- Email: [email protected]
Analyticity, scaling and renormalization for some complex analytic dynamical systems
Plrwt. Pergamon
Spuw SC;., Vol. 46, No. 1 I ‘12, pp. 1487- 1497. 1998 (“ 1998 Elsevier Science Ltd All rights reserved 0032-0633:‘981$ - see front matter
PII:SOO324633(98)OOO62-2
Analyticity, scaling and renormalization for some complex analytic dynamical systems Albert0 Berretti II Universita di Roma (Tor Vergata), Dipto di Matematica. Via della Ricerca Scientifica, 00133 Rome, Italy Received
I I July 97 : revised 7 April 1998 : accepted
15 April 1998
Abstract. We review some results about the analytic structure of Lindstedt series for some complex analytic dynamical systems: in particular, we consider Hamiltonian maps (like the standard map and its generalizations), the semi-standard map and Siegel’s problem of the linearization of germs of holomorphic diffeomorphisms of (C,O). The analytic structure of those series can be studied numerically using Padt approximants, and one can show the existence of natural boundaries for real, diophantine values of the rotation number; by complexifying the rotation number, we show how these natural boundaries arise from the accumulation of singularities due to resonances, providing a new intuitive insight into the mechanism of the breakdown of invariant KAM curves. Moreover, we study the Lindstedt series at resonances, i.e. for rational values of the rotation number, by suitably resealing to 0 the value of the perturbative parameter, and a simple analytic structure emerges. Finally, we present some proofs for the simplest models and relate these results to renormalization ideas. 0 1998 Elsevier Science Ltd. All rights reserved
1. Introduction The analytic structure of perturbation series in all branches of theoretical physics is a very important object of study for a variety of reasons; besides the issue of their convergence, from the study of their properties we gain valuable information about the analytic structure of the functions they represent : this is well known in many branches of modern theoretical and mathematical physics. from statistical mechanics to quantum field theory. Classical (Hamiltonian) mechanics poses some special challenges since the perturbative series which are here relevant are characterized by the occurrence of the so-
E-mail : berrettiicl mat.uniroma?.it
called small denominators. So the study of their convergence presents some additional difficulties of arithmetic nature, and their analytic properties (i.e. the nature of their singularities and their dependence on the parameters of the problem) reserve many surprises. In this article we review some of the results we obtained so far on some of the simplest model problems in which these difficulties arise: while of only indirect application to physically relevant concrete problems, we believe that an understanding of those models is a prerequisite to the study of anything more complex and realistic. We believe that this point of view is complementary to the one based on “renormalization” techniques (see Escande, 1985 ; MacKay, 1983). in much the same way as, in quantum field theory, perturbative renormalization (based on renormalized perturbation theory. running coupling constants and the Callan-Symanzik equation) completes the other, more “modern”. approach to renormalization (based on coarse-graining of the degrees of freedom and self-similarity). Actually, this would be very much in the spirit of taking the method of “tree expansions” of Eliasson (1988, 1989); [see also Vittot (1991): Feldman and Trubowitz ( 1992) ; Gallavotti ( 1994) : Chierchia and Falcolini (1994) ; Gentile and Mastropietro ( 1996) to name a few] as the analogous, for classical mechanics. of the Feynman graphs of quantum field theory.
1.1
Tk .strrtdmi map und its genernli=atiott.r
The generalized standard map is a discrete time. one dimensional dynamical system generated by the iteration of the area-preserving (symplectic) map T,,, of the cylinder U x R into itself given by : (1.1) where XE U, J’E R andf’(s) is a trigonometric polynomial. To make things simpler. we shall always assume thatJ’(s) is odd. If ,f(.u) = sins. than we have the standard map,
1488
A. Berretti : Analyticity, scaling and renormalization
henceforth denoted simply by T,.We refer the reader to the huge literature on the topic (see e.g. Mackay (1985) as a starting point) for further properties. A map of the above kind, with of replaced by a specific periodic function analytic around the real axis, has been considered in Willbrink (1987) and studied with (heuristic) renormalization methods. It is useful to reformulate the dynamics given by (1.1) in a “Lagrangian” form, by eliminating _t’to obtain a second order finite difference equation for s : .Y,,+, -2X,+X,,_
, = q(x,,).
