Non-abelian cohomology of abelian Anosov actions

30 Pages·2000·0.26 MB·English

Ergod.Th.&Dynam.Sys.(2000),20,259–288
PrintedintheUnitedKingdom (cid:13)c 2000CambridgeUniversityPress
Non-abelian cohomology of abelian
Anosov actions
ANATOLEKATOK†,VIORELNIT¸ICA˘‡§andANDREITO¨RO¨K‡¶
†DepartmentofMathematics,ThePennsylvaniaStateUniversity,UniversityPark,
PA16802,USA
(e-mail:katok [email protected])
‡InstituteofMathematicsoftheRomanianAcademy,P.O.Box1–764,
RO-70700Bucharest,Romania
(Received27January1998andacceptedinrevisedform15May1998)
Abstract. We develop a new technique for calculating the first cohomology of certain
classesofactionsofhigher-rankabeliangroups(Zk andRk,k (cid:21)2)withvaluesinalinear
Liegroup. Inthispaperweconsiderthediscrete-timecase. Ourresultsapplytococycles
of differentregularity, fromHo¨lder to smooth and real-analytic. The main conclusionis
that the corresponding cohomology trivializes, i.e. that any cocycle from a given class
is cohomologousto a constant cocycle. The principalnovelfeature of our method is its
geometric character; no global informationabout the action based on harmonic analysis
isused. Themethodcanbedevelopedtoapplytococycleswithvaluesincertaininfinite
dimensionalgroupsandtorigidityproblems.
1. Introduction
1.1. Basicdefinitions. LetGbeagroupactingonacompactboundarylessRiemannian
manifold M by (cid:11) V G(cid:2)M ! M, .g;x/ 7! (cid:11) .x/ (cid:17) gx. Let 0 be some topological
g
group.Acocycle(cid:12) overtheaction(cid:11)isacontinuousfunction(cid:12) VG(cid:2)M !0suchthat
(cid:12).g g ;x/D(cid:12).g ;g x/(cid:12).g ;x/; (1.1)
1 2 1 2 2
forallg ;g 2G,x 2M.
1 2
A geometricinterpretationof a cocycleis the following: considerthe trivialprincipal
0-bundle E D M (cid:2)0 over M. Then the cocycle (cid:12) described above corresponds to a
lift of the action (cid:11) to an action (cid:11)Q V G (cid:2)E ! E by principal bundle maps. Namely,
§Currentaddress:DepartmentofMathematics,UniversityofNotreDame,NotreDame,IN46556–5683,USA.
(e-mail:[email protected])
¶Currentaddress:DepartmentofMathematics,UniversityofHouston,Houston,TX77204–3476,USA.
(e-mail:[email protected])
260 A.Katoketal
g 2 Ginducesthemap(cid:11)Q V E ! E givenby.x;h/ 7! .(cid:11) .x/;(cid:12).g;x/h/. Thecocycle
g g
equation(1.1)isequivalenttothefactthat(cid:11)Q isanaction,i.e.(cid:11)Q (cid:11)Q D(cid:11)Q .
g1 g2 g1g2
If0 D Aut.F/forsomespaceF,thenacocycle(cid:12) V G(cid:2)M ! 0alsocorrespondsto
a liftof(cid:11) to anactionbybundlemapson thetrivialbundleM (cid:2)F. Inthiscase g 2 G
actsby.x;(cid:24)/ 7! .(cid:11) .x/;(cid:12).g;x/.(cid:24)//. Here‘Aut.(cid:1)/’hasthemeaningappropriateforthe
g
structureofF. ItcanbeGL.(cid:1)/forF alinearspace,orDiff.(cid:1)/forF amanifold.
Thenaturalequivalencerelationforcocyclesisthecohomology. Twococycles(cid:12) and
1
(cid:12) arecalledcohomologousifthereexistsacontinuousmapP VM !0suchthat
2
(cid:12) .g;x/DP.gx/(cid:12) .g;x/P.x/−1; (1.2)
1 2
forallg 2G,x 2M. SuchamapP iscalledatransfermap.
Acocycle(cid:12)iscohomologoustoaconstantcocycleifthereexistsacontinuousfunction
P VM !0andahomomorphism(cid:25) VG!0suchthat
(cid:12).g;x/DP.gx/(cid:25).g/P.x/−1:
Inparticular,if(cid:25) isthetrivialhomomorphism,(cid:12) issaidtobecohomologoustothetrivial
cocycle.Inorderforacocycle(cid:12) tobecohomologicallytrivial,ithastosatisfytheclosing
conditions:(cid:12).g;x/DId forallg 2Gandx 2M suchthatgx Dx.
