Annals of Mathematics - Semantic Scholar
Annals of Mathematics Asymptotic Cycles Author(s): Sol Schwartzman Source: Annals of Mathematics, Second Series, Vol. 66, No. 2 (Sep., 1957), pp. 270-284 Published by: Annals of Mathematics Stable URL: http://www.jstor.org/stable/1969999 Accessed: 20-05-2015 22:39 UTC
Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://www.jstor.org/page/ info/about/policies/terms.jsp JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected].
Annals of Mathematics is collaborating with JSTOR to digitize, preserve and extend access to Annals of Mathematics.
http://www.jstor.org
This content downloaded from 155.97.178.73 on Wed, 20 May 2015 22:39:17 UTC All use subject to JSTOR Terms and Conditions
ANNALS
OF MATHEMATICS
XVol.66, No. 2, September, 1957
Printedin U.S.A.
ASYMPTOTIC CYCLES* BY SOL
SCHWARTZMAN
(Received April 16, 1956) (Revised March 18, 1957)
1. Introduction[4]
Let X be a topologicalspace, G a topologicalgroup with identityelemente. Let f be a continuousmapping fromX X G into X. For simplicitywe will writef(x, g) as xg. If (xg1)g2= x(g9g2),and if we = x, G is said to act as a topological transformationgroup on X. Such transformationgroups have been studiedfrommany points of view. In this paper we will be concernedwith the viewpointfirstemployed systematicallyby George David Birkhoff.Here the concretemodel to be borne in mind is the flowarisingfroma systemof ordinary differential equations on a manifold,and in fact we will confineourselves to the case whereG is the additive group R of the real line with the usual topology. There is a strongconnectionbetweentopologicaldynamics,whichis the field we are concernedwithhere,and ergodictheory,anotherfieldin whichBirkhoff pioneered.Thus, in analogy with the Poincare recurrencetheoremthere are theorems concerningvarious kinds of topological recurrence.In addition, various incompressibility propertieshave been studied. These of course are a for kind of substitute the measure-preserving propertyusually required for in Other transformations ergodic theory. parallels exist-for example, sets of the firstcategoryplay much the same role in some theoremsthat one might expect fromsets of measure zero. Results of all these types can be found togetherwith many other theoremsin the book by Gottschalk and Hedlund cited above. The close parallelismbetweenthe two fieldshas tended to minimizethe attentiongivento questionswhichstronglyinvolvethe topologyon X. In particular algebraic topologyhas played no part in the developmentof the general theoremsin topologicaldynamics.In this paper a firstattemptis made to overcome this restriction,at least to the limitedextentof tryingto show the role it is not played by the firstBetti group. Since the orbitsare one-dimensional, surprisingthat the one-dimensionalgroup plays a leading role. It turns out that, except fora set of orbitshaving measure zero with respectto every invariantmeasure,we can associate with each orbitan elementof the firstBetti group of X. We call this the asymptoticcycle of the orbit. In a sense it tells * This paper represents results obtained in part at the Institute of Mathematical Sciences, New York University, under the sponsorship of the National Science Foundation, Contract NSF-G3050. Reproduction in whole or in part is permitted for any purpose of the United States Government. 270 This content downloaded from 155.97.178.73 on Wed, 20 May 2015 22:39:17 UTC All use subject to JSTOR Terms and Conditions
ASYMPTOTIC
271
CYCLES
how the orbitwinds around the space X. In the case of flowson the torus,the coefficients (withrespectto the naturalbasis) of the asymptoticcycleassociated withan orbitare preciselythe windingnumbersof Poincar6. We shall see that the study of the asymptoticcycles of orbitsis intimatelyconnectedwith the question of the existenceof surfacesof section,and the equally standardprobof the flow. lem of findingeigenfunctions Theorems in topological dynamics yield as special cases theorems about equations on manifolds.Thus far, apart fromconsequences of the differential fact that a Hamiltoniansystemhas an invariantmeasurewhichis positivefor open sets, thereis comparativelylittlein the generaltheorythat would enable equations to deduce additional informaone to use the formof the differential tion about the asymptoticbehaviorof the solutions.We will see below however that once the concept of the asymptoticcycle has been introducedwe will be able to performexplicit computations,particularlyfor Hamiltonian systems, whichwill tell us how an "average" orbitwindsaround the space. theory[5] 2. Kryloff-Bogoliouboff In additionto the parallel developmentthat has taken place betweenergodic theory and topological dynamics, methods have been developed which use resultsin one fieldto get theoremsin the other.For purposes of futurereference we collect here a numberof well-knowndefinitionsand theoremsof this type. A completeexpositionwhich contains proofsof these resultsand many otherswill be foundin the articleby Oxtoby cited above. Here and in all later paragraphsX will denote a compactmetricspace on whichthe real line acts as a topologicaltransformation group. A positive measure is a measure gi definedfor all Borel sets DEFINITION. S a X such that ,.(S) > 0 forall S. A normalizedmeasureis a positivemeasure forwhichtu(X)- 1. A measure gi definedon the Borel sets of X is called invariant DEFINITION. providedthat foreveryBorel set S and everyreal numbert, ji(St) = ji(s). A point p in X is called quasi-regularprovidedthat DEFINITION. limT_:O
1T
f T
f(pt) dt
existsforeveryreal-valuedcontinuousfunctionf(x) definedon X. For everyquasi-regularpointp thereexistsa uniquefinitemeasure THEOREM. Ap
suchthatforeverycontinuousf(x),
limrnT,
1/T
f(pt) dt
= Lf(x) dg,(x).
Moreover,gipis a positiveinvariantmeasuresuch thatjp (X) = 1. The set of pointswhichare not quasi-regularhas measurezerowith THEOREM. respectto everyfiniteinvariantmeasure. with respect to the A functionf(x) is said to be differentiable DEFINITION. flow provided that limAt~o (f(xAt) - f(x))/At exists uniformlyover X. This limitwill be denotedby f'(x) or (d/(dt))f(x).
This content downloaded from 155.97.178.73 on Wed, 20 May 2015 22:39:17 UTC All use subject to JSTOR Terms and Conditions
272
SOL
SCHWARTZMAN
The followingtheoremwas communicatedorally to the author by Professor Kakutani. It is this theoremthat will enable us to state our resultsfor arbitrary compact metric spaces, without restrictingourselves to differentiable manifolds. THEOREM
(Kakutaiii).
bAny
continuous function f(x) on X can be approxi-
nateduniformly byfunctionswhichare differentiable withrespectto theflow. PROOF.
Let gc(x)
=
ge(xiAt)-
1/
j
x
f(xt)dt. Then I
(p+At
1( 1
E Azt
fC+t
f(xt)dt
-
f(xt)dt)
f
f(xt)dt- _ At
f(xt)dt).