(1.2)
higher frequency in the trigonometric polynomial ,f is p,then the sum over k in (1.5) turns out to be limited to Ikl
As is well known from KAM theory, the homotopically non-trivial invariant curves %,,,,,with rotation number (tj of the map T,:,, may be determined by finding a change of coordinate on U : .Y =
D,~,(@,w) = sinH,
(1.3)
d+Zl(t),E,O),
x
where obviously 1 + u”(t), E, tu) > 0, such that in the new variable 8 the dynamics reduces to simple rotations by tr) :
We say that the coordinate transformation (1.3) conjugates the dynamics on the invariant curve to a rotation. and refer to the function u as the conjugating function. It is easily seen that it satisfies the functional equation :
D,?,u(-,E,LO) = M,u(',E,w),
(1.4)
where the operators Df,and IV,: act on functions of 6 as follows :
If we impose that the average of u over 8 be 0, then the solutions of (1.4) are formally unique and odd as functions of 8. To each (sufficiently smooth) solution to (1.4) there corresponds an invariant curve %‘*:.,,, given by the parametric equations : .Y =
6+U(&E,W),
.
1..u, .
c I' =
to+
U(H,E,W)-
L4(6-O~,E,CO);
it clearly has the same smoothness properties of u( *, E,oj).The existence problem for smooth curves of the (generalized) standard map is related to the same problem for the solutions of
1.2. Perturbation
as those invariant therefore eqn (1.4).
theory and small divisors
We can try to attack the problem perturbatively,
setting :
+
14,,,(B,w) . . . u,,,,(b).
o,, =
(1.6b)
PI
,,$,Lzu,,.do)sinkQ&“.
on the space of smooth, 2rr-periodic functions of 0. Note that the eigenvalues of D, on the space of 271periodic, zero-mean functions are given by 2isin (nkw), k # 0 ; the spectrum of D,,, therefore contains 0, either as an eigenvalue (in case co/2n is rational) or as an accumulation point of eigenvalues (when 0/27c is irrational). We then clearly run into problems when o/2n is rational, since then the operatorDz, is not invertible. On the other hand, if 0/2rc is irrational,DJ’ is unbounded and the coefficients of the Lindstedt series will be hard to controLtheir growth being determined by the arithmetic properties of 0/27r. This is the small divisors problem, for which the superconvergent methods of KAM theory are effective; these are not really necessary, however, as the recent developments of the methods based on tree expansions mentioned above-e.g. Gallavotti (1994) ; Chierchia and Falcolini (1994) ; Gentile and Mastropietro (1996)-are also effective-implies. The well-known scenario emerging from KAM theory can be summarized as follows. If w satisfies the usual diophantine condition :
This expansion is known (since Poincare, 1893 who cites Lindstedt, 1882) as the Lindstedt series for the problem and it is the main tool for investigating the analytic properties of u and, therefore, %‘>_‘,,,,. Incidentally, if the
(1.7)
then there are constants c,, c2 such that u is jointly analytic in 0, E for 1~1< c,,lIm 01 < c2. We note that the above diophantine condition (1.7) could be restated as follows : let : -_= 2rL
(1.5)
I I
K
Z-J! >--Iqj’+” 2n q
vp,qEz,q#0,35,K>o:
1
0
U(&E,O) = 1 u,z(&w)En n2 I =
c
We could also write (1.4) as a fixed point problem :
0’ = o+o.
v
II , +
(1.6a)
a,+
~ a?+:
= [a,.a:....]
