A higher-dimensional Poincaré–Birkhoff theorem without monotone twist

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Ordinary differential equations/Dynamical systems

A higher-dimensional Poincaré–Birkhoff theorem without monotone twist

Un théorème de Poincaré–Birkhoff en plusieurs dimensions sans torsion monotone

Alessandro Fonda

^a

, Antonio J. Ureña

^b

aDipartimentodiMatematicaeGeoscienze,UniversitàdiTrieste,P.leEuropa, 1,34127 Trieste,Italy bDepartamentodeMatemáticaAplicada,FacultaddeCiencias,UniversidaddeGranada,18071Granada,Spain

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received30October2015

Acceptedafterrevision1February2016 Availableonline21March2016 PresentedbytheEditorialBoard

We provide a simple proof for a higher-dimensional version of the Poincaré–Birkhoff theorem,whichapplies toPoincarétimemapsofHamiltoniansystems. Thesemapsare requiredneithertobeclosetotheidentitynortohaveamonotonetwist.

©^{2016Académiedessciences.PublishedbyElsevierMassonSAS.}All rights reserved.

r é s um é

Nousfournissonsunepreuvesimpled’uneversionenplusieursdimensionsduthéorèmede Poincaré–Birkhoffquis’applique auxapplicationsdePoincarédessystèmeshamiltoniens.

Ces applications ne sont tenues, ni d’être proches de l’identité, ni d’avoirune torsion monotone.

©^{2016Académiedessciences.PublishedbyElsevierMassonSAS.}All rights reserved.

1. Statementoftheresult

The aim of thisshort note is to give a simpleproof, followingthe ideas developed in [3,4], ofa higher-dimensional versionofthePoincaré–Birkhofftheorem,whichappliestoPoincarétimemapsofaHamiltoniansystem,say

(

HS

) ˙

z

=

^J

∇

^H

(

t

,

z

) .

Here, J

=

_{0 I}

−

^IN 0N

denotesthestandard 2N

×

^{2N symplecticmatrixand}

∇

^{stands forthegradientwithrespecttothe} z variables.TheHamiltonianfunction H

:

R

×

R^2N

→

R^{isassumedtobe T-periodicinitsfirstvariablet andC∞-smooth} withrespecttoallvariables.

E-mailaddresses:[email protected](A. Fonda),[email protected](A.J. Ureña).

http://dx.doi.org/10.1016/j.crma.2016.01.023

1631-073X/©^{2016Académiedessciences.PublishedbyElsevierMassonSAS.}All rights reserved.

Consequently,foreveryinitialposition

ζ ∈

R^2N,i.e.,

ζ = (ξ, η ) ∈

R^N

×

R^{N,thereisauniquesolution}Z(

· , ζ ) =

Z(

· , ξ, η )

of

(

HS

)

satisfyingZ(⁰

, ζ ) = ζ

.Letusfurther assumethat,for

η

insomeclosedball B

⊂

R^{N centered attheorigin,these} solutions canbe continuedtothewholetimeinterval

[

⁰

,

T

]

^{.Wecanthen considertheso-calledPoincarétimemap:thisis} thefunctionP

:

R^N

×

^B

→

R^N

×

R^N,definedby

P

(ζ ) =

Z

(

T

, ζ ) ,

whosefixedpointsgiverisetoT-periodicsolutionsof

(

HS

)

.

We usethe notation z

= (

x

,

y

)

, with x

= (

x₁

, . . . ,

x_N

) ∈

R^{N and y}

= (

y₁

, . . . ,

y_N

) ∈

R^{N, andassume that H}

(

t

,

x

,

y

)

is 2π^{-periodic ineach ofthevariables x}1

, . . . ,

xN. Then,once a T-periodic solutionz

(

t

) = (

x

(

t

),

y

(

t

))

has beenfound,many others appear by just adding an integer multiple of 2π ^{to some of the components x}i

(

t

)

; for thisreason, we will call geometricallydistincttwo T-periodicsolutionsto

(

HS

)

(ortwofixedpointsofP)whichcannot beobtainedfromeachother inthisway.

Theresultwewanttoproveisthefollowing.

Theorem1.1.Writing

P

(

x

,

y

) = (

x

+ ϑ(

x

,

y

), ρ (

x

,

y

)) , (

x

,

y

) ∈ R

^N

×

^B

,

assumethat,either

ϑ(

x

,

y

) / ∈ { α

y

: α ≥

⁰

} ,

for every

(

x

,

y

) ∈ R

^N

× ∂

B

,

(1)

or

ϑ(

x

,

y

) / ∈ {− α

y

: α _≥

0

} ,

for every

(

x

,

y

) ∈ R

^N

× ∂

B

.

