstablefoliation
loc
FcuwithWcs(p)foliatesWcs(p)bycenterarcs. Bycontinuity,anyofthese
loc loc
centerarcs,closeenoughtoT,hasapointofh−1(h(p)). Certainly,thesame
is true for p¯, a lift of p to universal cover. We choose x ,...,x ∈Wcs(p¯)
1 N loc
points that are in h¯−1(h¯(p¯)) and in different center curves. Let C > 0 be
such that diam(h¯−1(y))<C for every y∈T and e >0 so small that if two
points are at a distance less than e , then they are in a trivializing chart of
Fcu. Now, we choose N in such a way that if N points are contained in a
setofdiameterC then,at leasttwoofthemareadistancelessthane . Since
CENTER-UNSTABLEFOLIATIONSDONOTHAVECOMPACTLEAVES 9
x
j
a ⊂Ws(x )
2 j
a ⊂Wc(x)
1 i
x
i
J
p¯
FIGURE 1. Wcs(p¯)
f¯n(h¯−1(h¯(p¯)))=h¯−1(h¯(f¯n(p¯))) we have that diam{f¯n(x ),..., f¯n(x )}<
1 n
C, ∀n ∈ Z. Then, there exists a subsequence n → −¥ and two different
k
points xi and xj such that dist(f¯nk(xi), f¯nk(xj))< e , ∀k >0. Now, take an
arc a joiningx and x that consistsof two sub-arcs, a beginningat x and
i j 1 i
tangent to the center bundle and a ending at x and tangent to stable one
2 j
(seeFigure1).
For k large enough we obtain that f¯nk(a 2) is a very long stable curve,
f¯nk(a 1) is contained in a leaf of F˜ cu and the extremes of f¯nk(a ) are a dis-
tancelessthan e (seeFigure2).
f¯n(a ) F¯cu(f¯n(x ))
2 1
f¯n(a )
1
f¯n(x )
1
f¯n(x )
2
FIGURE 2. f¯n(a )
Standard arguments offoliationtheory implythat there is a closed curve
transverse to F˜ cu which implies the existence of a Reeb component which
is, as it is well known (see for instance [3, 14]), impossible. This finishes
theproofofTheorem1.1.
10 F.RODRIGUEZHERTZ,J.RODRIGUEZHERTZ,ANDR.URES
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