1 1 .
be the continued fraction expansion of w/271,and let pk/qr be the rational approximants-the convergents-to 0)/2x obtained by truncating this expansion to order k: then the diophantine condition takes the form :
A. Berretti : Analyticity, scaling and renormalization 44 i-l
=
1489 Conjecture
0(4X)
with ;’ 3 2 ; see e.g. Hardy and Wright (1979) for details. As c: grows, a “breakdown threshold” is reached, and the invariant curve ceases to exist as a smooth object, leaving place to more complex invariant sets. Quantitatively, we can define the analytic breakdown threshold--also called the critical function-as follows : E,(~O)= sup it:’ 3 0 :
v E” < E’ 3 ‘G’;.,,<,,],
that is the supremum of those real. nonnegative E’ for which there is always an analytic invariant curve with rotation number tr) at lower values of E. We remark that the radius of convergence of the Taylor series in ( 1.5). given by :
is in principle a different object than E,.(W); obviously ~~((0) > p(tu), but the inequality could be strict, depending on the analytic structure of u in the complex E plane. We believe that the relation between these two quantities is an important issue to clarify : as we shall see, the numerical results suggest that the relation depends strongly on the nature of the trigonometric polynomialf: Finally, observe that if ~o/27c is rational then E,(CU)= p(cv) = 0. So they are zero on a dense subset of the interval [0, I] and nonzero on another dense subset of [0, I] (the diophantine irrationals).
1.3. Brjuno’s imerpolutiow In the standard map case. the radius of convergence p(w), defined in (I .X). has some interesting arithmetic properties which we are going to illustrate ; since most of them hold only for the standard map. in this paragraph we shall assume that .f’(.\-) = sins. In fact, we have that p(w) is related to an arithmetic function. introduced by Yoccoz (1995), and named, after the work of Brjuno ( 197 I), Brjuno’s function : B(t) = -logt+tB(t~‘).
for ~EIR\Q,
0 < I
I. 1. Let p E IR and : C(a) = ~logp(i:,++)~:
(1.10)
then 38 > 0 such that C(to) is bounded, and actually strictly positive. continuous function of C~J.
a
That is, Brjuno’s function captures the “most singular” behaviour of the radius of convergence of the Lindstedt series. The exponent fl is the critical exponent of the radius of convergence.
Brjuno’s interpolation (Cor!jecturr 1. I ) for the standard map was introduced in Marmi and Stark (1992). Davie ( 1994) actually proved that log p(o)) + 2B(&27r) is bounded from above: this implies that Brjuno’s condition is necessary for the convergence of the Lindstedt series. Brjuno’s interpolation relates also to the modular smoothing technique of Buric et ul. (1991): Buric and Percival( 1991), as shown in Marmi and Stark ( 1992). Of course, one could make analogous conjectures for E,(V)): there is II priori just no reason for the critical exponent for the breakdown threshold to be the same as for the radius of convergence. As we shall see in a moment, the numerical results show that for generalized standard maps the situation is different : therefore an important issue would be to find a substitute of conjecture I.1 when the perturbation contains several frequencies- -as is the case in practically any concrete model.
1.4. Sculiny of‘ the c,or?ju~yating,filnc.tio77 Brjuno’s interpolation implies the following formula for the asymptotic behaviour of the radius of convergence of the Lindstedt series as o tends to a rational valuep,‘y (e.g. through a sequence of diophantine (1)‘s): (1.11)
(1.9a) B(r) = B(t+ 1) = B( -t). It is easily seen that B(t) diverges is finite if and only if the series :
if t is rational.
(1.9b) and it
where prlqa is the sequence to convergents to t, is convergent-this is Brjuno’s condition. It is also not too hard to prove that formulas (1.9a), (1.9b) define B(t) uniquely on each L,,. 1 < p < K, by a contraction argument. For more results on the properties of Brjuno’s function see Marmi et N/. (1995) and references quoted therein. The relation between p(to) and B(t) is stated in the following conjectures.