0
One of the central questions in studying cocycles over group actions is: When is a
cocyclecohomologoustoaconstant(trivial)cocycle?
Remark. For brevity, we use the term small for a cocycle whose values are close to the
identity,onacompactgeneratingsetinthegroupwhoseactionweconsider.
Werecallthedefinitionofapartiallyhyperbolicdiffeomorphism.
Let M be a compact manifold. A C1 diffeomorphism T V M ! M is called
partially hyperbolic if there is a continuous invariant splitting of the tangent bundle
TM D Es.T/(cid:8)E0.T/(cid:8)Eu.T/andconstantsC D C.T/,(cid:21)(cid:6) D (cid:21)(cid:6).T/,e(cid:21)(cid:6) De(cid:21)(cid:6).T/,
C >0,0<(cid:21)− <e(cid:21)− (cid:20)e(cid:21)C <(cid:21)C,(cid:21)− <1<(cid:21)C,suchthatforn2Z,n(cid:21)0:
kDTnvsk(cid:20)C(cid:21)n−kvsk; vs 2Es.T/;
kDT−nvuk(cid:20)C(cid:21)−Cnkvuk; vu 2Eu.T/;
kDT−nv0k(cid:20)Ce(cid:21)−−nkv0k; v0 2Es.T/;
kDTnv0k(cid:20)Ce(cid:21)nCkv0k; v0 2Es.T/:
IfE0 Df0gthenthediffeomorphismT iscalledAnosov.
The sub-bundles Es.T/ and Eu.T/ are called the stable and, respectively, unstable,
distributions. Thesedistributionsareintegrable. We denotebyWs.xIT/ andWu.xIT/,
respectively,thestableandunstablemanifoldsofthepointx 2M. Thestableandunstable
foliationsareHo¨lderfoliations.IfthediffeomorphismT 2CK.M/,thentheleavesofthe
stableandunstablefoliationsareCK too.
1.2. Historic remarks and outline. The study of cocycles over (transitive) Anosov
diffeomorphismsandflows(i.e.actionsofZandR,respectively)wasstartedintwopapers
Non-abeliancohomologyofabelianAnosovactions 261
byLivsic(moreappropriatelyspelledLivshits)[Li1,Li2],whichbecameveryinfluential
andgeneratedextensiveliterature.
Livsic provedthat a real-valuedHo¨lder cocyclewhich satisfies the closing conditions
is cohomologous to the trivial cocycle [Li1]. He also proved a similar result for small
cocycleswithvaluesinafinite-dimensionalLiegroup[Li2]. Inthesamepaperheclaims
a globalresult for cocycleswith values in arbitraryLie groups. His argumentworks for
solvablegroupsbutitismistakenforthegeneralcase. Thisquestionisstillopen.
Some early applications of Livsic’s results appeared in [LS], where, in particular, a
necessaryandsufficientconditionforthe existenceofan absolutelycontinuousinvariant
measureforanAnosovsystemisgiven.
Severalmajor developmentsfollowedthe work of Livsic. One directionis concerned
withtheregularityofthe(essentiallyunique,ifitexists)solutionP ofthecohomological
equation
(cid:12).g;x/DP.gx/P.x/−1:
Livsic showed that if a real-valued cocycle (cid:12) over an Anosov system is C1, then the
transfermap P is also C1 [Li1]. For some linear actionson a torushe also showed that
ifthecocycleisC1, respectivelyC!, thensoisthesolution[Li2]; thiswasobtainedby
studyingthedecayoftheFouriercoefficients.
1
LaterGuilleminandKazhdan[GK1,GK2]showedtheC regularityofthesolutions
inthecaseofgeodesicflowsonnegativelycurvedsurfaces. Collet,EpsteinandGallavotti
[CEG]provedaC! versionforgeodesicflowsonsurfacesofconstantnegativecurvature.
