May 7, 2003 15:6 WSPC/148-RMP 00165 EXPONENTIALLY SMALL SPLITTING AND ARNOLD

47 Pages·2015·0.45 MB·English

May 7, 2003 15:6 WSPC/148-RMP 00165
ReviewsinMathematical Physics
Vol.15,No.4(2003)1{47
c WorldScienti(cid:12)c PublishingCompany
(cid:13)
EXPONENTIALLY SMALL SPLITTING AND ARNOLD
DIFFUSION FOR MULTIPLE TIME SCALE SYSTEMS
MICHELAPROCESI
Received8May2002
Revised10February2003
WeconsidertheclassofHamiltonians:
1n(cid:0)1 1 p2 n
2 Ij2+ 2"In2+ 2 +"[(cosq(cid:0)1)(cid:0)b2(cos2q(cid:0)1)]+"(cid:22)f(q) sin( i);
jX=1 Xi=1
where 0 b < 1, and the perturbing function f(q) is a rational function of eiq. We
(cid:20) 2
proveupperandlowerboundsonthesplittingforsuchclassofsystems,inregionsofthe
phase space characterized by one fast frequency. Finally using an appropriate Normal
Formtheoremweprovetheexistence ofchainsofheteroclinicintersections.
Keywords:
Contents
1. Presentation of the Model andMain Theorems 2
2. PerturbativeConstruction of the Homoclinic Trajectories 6
2.1 Whisker calculus, the \primitive" =t 8
2.2 The recursive equations 10
3. Proofs of the Theorems 12
3.1 The formal linear equation 12
3.2 Lower boundson the Melnikov term 15
3.3 Heteroclinic intersection for systems with one fast frequency 16
4. Tree Representation 21
4.1 De(cid:12)nitions of trees 21
4.2 Admissible trees 24
4.3 Values of trees 26
4.4 Tree identities 28
4.4.1 Mark addingfunctions 28
4.4.2 Fruitadding functions 29
4.4.3 Changing the (cid:12)rstnode 32
4.5 Upperboundson the values of trees 34
A. Appendix 39
A.1 Proof of Proposition 4.16 39
A.2 Normal form theorem 40
A.3 Proof of Lemma 4.23 45
References 46
1
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2 M.Procesi
1. Presentation of the Model and Main Theorems
ThegeneralsettingofthispaperistheproblemofhomoclinicsplittingandArnol’d
di(cid:11)usion in a priori stable systems with three or more relevant time scales. The
generalstrategyis the oneproposedin [1] and[2] andin particularthe application
to a priori stable systems proposed in [3] and further developed in [4]. More pre-
cisely we consider a class of close to integrable n degrees of freedom Hamiltonian
systemsforwhichonecanprovetheexistenceof(n 1)-dimensionalunstableKAM
(cid:0)
tori together with their stable and unstable manifolds. We use a perturbative dia-
grammaticconstruction(proposedanddevelopedin [3], [4]and[5]) toproveupper
boundsontheanglesofintersectionofthestableandunstablemanifoldsofaKAM
torus (homoclinic splitting). Such bounds are generally exponentially small in the
perturbation parameter and depend on the chosen torus and in particular on the
number of fast degrees of freedom. For systems with one fast degree of freedom
we prove as well lower bounds on the homoclinic splitting through the mechanism
of Melnikov dominance. Finally for such systems we prove the existence of \long"
chainsof heteroclinic intersections;namelyweproducealist of unstableKAM tori
;:::; suchthat ; areat distances of orderone in the action variablesand
1 h 1 h
T T T T
the unstable manifold of each intersects the stable manifold of . This paper
i i+1
T T
is a generalization of the results of [4], [5], [6], therefore in proving our claims we
will rely heavily on intermediate results proved in the latter papers which we will
not prove again.
Consider the class of Hamiltonians
1n(cid:0)1I~2+ 1"I~2+ p~2 +" (cosq~ 1) 1(cid:0)c2(cos2q~ 1) +"(cid:22)f(q~) n sin ~ ;
2 2 n 2 (cid:0) (cid:0) 4 (cid:0) i
j=1 (cid:20) (cid:21) i=1
X X
(1.1)
where the pairs I~ Rn, ~ Tn and p~ R, q~ T are conjugate action-angle
2 2 2 2
coordinates, 0 < c 1, f(q~) is odd and analytic on the torus and (cid:22), " are small
(cid:20)
parameters.We will considerthem independent andthen provethat onecanprove
Arnold Di(cid:11)usion for (cid:22) "P, for an appropriate P.