Since, as t goes to to, f(xt) approachesf(xto)uniformly, it followsthat gE(x) is differentiable with respectto the flow,and in fact g'(x) = (1/c)(f(xe) - f(x)). It is evidentthat as ? goes to zero,g,(x) approachesf(x) uniformly, so the theoremis proved. 3. The theoryof Eilenbergand Bruschlinsky[1], [3] Because of the relation between quasi-regularpoints and continuousfunctions, the theoryreferredto in the title of this section enables us to connect propertiesof the flow with the firstBetti group of the space. We will summarize those resultswhich will be needed later on, referring the reader to the articlescited above forproofs.As in the restof this paper, the space X referred to below is assumed to be compactmetric. Throughoutthe rest of this paper the homologyand cohomologygroupswe referto are those obtained fromthe Cech theoryusing real coefficients. When we speak of an integralcycle we will mean an elementof this homologygroup whicharises fromintegralcycles in the ordinarysense when we representour space as an inverseprojectivelimit of polyhedra.By a cocycle with integral periodswe will mean an elementof the cohomologygroup which,when consideredas a functionalon the homologygroup,assigns integervalues to each integralcycle. Actuallywe will use only the one-dimensionalgroups.It should be bornein mindthat any closed parametrizedcurveK in X determinesan element of the firsthomologygroupin the Cech theory,since a map of X into a polyhedronsends K into a curve of the polyhedron. The set of all continuouscomplex-valuedfunctionsof absolute DEFINITION. value one definedon X will be denotedby C(X). The subset of C(X) consisting of those functionswhichcan be writtenin the formexp (2iriH(x)), whereH(x) is continuousand real valued, will be denoted by R(X). If K is any parametrizedcurvein X and f(X) C(X), AK argf(x) will mean the changein the angular variableforf(x) along K. Notice that whenf(x) = exp (27riH(x)), (1/27r)AK argf(x) = H(p2) -H(P), wherepi and P2 are the initialand terminalpointsof K.
This content downloaded from 155.97.178.73 on Wed, 20 May 2015 22:39:17 UTC All use subject to JSTOR Terms and Conditions
273
ASYMPTOTICCYCLES
THEOREM.If f(x),-C(X) and K is a closedparametrized curve,(1/27r)AKargf is an integerdependingonlyon theelementof thefirsthomology groupcorresponding to K and thecosetdetermined byf(x) in C(X)/R(X). It is obvious that if X were a polyhedronthe above theoremwould immediately enable us to associate witheach elementof C(X)/R(X) an elementof the firstcohomologygroup which,as a functionalon the homologygroup, would assignto each cycleK the integermentionedin the above theorem.It turnsout to be possibleto generalizethis procedurein a naturalway so that it applies to any compactmetricspace X. THEOREM. The mapping indicatedabove, sending C(X)/R(X) into thefirst ontoall Cech,cocycleswith.integral cohomology group,is an algebraicisomorphism periods. THEOREM. If fi(x) and f2(x) belongto C(X) and fI(x) - f2(x)i < 1 thenthese twofunctionsdetermine thesame elementmod R (X). manifoldand f(x)eC(X) is a differFinally, suppose X is a differentiable entiable function.In a suitable local coordinatesystemf(x) = exp (2iriH(x)) where H(x) is differentiable and uniquely determinedup to an additive constant.If H1 and H2 are determinedas above in two different coordinatesystems, dHj1and dH2 agree in the regionof overlapping.Thus f(x) determinesa closed one-formover X. This mapping, sendingthe differentiable elementsof C(X) into closed one-forms,induces the natural mapping of C(X)/R(X) onto the elementswithintegralperiodsin the quotientgroupof closed one-forms modulo boundingone-forms. 4. Asymptoticcycles Let f(x) be an elementof 0(X) and let p be any point in X. By argf we mean the change in the angular variable off(x) along the orbit goingfromp to pt. LEMMA. Let p be any point of X and supposefi(x) and f2(x) are twoelements thesame-elementof C(X)/R(X). Then of C(X) whichdetermine DEFINITION.
A(p,pt)
(1/27r)A(,ppt)
argf1 -
(1/27r)A(p,pt) argf2
0(1).
PROOF. We knowthat forsome continuousH(x), fi(x) = f2(x) exp (2iriH(x)), Hence (1/27r)A(pgpt) argf1 - (1/2r)A(p,pt) argf2 = H(pt) - H(p). Since H(x)
mustbe bounded,the lemma is proved. THEOREM. Let p be any quasi-regularpoint. Then for any f(x) eC(X) lim,.b (1/(2tt))A(p,pt)argf exists.Moreover,this limitdependsonly on theelement of C(X)/R(X)
determined by f(x). The induced mapping of C(X)/R(X)
intothereal line is a grouphomomorphism. PROOF. By the above lemma,if this limitexistsforany fj(x) in 0(X) it exists and has the same value forany f2(x) equivalent to fj(x) mod R(X). Given any f2(x) in C(X) we know we can approximatef2(x) as closely as we wish by a withrespectto the flow.(A trivial functionfj(x) in C(X) whichis differentiable supplementto Kakutani's theoremin Section two shows this.) If we choose This content downloaded from 155.97.178.73 on Wed, 20 May 2015 22:39:17 UTC All use subject to JSTOR Terms and Conditions
274
SOL SCHWARTZMAN
fi(x) so that Ifi(x) - f2(x)I 0 such that any arc in X of diameterless than - has the propertythat the total change in the angular variable forf(x) over that arc is less than one radian. Divide K into
subares K1,
...
,
K. such that each subarc has diameter less than ?/2. For
any real numbert, Krt will mean the arc that K, goes into after time t has elapsed. thereexistsa sequence ti} going Since we are assumingthe flowis recurrent, to infinity such that lim1 Osupzxpp(x, xti) = 0. Then foreach r and sufficiently largevalues i, the diameterof Krti is less than t. Thereforeforsufficiently large values i the total changein the angularvariable off(x) along Kti is less than n radians,wheren is the numberof subares into whichK was divided. The same statementobviouslyholds forK itself. Now foreach i we will definea map 0i sendingthe rectanglein the (U, V)
ti, U = 0 and U = 1 into X.
plane bounded by the straight lines V = 0, V =
Any point (U, 0) on the unit intervalalong the U-axis is to be sent by (P into the curveK in such a way that(p taken on thisintervalsimplygivesthe original parametrizationof the parametrizedcurveK. An arbitrarypoint (U, V) in the rectangleis to be sent by (p into the point that 0j(U, 0) goes into under the flowaftertime V. Thus 0i sends vertical lines in the rectangleinto orbitsin the space X. In particularthe segmentof the boundaryof the rectanglelying on the line U = 0 gets sent into the orbitgoingfromxi to xit , and the corre1 goes into the orbit fromx2 to x2ti. The spondingsegmenton the line U portionof the boundaryalong the line V = 0 goes into K, and the part lying on the line V = tj goes into Kti . The functionf(x) on X now gives a functionf(oi(U, V)) on the rectangle. Since the firstBetti numberof a two-cellis zero, this functionon the rectangle has a logarithm,so the total change in the angularvariable around the boundary of the rectangleis zero. This can now be applied to curvesin X whichare the image of the boundary.Therefore AK
argf +
argf A(X2,X2ti)
AKtjargf
-
argf = 0 A(xj,xjtj)
or argf AA(X2,x2tj)
-
A(xl,zlti) argf =
AKti
argf -
AK
argf.