this follows from elementary properties of Brjuno’s function obtained directly from its definition (Marmi and Stark. 1992). One may ask for more, and in particular if the whole conjugating function II scales as ~j.3~ tends to a rational value. We make this statement precise in the following conjecture. Conjecture 1.3. There exists a real. positive number /j such that for every rational number r = p/y in the interval [O. I] there is a function u~‘~’ (8, e), jointly analytic in (U, C) in a strip around the real axis and E in a disk around the origin, such that:
A. Berretti : Analyticity,
1490 The exponent fl is the critical exponent for the radius of convergence, and it is equal to two. In other words, we keep the radius of convergence fixed as w tends to resonant values and see what happens to the resealed conjugating function. One might even wonder whether it is possible to calculate explicitly these functions, and get exact values for their singularities on the complex 8 and E planes: note that a priori this is far from being obvious or simple. 1.5. Complex$cation
of’w
Before we get any further, we must take a fundamental step. As we are looking at analytic properties of the function CI(and of the curves %“,,,,,),we must take all variables to be complex, i.e. we must regard the dynamical system defined by the iteration of the map T,:3fas acting not on the real cylinder U x R but on C’. At this point, there is just no reason to consider only real rotation numbers, so we will take the liberty of considering any complex value of w. For the generalized standard maps there is an obvious symmetry (related to their Hamiltonian origin and, ultimately, to time reversal symmetry) so that changing o into its complex conjugate w* preserves the form of the conjugating function II, so we can take w to be in the complex upper half plane X = .(m E @I Im (0) > O)-or rather in its closure, to include also real o’s Note that if Q E X’, then the small denominators disappear as the spectrum of D,,, does not contain 0 so that the convergence of the Lindstedt series may be proved directly by simple estimates on its coefficients and using the smallness of 1~1. Taking the limits 0/2rr -p/q now is much simpler, as we may take any continuous path in X to a rational point on the real axis, with some “regularity”condition (e.g. non-tangentiality). On the other hand, the definition of limit in (1.12) and similar equations is plagued by the fact that we must take a sequence of diophantine values of to/2n converging to a rational value, and one is left with the task of proving that such a limit is independent on the choice of the sequence. One may also think of the complexification of w as a new way to approach the problem of existence of KAM invariant curves for the standard map and its generalizations. In fact, if w E 2 we have one complex analytic invariant curve parameterized by w-and this is easy to prove, since small denominators do not exist. We may therefore regard the imaginary part added to Edas a “regularization”, and consider the problem of obtaining a real analytic curve in the limit o + co,,, 0,/27r being real and satisfying a diophantine (or more generally Brjuno’s) condition, as a problem of “removing the regularization”, dealt with by uniform estimates for ~0 in a cone in fl over w,, (see Carletti and Marmi (in preparation) for further details). Finally, we remark that it would be interesting to prove some regularity properties for p(a) when w E X-or at least to provide a strong numerical analysis. For example, to prove that p is harmonic in X. 2. Numerical results In this section we review the main numerical results we obtained on the problem of the singularity structure of u.
scaling and renormalization
We shall comment briefly on these results in a later section, after a discussion on the main analytical results obtained so far. First of all, we note that the most trivial thing one could imagine to do, that is : (1) compute numerically the coefficients u,,(e, tn) of the Lindstedt expansion (1.5) for some “random” values of 0 # 0, rr and diophantine (I) (e.g. the golden mean) (2) use the root criterion to obtain numerical estimates of P(to)
is not effective. In fact, due to the presence of small denominators, the coefficients u,(~,o) vary in a very erratic way, so that the root criterion fails to give a decent approximation to p(w) even when high orders in the Lindstedt series are used. Take a look at Fig. 1 to understand the scale of the phenomenon (and see Berretti and Chierchia (1990) for more details); here o is the golden mean, that is (0, = (& I)/2 = [I’], and up to order 440 in F the root criterion still fails to relax to the correct value (computed in this case with Greene’s (1979) criterion, represented for comparison by the horizontal line in the figure. On the other hand, the use of Greene’s criterion to extensively compute p(w) for several values of o and several kind of maps (beyond just the standard map) is computationally too hard to