(cid:20)
This class of Hamiltonians is a model for a near to integrable system close to a
simple resonancewhere the dependence on the hyperbolic variables is not through
the standard pendulum, but still maintains various qualitative properties of the
pendulum. Namely we have a \generalized pendulum",
p~2 1 c2
+" (cosq~ 1) (cid:0) (cos2q~ 1)
2 (cid:0) (cid:0) 4 (cid:0)
(cid:20) (cid:21)
which has an unstable (cid:12)xed point in p~=q~=0 with Lyapunov exponent (cid:21)=cp".
Generally one rescales the time and action variables so that the Lyapunov
exponent is one:
I~ t t p~ t t
I(t)= cp" ; (t)= ~ ; p(t)= cp" ; q(t)=q~ :
(cid:0)cp"(cid:1) cp" (cid:0)cp"(cid:1) cp"
(cid:18) (cid:19) (cid:18) (cid:19)
(1.2)
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Exponentially Small Splittingand Arnold Di(cid:11)usion 3
Such rescaling sends Hamiltonian 1.1 in
(I;A(")I) p2 1 1 c2 n
+ + (cosq 1) (cid:0) (cos2q 1) +(cid:22)f(q) sin( ) (1.3)
2 2 c2 (cid:0) (cid:0) 4 (cid:0) i
(cid:20) (cid:21) i=1
X
where A(") is the diagonal matrix with eigenvalues a = 1 for i = 1;:::;n 1
i
(cid:0)
and a = ". So from now on we will work on Hamiltonian (1.3) and turn back to
n
Hamiltonian (1.1) only to prove the existence of heteroclinic chains. The system
(1.3) is integrable for (cid:22) = 0. It represents a list of n uncoupled rotators and a
generalizedpendulum(dependingontheparameterc).Wewilldenotethefrequency
of the rotators (which determines the initial data I(0)) by ! so that
I(t)=I(0)=A 1!; (t)= (0)+!t:
(cid:0)
The initial data are chosen in an appropriate domain (physically interesting in the
variables I~) so that there are at least three characteristic orders of magnitude for
the frequencies of the unperturbed system.
De(cid:12)nition 1.1. In frequency space we (cid:12)rst consider the ellipsoid
n
(cid:6):= x Rn : x2=a =2E
( 2 i i )
i=1
X
where E is an order one constanta E O (1).
"
(cid:24)
Fornotationalconveniencewesplitthefrequency! intwovectorialcomponents:
! = (p!1";"(cid:11)!2) with !1 2 Rm; !2 2 Rn(cid:0)m; and 0 (cid:20) (cid:11) (cid:20) 12: Finally; given two
suitable order one constants R; r O (1); we consider the region
"
(cid:24)
(cid:10) ! Rn :p"! (cid:6); r < ! <R and r < ! <R; "(cid:11) ! p";
1;i 2 2;i
(cid:17)f 2 2 j j j j j j(cid:21)
"(cid:11) ! p" :
2;n m
j (cid:0) j(cid:24) g
Wehavechosenthegeneralizedpendulumsothatitsdynamicsontheseparatrix
is particularly simple,b namely
1 sinh( t)+ic
q(t)=2 arccotg sinh( t) ; eiq(t) = (cid:6) : (1.4)
c (cid:6) sinh( t) ic
(cid:18) (cid:19) (cid:6) (cid:0)
ThereareatleastthreecharacteristictimescalesO"("(cid:0)12),O"("(cid:11)),O"(p")(coming
from the degenerate variable I ) and 1 which is the Lyapunov exponent of the
n
unperturbed pendulum.
We will call ;:::; the fast variables and we will sometimes denote them
1 m
as Tm. Converselywe will call ;:::; the slow variables Tn m.