(2)

Then,P^hasatleastN

+

¹geometricallydistinctfixedpointsinR^N

×

^{B.Moreover,ifallfixedpointsare}nondegenerate,thenthereare atleast2^Nofthem.

This isa specialcaseof[4, Theorem 2.1],where amuch moregeneralsituationwas considered. However, webelieve that thesimple proofproposed belowwillclarify themain ideas andhelp theinterested readertowardspossiblefurther generalizations.

2. Theproof

In order to fix ideas, we assume that B is the open unit ball in R^{N and that (1) holds. As before, for}

ζ = (ξ, η ) ∈

R^N

×

R^N,wedenotebyZ(t

, ζ )

thevalueattimet ofthesolutionz of

(

HS

)

withz

(

0

) = ζ

.TheHamiltonianH

(

t

,

x

,

y

)

being 2π^{-periodic in the variables x}i,the continuous image by Z ^of

[

⁰

,

T

] × (R

^N

/

2πZ^N

) ×

^{B willbe bounded inthe cylinder}

(

R^N

/

2πZ^N

) ×

R^{N and,after}multiplying Hbyasmoothcutofffunctionof y,thereisnolossofgeneralityinassumingthat:

(•)

^{thereissome R}

≥

^{2 suchthat H}

(

t

,

x

,

y

) =

^0,if

|

^y

| ≥

^R.

Inparticular,theC^∞-smoothmapZ

:

R

×

R^2N

→

R^{2N isnowgloballydefined.Foranyt,wewrite}Zt

:=

Z

(

t

, · ) :

R^2N

→

R^{2N, and denote by} Xt

,

Yt

:

R^2N

→

R^{N the} corresponding components, i.e., Zt

= (X

t

,

Yt

)

. The following assertions are standardconsequencesfromourassumptions.

(i) Z0istheidentitymapinR^2N;

(ii) Zt

(ζ +

^p

) =

Zt

(ζ ) +

^p,ifp

∈

²πZ^N

× {

⁰

}

^;

(iii) eachZtisa(C^∞-smooth)canonicaltransformationofR^2Nonitself1; (iv) Z

(

t

, ξ, η ) = (ξ, η )

,if

| η _{| ≥}

R;

(v) thereissomeconstant

∈ ]

⁰

,

1

[

^suchthat

XT

(ξ, η ) − ξ ∈ { αη _: α _≥

0

} ,

if 1

≤ | η _{| ≤}

1

+ .

ChoosenowaC^∞-function

γ : [

⁰

, +∞[ →

R^,with

[

^h

]

γ (

s

) =

^{0 on}

[

⁰

,

1

] , γ

(

s

) ≥

^{0 on}

]

¹

,

1

+ [ , γ

(

s

) ≥

^{1 on}

[

¹

+ ,

2

] , γ (

s

) =

^s2on

[

²

, +∞[ ,

andlet

λ >

0 beaparameter,tobefixedlater.WedefinethefunctionR_λ

:

R^2N

→

R^as

1 ThismeansthateachZ^t:R^2N→R^2Nisadiffeomorphismand(∂Z/∂ζ )^∗J(∂Z/∂ζ )=^{Jatanypoint.See,e.g.,}Proposition3(p. 4)in[2].

R

_λ

(ξ, η ) := − λ γ ( | η _| ) ,

andthefunctionR_λ

:

R

×

R^2N

→

R^by R_λ

(

t

, ·) := R

_λ

◦

Z_t⁻¹

,

if 0

≤

t

<

T

,

extendedbyT-periodicityint.Now,set

H_λ

(

t

,

z

) :=

^H

(

t

,

z

) +

^Rλ

(

t

,

z

) .

ThisfunctionH_λ

:

R

×

R^2N

→

R^{willbereferredtoas‘themodified}Hamiltonian’,andoneeasilychecksthat:

(vi) Hλ

(

t

,

z

) =

Hλ

(

t

+

^T

,

z

) =

Hλ

(

t

,

z

+

^p

)

,ifp

∈

²πZ^N

× {

⁰

}

^; (vii) H_λ

(

t

,

x

,

y

) = −λ|

^y

|

^2,if

|

^y

| ≥

^R;

(viii) H_λandH coincideontheopenset

(

t

,

Z

(

t

, ξ, η )) :

⁰

<

t

<

T

, η ∈

^B .

Atthispoint,wewouldliketoapplyeither[1,Theorem3],[5,Theorem4.2],or[6,Theorem8.1];themainassumptions of these results are ensured by (vi) and (vii) above. They would provide the existence of at least N

+

1 geometrically distinct T-periodicsolutionstotheHamiltoniansystem

(

HS

)

_λassociatedwiththemodifiedHamiltonian,and2^N ofthemif nondegenerate.