be feasible : in fact, one has to go quite close to the resonance to be able to see the asymptotic behaviour conjectured, and this proves to be just too hard to do (see Marmi and Stark (1992) for an attempt to do exactly this). Moreover one would have to define and implement a Green’s criterion for complex rotation numbers : and it is not quite clear how to do it. In de la Llave and Falcolini (1992) nevertheless Greene’s criterion was indeed used to compute breakdown thresholds for complex values of E, though only for real values of (rj. As we are interested in the more general case in which everything is complex, we stick to the method of Pade approximants, introduced in this context in Berretti and Chierchia (1990) and Berretti et ul. ( 1992). We shall not present here the theory of PadC approximants, as this would be completely off topic in the context of this review, and many fine monographs exist on the matter (see e.g. Baker, 1975; Baker and Graves-Morris, 1981). It will be enough to recall that, given a function F(Z), analytic near the origin, its Pade approximant [M/M is a rational function P(;)/Q(=), with A4 and N being respectively the degrees of the P and Q, such that Taylor expansion coincides with the Taylor expansion of F up to order M + N; formal uniqueness is obtained by imposing e.g. that Q(0) = 1. Under suitable non-degeneracy conditions on F such approximants are shown to exist, are unique and can be recursively computed from the first M+ Nf 1 coefficients of the Taylor expansion of F. For meromorphic functions F, their Pade approximants may be proved to converge to F as their order increases. uniformly on compact sets not containing poles, and the singularities of the Pade approximants may be used to approximate the singularities of the function F. For functions F not just meromorphic, but presenting a more complicated pattern of singularities (e.g. branch point or essential singularities) much less is proved. but
A. Berretti : Analyticity, scaling and renormalization
1491
nevertheless one can use the location of the singularities of the Padl approximant as an estimate for the location of the singularities of F (Nuttal, 1977). Of course, many spurious pole/zero pairs will be created in the process, and these pairs can be detected computing the residue of each pole found and eliminating those which have an extraordinary low residue (how low is low enough depends on the scales of the problem). See Baker (1975) more information. for references and for the practical formulas used to compute numerically the coefficients of the polynomials P and Q. See also Berretti er al. (1992) for details on the techniques used in this context.
is the golden mean: we observe a power law behaviour, with an exponent r approximately one. The exact value of this critical exponent should be related (de la Llave, 1992) to the exponent 6 of MacKay’s fixed point of the renormalization group for the standard map. The exponent 6 is computed in Ketoja and MacKay (1989) to be (5 2 1.6280, and according to de la Llave (1992) c( = - logto/log6, which for UJ equal to the golden mean gives x 2 0.9874. which is well within numerical errors.
We consider now the standard map, and look numerically, using Padl approximants, to the analytic structure of the conjugating function U. We compute numerically the coefficients u,,(H, Q) for selected values of 0 and for some real, diophantine values of OJ, and then we compute Padl approximants; by looking at the roots of their denominators. we can locate the singularities of 11. We find (Berretti and Chierchia, 1990) that, independently of the values of 0 # 0,~ there exists on the complex t: plane a natural boundary, i.e. a dense set of singularities which precludes analytic continuation. The size and shape of the natural boundary does not seem to depend on 0 (except, of course, in the trivial cases t3 = 0,7c in which u = 0 identically!). The shape of the natural boundary, at least for diophantine (o/271 “far” from a rational number, appears to be circular within numerical errors. so that this would suggest that E,.= p-but the situation could be different as we move close to resonances : in fact Davie (1995) claims that as the rotation number tends to 0 (staying real. so through a sequence of diophantine values of to127r)E,.and p tend to zero with a power law characterized by different critical exponents. As we shall see, this is coherent with our findings, both numeric and analytic. about the scaling of the conjugating function 11. In Fig. 2 we see the poles of the Padtt approximants [35/35], [50’50]. [60/60] and [85/85] for ~(H,E,o), when 0 = 1, and (I) is the golden mean : we clearly see the coalescence of poles of the Padl approximant to form the natural boundary. All the poles corresponding to spurious pole/zero pairs have been deleted from the plot. It is interesting to look also to the singularities in the complex 0 plane : here any structure must be symmetricin respect to the real 0 axis. It is convenient to use the variable <=e “I; on the complex [ plane we see a pair of circular natural boundaries, of radius respectively 1 +a and I/( I +o), with CJ+ 0 as i-:+ c,((o): we took the golden mean as a rotation number. In the complex 6, plane therefore the analyticity region is a strip, symmetric around the real axis. bounded by a natural boundary; the width of the strip tends to 0 as E tends to the breakdown threshold. In Fig. 3 we plot the poles of the Padt: approximant [SO/SO]in the complex < plane for the function U, at E = 0.8 to the outer and (0 = (II~: only the poles corresponding boundary have been plotted. In Fig. 4 we plot CTvs F, for the standard map, when o)