However the righthand side of this last equation is bounded and in fact is smallerthan 2n radians. Therefore,dividingby 2irtiand lettingi go to infinity, argf = limbo (1/(27rti))A(X2,X2tj) arg f. Since f(x) was liming(1/(27rti))A(x1lxti)
This content downloaded from 155.97.178.73 on Wed, 20 May 2015 22:39:17 UTC All use subject to JSTOR Terms and Conditions
280
SOL
SCHWARTZMA N
an arbitraryfunctionin C(X) this shows that the asymptoticcycle associated with xl is the same as that associated with x2. Thus the flowis spectrallydeterminateand the asymptoticcycle is definedand has the same value for all orbits,whetherquasi-regularor not. In particularthen fora recurrentflowany two closed orbits determineelementsof the firstBetti group which are rationallydependent. 7. Cross sections For the purposesof this paper, a subset K is said to be a crosssectionof X providedK is closed and the mapping4) of K X R into X sending (p, t) into pt is a local homeomorphism onto X. This is the same as what was called a "surfaceof section" by G. D. Birkhoffin his book "Dynamical Systems". Up to the pointwhereasymptoticcyclesenterthe picture,the discussion givenbelow followsBirkhoff(p. 144). It is easy to see that if K is any closed set and the mapping4 definedabove is onto all of X, then a necessaryand sufficient conditionthat K be a crosssection is that for some - > 0, 4 maps the Cartesianproduct of K with the open interval(-At, t) homeomorphically onto an open subset of X. Next, suppose that K is any crosssectionof X. For any pointx E X, let txbe the smallest non-negativereal numbert such that thereexistsa point p in K forwhichx = pt. Since the transformation sendingx into xt is a homeomorphism thereis only one p forwhichx = ptx. Call this point px . Next, let Tx be the smallestpositive value of T for which pxT belongs to K. (Clearly, such a Tx exists.) We now definefK(X) to equal exp (2iritx/Tx). It is easy to see thatfK(X) is in C(X). LEMMA. Let f(x) be in C(X) and supposef(x) is differentiable withrespectto theflow,and that (f'(x))/(2irif(x)) > 0 for all x. If we let K be the set of pointsfor whichf(x) = 1, thenK is a cross-section and fK(x) and f(x) are congruentmod R(X). PROOF. Let 4 be the mappingof K X R into X sending (p, t) into pt. Obviously4 is a continuousmappingonto all of X. Let (p, t) be a point of K X R and suppose (pi, Tj) and (qi, Si) are two sequences of points in K X R convergingto (p, t). Suppose piT, = qjSi. Then pi(Ti - Si) = qi. Therefore argf is an integer,i.e. f(pi(T, - Si)) = 1 so that (1/(2r))A(pi,pi(Ti(Si)) DEFINITION.
T.-Si
a(f'(p
it))/(27rif(p it)) dt
is an integer. Since the integrandis positiveand Ti - Si goes to zero, forsufficiently large values of i, Ti = Si and pi = qi. This showsthat 4 is locally one-to-one. Next let 0 be an open subset of K X R and suppose (p, t) belongsto 0. Let {qi} convergeto pt in X. Then {f(qi(-t)) } convergesto f(p) 1. Let m = inf (f'(x))/(27rif(x)). Then I(1/(27r))A(xtjxt2) arg f i ? mi t1 t2[ for any x E X and any t1, t2. From thisit followsthat it is possibleto choose a sequence of real numbersti such that ti convergesto - t and f(qiti) = 1 forall i. There-
This content downloaded from 155.97.178.73 on Wed, 20 May 2015 22:39:17 UTC All use subject to JSTOR Terms and Conditions
281
ASYMPTOTIC CYCLES
foreqjtjbelongsto K foreveryintegeri. Since tj convergesto -t, qjtjconverges to p and for sufficiently large values of i, (qjtj, - ti) belongs to 0. However = qi . This shows that 4 is an open mapping. Since we already know ti) 0(qjtj2that p is locally one-to-oneit followsthat 4 is a local homeomorphismand therefore K is a cross-section. Next, let s' be the mappingof [0, 1] X K into X whichsends (a, p) into p(aT ).
Since Tp is a continuousfunctionof p in K,
AVis
a continuousmappingonto all
of X. If we let g(a, p) = exp (2iria), then g(a, p) = fK(V/(a,p)). Now it is easy
to see that there is one and only one real-valuedfunctionH(a, p) such that H(O, p) = 0 and f(i/(a, p)) = exp (2iriH(a, p)). Obviously H(a, p) is continuousand H(1, p) = 1 forall p in K. Then fK6(/(a,
p))/f(V/(a, p))
=
exp (2ri(a
-
H(a, p))).
Since a - H(a, p) = 0 fora = 0 or 1, thereis a real valued functionh(x) definedon X forwhicha - H(a, p) = h(VI(a,p)). Obviouslyh(x) is continuous and fK(x)/f(x)
= exp (2irih(x)). Therefore fK(x) is congruent to f(x) mod R(X)
and the proofof the lemmais completed. THEOREM. Let f(x) E C(X). A necessaryand sufficient conditionthat there existsa cross-section K for whichfK(X) is congruentto f(x) mod R(X) is that A,,(f) > 0 foreverypositiveinvariantmeasureju. PROOF. First suppose that such a cross-section K exists. If we let M suppEKT, and if x is any quasi-regularpoint, Ax(fK)
=
limToo
=
(1/(27rT))A(x,xT) argfk ? 1/M.
Thus forany positiveinvariantmeasure u,A,(f) = A,(fK)
=
f
A,(f) d1i(x) >
0.
Next suppose that A,>(f)> 0 for every positive invariantmeasure Al.There is no loss in generalityin assumingthatf is differentiable withrespectto the flow, since if this were not the case we could replacef by a functionequivalent to f mod R(X) and havingthisproperty.Now let B(X) be the space ofall continuous real-valuedfunctionsg(x) on X with 11g(x) 11= supe I g(x) 1. As in the previous sectionlet D be the space of functionswhichare derivativesof real valued with respectto the flow.The statementthat A,;(f) >0 functionsdifferentiable for all positive 4 says that
f
(f'(x))/(2irif(x))dyu(x)> 0 for all positive ,u.
If D 0 {J(f'(x))/(27rif(x))} did not intersectthe cone of positive functionsin B(X) it would followfromthe Hahn-Banach theoremthat forsome positive,u, (f'(x))/(2irif(x))d,4(x) = 0. Thus thereis a functionH(x) whichis differentiable with respect to the flow and has the propertythat H'(x) + (f'(x))/ (27rif(x))> 0. Let f(x) = f(x) exp (2iriH(x)). Then f'(x) existsand f'(x)/2rif(x) This content downloaded from 155.97.178.73 on Wed, 20 May 2015 22:39:17 UTC All use subject to JSTOR Terms and Conditions
282
SOL
SCHWARTZMAN
is positive,and so by the lemmaprovedabove, ifwe let K be the set of pointsx for whichf(x) = 1, K is a cross-sectionand fK(x) is equivalent to f(x). Since f(x) is equivalent to the originalf(x) mod R(X), the theoremis proved. Now recall that the conjugate space of B(X) is the space of all finiterealvalued measuresA definedon the Borel sets of X. If we put the weak* topology on this space of functionals,it is well known that the subspace consistingof normalizedpositivemeasuresinvariantwithrespectto the flowis compact,and is in fact the convex closure in the weak* topologyof the set of measures 4, arisingfromquasi-regularpointsp. Since forany f(x) E 0(X), the map sendingj intoA,(f) is continuousin the weak* topology,the same can be said forthe map into the firstBetti grouptaking,uinto A, . From this it followsthat the subset of the firstBetti group consistingof all elementsA, is a compact convex set 21which is the convex closure of those elementswhich are of the type A, for some quasi-regularpoint p. From the previous theoremit now followsthat our flowadmits of a crosssectionifand onlyif 21does not containthe zero elementofthe firstBetti group. From this therefollowsa practicalcriterionforprovingin certaincases that a , cross-sectiondoes not exist. Suppose that forsome finiteset of points pi, , A, (In particularthis pk we can findthe associated asymptoticcyclesA , will be possible wheneverwe have located some periodicorbits.) The smallest convexset containingA1,, - , Apk is clearlycontainedin 21,so if this set contains the origin there can be no cross-section. Finally,we note that ifthe flowis spectrallydeterminatea cross-sectionexists fromzero. In particular, ifand onlyifthe asymptoticcycleofthe flowis different of the closed orbits is each if the flowis periodica cross-sectionexists unless X This is so because a periis arewise connected. homologousto zero, provided is flowis spectrally and we have that a recurrent proved odic flow recurrent determinate. ...