There is, however, a difficulty: these three theorems also assume that the Hamiltonian function is continuous in all variables,butourmodifiedHamiltonianH_λwillprobablybediscontinuouswhentisanintegermultipleofT.Nevertheless, one observes that the restriction of Hλ to

]

⁰

,

T

[×R

^{2N can be} continuously extended to

[

⁰

,

T

] ×

R^{2N (just by the same} formula

(

t

,

z

) →

^H

(

t

,

z

) +

Rλ

◦

Z_t⁻¹), and this extension is now C^∞-smooth on

[

⁰

,

T

] ×

R^{2N. Under this condition, the} proofsoftheresultsabovekeeptheirvalidity.LetusbrieflyjustifythisassertioninthecaseofSzulkin’sresults[5,6],which aresufficientforourpurposes.

Inbroadterms,Szulkin’sargumentsarebasedonthestudyofthefunctional

(

z

) =

¹ 2

T

0

˙

z

(

t

)

^∗J z

(

t

)

dt

−

T

0

H

(

t

,

z

(

t

))

dt

,

z

∈

^H1/2

( R /

T

Z , R

^2N

) ,

whosecritical points arethe T-periodicsolutions to

(

HS

)

.The periodicityof H inthe xi variables canbe used tosee

asbeingdefinedonthe productofthe N-torus R^N

/

2πZ^{N anda suitableHilbertspace, alongwhich}

hasthegeometry ofa (stronglyindefinite)saddle. Now,it iswell knownthat asmooth function onthe N-torushasatleast N

+

^{1 critical} points(Lusternik–Schnirelmann) and2^N ifnondegenerate(Morse).Thisresulthasaninfinite-dimensionalanalogue;under assumptions (vi)–(vii), ourfunctionalhas atleast N

+

^{1 criticalpoints and 2N in the}nondegenerate case. Thecontinuity inthetimevariableof H,whichwasanaturalassumptioninSzulkin’swork,is,infact,notusedinthediscussion.These differentcriticalpointsaregeometricallydistinctT-periodicsolutionsto

(

HS

)

_λ.

Asaconsequenceof(viii),theHamiltoniansystems

(

HS

)

and

(

HS

)

λhavethesameT-periodicsolutionsz

(

t

) = (

x

(

t

),

y

(

t

))

departingwithy

(

0

) ∈

^{B.Thus,inordertocompletetheproofof}Theorem 1.1,itwillsufficetocheckthefollowing

Proposition.If

λ >

0islargeenough,then

(

HS

)

_λdoesnothaveT -periodicsolutionsz

(

t

) = (

x

(

t

),

y

(

t

))

departingwithy

(

0

) ∈

^B.

Proof. Inviewof

(•)

^{,wemaychoosesomeconstantc}

>

0 suchthat

∂

H

∂

y

(

t

,

x

,

y

)

≤

^c

,

for every

(

t

,

x

,

y

) ∈ R × R

^N

× R

^N

,

andobservethat,consequently,

|

XT

(ξ, η ) − ξ | ≤

^cT

,

for any

ξ, η _{∈ R}

^N

.

(3)

Itwillbeshownthat,if

λ >

c

,

(4)

thentheconclusion holds.We provethisresultbya contradictionargumentandassumeinsteadthat z

= (

x

,

y

) : [

⁰

,

T

] →

R^N

×

R^{N is a solution to}

(

HS

)

λ for such a value of

λ

, with z

(

0

) =

^z

(

T

)

, departing with

|

^y

(

0

) | ≥

^{1. We consider the} C^∞-function

ζ : [

⁰

,

T

] →

R^{2N,definedby}

ζ (

t

) :=

Z_t⁻¹

(

z

(

t

)) .

Claim.

ζ ˙ =

^J

∇

R_λ

(ζ )

.

ProofoftheClaim. Differentiatingintheequalityz

(

t

) =

Z(t

, ζ (

t

))

,wefind

˙

z

= ∂

Z

∂

t

(

t

, ζ ) + ∂

Z

∂ζ (

t

, ζ )˙ ζ ,

sothat

∂

Z

∂ζ (

t

, ζ ) ζ ˙ =

^J

∇

H_λ

(

t

,

z

) −

^J

∇

^H

(

t

,

z

) =

^J

∇

^Rλ

(

t

,

z

) .

(5)

By (iii)above,Zt iscanonical,sothat

∂

Z

∂ζ (

t

, ζ (

t

))

^∗J

∂

Z

∂ζ (

t

, ζ (

t

)) =

^J

,

for everyt

∈ [

⁰

,

T

] .