F m+1 n S (cid:0)
2 2
aNowandinthefollowingwewillsaya(")(cid:24)O"(f("))iflim"!0+ fa(("")) =L6=0.
bThe motion on the separatrix can be easily obtained by direct computation; the main feature
is that the motion on the separatrix is such that eiq(t) is a rational function of et. Here we are
consideringthesimplestclassofexamples,whichcontainsthestandardpendulumc=1.
May 7, 2003 15:6 WSPC/148-RMP 00165
4 M.Procesi
The perturbing function is a trigonometric polynomial of degree one in the
rotators and a rational functionc in eiq. We havedecoupled the dependence of
and q only to simplify the computations. Foreach ! Rn the unperturbed system
2
has an unstable (cid:12)xed torus,
p(t)=q(t)=0; I(t)=I(0)=A 1!; (t)= (0)+!t:
(cid:0)
The stable and unstable manifolds of such tori coincide and can be expressed as
graphs on the angles.
De(cid:12)nition 1.2. Given any (cid:13) R; "<(cid:13) O("12) and a (cid:12)xed (cid:28) >n 1; we de(cid:12)ne
2 (cid:20) (cid:0)
the set
(cid:13)
(cid:10) ! (cid:10): ! l > ; l Zn= 0
(cid:13) (cid:17) 2 j (cid:1) j l (cid:28) 8 2 f g
(cid:26) j j (cid:27)
of (cid:13); (cid:28) Diophantine vectors in (cid:10): Now we consider
1 1
(cid:10) (cid:10) ;
(cid:3)(cid:13) (cid:17) (cid:13) (cid:2) (cid:0) 2 2
(cid:18) (cid:19)
and for all (!;(cid:26)) (cid:10) we set ! =(1+(cid:26))!:
2 (cid:3)(cid:13) (cid:26)
For all (!;(cid:26)) (cid:10) and for all l Zn= 0 ! l > (cid:13) ; ! (cid:10) implies that !
2 (cid:3)(cid:13) 2 f gj (cid:26)(cid:1) j 2l(cid:28) 2 (cid:13) 1
and ! are Diophantine as well; we will call (cid:28) and (cid:28) jtjheir exponents:
2 F S
KAM like theorems (see [2], [5]) imply that there exists (cid:22) (";(cid:13)) "2 such
0
(cid:24)
thatif (cid:22) (cid:22) andif(!;(cid:26)) (cid:10) ,thereexistsoneandonlyonen-dimensionalH -
j j(cid:20) 0 2 (cid:3)(cid:13) (cid:22)
invariantunstabletorusT (!;(cid:26))whoseHamiltonian(cid:13)owisanalyticallyconjugated
(cid:22)
to the (cid:13)ow Tn # #+! t. Moreover one can parameterize the stable and
(cid:26)
3 !
unstablemanifoldsof T (!;(cid:26))byfunctions I (!;’;q;(cid:22)),analyticin the lastthree
(cid:22) (cid:6)
arguments, with ’;q Tn [ 3(cid:25);3(cid:25)]. Namely given
2 (cid:2) (cid:0)2 2
z(cid:6)(!;’;q;(cid:22))=(I(cid:6)(!;’;q;(cid:22));p(cid:6)(!;’;q;(cid:22));’;q);
where the pendulum action is derived by energy conservation, the trajectoryd:
(cid:8)t z+(!;’;q;(cid:22)) if t>0
H
z(!;’;q;(cid:22);t)=
((cid:8)tHz(cid:0)(!;’;q;(cid:22)) if t<0
tends exponentially to a quasi-periodic function of frequency !.
Remark 1.3. We have introduced the variable (cid:26) in order to (cid:12)x the energy of the
perturbedsystem,e namelygivenalistof! (cid:10) onecan(cid:12)nd(cid:26)(! ;(cid:22))suchthatall
i (cid:13) i
2
thecorrespondingwhiskeredtoriareonthesameenergysurface,seeforinstance[5].
cActually it is su(cid:14)cient that the singularity of f( (t);q(t)), which is nearest to the real axis is
polarandisolated.
d(cid:8)t istheevolutionattimetoftheHamiltonian(cid:13)ow(1.3).