1
ThecompletesolutionfortheC caseappearsinthepaperbydelaLlave,Marcoand
1
Moriyon[LMM]. Theyshowed thatif a real-valuedcocycleovera C Anosovsystem
1 1
is cohomologically trivial and C , then the transfer map is C . This follows from a
generaltheoremfromharmonicanalysiswhich asserts thatif a functionis smoothalong
twotransversefoliationswhichareabsolutelycontinuousandwhoseJacobianshavesome
regularity properties, then it is smooth globally. This theorem was proved in [LMM]
using properties of elliptic operators. Later a more general result was proved by Journe´
[J], relying mainly on Taylor expansions and the estimate of the error: if a function is
CKC(cid:11) alongtheleavesoftwotransversefoliationswithuniformlysmoothleaves,thenthe
function is CKC(cid:11), (0 < (cid:11) < 1, K D 1;2;:::;1). Another approach is presented by
HurderandKatok[HK], basedonanunpublishedideaofC. Toll, in whichthe decayof
the Fourier coefficients is used to characterize smoothness. The method can be applied
forspanningfamiliesoffoliationswhichhavethesamepropertyasthoseusedin[LMM].
Note that foliationsarising from Anosovdiffeomorphismshave this property. Using the
approachin[HK],delaLlaveprovedtheanalyticcasein[Ll1].
In [NT1] the second and third authors provedthat a small cocycle with values in the
diffeomorphismgroupofacompactmanifoldwithtrivialtangentbundleiscohomologous
to the trivial cocycle, provided the closing conditions hold. The regularity results were
extendedtococycleswithvaluesinDiffandLiegroupsin[NT2,NT3]. SeeTheorem5.5
laterforsuchastatement. Theresultsin[NT3]areoptimal,asfarastheinitialregularity
ofthetransfermapisconcerned. Severalimprovementsof[NT1]arepresentedin[Ll2],
aswellasadifferenttreatmentofLivsic’sresults.
262 A.Katoketal
ForAnosovactionsofgroupsotherthanZandRthesituationmaybequitedifferent.
While in the above cases the closing conditions imply infinitely many independent
obstructionstotrivialization,foractionsofmanyothergroupsvariousrigidityphenomena
appear. For ‘large’ groups, such as lattices in higher-rank Lie groups, this is related to
thesuper-rigiditytheoremofZimmer[Z]andarenotaconsequenceofhyperbolicity.For
other groups (e.g. free non-abelian groups) rigidity does not take place. Note, however,
thatforgenericactionsofanygroupthatcontainatransitiveAnosovelement,theclosing
conditionsstillimplytrivialityofthecocycle(see[NT1,§5]).
Ontheotherhand,foractionsofhigher-rankabeliangroups(e.g.Zk andRk fork (cid:21)2),
cocyclerigidityappearsinconnectionwithhyperbolicbehavior.
Nevertheless, the proofs of these rigidity results relied on harmonic analysis (abelian
and non-abelian), more specifically on the exponential decay of Fourier coefficients for
smooth functions on a torus and exponentialdecay of matrix coefficients for irreducible
representations of semisimple Lie groups. Using these methods, Katok and Spatzier
showed in [KSp1, KSp2, KSp3] that real-valued cocycles over certain Anosov Rk
and Zk actions, k (cid:21) 2, are cohomologous to constant cocycles. Related results for
expansiveZk actionsbyautomorphismsofcompactabeliangroupswerefoundbyKatok
and Schmidt [KSch], and for higher-dimensional shifts of finite type were found by
Schmidt[Sch1, Sch2]. KatokandKatokprovedin [KK] similar resultsforhigher-order
cohomologies.
AdifferentapproachwassuggestedbyKatokinthespringof1994,basedonthenotion
ofTNS(i.e.totallynon-symplectic)Zk action.Hisoriginalargumentprovidedageometric
(i.e. independent of harmonic analysis) proof for some of the results in [KSp1, KSp2].
This method does not require algebraicity of the action, but assumes a special structure
of the stable and unstable manifoldsof variouselementsof the action. Using the notion
of TNS actions, Ni¸tica˘ and To¨ro¨kprovedcocyclerigidity for some Diff- and Lie-valued
cocycles.
Thecurrentpaperrepresentsanaccountofthesedevelopments.Werestrictourselvesto
thecaseofRandLie-groupvaluedcocycles.Ourresultsgiveapartialanswertoaquestion
askedbyKatokandSpatzierintheintroductionof[KSp1]aboutthegeneralizationoftheir
rigidityresultstococycleswithvaluesinnon-abeliangroups.Theresultsforcocycleswith
valuesindiffeomorphismgroupsarebepresentedelsewhere[NT4]. Theseareusedinthe
study of partially hyperbolic actions of higher-rank abelian groups, and to prove local
rigidityofsomepartiallyhyperbolicactionsoflatticesinhigher-rankLiegroups.