8. Hamiltoniansystems' [2] manifoldofclass C2 and supposeX is a closed Let X be a compactdifferentiable non-singulartwo formof class C1 definedover X. Since X is non-singularthe dimensionof X must be even. By definitiona canonical systemof coordinatesis a local coordinatizationfroma sphere in 2n-dimensionaleuclidean space with coordinates(pi, qi) such that w = E dpi A dqi . The fact that in generalit is possible to cover X by such canonical coordinatesystemsfollowsfroma wellknowntheoremof Pfaff. Next let a be a closed one-formof class C' definedover X. In any canonical coordinatesystemone can finda functionH(pi, qi) such that a = dH. Since H equations is uniquelydeterminedto withinan additive constant,the differential dqi dt
aH apt
dp dt
_
aH aqi
1 Hamiltonian in thelargerecently by G. systemsofthistypehavebeenconsidered Reeb. This content downloaded from 155.97.178.73 on Wed, 20 May 2015 22:39:17 UTC All use subject to JSTOR Terms and Conditions
ASYMPTOTIC
283
CYCLES
are determinedby the closed one-forma. Moreoverfromthe classical theoryof Hamiltoniansystemsit followsthat the differential equations obtained in and two canonical coordinatesystemsare compatible whereverthe coordinatized regionsoverlap. Thus we obtain fromthese differential equations a flowon the manifold.We denote the differentiation associated with the flowby Dax. Next, let a and A be any two givendifferentiable closed one-forms. In a canonical coordinatesystemlet a and = dH2 . The classical Poisson bracket dHj of H1 and H2 is
aH1aH2 _ aH1 aH2 a0q ape api aqi It is obvious that this expressionwould be the same forany local integralsH1 a and A; moreoverit is a standardresultthat this exand H2 of the one-forms pressionhas the same value in any canonical coordinatesystem.Thus a and A determinea functionon X by the above formula;we denote this functionby [a, A] and notice that [a, 3] - [A,a]. We now introducethe measure,udeterminedby the 2n-form whichis obtained by taking the exteriorproduct of X with itselfn times. We assume that X is such that ,u(X) = 1. It followsfroma well knowntheoremof Liouville's that forany closedone-form a, theflowwe have associatedwitha by meansofHamilton's equation has ,uas an invariantmeasure. Notice next that if A = dH2A whereH2 is a functiondefinedover the entiremanifoldX, then [a, A] = DaH2. Since u is an invariantmeasure,
[a, dH2(x)]
du(x) =
f
Da(H2(X)) du(x) =
of the Poisson bracketit followsthat if a = dH,(x) 0. By the skew-symmetry
f f
and : is any closed one-form, [dH1,73] dlu(x) Poisson bracketit followsthat
= 0. By the bilinearityof the
[a, 0] dbt(x) is completelydeterminedby the
cohomologyclasses to whicha and 3 belong. Next let f(x) E C(X) and suppose f(x) is of class C2. We know that there is such an f(x) in everyequivalenceclass mod R(X). It was pointedout in Section threethat if we representf(x) locally as exp (27riH2(x))thendH2 gives a unique closed one-formof definedover all of X, and the map sendingf into Ofinduces the naturalmappingof C(X)/R(X) into the quotientof the space of closed oneformsmodulo the boundingone-forms.Now let a be a fixedone-formand f a willmean differentiation fixedelementof C(X). Fromnow on differentiation with respectto the flowdeterminedby a. It is clear that since locally f(x) exp (27riH2(X)), (1/(2iri))(f'(x))1(f(x)) = H2'(x) = DaH2(X) = [a, Of] Thus
=
I /(27ri)
f
(f'(x)),/(f(x)) dlu(x)
=
f[a,
of] do.
The factthat theintegralofthisPoisson bracketdependsonlyon thecohomology class of of is in accordancewith the theorydeveloped previously;it simplyreThis content downloaded from 155.97.178.73 on Wed, 20 May 2015 22:39:17 UTC All use subject to JSTOR Terms and Conditions
284
SOL
minds us that 1/(2ri)
SCHWARTZMAN
f
(f'(x))/(f(x)) dyi(x)depends only on the equivalence
class to whichf(x) belongsmod R(X). The fact that this integraldepends only on the cohomologyclass to which a belongs,togetherwith the fact that the Poisson bracketis bilinearyieldssomethingmoreinteresting, however. THEOREM. The ,u-asymptotic cycleA., associatedwiththeflow obtainedfroma a is completely closedone-form determined by thecohomology class to whicha belongs.Moreover,themappingwhichsends cohomology classes into theassociated ,u-asymptotic cyclesis a linearmappingof thefirstcohomology groupinto thefirst homology group. As a special case we considerHamiltonianflowson even-dimensionalmultitori. Let H(pi, qi) be a functionof class C2 in euclidean 2n-space and suppose that in the Hamiltonianequations dqi dt
aH 'api
dpi dt
___H
fti
the functionson the righthand side of theseequationsare periodicofperiodone withrespectto each ofthe variablespi, qi . Then thereexistsa functionK(pi , qi) whichis also of period one with respectto each of the variables,togetherwith constantsa, , b, such that H(pi, qi) = K(pi , q) + aipi + biqj. If we considertheseequationsas inducinga flowon the multi-torus, and let ,ube ordinary on euclideanmeasure thismulti-torus, the theoremprovedabove showsthatthe cycle for the Hamiltonian H(pi, qi) is the same as the (,u)(p)-asymptotic asymptoticcycle we would get usingE aipi + biqi as our Hamiltonian. However, the equations of motion for this Hamiltonian are
Z
dq2
dt =
,
dp2 _
b_ dt =-s
and it is obvious fromgeometricconsiderationsthat the (p)-asymptoticcycle is E (azgqj - bigpt);where g, and g,, are the fundamentalcycles associated withthe variablesqi and pa. As a finalremark,we note that in the case of Hamiltonianflowson the two dimensionaltorus,thistellsus immediatelywhat thewindingnumberofPoincar6 are. Also,even on themulti-torus, we can apply the resultsoftheprevioussection on continuouseigenfunctions to see which eigenvaluesare possible. JOHNS HOPKINS UNIVERSITY AND INSTITUTE OF MATHEMATICAL SCIENCES NEW YORK UNIVERSITY BIBLIOGRAPHY
[1] N.
Stetige Abbildungen und Bettische Gruppen, Math. Ann., Vol. 109 (1934), pp. 525-537. [21E. CARTAN, Legons sur les Invariants Integraux, Hermann, Paris, 1922. Sur les Transformations d'Espaces Ml1triquesen Circonfrrence,Fund. [31 S. EILENBERG, Math., Vol. 24 (1935), pp. 160-176. [4] W. H. GOTTSCHALK and G. A. HEDLUND, Topological Dynamics, New York, 195!. [51 J. C. OXTOBY, Ergodic Sets, Bull. Amer. Math. Soc., Vol. 58 (1952), pp. 116-136. BRUSCHLINSKY,
This content downloaded from 155.97.178.73 on Wed, 20 May 2015 22:39:17 UTC All use subject to JSTOR Terms and Conditions
Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://www.jstor.org/page/ info/about/policies/terms.jsp JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected].