H
eThe (cid:12)nal goal is to (cid:12)nd heteroclinic intersections on the (cid:12)xed energy surface, and so \Arnold
di(cid:11)usion", but in the following sections we will discuss only homoclinic intersections and so we
willdroptheparameter(cid:26).
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Exponentially Small Splittingand Arnold Di(cid:11)usion 5
De(cid:12)nition 1.4. We will study the di(cid:11)erence between the stable and unstable man-
ifolds on an hyper-plane transverse to the (cid:13)ow (a Poincar(cid:19)e section); we choose the
hyper-plane q =(cid:25) and consequently drop the dependence on q: We call
1
G0(’;!)= a (I (’;!;0 ) I (’;!;0+))
j 2 j j (cid:0) (cid:0) j
thesplittingvectorandprovethatG0(’=0;!)=0:Ameasureofthetransversality
j
is
(cid:1)0 =@ G0(’)
ij ’j i j’=0
called splitting matrix:
We will prove the following theorems:
Theorem 1. The splitting matrix (cid:1)0 satis(cid:12)es the formal power series relationf:
(cid:1)0 AD0B
(cid:24)
where A; B are close to identity matrices and D0 is the \holomorphic part" of the
splitting matrix; namely its entries are expressed as integrals over R of analytic
functions: Moreover the formal power series involved are all asymptotic.g
This statement was posed as a conjecture in [7] Paragraph3.
Corollary 1.5. The preceding Theorem implies that Hamiltonian (1:3); in regions
of the action variables corresponding to m = 0 fast time scales; has exponentially
6
small upper bounds on the determinant of the splitting matrix:
1
det(cid:1)0 Ce(cid:0)"cb ; with b= ;
j j(cid:20) 2m
provided that (cid:22)<"1+2mn:
Notice that Theorem 1 can be proved for much more general systemsthan model
(1:3):
Theorem 2. Consider Hamiltonian (1:3) in regions of the action variables corres-
ponding to m = 1 fast variables and for perturbing functions f(q) such that the
pole f(q(t)) closest to the imaginary axis; say t(cid:22); is such that Imt(cid:22) =d arc sinc:
j j (cid:20)
Setting (cid:22) "P with P = p=2+8+4n where p is the degree of the pole of f(q(t))
(cid:20)
in t(cid:22)we prove that
C1"(cid:0)p1e(cid:0)djp!"1j det(cid:1)0 C2"(cid:0)p2e(cid:0)djp!"1j
(cid:20)j j(cid:20)
where C ;C ;p ;p are appropriate order one constants.
1 2 1 2
fWedenote formalpowerseriesidentitieswiththesymbolA B.
gAformalpower series (cid:22)nan(")isasymptotic ifforall q>(cid:24)0there existsQ>0suchthat for
alln "(cid:0)q thenan(") "(cid:0)Qn.
(cid:20) (cid:20)P
May 7, 2003 15:6 WSPC/148-RMP 00165
6 M.Procesi
Corollary 1.6. Under the conditions of Theorem 2 the Hamiltonian (1:1) has
heteroclinic chains; namely a set of N 1 trajectories z1(t);:::;zN(t) together
(cid:21)
with N +1 di(cid:11)erent minimal setsh ;:::; such that for all 1 i N
0 N
T T (cid:20) (cid:20)
lim dist(zi(t); )=0= lim dist(zi(t); ):
i 1 i
t T(cid:0) t T
!(cid:0)1 !1
Moreover one can construct such chains between tori (!a;(cid:22)); (!b;(cid:22)) such that
T T
!a; !b (cid:10)(cid:22) (cid:10) and
(cid:13)
2 (cid:26)
j"(cid:0)12(!na (cid:0)!nb)j(cid:24)O"(1):
The techniques used for proving the Theorems are those proposed in [3] and
developedin[4]forpartiallyisochronousthreetimescalesystemswiththreedegrees
of freedom. In this paper, particular attention is given to the formalization of the
treeexpansionsandofthe\Dysonequation"andrelativecancellationsproposedin
[4].ThisenablesustoextendTheorem1tosystemswithndegreesoffreedomandat
leasttwotime scales;moreoverthe proofisde(cid:12)nitelysimpli(cid:12)edandquitecompact.