It is natural to look numerically at what happens to the singularities of II as OJ becomes complex, to try to understand the mechanisms leading to the formation of the natural boundary (Berretti and Marmi, 1992). As small denominators disappear, so does the natural boundary. The PadC approximants show radial lines of poles alternating with zeros-typically representing the presence of a branch point for u at the end of the line closer to the origin. and as Im (TV)+ 0. while Re (to)/h tends to a diophantine, the number of branch points grows. building up the natural boundary as OJ becomes real ; see Fig. 5 where we plot the poles of the Padt approximants [40/40]. [40/40] and [30/30], at H = 1, respectively for OJ&T = (0, +0.3i,cu, +o.o%, UJ, +o.oii. where OJ, is the golden mean. Taking into account (1.11) and Co+c.tur.r 1.3. we can get a better insight to the mechanism of breakdown of the invariant curves by looking at the structure of singularities of 11as ~/2x tends, from the complex upper half plane ,Y?‘. to rational values: if the imaginary part of (1~ is chosen small enough, so as that only the terms “in resonance” with the given rational value play a significant role in the Lindstedt series, and OJ -+ 2zppiy, then we find that the Pad6 approximants have exactly 2q radial lines of alternating poles and zeros, accumulating on the circle of convergence of the series. These lines of singularities can therefore be interpreted as branch points for 14. and those which appear when Re ((c))/27( is diophantine can be interpreted as the contributions of the leading resonances corresponding to the best rational approximations given by truncated continued fraction expansions of Re (01);271: as Im (0) tends to 0,resonances of higher and higher order contribute to U. adding singularities on the circle of convergence and creating the natural boundary. An analogous mechanism happens if we look at singularities on the complex (1plane. In Fig. 6 we see poles of the Pad& approximants [X$/28]. [40/40] and [48/48], at 0 = I. respectively for CJJ/?~~ =
I /2
+
0.0 Ii, 215 + O.OOOOi.3% + 0.00005i.
We can use Pad& approximants also to check the conjectures stated in the preceeding section. In particular, we can compute the radius of convergence p(to) as the distance from the origin of the closest singularity of II, whose location is estimated using PadC approximants. While crude in terms of numerical errors (which can be roughly estimated of the order of magnitude of about 1%). we can
A. Berretti : Analyticity, scaling and renormalization
1492 get quite deeply into the asymptotic region, i.e. quite close to the resonance. In the following table, taken from Berretti and Marmi (1994), we show the results of the least squares fit of the empirical data for p(o) to the scaling law ~(01) = K(o)lcc, - 2xp/qlP1yas Q + 27cp/q. P/q
B
O/l l/2
2.01 2.04 2.03 2.00 2.02 2.00 2.00 2.00 2.00 2.00
l/3 l/5 215 l/3 l/3 O/l l/2 O/l
exponent is no longer Z/q-actually it is no longer of the form /?/q for some p E [w! We present here the results of some calculations done for the same map as the example of Fig. 8, that is for f(s) = sin x+ l/20 sin 2.x. In this case the following scaling law seems to hold :
Comments 34.9 26.3 14.0 5.9 7.43 13.5 13.4 32.2 23.0 32.4
In rows (a) o tends to the resonance through a vertical straight line in the complex plane; in row (b) cu tends to the resonance through an arc of parabola ; in row (c) cc) tends to the resonance through a curve with an infinite order of contact to the real axes; in rows (d), (e) and (f) o tends to the resonance through the real, diophantine sequences cl,, = [n, l”], o, = [2, n, l”] and w,, = [nr], respectively. The conjectured scaling law for the radius of convergence is therefore verified quite well. Figure 7 instead show the plots of p(27cp/q+ iAw) vs Aol, with p/q = O/l, l/2, l/3, l/S, 2/5 ; we can easily verify here the power law behaviour characterized by an exponent 2/q.
where r.yl denotes the lowest integer greater or equal to x and again p seems to be two, as the data from the following table suggests. Pirl
B
O/l
2.001 2.000 1.998 2.008 2.015
l/2 l/3 l/5 215
Clearly Brjuno’s interpolation does not hold, at least with the same Brjuno’s function we use for the standard map. Work is in progress to strengthen the numerical analysis and to extend it to more maps.