Annals of Mathematics is collaborating with JSTOR to digitize, preserve and extend access to Annals of Mathematics.
http://www.jstor.org
This content downloaded from 155.97.178.73 on Wed, 20 May 2015 22:39:17 UTC All use subject to JSTOR Terms and Conditions
ANNALS
OF MATHEMATICS
XVol.66, No. 2, September, 1957
Printedin U.S.A.
ASYMPTOTIC CYCLES* BY SOL
SCHWARTZMAN
(Received April 16, 1956) (Revised March 18, 1957)
1. Introduction[4]
Let X be a topologicalspace, G a topologicalgroup with identityelemente. Let f be a continuousmapping fromX X G into X. For simplicitywe will writef(x, g) as xg. If (xg1)g2= x(g9g2),and if we = x, G is said to act as a topological transformationgroup on X. Such transformationgroups have been studiedfrommany points of view. In this paper we will be concernedwith the viewpointfirstemployed systematicallyby George David Birkhoff.Here the concretemodel to be borne in mind is the flowarisingfroma systemof ordinary differential equations on a manifold,and in fact we will confineourselves to the case whereG is the additive group R of the real line with the usual topology. There is a strongconnectionbetweentopologicaldynamics,whichis the field we are concernedwithhere,and ergodictheory,anotherfieldin whichBirkhoff pioneered.Thus, in analogy with the Poincare recurrencetheoremthere are theorems concerningvarious kinds of topological recurrence.In addition, various incompressibility propertieshave been studied. These of course are a for kind of substitute the measure-preserving propertyusually required for in Other transformations ergodic theory. parallels exist-for example, sets of the firstcategoryplay much the same role in some theoremsthat one might expect fromsets of measure zero. Results of all these types can be found togetherwith many other theoremsin the book by Gottschalk and Hedlund cited above. The close parallelismbetweenthe two fieldshas tended to minimizethe attentiongivento questionswhichstronglyinvolvethe topologyon X. In particular algebraic topologyhas played no part in the developmentof the general theoremsin topologicaldynamics.In this paper a firstattemptis made to overcome this restriction,at least to the limitedextentof tryingto show the role it is not played by the firstBetti group. Since the orbitsare one-dimensional, surprisingthat the one-dimensionalgroup plays a leading role. It turns out that, except fora set of orbitshaving measure zero with respectto every invariantmeasure,we can associate with each orbitan elementof the firstBetti group of X. We call this the asymptoticcycle of the orbit. In a sense it tells * This paper represents results obtained in part at the Institute of Mathematical Sciences, New York University, under the sponsorship of the National Science Foundation, Contract NSF-G3050. Reproduction in whole or in part is permitted for any purpose of the United States Government. 270 This content downloaded from 155.97.178.73 on Wed, 20 May 2015 22:39:17 UTC All use subject to JSTOR Terms and Conditions
ASYMPTOTIC
271
CYCLES
how the orbitwinds around the space X. In the case of flowson the torus,the coefficients (withrespectto the naturalbasis) of the asymptoticcycleassociated withan orbitare preciselythe windingnumbersof Poincar6. We shall see that the study of the asymptoticcycles of orbitsis intimatelyconnectedwith the question of the existenceof surfacesof section,and the equally standardprobof the flow. lem of findingeigenfunctions Theorems in topological dynamics yield as special cases theorems about equations on manifolds.Thus far, apart fromconsequences of the differential fact that a Hamiltoniansystemhas an invariantmeasurewhichis positivefor open sets, thereis comparativelylittlein the generaltheorythat would enable equations to deduce additional informaone to use the formof the differential tion about the asymptoticbehaviorof the solutions.We will see below however that once the concept of the asymptoticcycle has been introducedwe will be able to performexplicit computations,particularlyfor Hamiltonian systems, whichwill tell us how an "average" orbitwindsaround the space. theory[5] 2. Kryloff-Bogoliouboff In additionto the parallel developmentthat has taken place betweenergodic theory and topological dynamics, methods have been developed which use resultsin one fieldto get theoremsin the other.For purposes of futurereference we collect here a numberof well-knowndefinitionsand theoremsof this type. A completeexpositionwhich contains proofsof these resultsand many otherswill be foundin the articleby Oxtoby cited above. Here and in all later paragraphsX will denote a compactmetricspace on whichthe real line acts as a topologicaltransformation group. A positive measure is a measure gi definedfor all Borel sets DEFINITION. S a X such that ,.(S) > 0 forall S. A normalizedmeasureis a positivemeasure forwhichtu(X)- 1. A measure gi definedon the Borel sets of X is called invariant DEFINITION. providedthat foreveryBorel set S and everyreal numbert, ji(St) = ji(s). A point p in X is called quasi-regularprovidedthat DEFINITION. limT_:O
1T
f T
f(pt) dt
existsforeveryreal-valuedcontinuousfunctionf(x) definedon X. For everyquasi-regularpointp thereexistsa uniquefinitemeasure THEOREM. Ap
suchthatforeverycontinuousf(x),
limrnT,
1/T
f(pt) dt
= Lf(x) dg,(x).
Moreover,gipis a positiveinvariantmeasuresuch thatjp (X) = 1. The set of pointswhichare not quasi-regularhas measurezerowith THEOREM. respectto everyfiniteinvariantmeasure. with respect to the A functionf(x) is said to be differentiable DEFINITION. flow provided that limAt~o (f(xAt) - f(x))/At exists uniformlyover X. This limitwill be denotedby f'(x) or (d/(dt))f(x).
This content downloaded from 155.97.178.73 on Wed, 20 May 2015 22:39:17 UTC All use subject to JSTOR Terms and Conditions
272
SOL
SCHWARTZMAN
The followingtheoremwas communicatedorally to the author by Professor Kakutani. It is this theoremthat will enable us to state our resultsfor arbitrary compact metric spaces, without restrictingourselves to differentiable manifolds. THEOREM
(Kakutaiii).
bAny
continuous function f(x) on X can be approxi-
nateduniformly byfunctionswhichare differentiable withrespectto theflow. PROOF.
Let gc(x)
=
ge(xiAt)-
1/
j
x
f(xt)dt. Then I
(p+At
1( 1
E Azt
fC+t
f(xt)dt
-
f(xt)dt)
f
f(xt)dt- _ At
f(xt)dt).
