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Cohomological equations and invariant distributions on a Lie group

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Niveau: Supérieur, Doctorat, Bac+8
Cohomological equations and invariant distributions on a Lie group by Aziz El Kacimi Alaoui & Hadda Hmili (August 2011) Abstract. This paper deals with two analytic questions on a connected compact Lie group G. i) Let a ? G and denote by ? the diffeomorphism of G given by ?(x) = ax (left translation by a). We give necessary and sufficient conditions for the existence of solutions of the cohomological equation f ? f ? ? = g on the Frechet space C∞(G) of complex C∞ functions on G. ii) When G is the torus Tn, we compute explicitly the distributions on Tn invariant by an affine automorphism ?, that is, ?(x) = Ax+ a with A ? GL(n,Z) and a ? Tn. iii) We apply these results to describe the infinitesimal deformations of some Lie foliations. 0. Preliminaries Let M be a manifold and ? a diffeomorphism of M . Usually, the couple (M,?) is called a discrete dynamical system. Natural question : What are the geometric objects invariant under the action of ?? Formulated as such, this question is far to be trivial. However one can answer it in special situations for a given manifold if we specify the diffeomorphism ? and the nature of the geometrical objects. It has been customary, in the theory of dynamical systems, to seek an invariant measure.

  • invariant distributions has

  • frechet space

  • arbitrary compact

  • m?zn

  • compact lie

  • lie group


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COHOMOLOGICALEQUATIONSANDINVARIANTDISTRIBUTIONSONACOMPACTLIEGROUPybAzizElKacimiAlaoui&HaddaHmili(ToappearinHokkaidoMathematicalJournal)(Revised:June2012)Abstract.ThispaperdealswithtwoanalyticquestionsonaconnectedcompactLiegroupG.i)LetaGanddenotebyγthediffeomorphismofGgivenbyγ(x)=ax(lefttranslationbya).Wegivenecessaryandsufficientconditionsfortheexistenceofsolutionsofthecohomologicalequationffγ=gontheFre´chetspaceC(G)ofcomplexCfunctionsonG.ii)WhenGisthetorusTn,wecomputeexplicitlythedistributionsonTninvariantbyanaffineautomorphismγ,thatis,γ(x)=A(x+a)withAGL(n,Z)andaTn.iii)WeapplytheseresultstodescribetheinfinitesimaldeformationsofsomeLiefoliations.0.PreliminariesLetMbeamanifoldandγadiffeomorphismofM.Usually,thecouple(M,γ)iscalledadiscretedynamicalsystem.Naturalquestion:Whatarethegeometricobjectsinvariantundertheactionofγ?Formulatedassuch,thisquestionisfartobetrivial.Howeveronecanansweritinspecialsituationsforagivenmanifoldifwespecifythediffeomorphismγandthenatureofthegeometricalobjects.Ithasbeencustomary,inthetheoryofdynamicalsystems,toseekaninvariantmeasure.Butthisproblemisveryhardingeneral.Insteadofthis,itismoreeasiertoseekaninvariantdistribution.Thisleadssystematicallytosolvingcertainequations(calledcohomologicalequations)ontheFre´chetspaceC(G)ofcomplexCfunctionsonG.Regardlessofthis,theseequationsconstituteathemecurrentlybooming.Thepurposeofthispaperistoanswerthesequestionsforsomediffeomorphismsofa(connected)LiegroupG:First,Gisarbitrarycompactandγisatranslationandthen,GisthetorusTnandγisanaffineautomorphism.LetGbeaconnectedcompactLiegroupofdimensionn.WedenotebyC(G)thespaceofcomplexC-functionsonGequippedwiththeC-topology.Thistopologycanbedefinedasfollows.Let{U1,∙∙∙,Uk}beanopencoverofGsuchthat,foreachi∈{1,∙∙∙,k},thereexistsaC-diffeomorphismφi:Rn−→Ui.Let{ρ1,∙∙∙k}beaC-partitionof1suchthatthesupport(whichiscompact)ofeachρiiscontainedinUi.PThen,iffC(G),onemaywritef=ik=1ρif.——————————————MathematicsSubjectClassification2000:53C12,37A05,37C10,58A30.KeyWords:Liegroup,cohomologicalequation,distribution,foliation,deformation.1