In this article we have considered completely anisochronous systems only to (cid:12)x
an example; generalizing to partially (or totally, thus recovering the results of [8])
isochronoussystemsiscompletelytrivial.IndeedTheorem1andhenceCorollary1.5
can be proved for very general systems, as we will show in a forthcoming paper.
Moreover we have generalized the class of perturbing functions and the \pen-
dulum" (the literature considers only trigonometric polynomials and the standard
pendulum);thelattergeneralizationsarequitetechnicalbutneverthelessnon-trivial
andinteresting,wethink,asthe techniquesweproposeareeasilygeneralizableand
giveaclearpictureofthelimitsofprovingArnolddi(cid:11)usionviaMelnikovdominance.
2. Perturbative Construction of the Homoclinic Trajectories
One can use perturbation theory to (cid:12)nd the (analytic for (cid:22) (cid:22) ) trajectories on
0
(cid:20)
the S/U manifolds of Hamiltoniani (1.3)
z(’;!;t)= ((cid:22))kzk(’;!;t):
k
X
Namely we insert the expansion in (cid:22) in the Hamilton equations of system (1.3),
I_ = ((cid:22))cos f(q); _ =a I ;
j j j j j
(cid:0)
1 n df (2.1)
p_ = sinq(1 (1 c2)cosq) ((cid:22)) sin (q); q_ =p;
c2 (cid:0) (cid:0) (cid:0) idq
i=1
X
hAclosed subset ofthe phase space iscalled minimal(withrespectto aHamiltonian(cid:13)ow(cid:30)t)if
h
itisnon-empty,invariantfor(cid:8)t andcontainsadenseorbit.Inourcasetheminimalsetswillbe
h
unstable tori (I)with!(I)Diophantine.
iNotice that tThe apex k on the functions I, represents the order in the expansion in (cid:22) NOT
an exponent. To avoid confusion, when we need to exponentiate we always set the argument in
parentheses.
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Exponentially Small Splittingand Arnold Di(cid:11)usion 7
and (cid:12)nd initial data I(!;’;(cid:22);0 ) (and consequently p(!;’;(cid:22);0 )) such that the
(cid:6) (cid:6)
solution of (2.1) tends exponentially to a quasi-periodic function of frequency !.
Inserting in the Hamilton equations the convergentpower series representation:
1 1
I(t;’;(cid:22))= ((cid:22))kIk(t;’); (t;’;(cid:22)) = ((cid:22))k k(t;’);
k=0 k=0
X X
1 1
p(t;’;(cid:22))= ((cid:22))kpk(t;’); q(t;’;(cid:22)) =q0(t)+ ((cid:22))k k(t;’)
0
k=0 k=1
X X
we obtain, for k >0, the hierarchy of linear non-homogeneousequations,j
I_k =Fk( h ); _k =a Ik; for j =1;:::;n;
j j f igi=0;:::;n j j j
h<k
(2.2)
1
p_k = (cos(q0(t)) (1 c2)cos(2q0(t))) k +Fk( h ); _k =pk;
c2 (cid:0) (cid:0) 0 0 f igi=0;:::;n 0
h<k
where the functions Fk are de(cid:12)ned as follows. Set: [] = 1 dk ( ) ; we have
i (cid:1)k k!d(cid:22)k (cid:1) j(cid:22)=0
k 1
Fk(t)= @ f1 (cid:0) ((cid:22))h ~h(t)
j (cid:0)" j !#
hX=1 k(cid:0)1
k 1
(cid:0)
(cid:14) @ f0 ((cid:22))h h(t) ; j =0;:::;n
(cid:0) j0" 0 0 !#
hX=1 k
where ~h(t) is the vector h(t);:::; h(t),
0 n
n 1 1 c2
f1( ~)= sin f( ); f0( )= (cos 1)+ (cid:0) sin2 ;
i 0 0 c2 0(cid:0) 2 0
i=1 (cid:18) (cid:19)
X
(cid:12)nally (cid:14) denotes the Kronecker delta. For k = 0 we obtain the unperturbed
ji
homoclinic trajectory:
z0(t)=(A 1!;p0(t);’+!t;q0(t));
(cid:0)
(q0(t);p0(t)) is the lower branch of the pendulum separatrix starting at q = (cid:25)
written in Eq. (1.4).