Since, as t goes to to, f(xt) approachesf(xto)uniformly, it followsthat gE(x) is differentiable with respectto the flow,and in fact g'(x) = (1/c)(f(xe) - f(x)). It is evidentthat as ? goes to zero,g,(x) approachesf(x) uniformly, so the theoremis proved. 3. The theoryof Eilenbergand Bruschlinsky[1], [3] Because of the relation between quasi-regularpoints and continuousfunctions, the theoryreferredto in the title of this section enables us to connect propertiesof the flow with the firstBetti group of the space. We will summarize those resultswhich will be needed later on, referring the reader to the articlescited above forproofs.As in the restof this paper, the space X referred to below is assumed to be compactmetric. Throughoutthe rest of this paper the homologyand cohomologygroupswe referto are those obtained fromthe Cech theoryusing real coefficients. When we speak of an integralcycle we will mean an elementof this homologygroup whicharises fromintegralcycles in the ordinarysense when we representour space as an inverseprojectivelimit of polyhedra.By a cocycle with integral periodswe will mean an elementof the cohomologygroup which,when consideredas a functionalon the homologygroup,assigns integervalues to each integralcycle. Actuallywe will use only the one-dimensionalgroups.It should be bornein mindthat any closed parametrizedcurveK in X determinesan element of the firsthomologygroupin the Cech theory,since a map of X into a polyhedronsends K into a curve of the polyhedron. The set of all continuouscomplex-valuedfunctionsof absolute DEFINITION. value one definedon X will be denotedby C(X). The subset of C(X) consisting of those functionswhichcan be writtenin the formexp (2iriH(x)), whereH(x) is continuousand real valued, will be denoted by R(X). If K is any parametrizedcurvein X and f(X) C(X), AK argf(x) will mean the changein the angular variableforf(x) along K. Notice that whenf(x) = exp (27riH(x)), (1/27r)AK argf(x) = H(p2) -H(P), wherepi and P2 are the initialand terminalpointsof K.
This content downloaded from 155.97.178.73 on Wed, 20 May 2015 22:39:17 UTC All use subject to JSTOR Terms and Conditions
273
ASYMPTOTICCYCLES
THEOREM.If f(x),-C(X) and K is a closedparametrized curve,(1/27r)AKargf is an integerdependingonlyon theelementof thefirsthomology groupcorresponding to K and thecosetdetermined byf(x) in C(X)/R(X). It is obvious that if X were a polyhedronthe above theoremwould immediately enable us to associate witheach elementof C(X)/R(X) an elementof the firstcohomologygroup which,as a functionalon the homologygroup, would assignto each cycleK the integermentionedin the above theorem.It turnsout to be possibleto generalizethis procedurein a naturalway so that it applies to any compactmetricspace X. THEOREM. The mapping indicatedabove, sending C(X)/R(X) into thefirst ontoall Cech,cocycleswith.integral cohomology group,is an algebraicisomorphism periods. THEOREM. If fi(x) and f2(x) belongto C(X) and fI(x) - f2(x)i < 1 thenthese twofunctionsdetermine thesame elementmod R (X). manifoldand f(x)eC(X) is a differFinally, suppose X is a differentiable entiable function.In a suitable local coordinatesystemf(x) = exp (2iriH(x)) where H(x) is differentiable and uniquely determinedup to an additive constant.If H1 and H2 are determinedas above in two different coordinatesystems, dHj1and dH2 agree in the regionof overlapping.Thus f(x) determinesa closed one-formover X. This mapping, sendingthe differentiable elementsof C(X) into closed one-forms,induces the natural mapping of C(X)/R(X) onto the elementswithintegralperiodsin the quotientgroupof closed one-forms modulo boundingone-forms. 4. Asymptoticcycles Let f(x) be an elementof 0(X) and let p be any point in X. By argf we mean the change in the angular variable off(x) along the orbit goingfromp to pt. LEMMA. Let p be any point of X and supposefi(x) and f2(x) are twoelements thesame-elementof C(X)/R(X). Then of C(X) whichdetermine DEFINITION.
A(p,pt)
(1/27r)A(,ppt)
argf1 -
(1/27r)A(p,pt) argf2
0(1).
PROOF. We knowthat forsome continuousH(x), fi(x) = f2(x) exp (2iriH(x)), Hence (1/27r)A(pgpt) argf1 - (1/2r)A(p,pt) argf2 = H(pt) - H(p). Since H(x)
mustbe bounded,the lemma is proved. THEOREM. Let p be any quasi-regularpoint. Then for any f(x) eC(X) lim,.b (1/(2tt))A(p,pt)argf exists.Moreover,this limitdependsonly on theelement of C(X)/R(X)
determined by f(x). The induced mapping of C(X)/R(X)
intothereal line is a grouphomomorphism. PROOF. By the above lemma,if this limitexistsforany fj(x) in 0(X) it exists and has the same value forany f2(x) equivalent to fj(x) mod R(X). Given any f2(x) in C(X) we know we can approximatef2(x) as closely as we wish by a withrespectto the flow.(A trivial functionfj(x) in C(X) whichis differentiable supplementto Kakutani's theoremin Section two shows this.) If we choose This content downloaded from 155.97.178.73 on Wed, 20 May 2015 22:39:17 UTC All use subject to JSTOR Terms and Conditions
274
SOL SCHWARTZMAN
fi(x) so that Ifi(x) - f2(x)I 0 such that any arc in X of diameterless than - has the propertythat the total change in the angular variable forf(x) over that arc is less than one radian. Divide K into
subares K1,
...
,
K. such that each subarc has diameter less than ?/2. For
any real numbert, Krt will mean the arc that K, goes into after time t has elapsed. thereexistsa sequence ti} going Since we are assumingthe flowis recurrent, to infinity such that lim1 Osupzxpp(x, xti) = 0. Then foreach r and sufficiently largevalues i, the diameterof Krti is less than t. Thereforeforsufficiently large values i the total changein the angularvariable off(x) along Kti is less than n radians,wheren is the numberof subares into whichK was divided. The same statementobviouslyholds forK itself. Now foreach i we will definea map 0i sendingthe rectanglein the (U, V)
ti, U = 0 and U = 1 into X.
plane bounded by the straight lines V = 0, V =
Any point (U, 0) on the unit intervalalong the U-axis is to be sent by (P into the curveK in such a way that(p taken on thisintervalsimplygivesthe original parametrizationof the parametrizedcurveK. An arbitrarypoint (U, V) in the rectangleis to be sent by (p into the point that 0j(U, 0) goes into under the flowaftertime V. Thus 0i sends vertical lines in the rectangleinto orbitsin the space X. In particularthe segmentof the boundaryof the rectanglelying on the line U = 0 gets sent into the orbitgoingfromxi to xit , and the corre1 goes into the orbit fromx2 to x2ti. The spondingsegmenton the line U portionof the boundaryalong the line V = 0 goes into K, and the part lying on the line V = tj goes into Kti . The functionf(x) on X now gives a functionf(oi(U, V)) on the rectangle. Since the firstBetti numberof a two-cellis zero, this functionon the rectangle has a logarithm,so the total change in the angularvariable around the boundary of the rectangleis zero. This can now be applied to curvesin X whichare the image of the boundary.Therefore AK
argf +
argf A(X2,X2ti)
AKtjargf
-
argf = 0 A(xj,xjtj)
or argf AA(X2,x2tj)
-
A(xl,zlti) argf =
AKti
argf -
AK
argf.