We describethe necessary notionsand formulatethe resultsin §2. In §3 we consider
1
in detailthe case of real-valuedC cocycles. This emphasizesthe maingeometricidea
of the method, namely that the expected solution of the cohomological equation is first
constructedas a differential1-form,the TNS conditionimplyingthat the formis closed.
The fact that this form is exact follows from the hyperbolicity of the induced action on
homology. This method can be extended to some situations where the TNS condition
does not hold, e.g. Weyl chamber flows [FK]. In that case the constantcocycles do not
correspondtoclosedformsanymore,however,theirexteriorderivativesareofaparticular
form, and one can show that for an arbitrary sufficiently smooth cocycle the exterior
derivativeofthecorrespondingformisalsoofthisspecialform.
Non-abeliancohomologyofabelianAnosovactions 263
In §4 we gathered some general results, independentof the TNS property, which are
usedforthecaseofLie-groupvaluedorHo¨ldercocycles.Asasubstituteforthedifferential
1-form,oneconstructsaninvariantfoliationbyputtingtogetherthe‘stable’and‘unstable’
foliationsof the generatorsof the skew-productaction determinedby the cocycle. In §5
wecompletetheproofsofthetheoremsgivenin§2. Duetocertaintechnicaldifficulties,
forHo¨ldercocycleswerestrictourselvestothecaseofanactiononatorus.Theresultsfor
smoothcocyclesareprovenforactionsoninfranilmanifolds.
In §6 we use the main result to show that the derivative cocycle of a small C1
perturbationofa linearTNS Zk actionona torusis cohomologoustoa constantcocycle
viaaHo¨ldertransfermap. Thederivativecocycleofsuchanactionisnotasmallcocycle,
butonecanreduceittothatcasebyconsideringthesplittingintoLyapunovspaces. Some
examplesandrelatedquestionsareincludedin§7.
2. Themainresults
The only manifolds which are known to admit Anosov diffeomorphisms are tori,
nilmanifoldsandinfranilmanifolds. Itisanoutstandingconjecturethatthesearetheonly
onessupportingAnosovdiffeomorphisms(see[F2]).
A nilmanifoldisthequotientofaconnected,simplyconnectednilpotentLiegroupN
byalattice0. Allsuchlatticesarecocompact,torsionfreeandfinitelygenerated(see[Ra,
Theorems2.1and2.18]). Aninfranilmanifoldisfinitelycoveredbyanilmanifold. More
precisely, let N be a connected, simply connected nilpotent Lie group and C a compact
groupofautomorphismsofN. Let0beatorsion-freecocompactdiscretesubgroupofthe
semi-directproductNC. Recallthatanelement.x;c/ofNC (wherex 2 N andc 2 C)
actsonN byfirstapplyingcandthenlefttranslatingbyx. ByaresultofAuslander(see
[A]),0\N isacocompactdiscretesubgroupofN and0\N hasfiniteindexin0. The
quotientspaceN=0isacompactmanifoldcalledaninfranilmanifold.
Anosov diffeomorphisms on nilmanifolds and infranilmanifolds were introduced in
[Sm,F2,Sh]. LetfNVNC !NC beanautomorphismforwhichfN.0/D0,fN.N/DN.
Then fN induces a diffeomorphism f V N=0 ! N=0, called an infranilmanifold
automorphism. If the derivative DfNj at the identity is hyperbolic, i.e. has all the
N
eigenvalues of absolute value different from one, then f is an Anosov diffeomorphism.
Notethatinthiscasethestableandunstabledistributionsaresmooth.
InthesequelweconsiderZk actionsonlyoninfranilmanifolds.
Definition. Wecallanactionlinearifitisgivenbyinfranilmanifoldautomorphisms.
RecalltheFranks–ManningclassificationofAnosovdiffeomorphismsoninfranilman-
ifolds(see [F1, Man] for the case of a Z action and [H1, proof of Proposition2.18] for
thecaseofaZk-action). LetM beaninfranilmanifoldand(cid:11) V Zk (cid:2)M ! M anabelian
C1 action containing an Anosov diffeomorphism. Assume that (cid:11) has a fixed point x .
0
Then the action (cid:11) is Ho¨lder conjugate to the linear Zk action (cid:11)N V Zk (cid:2)M ! M given
byautomorphismsinducedbythemapinhomotopy(cid:11)(cid:3) V Zk (cid:2)(cid:25)1.M;x0/ ! (cid:25)1.M;x0/.