Fork >0wehavealinearnon-homogeneousODEthatwecansolvebyvariation
ofconstants.Thefundamentalsolutionofthelinearizedpendulumequationisgiven
by,
W(t)= w_0 x_00 ; w = 1(cid:27)(t)x1 where (cid:27)(t)=sign(t)
w x0 0 2 0
(cid:12) 0 0(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
jWhen it is not strictly necessary we will omit the pre(cid:12)xed initial data of the angles ’ =
1(0);:::; n(0); 0(0)=(cid:25).
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8 M.Procesi
c2cosh(t)
x0 = ;
0 c2+sinh(t)2
(cid:27)(t)x0
x1 = 0(2( 3+4c2)t+sinh(2t)+4( 1+c2)2 tanh(t)): (2.3)
0 2c4 (cid:0) (cid:0)
It is easily seen (see [3] or [5]) that one can choose an appropriate \primitive"
in the right hand side of the (cid:12)rst column of Eqs. (2.2) so that the solutions are
exponentially quasi-periodic.
2.1. Whisker calculus, the \primitive" =t
Let us (cid:12)rst de(cid:12)ne the function spaces on which we work, all the de(cid:12)nitions and
statements of this Subsection and of the following one are proposed and explained
in detail in [3], we are simply reformulating them to suit our needs.
De(cid:12)nition 2.1. (i) H is the vector space (on C) generated by monomials of the
form
tj
m=(cid:27)(t)aj j xhei(’+!t)(cid:1)(cid:23) where h Z; (cid:23) Zn; j N;
j! 2 2 2
x=e t ; a=0;1; (cid:27)(t)=sign(t): (2.4)
(cid:0)jj
(ii) Given two positive constants b and d; H(b;d) is the subset of functions f(t)
analytic on the real axis in t = 0 that admit; separately for t > 0 and t < 0; a
6
(unique) representation;
k tj
f(t)= j j M(cid:27)(t)(x;’+!t); (2.5)
j! j
j=0
X
with M(cid:27)(t)(x;’) trigonometric polynomials in ’ and the function M(cid:27)(t) not iden-
j k
tically zero:
The Fourier coe(cid:14)cients M(cid:27)(t)(x) are all holomorphic in the x-plane in a region
j(cid:23)
0< x <e b argx <d
(cid:0)
f j j g[fj j g
and have possible polar singularities at x=0: k is called the t degree of f:
In Fig. 1 we have represented a possible domain of analyticity for the M .
(cid:23)j
Notice that H is contained in all the spaces H(b;d); moreover if t > b, f(t) can
j j
be represented as an absolutely convergent series of monomials of the type m,
separately for t>b and t< b. One can easily check that the functional that acts
(cid:0)
on monomials m of the form (2.4) as
j tj p
(cid:27)a+1xhei( +!t)(cid:23) j j (cid:0) if h + (cid:23) =0
8(cid:0) (cid:1) (j p)!(h i(cid:27)! (cid:23))p+1 j j j j6
=t(m)=>>>>< (cid:27)a+1 tj+1 Xp=0 (cid:0) (cid:0) (cid:1) (2.6)
j j if h + (cid:23) =0
(cid:0) (j+1)! j j j j
is a primi>>>>:tive of m.
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Exponentially Small Splittingand Arnold Di(cid:11)usion 9
b
e
d
Fig.1.
We can extend t, with t > b, to a primitive on functions f H(b;d) by
= j j 2
expanding f in the monomials m (we obtain absolutely convergent series) and
applying (2.6). Then if t b we set
j j(cid:20)
t
t 2(cid:27)(t)b+ ; (2.7)
= (cid:17)=
Z2(cid:27)(t)b
obviously the choice of 2b is arbitraryand this is still the same primitive of f.
In H(b;d) we can extend t to complex values of t such that t C(b;d) where
= 2
C(b;d):= t C: Im t d; Re t b t C: Im t 2(cid:25); Re t >b ;
f 2 j j(cid:20) j j(cid:20) g[f 2 j j(cid:20) j j g
is the domain in Fig. 1 in the t variables.