However the righthand side of this last equation is bounded and in fact is smallerthan 2n radians. Therefore,dividingby 2irtiand lettingi go to infinity, argf = limbo (1/(27rti))A(X2,X2tj) arg f. Since f(x) was liming(1/(27rti))A(x1lxti)
This content downloaded from 155.97.178.73 on Wed, 20 May 2015 22:39:17 UTC All use subject to JSTOR Terms and Conditions
280
SOL
SCHWARTZMA N
an arbitraryfunctionin C(X) this shows that the asymptoticcycle associated with xl is the same as that associated with x2. Thus the flowis spectrallydeterminateand the asymptoticcycle is definedand has the same value for all orbits,whetherquasi-regularor not. In particularthen fora recurrentflowany two closed orbits determineelementsof the firstBetti group which are rationallydependent. 7. Cross sections For the purposesof this paper, a subset K is said to be a crosssectionof X providedK is closed and the mapping4) of K X R into X sending (p, t) into pt is a local homeomorphism onto X. This is the same as what was called a "surfaceof section" by G. D. Birkhoffin his book "Dynamical Systems". Up to the pointwhereasymptoticcyclesenterthe picture,the discussion givenbelow followsBirkhoff(p. 144). It is easy to see that if K is any closed set and the mapping4 definedabove is onto all of X, then a necessaryand sufficient conditionthat K be a crosssection is that for some - > 0, 4 maps the Cartesianproduct of K with the open interval(-At, t) homeomorphically onto an open subset of X. Next, suppose that K is any crosssectionof X. For any pointx E X, let txbe the smallest non-negativereal numbert such that thereexistsa point p in K forwhichx = pt. Since the transformation sendingx into xt is a homeomorphism thereis only one p forwhichx = ptx. Call this point px . Next, let Tx be the smallestpositive value of T for which pxT belongs to K. (Clearly, such a Tx exists.) We now definefK(X) to equal exp (2iritx/Tx). It is easy to see thatfK(X) is in C(X). LEMMA. Let f(x) be in C(X) and supposef(x) is differentiable withrespectto theflow,and that (f'(x))/(2irif(x)) > 0 for all x. If we let K be the set of pointsfor whichf(x) = 1, thenK is a cross-section and fK(x) and f(x) are congruentmod R(X). PROOF. Let 4 be the mappingof K X R into X sending (p, t) into pt. Obviously4 is a continuousmappingonto all of X. Let (p, t) be a point of K X R and suppose (pi, Tj) and (qi, Si) are two sequences of points in K X R convergingto (p, t). Suppose piT, = qjSi. Then pi(Ti - Si) = qi. Therefore argf is an integer,i.e. f(pi(T, - Si)) = 1 so that (1/(2r))A(pi,pi(Ti(Si)) DEFINITION.
T.-Si
a(f'(p
it))/(27rif(p it)) dt
is an integer. Since the integrandis positiveand Ti - Si goes to zero, forsufficiently large values of i, Ti = Si and pi = qi. This showsthat 4 is locally one-to-one. Next let 0 be an open subset of K X R and suppose (p, t) belongsto 0. Let {qi} convergeto pt in X. Then {f(qi(-t)) } convergesto f(p) 1. Let m = inf (f'(x))/(27rif(x)). Then I(1/(27r))A(xtjxt2) arg f i ? mi t1 t2[ for any x E X and any t1, t2. From thisit followsthat it is possibleto choose a sequence of real numbersti such that ti convergesto - t and f(qiti) = 1 forall i. There-
This content downloaded from 155.97.178.73 on Wed, 20 May 2015 22:39:17 UTC All use subject to JSTOR Terms and Conditions
281
ASYMPTOTIC CYCLES
foreqjtjbelongsto K foreveryintegeri. Since tj convergesto -t, qjtjconverges to p and for sufficiently large values of i, (qjtj, - ti) belongs to 0. However = qi . This shows that 4 is an open mapping. Since we already know ti) 0(qjtj2that p is locally one-to-oneit followsthat 4 is a local homeomorphismand therefore K is a cross-section. Next, let s' be the mappingof [0, 1] X K into X whichsends (a, p) into p(aT ).
Since Tp is a continuousfunctionof p in K,
AVis
a continuousmappingonto all
of X. If we let g(a, p) = exp (2iria), then g(a, p) = fK(V/(a,p)). Now it is easy
to see that there is one and only one real-valuedfunctionH(a, p) such that H(O, p) = 0 and f(i/(a, p)) = exp (2iriH(a, p)). Obviously H(a, p) is continuousand H(1, p) = 1 forall p in K. Then fK6(/(a,
p))/f(V/(a, p))
=
exp (2ri(a
-
H(a, p))).
Since a - H(a, p) = 0 fora = 0 or 1, thereis a real valued functionh(x) definedon X forwhicha - H(a, p) = h(VI(a,p)). Obviouslyh(x) is continuous and fK(x)/f(x)
= exp (2irih(x)). Therefore fK(x) is congruent to f(x) mod R(X)
and the proofof the lemmais completed. THEOREM. Let f(x) E C(X). A necessaryand sufficient conditionthat there existsa cross-section K for whichfK(X) is congruentto f(x) mod R(X) is that A,,(f) > 0 foreverypositiveinvariantmeasureju. PROOF. First suppose that such a cross-section K exists. If we let M suppEKT, and if x is any quasi-regularpoint, Ax(fK)
=
limToo
=
(1/(27rT))A(x,xT) argfk ? 1/M.
Thus forany positiveinvariantmeasure u,A,(f) = A,(fK)
=
f
A,(f) d1i(x) >
0.
Next suppose that A,>(f)> 0 for every positive invariantmeasure Al.There is no loss in generalityin assumingthatf is differentiable withrespectto the flow, since if this were not the case we could replacef by a functionequivalent to f mod R(X) and havingthisproperty.Now let B(X) be the space ofall continuous real-valuedfunctionsg(x) on X with 11g(x) 11= supe I g(x) 1. As in the previous sectionlet D be the space of functionswhichare derivativesof real valued with respectto the flow.The statementthat A,;(f) >0 functionsdifferentiable for all positive 4 says that
f
(f'(x))/(2irif(x))dyu(x)> 0 for all positive ,u.
If D 0 {J(f'(x))/(27rif(x))} did not intersectthe cone of positive functionsin B(X) it would followfromthe Hahn-Banach theoremthat forsome positive,u, (f'(x))/(2irif(x))d,4(x) = 0. Thus thereis a functionH(x) whichis differentiable with respect to the flow and has the propertythat H'(x) + (f'(x))/ (27rif(x))> 0. Let f(x) = f(x) exp (2iriH(x)). Then f'(x) existsand f'(x)/2rif(x) This content downloaded from 155.97.178.73 on Wed, 20 May 2015 22:39:17 UTC All use subject to JSTOR Terms and Conditions
282
SOL
SCHWARTZMAN
is positive,and so by the lemmaprovedabove, ifwe let K be the set of pointsx for whichf(x) = 1, K is a cross-sectionand fK(x) is equivalent to f(x). Since f(x) is equivalent to the originalf(x) mod R(X), the theoremis proved. Now recall that the conjugate space of B(X) is the space of all finiterealvalued measuresA definedon the Borel sets of X. If we put the weak* topology on this space of functionals,it is well known that the subspace consistingof normalizedpositivemeasuresinvariantwithrespectto the flowis compact,and is in fact the convex closure in the weak* topologyof the set of measures 4, arisingfromquasi-regularpointsp. Since forany f(x) E 0(X), the map sendingj intoA,(f) is continuousin the weak* topology,the same can be said forthe map into the firstBetti grouptaking,uinto A, . From this it followsthat the subset of the firstBetti group consistingof all elementsA, is a compact convex set 21which is the convex closure of those elementswhich are of the type A, for some quasi-regularpoint p. From the previous theoremit now followsthat our flowadmits of a crosssectionifand onlyif 21does not containthe zero elementofthe firstBetti group. From this therefollowsa practicalcriterionforprovingin certaincases that a , cross-sectiondoes not exist. Suppose that forsome finiteset of points pi, , A, (In particularthis pk we can findthe associated asymptoticcyclesA , will be possible wheneverwe have located some periodicorbits.) The smallest convexset containingA1,, - , Apk is clearlycontainedin 21,so if this set contains the origin there can be no cross-section. Finally,we note that ifthe flowis spectrallydeterminatea cross-sectionexists fromzero. In particular, ifand onlyifthe asymptoticcycleofthe flowis different of the closed orbits is each if the flowis periodica cross-sectionexists unless X This is so because a periis arewise connected. homologousto zero, provided is flowis spectrally and we have that a recurrent proved odic flow recurrent determinate. ...