Notethattheactionalwayshasaperiodicpoint.Ingeneral,theactionisHo¨lderconjugate
to an affine action, whose restriction to a subgroup of Zk of finite index is an action by
264 A.Katoketal
linearautomorphisms. RecallthatHurderconstructedin[H2]abelianAnosovactionson
thetorusbyaffinemapswithoutfixedpoints.
Let(cid:11) V Zk (cid:2)M ! M beanabelianCK action. View(cid:11) asahomomorphismfromZk
intoDiffK.M/anddenotebyA(cid:26)DiffK.M/itsimage.
InordertoobtaintherigidityresultsaboutcocyclesoverZk actions,weintroducethe
following.
Definition. We say that an action (cid:11) is TNS, if there is a family S of partially hyperbolic
elements in A and a continuoussplitting of the tangent bundle TM D (cid:8)m E into A-
iD1 i
invariantdistributionssuchthat:
(i) the stable and unstable distributions of any element in S are direct sums of a sub-
familyoftheE ’s;
i
(ii) anytwodistributionsE andE ,1(cid:20)i,j (cid:20)m,areincludedinthestabledistribution
i j
ofsomeelementinS.
1
If, moreover, the action (cid:11) is C and each distribution E is smooth, we say that the
i
actionissmoothly-TNS.
Remarks.
(1) It is easy to see that given a TNS action, one can assume that S consists only of
Anosovelements.
(2) Given a TNS action described by S (cid:26) A with all elements of S Anosov and a
splitting TM D (cid:8)m E , one can replace the distributions fE g by the non-zero
T iD1 i i
intersections E(cid:27).a/.a/,where(cid:27).a/ 2 fu;sg. Indeed,denotethenewsplitting
a2S
by TM D (cid:8)k F . It obviously satisfies (i), and (ii) can be checked as follows:
iD1 i
given Fi and Fj, there are 1 (cid:20) i0;j0 (cid:20) m such that Ei0 (cid:26) Fi and Ej0 (cid:26) Fj and
a 2 S suchthatEi0;Ej0 (cid:26) Es.a/; thenFi;Fj (cid:26) Es.a/, bythechoiceofthenew
splitting.
Iftheoriginalsplittingwassmooth,sowillbethenewone.
(3) Inviewoftheabove,onecanalwaysassumethatthedistributionsE areintegrable.
i
(4) By Remarks (1) and (2), any linear TNS action on an infranilmanifold is actually
smoothly-TNS.Ifthelinearactionisonatorus,onecanassumethatthedistributions
E areconstant(i.e.givenbytranslatesofsomefixedvectorsubspaces).
i
(5) Consider a TNS Zk action (cid:11) on an infranilmanifoldM. Since it contains Anosov
elements, thereis a subgroup0 (cid:26) Zk offinite indexacting witha fixedpoint, say
x , andbytheFranks–Manningclassification(cid:11)j isconjugatedtothelinearaction
0 0
(cid:11)N VD.(cid:11)j0/(cid:3)inducedon(cid:25)1.M;x0/. UsingRemark(1)andthefactthattheelements
of S (cid:26) Zk can be replaced by their powers, one can assume that S (cid:26) 0 and S
consistsofAnosovelementsonly.Then,byRemark(2),theaction(cid:11)N isTNSaswell,
becausetheTNSpropertycanbedescribedintermsoftheintersectionsofthestable
andunstablefoliationsoftheelementsofS.
LetG (cid:26) GL.d;R/beaclosedsubgroup,withthemetricinducedbythematrixnorm
onGL.d;R/.
The following theorems apply for G-valued cocycles that are small. However, the
smallnessassumptionisnotnecessaryintheproofifGDR. SinceanyR-valuedcocycle
Non-abeliancohomologyofabelianAnosovactions 265
can be made arbitrarily small by multiplying it by some non-zero number, we will not
makethisdistinctioninthesequel.
ForHo¨ldercocycles,ourresultisasfollows.
THEOREM2.1. LetMbeatorusand(cid:11) VZk(cid:2)M !MaTNSaction.Let(cid:12) VZk(cid:2)M !
Gbeasmall(cid:14)-Ho¨ldercocycleover(cid:11). Then(cid:12) iscohomologoustoaconstantcocycle,i.e.
thereisa(cid:14)-Ho¨lderfunctionP VM !Gandarepresentation(cid:25) VZk !Gsuchthat
(cid:12).a;x/DP.ax/−1(cid:25).a/P.x/:
Moreover, if (cid:11) and (cid:12) are CK, K D 1;2;:::;1;!, then P is CK−", for any small
" >0. .K−" DK forK 2f1;1;!g/.