3. Analytical
results
Here we summarize some of the (few) analytic results we have obtained about the analytic structure of the conjugating functions for the generalized standard map. We also present some results obtained for simpler models, like the so-called semi-standard map (Green and Percival, 1981) or Siegel’s problem (see Yoccoz (1995) and the references therein).
2.3. Numerical results_for the generalized standard maps We also performed a similar numerical study on the more general maps of the type T,,( defined in (1.1). In Berretti et al. (1992) we looked at the analytic structure for u for real, diophantine rotation numbers. We found a number of surprising results. First of all, we still have a natural boundary, but in general its shape this implies that in some cases is not circular: E,.(Q) > p(tu), even significantly; in fact, E,.is given by the intersection of the natural boundary with the real, positive E axis, and in some cases the natural boundary elongates in that direction, so that it distance from the origin is strictly smaller. An example is given in Fig. 8, where we plot the poles (after removal of ghost pole/zero pairs) of the PadC approximant [70/70] in E for the map with ,f(s) = sins+ 1/20sin2x, rotation number equal to the golden mean and 0 = 1 : here F, and p differ by about 30%. More examples of this kind have been given in de la Llave and Falcolini (1992) where the boundary of the analyticity region is computed by a complex version of Greene’s criterion, at diophantine rotation numbers, and compared with Pade approximants. The situation is more intriguing if we look at the scaling of p(w) as (1) tends to resonant values. Indeed, we found (Berretti and Marmi, 1995) that a scaling law of the type we have for the standard map still holds, while the
3.1. Scaling near resonances.for the standard map : O/l and 112 In Berretti and Marmi (1994) we have been able to prove indeed that eqn (1.12) holds for the resonances O/l and l/2. Theorem 3.1. Let x2 be the space of 2rr-periodic functions f(0) analytic and bounded in a neighbourhood of the complex strip :
where];. are the Fourier coefficients closed complex cone defined by : %;. = {~E@Jlrn~
> 0.
off, and let g:. be the
[Rev] d IjIrny),
7 > 0.
Then if v]E g; and p/q = O/1 or l/2 the limit :
A. Berretti : Analyticity. scaling and renormalization
I403
exists in 11- II1 norm, 0 < x’ < z, uniformly in 1~1< I’ = const. em I. Moreover, they satisfy the differential equations : I$, ’ ‘(8, c) = c sin (H + zr’“’“(0, E)), d” I’( 0. E) = 11”’“(27L E) = 0,
kK(k) = 7~~:i..
(3.2)
See e.g. Whittaker and Watson (1927) and Chandrasekharan (1985) for details and for an elementary introduction to the theory of elliptic functions from the complex-analytic point of view. It is then immediately found that the singularities of .Y in t closer to the real axis are branch points of infinite order at :
II” “(0. F) = II” y27c. E) = 0. Notice that : where K’ = K(k’) lished notation :
The proof comes in two parts: first one shows the existence of the limit, then we compute it explicitly by showing that it satisfies the differential equation. Full details may be found in Berretti and Marmi (1994). While the case (0 -+ 0 is very simple, the case (u + II (p/q = 11’2)is slightly tricky as it uses some special symmetries of the standard map, specific to the resonance l/3 case. The basic idea is to prove that the 112 resonance corresponds, loosely speaking, to a O/l resonance for the map iterated twice. We believe that the full formalism of trees is necessary to extend the proof to all resonances. We therefore make the following conjecture. Conjrcturr
3.1. u”’ ‘I’ satisfies : 2& “’ = C, /
for some real constant
sin q(8 + dp "I).
C, ‘i. with boundary
11”’“‘(0, 8) = u”‘qh,
:
conditions
e) = 0.