8. Hamiltoniansystems' [2] manifoldofclass C2 and supposeX is a closed Let X be a compactdifferentiable non-singulartwo formof class C1 definedover X. Since X is non-singularthe dimensionof X must be even. By definitiona canonical systemof coordinatesis a local coordinatizationfroma sphere in 2n-dimensionaleuclidean space with coordinates(pi, qi) such that w = E dpi A dqi . The fact that in generalit is possible to cover X by such canonical coordinatesystemsfollowsfroma wellknowntheoremof Pfaff. Next let a be a closed one-formof class C' definedover X. In any canonical coordinatesystemone can finda functionH(pi, qi) such that a = dH. Since H equations is uniquelydeterminedto withinan additive constant,the differential dqi dt
aH apt
dp dt
_
aH aqi
1 Hamiltonian in thelargerecently by G. systemsofthistypehavebeenconsidered Reeb. This content downloaded from 155.97.178.73 on Wed, 20 May 2015 22:39:17 UTC All use subject to JSTOR Terms and Conditions
ASYMPTOTIC
283
CYCLES
are determinedby the closed one-forma. Moreoverfromthe classical theoryof Hamiltoniansystemsit followsthat the differential equations obtained in and two canonical coordinatesystemsare compatible whereverthe coordinatized regionsoverlap. Thus we obtain fromthese differential equations a flowon the manifold.We denote the differentiation associated with the flowby Dax. Next, let a and A be any two givendifferentiable closed one-forms. In a canonical coordinatesystemlet a and = dH2 . The classical Poisson bracket dHj of H1 and H2 is
aH1aH2 _ aH1 aH2 a0q ape api aqi It is obvious that this expressionwould be the same forany local integralsH1 a and A; moreoverit is a standardresultthat this exand H2 of the one-forms pressionhas the same value in any canonical coordinatesystem.Thus a and A determinea functionon X by the above formula;we denote this functionby [a, A] and notice that [a, 3] - [A,a]. We now introducethe measure,udeterminedby the 2n-form whichis obtained by taking the exteriorproduct of X with itselfn times. We assume that X is such that ,u(X) = 1. It followsfroma well knowntheoremof Liouville's that forany closedone-form a, theflowwe have associatedwitha by meansofHamilton's equation has ,uas an invariantmeasure. Notice next that if A = dH2A whereH2 is a functiondefinedover the entiremanifoldX, then [a, A] = DaH2. Since u is an invariantmeasure,
[a, dH2(x)]
du(x) =
f
Da(H2(X)) du(x) =
of the Poisson bracketit followsthat if a = dH,(x) 0. By the skew-symmetry
f f
and : is any closed one-form, [dH1,73] dlu(x) Poisson bracketit followsthat
= 0. By the bilinearityof the
[a, 0] dbt(x) is completelydeterminedby the
cohomologyclasses to whicha and 3 belong. Next let f(x) E C(X) and suppose f(x) is of class C2. We know that there is such an f(x) in everyequivalenceclass mod R(X). It was pointedout in Section threethat if we representf(x) locally as exp (27riH2(x))thendH2 gives a unique closed one-formof definedover all of X, and the map sendingf into Ofinduces the naturalmappingof C(X)/R(X) into the quotientof the space of closed oneformsmodulo the boundingone-forms.Now let a be a fixedone-formand f a willmean differentiation fixedelementof C(X). Fromnow on differentiation with respectto the flowdeterminedby a. It is clear that since locally f(x) exp (27riH2(X)), (1/(2iri))(f'(x))1(f(x)) = H2'(x) = DaH2(X) = [a, Of] Thus
=
I /(27ri)
f
(f'(x)),/(f(x)) dlu(x)
=
f[a,
of] do.
The factthat theintegralofthisPoisson bracketdependsonlyon thecohomology class of of is in accordancewith the theorydeveloped previously;it simplyreThis content downloaded from 155.97.178.73 on Wed, 20 May 2015 22:39:17 UTC All use subject to JSTOR Terms and Conditions
284
SOL
minds us that 1/(2ri)
SCHWARTZMAN
f
(f'(x))/(f(x)) dyi(x)depends only on the equivalence
class to whichf(x) belongsmod R(X). The fact that this integraldepends only on the cohomologyclass to which a belongs,togetherwith the fact that the Poisson bracketis bilinearyieldssomethingmoreinteresting, however. THEOREM. The ,u-asymptotic cycleA., associatedwiththeflow obtainedfroma a is completely closedone-form determined by thecohomology class to whicha belongs.Moreover,themappingwhichsends cohomology classes into theassociated ,u-asymptotic cyclesis a linearmappingof thefirstcohomology groupinto thefirst homology group. As a special case we considerHamiltonianflowson even-dimensionalmultitori. Let H(pi, qi) be a functionof class C2 in euclidean 2n-space and suppose that in the Hamiltonianequations dqi dt
aH 'api
dpi dt
___H
fti
the functionson the righthand side of theseequationsare periodicofperiodone withrespectto each ofthe variablespi, qi . Then thereexistsa functionK(pi , qi) whichis also of period one with respectto each of the variables,togetherwith constantsa, , b, such that H(pi, qi) = K(pi , q) + aipi + biqj. If we considertheseequationsas inducinga flowon the multi-torus, and let ,ube ordinary on euclideanmeasure thismulti-torus, the theoremprovedabove showsthatthe cycle for the Hamiltonian H(pi, qi) is the same as the (,u)(p)-asymptotic asymptoticcycle we would get usingE aipi + biqi as our Hamiltonian. However, the equations of motion for this Hamiltonian are
Z
dq2
dt =
,
dp2 _
b_ dt =-s
and it is obvious fromgeometricconsiderationsthat the (p)-asymptoticcycle is E (azgqj - bigpt);where g, and g,, are the fundamentalcycles associated withthe variablesqi and pa. As a finalremark,we note that in the case of Hamiltonianflowson the two dimensionaltorus,thistellsus immediatelywhat thewindingnumberofPoincar6 are. Also,even on themulti-torus, we can apply the resultsoftheprevioussection on continuouseigenfunctions to see which eigenvaluesare possible. JOHNS HOPKINS UNIVERSITY AND INSTITUTE OF MATHEMATICAL SCIENCES NEW YORK UNIVERSITY BIBLIOGRAPHY
[1] N.
Stetige Abbildungen und Bettische Gruppen, Math. Ann., Vol. 109 (1934), pp. 525-537. [21E. CARTAN, Legons sur les Invariants Integraux, Hermann, Paris, 1922. Sur les Transformations d'Espaces Ml1triquesen Circonfrrence,Fund. [31 S. EILENBERG, Math., Vol. 24 (1935), pp. 160-176. [4] W. H. GOTTSCHALK and G. A. HEDLUND, Topological Dynamics, New York, 195!. [51 J. C. OXTOBY, Ergodic Sets, Bull. Amer. Math. Soc., Vol. 58 (1952), pp. 116-136. BRUSCHLINSKY,
This content downloaded from 155.97.178.73 on Wed, 20 May 2015 22:39:17 UTC All use subject to JSTOR Terms and Conditions