ThemainpartoftheproofistodealwithHo¨ldercocyclesoveralinearaction.
THEOREM2.2. Let M be a torus and (cid:11) V Zk (cid:2) M ! M a linear TNS action. Let
(cid:12) V Zk (cid:2)M ! G be a small (cid:14)-Ho¨lder cocycle over (cid:11). Then (cid:12) is cohomologousto a
constantcocyclethrougha(cid:14)-Ho¨ldertransfermap.
The reductionto this case essentially involvesthe Franks–Manningclassification and
previousregularityresults.
1
ForC cocycleswedonothavetorequirethatthemanifoldbeatorus.
THEOREM2.3. Let M be an infranilmanifold and (cid:11) V Zk (cid:2)M ! M a smoothly-TNS
action. Let(cid:12) V Zk (cid:2)M ! GbeasmallC1 cocycleover(cid:11). Then(cid:12) iscohomologousto
1
aconstantcocyclethroughaC transfermap.
Moreover,if(cid:11)and(cid:12) areC!,thenthetransfermapisC!.
Remark. As can be seen fromthe proof, a similar result holdsfor cocyclesthat are only
finitelysmooth.Inthatcasethereisalossofregularityforthetransfermap.
Wenowintroducesomenotationwhichwillbeusedinthesequel.
Let a be a partially hyperbolicdiffeomorphism. We denote by (cid:21)(cid:6).a/ the contraction
andexpansioncoefficientsofa,definedby
(cid:21)−.a/VDnl!im1kD.na/jEs.a/k1=n;
(2.1)
(cid:21)C.a/VDnl!im1kD.na/−1jEu.a/k−1=n:
Let(cid:12) VZk(cid:2)M !GL.d;R/beacocycleanda 2Zk.Wedenoteby(cid:22)(cid:6).a/D(cid:22)(cid:6).(cid:12);a/
thecontractionandexpansioncoefficientsfor(cid:12)jhai,definedby
(cid:22)−.a/VD lim inf k(cid:12).na;x/−1k−1=n;
n!1x2M
(2.2)
(cid:22)C.a/VD lim supk(cid:12).na;x/k1=n:
n!1x2M
Notethatinfx2Mk(cid:12).a;x/−1k−1 (cid:20)(cid:22)−.a/(cid:20)(cid:22)C.a/(cid:20)supx2Mk(cid:12).a;x/k.
If the cocycle takes values in R (which we see as the additive group) then (cid:22)(cid:6) D 1
because(cid:12) actsbytranslations.
IfW isafoliationofM andx 2 M,denotebyW .x/thepath-connectedcomponent
loc
offy 2W.x/jdist .x;y/<(cid:14) gwhichcontainsx,where(cid:14) >0issmallandfixed. The
M 0 0
constant(cid:14) iscalledthesizeofthelocalfoliation.
0
266 A.Katoketal
3. Proofforreal-valuedcocycles
WeprovehereaspecialcaseofTheorem2.3inordertoillustratethemaingeometricidea
ofthemethod(asmentionedintheintroduction).
THEOREM3.1. Let M be an infranilmanifoldand (cid:11) V Zk (cid:2)M ! M a linear TNS Zk
action. Let (cid:12) V Zk (cid:2)M ! R be a C1 cocycle over (cid:11). Then (cid:12) is cohomologousto a
1
constantcocyclethroughaC transfermap.
ByRemark(4)in§2,wecanassumethatthedistributionsE aresmooth.
i
Theproofwillfollowfromasequenceoflemmas. TheTNSpropertyisrequiredonly
forLemma3.3.
Assumethat(cid:12) V Zk (cid:2)M ! Risareal-valuedcocycleovertheTNSlinearaction(cid:11),
i.e.
(cid:12).a Ca ;x/D(cid:12).a ;a x/C(cid:12).a ;x/; foralla ;a 2Zk; x 2M:
1 2 1 2 2 1 2
We want to show that, under certain regularity conditions, (cid:12) is cohomologous to a
constantcocycle,i.e.thereisafunctionP V M ! Randahomomorphism(cid:25) V Zk ! R
suchthat
(cid:12).a;x/DP.ax/C(cid:25).a/−P.x/:
1
TheideaoftheproofistoconstructaC 1-formonM whichisclosedanddetermines
aZk-invariantclassincohomology.Sincetheactioninducedincohomologyishyperbolic,
the aboveformhasto be exact. This allowsus to recoverthe homomorphism(cid:25) and the
transfermapP.