Hence,ifwemultiplybothsidesof(5)by

−

^J

(∂Z/∂ζ )

^∗J,weget

ζ ˙ =

^J

∂

Z

∂ζ (

t

, ζ )

^∗

∇

^Rλ

(

t

,

z

) =

^J

∇ R

_λ

(ζ ) ,

thelastequalitycomingfromthefactthat R_λ

(

t

,

Z

(

t

, ζ )) =

R_λ

(ζ )

.ThisfinishestheproofoftheClaim. 2

LetusnowcompletetheproofofourProposition.Wewrite

ζ (

t

) = (ξ(

t

), η (

t

))

; combiningtheClaimandthedefinition ofR_λ,wehave

ξ ˙ = −λ γ

(| η |) η

| η | , η ˙ =

⁰

,

andconsequently,recalling(i),

η (

t

) = η (

0

) =

y

(

0

) , ξ(

t

) =

x

(

0

) −

t

λ γ

(|

y

(

0

)|)

^y

(

0

)

|

y

(

0

)| ,

foreveryt

∈ [

⁰

,

T

]

^.Inparticular, x

(

T

) =

XT

ξ(

T

), η (

T

)

=

XT

x

(

0

) −

^T

λ γ

(|

^y

(

0

)|)

^y

(

0

)

|

^y

(

0

)| ,

y

(

0

)

.

(6)

Inordertoobtainthedesiredcontradiction,weshallshowthatx

(

T

) =

^x

(

0

)

.Wedistinguishthreecases:

Case I: 1

≤ |

^y

(

0

) | <

1

+

^.Since

γ

( |

^y

(

0

) | ) ≥

^0,by

[

^h

]

^,thecombinationof(6)and(v)gives x

(

T

) −

^x

(

0

) +

^T

λ γ

( |

^y

(

0

) | )

^y

(

0

)

|

^y

(

0

) | ∈ { α

y

(

0

) : α _≥

0

} ,

implyingthatx

(

T

) =

^x

(

0

)

.

Case II: 1

+ ≤ |

^y

(

0

)| ≤

^{R.Bythetriangle}inequality,

|

x

(

T

) −

x

(

0

)| ≥

T

λ γ

(|

y

(

0

)|) −

^x

(

T

) −

x

(

0

) +

T

λ γ

(|

y

(

0

)|)

^y

(

0

)

|

y

(

0

)|

,

andrememberingthat

γ

( |

^y

(

0

) | ) ≥

^1,by

[

^h

]

^{,thejointactionof(3),(4)and(6)gives}

|

^x

(

T

) −

^x

(

0

) | ≥

^T

λ −

^{T c}

>

0

,

implyingagainthatx

(

T

) =

^x

(

0

)

.

Case III:

|

^y

(

0

) | >

R. Now

γ

( |

^y

(

0

) | ) =

²

|

^y

(

0

) |

^{, by}

[

^h

]

^{; combining (6) and (iv), we have that x}

(

T

) =

^x

(

0

) −

^2T

λ

y

(

0

)

. In particular,x

(

T

) =

^x

(

0

)

alsointhiscase.

Theproofiscomplete. 2

Remark.Eventhoughwehavealwaysassumed,forthesakeofsimplicity,thatHisC^∞-smoothwithrespecttoallvariables, everythingintheproofworksjustthesamebyassumingthatthisdependenceismerelyofclassC²(fortheN

+

1 solutions) orC³ (forthe2^N solutions)withrespecttothestatevariablez.

References

[1]C.C.Conley,E.J.Zehnder,TheBirkhoff–LewisfixedpointtheoremandaconjectureofV.I. Arnold,Invent.Math.73(1983)33–49.

[2]I.Ekeland,ConvexityMethodsinHamiltonianMechanics,Springer-Verlag,Berlin,1990.

[3] A.Fonda,A.J.Ureña,OnthehigherdimensionalPoincaré–BirkhofftheoremforHamiltonianflows,1: theindefinitetwist,preprint,availableonlineat www.dmi.units.it/~fonda/2013_Fonda-Urena.pdf,2013.

[4] A.Fonda,A.J.Ureña,OnthehigherdimensionalPoincaré–BirkhofftheoremforHamiltonianflows,2:theavoidingrayscondition,preprint,available onlineatwww.dmi.units.it/~fonda/2014_Fonda-Urena.pdf,2014.

[5]A.Szulkin,Arelativecategoryandapplicationstocriticalpointtheoryforstronglyindefinitefunctionals,NonlinearAnal.15(1990)725–739.

[6]A.Szulkin,CohomologyandMorsetheoryforstronglyindefinitefunctionals,Math.Z.209(1992)375–418.