An equivalent (and quite useful) de(cid:12)nition of t is
=
du t
tf = e (cid:27)((cid:28))u(cid:28)f((cid:28))d(cid:28); (2.8)
(cid:0)
= 2i(cid:25)u
I Z(cid:27)(t)1+is
where(cid:27)(t)=sign(Ret),t=t +is,with t ,s Randthe integralisperformedon
1 1
2
the lineIm(cid:28) =s;(cid:12)nallythe integralsinuhavetobeconsideredtobethe analytic
continuation on u from u positive and large.
This de(cid:12)nition is clearly compatible with the formal de(cid:12)nition givenaboveand
one easily sees that H(b;d) is closed under the application of t.
=
De(cid:12)nition2.2. H (b;d)isthesubspaceofH(b;d)offunctionsthatcanbeextended
0
to analytic functions in C(b;d):
Notice that f is in H (b;d) if it is in H(b;d) and f(t) is analytic at t=0.
0
Remark 2.3. Iff H (b;d)then generally f = H (b;d)andhasadiscontinuity
0 0
2 = 2
in t=0: For instance if f L is positive, then
1
2
(f):=( 0(cid:0) 0+)f = 1 f =0:
= = (cid:0)= 6
Z(cid:0)1
May 7, 2003 15:6 WSPC/148-RMP 00165
10 M.Procesi
We can construct operators which preserve H (b;d); let = 0(cid:0) 0+ and
0
= = (cid:0)=
t if t 0 t if t 0
t = = (cid:21) t = = (cid:20)
=+ ( t if t<0; =(cid:0) ( t+ if t>0:
= (cid:0)= = =
The operator
1 1
t = t (cid:27)(t)
2 =(cid:26) = (cid:0) 2 =
(cid:26)= 1
X(cid:6)
preserves the analyticity.
Now let us cite two important properties of H (b;d), proved in [3].
0
Lemma 2.4. In H (b;d) we have the following shift of contour formulas:
0
f H (b;d) and for all d>s R;
0
8 2 2
(i) f((cid:28))= f((cid:28) +is);
= =
dR t
(ii) t+isf((cid:28))= e R(cid:27)((cid:28))((cid:28)+is)f((cid:28) +is)d(cid:28):
=(cid:26) 2i(cid:25)R (cid:0)
(cid:26)X=(cid:6)1 I (cid:26)X=(cid:6)1Z(cid:26)1
2.2. The recursive equations
One can easily verify that f1( (t);q (t)) and f0(q (t)) are in H (a;d) (and
0 0 0 0
bounded at in(cid:12)nity) for some \optimal" values a;d corresponding respectively to
the maximal distance from the imaginary axis and the minimal distance from the
realaxisofthepolesofsuchfunctions.Onecanprovebyinduction,see[3]or[5]for
thedetails,thatthesolutionsofEqs.(2.2)tendtoquasi-periodicfunctionsprovided
that the initial data are chosen to be:
Ijk(’;!;0(cid:6))= (cid:22)k=0(cid:6)Fjk; p(’;!;0(cid:6))= (cid:22)k=0(cid:6)x00F0k:
k k
X X
Moreoveronecanprovethat Fk(’;!;t) hasnoconstantcomponent.Consequently
j
it is convenient to express the trajectories in terms of the \primitives" t in the
=
form (a =1):
0
((cid:22))k k(’;t)=((cid:22))ka QtFk+x0G1k+x1G0k
j j j j j j j j
where x0 =1, x1 = t for j =0 while the xi are de(cid:12)ned in Eq. 2.3,
j j j j 6 0
1 1
Qt[f]= ( t + t )[(x0(t)(cid:27)((cid:28))x1((cid:28)) x1(t)(cid:27)(t)x0((cid:28)))f((cid:28))]; Gik=((cid:22))k a xiFk:
j 2 =+ =(cid:0) j j (cid:0) j j j 2 j= j j
For the proofs of these assertions see [3] or [5].
Notice that by our de(cid:12)nitions,
Ij(’;0(cid:0))(cid:0)Ij(’;0+)=2a(cid:0)j1 G0jk (cid:17)2a(cid:0)j1G0j ; 2G00 =p(’;0(cid:0))(cid:0)p(’;0+):
k
X