We alsomentionasecondargument,whichwillbedevelopedindetailforthecaseof
Lie-group valued cocycles. Namely, since the form is closed, it describes a foliation of
M(cid:2)RwithleavesofdimensionmDdimM. Consideringtheholonomyofthisfoliation,
onecanshowthattheleavesareclosedandcoverMsimply(i.e.theformisactuallyexact).
Leta 2 Zk beahyperbolicelement. Assumethatx 2 M andy isinthestableleafof
a throughx, Ws.xIa/. Then the followingsum is convergentin C1 (see, for example,
[LMM, proof of Lemma 2.2]; note that if (cid:12) is only Ho¨lder then the sum still converges
inC0)
X1
P−.yIx/VD− T(cid:12).a;.na/y/−(cid:12).a;.na/x/U;
a
nD0
andwecandefinea1-form!−onEs.a/bytakingthedifferentialofP−inthey-variable
a x a
alongthestableleaf. Actually,thedifferentialofPa−.(cid:1)Ix/jWs.xIa/definestheformonthe
wholeTWs.xIa/anditdoesnotdependonthepointx chosenonthestableleaf.
Similarly,forx 2 M andz 2 Wu.xIa/D Ws.xI−a/,letthe1-form!C onEu.a/be
a x
definedasthez-differentialofP−−a.zIx/alongtheunstableleafofa. Considertheform
! D!C(cid:8)!−onT M DEu.a/(cid:8)Es.a/.
a a a x x x
We will showthatfor a largesetof hyperbolicelementsin Zk the aboveconstruction
leadstothesameform. Moreover,thisformissmoothandclosed. Weintroducefirstthe
notionsoftheLyapunovexponentandtheWeylchamber,whichweuseonlyinthissection
(see[KSp4]formoredetails).
The action of the derivative(cid:11)(cid:3) of the action (cid:11) on the tangentbundle of the universal
coverofM isdeterminedbycommutinginvertiblematrices. Therearelinearfunctionals
Non-abeliancohomologyofabelianAnosovactions 267
L V Zk ! R, called Lyapunov exponents, whose values for each a 2 Zk are given
j
by the logarithms of the absolute values of the eigenvalues of the matrix corresponding
to the derivative of (cid:11).a/. Each Lyapunov exponent can be extended to a linear map
L V Rk ! R, also called the Lyapunov exponent. There is a splitting of the tangent
j
bundle into Zk-invariantsub-bundlesTM D (cid:8) F such that the Lyapunovexponentof
j j
v 2F withrespectto(cid:11).a/isgivenbyL .a/. WecallF aLyapunovspaceorLyapunov
j j j
distributionfortheaction. ThekernelofeachLyapunovexponentisahyperplaneH in
j
Rk. WedenotebyH−thehalf-spacewhereL isnegative.Theconnectedcomponentsof
j j
Rk−[H arecalledWeylchambers.
j
NotethatusingLyapunovexponents,theTNSpropertycanbecharacterizedby
L DcL forsomeconstantc H) c >0:
j i
LEMMA3.2. Consider a linear Zk action (cid:11) which contains an Anosov element. Then
there is a subset S (cid:26) Zk of hyperbolic generators of Zk which contains elements from
eachWeylchamber,andwiththepropertythatifa;b2S then
! D! :
a b
Proof. Let(cid:21) VD exp(cid:14)L V Zk ! T0;1/,anddenotebyF thefoliationcorresponding
j j j
toF .
j
Assume firstthat a;b 2 Zk arepartiallyhyperbolic,F (cid:26) Es.a/\Es.b/and(cid:21) .b/,
j j
the contractioncoefficientalong F , is smaller than the inverseof the Lipschitz normof
j
(cid:11).a−b/. Letz2F .x/(cid:26)Ws.xIa/\Ws.xIb/.Usingthecocyclerelationwefindthat
j
Xn−1
(cid:12).a;.ka/z/D(cid:12).na;z/;
kD0
(cid:12).na;z/−(cid:12).nb;z/D(cid:12).n.a−b/;.nb/z/;
andsimilarly for x instead of z. Therefore, in orderto show that P−.zIx/ D P−.zIx/,
a b
andconsequentlythat! j D! j ,itisenoughtoshowthat
a Fj b Fj
lim .(cid:12).n.a−b/;.nb/z/−(cid:12).n.a−b/;.nb/x//D0:
n!1
But
j(cid:12).n.a−b/;.nb/z/−(cid:12).n.a−b/;.nb/x/j
(cid:12) (cid:12)
(cid:12)Xn−1 (cid:12)
D(cid:12)(cid:12) .(cid:12).a−b;TnbCk.a−b/Uz/−(cid:12).a−b;TnbCk.a−b/Ux//(cid:12)(cid:12)
kD0
(cid:20) (cid:21)
Xn−1
(cid:20)k(cid:12).a−b;(cid:1)/k dist .(cid:11).nbCk.a−b//.z/;(cid:11).nbCk.a−b//.x//(cid:14)
Ho¨lder M
kD0
Xn−1
(cid:20)k(cid:12).a−b;(cid:1)/k (cid:1)(cid:21) .b/n(cid:14)(cid:1)C(cid:1).dist .z;x//(cid:14) k(cid:11).a−b/kk(cid:14) ;
Ho¨lder j M Lip
kD0
whereCisaconstantthatisindependentofn. Since(cid:21) .b/<1and(cid:21) .b/(cid:1)k(cid:11).a−b/k
j j Lip
<1,theconclusionfollows. 2
268 A.Katoketal
We now construct the set S (cid:26) Zk. Consider first a finite set F of elements in Zk
closetotheorigin,whichcontainsaZbasisofZk. Thereisa constantM > 1suchthat
k(cid:11).c/k (cid:20) M,forallc 2 F. LetL V Rk ! Rbethej’sLyapunovexponentandH
Lip j j
thehyperplaneinRk determinedbythekernelof(cid:21) . ThenthereexistaballB aroundthe
j
originandconesC.H / (cid:26) H− intersectingallWeylchambersinH−,suchthatforeach
j j j
j andanyelementb2C.H /\.Zk −B/wehave
j
L <−logM;
j
andtherefore
(cid:21) .b/<M−1: (3.1)
j
Considertwoelementsa;b 2 C.H /\.Zk −B/. Wecanjoina andbbyasequence
j
ofelementsinC.H /\.Zk −B/addingateachstepanelementfromF. Formula(3.1)
j
allowsustoapplythefirstpartoftheproofrepeatedlyanddeducethat
! j D! j : (3.2)
a Fj b Fj
Bytheconstructionofthe1-form,(3.2)stillholdsifaandbareintheunionofC.H /
j
withtheoppositecone,−C.H /.
j
DefinethesetS tobe
(cid:20) (cid:21)
\m
S D .C.H /[.−C.H /// \.Zk −B/:
j j
jD1
LEMMA3.3. If the linear Zk action is TNS then the form ! (cid:17) !a, a 2 S constructed
aboveissmoothandclosed.
Proof. DenotemDdimM. LetU (cid:26)M beasmall-enoughopenset.
Since the distributions E are smooth, one can find a frame of smooth vector fields
i
fXjgjD1;m overU suchthateachfieldXj iscontainedinsomeEi. Letf(cid:17)jgjD1;m bethe
dualframeof1-formsoverU,andwrite
Xm
!j D f (cid:17) ; wheref D!.X /:
U j j j j
jD1
We will show that each function f is smooth along all the distributions E and the
j i
derivativesare continuouson U. However, this impliesthat each f is smooth on U (in
j
somecasesonecanusethecharacterizationofsmoothnessviaaFouriertransform,orthe
theoremofJourne´;ingeneral,oneneeds[HK,Theorem2.6]).
Indeed,inordertoshowthatf issmoothalongE ,pickanAnosovelementa 2Ssuch
j i
thatX ;E (cid:26) Es.a/. Since P−.(cid:1);x/ is smoothalongWs.xIa/andvariescontinuously
i j a
intheC1 topologywith x 2 M, oneconcludesthat!a−jEs.a/ iscontinuouslyC1 along
Ws.a/. ByLemma3.2,thisprovesourassertion.
To show that ! is closed, use again the TNS condition and Lemma 3.2. Clearly
!a−jWs.xIa/ is exact, hence, using the fact that pull-back and exterior differentiation
commute,
.d!/jWs.xIa/Dd.!jWs.xIa//D0 fora 2S:
Since over U any two directionsX and X are includedin the stable subspace of some
i j
hyperbolicelementa 2S,weobtainthat.d!/j D0. 2
U