Note that, for p/q = O/1 or l/2 and conjecturally for all differential equations permit us to compute explicitly the singularities in 8, E or u@“I),since they can be solved exactly in terms of Jacobian elliptic functions. In fact. let :
p/q, these
x(t) = q(o+Lf'"Y'(&E)). f = q&
i"= qCryeY.
and
t=ip.
k”+k’
k” K
= 0. lJsing
_
z-c
r,i
a well-estab-
(3.3)
we see that the singularities are at I = rc(I Is) or. in the variable [ = e”, at --_“. - 1;:. Using (3.1) we read off the singularities for 11(““’ in 8, and explain-also in a quantitative way (Berretti and Marmi, 1994)-the pattern of singularities found for the Pad6 approximants when (!I is close to a resonance. Concerning the singularities in I:. it is useful to expand .Yin Fourier series :
which converges and is analytic for /:I < 1 (i.e. Im (r) > 0). Notice that as s(t) depends on ELonly through z (because of the boundary conditions) this means that (1) the singularities of s(t) in /. do not depend on t. except for the trivial cases t = 0. TTand (2) they are found as the singularities of z as a function of Jb. As one can write i. in terms of S-functions of z- -actually, using the notations of Whittaker and Watson ( 1927). 1. = 3:(0, z)/4-we can locate the singularities closest to 0 to r as a function of i. as two branch points on the imaginary i. axis : i. = k ii,. Again using (3.1) we found a complete qualitative explanation of the numerical observations, and also a quantitative agreement with /)((I,) as crj -+ 2xpP;q (Berretti and Marmi. 1994).
(3.1) so that .Ysatisfies : X” = i. sin s,
s(0) = 0,
s( 271) = 27l;
Before attacking the problem of proving Conja-tzrrc 3.1 for the standard map, one may first try to study the simplest semi-standard map :
then : .y(t) = n-?am(F(t-n).k)
s, : where am( -, k) is the Jacobian amplitude K(k) is the quarted-period function :
of module
k,
K(k) = and k is determined boundary conditions tion) :
I’ = .v+ 1.’ i J.’ =
j’ +
this map can be put in Lagrangian
i:
e”
form as :
s,,_ , - 2s,, + .v,~ , = i: e”“. by the condition (coming from the imposed to the differential equa-
Note that is an intrinsically complex dynamical system : s’, J.’ are not real for real .Y.~3. The function ld which conjugates the dynamics to rotations by (1)satisfies :
1494
A. Berretti : Analyticity, scaling and renormalization D,;,u
=
Ee”“+“’
(3.4)
It is easily seen that u(~,E,w) is actually a function of sein and w only ; this suggests to use the following variables :
where
: uk = u(eA, L co), e, = e + kw, p = “,p,
&Z, 3”) = iU(0, E, 0) ;
(0 = 27tp/q+rj,
then 4 satisfies :
a:“’ are given by :
and the coefficients a),“’= 1,
a?’ = - 2,
a?’ = 0
otherwise.
We shall apply q- 1 times a transformation three steps :
and if:
defined in
0 write everything in terms of r&+, - hk + u/,~, apply the functional eqn (3.4) 0 linearize : l
(3.5) determines a recursion relation for the coefficients c$,, of the Taylor expansion of 4. It is easy to see that if o is real, then (3.6) is a power series with positive coefficients. This map has been introduced by Green and Percival (1981). Davie (1994) was able to prove a weak form of Brjuno‘s interpolation; in fact, denoting with pSS(i) the radius of convergence of (3.6) he proved that :
is bounded. Since the Lindstedt series (3.6) for the semistandard map is so much simpler, we could actually prove the equivalent of conjecture 1.1 in the following theorem. Theorem 3.2. Let $(z,l) be a solution of the functional eqn (3.5) satisfying &O, 1) = 0, and let 111< 1 ; let 1 be a primitive qth root of unity (i.e. x4 = 1 but Xk # 1 for all 0 < k < q). Then the following limit exists :
and is analytic in a neighbourhood of the origin, provided the path taken by 3. is non-tangential to the boundary of the unit disk. The proof and is based estimate of onances. Of course
Applying this transformation iteratively we obtain a sequence of expansions :
2:
(Se ‘(“~~+“~~))y~‘(a/“~, +ab”u,+a(A’,u_,)
In Berretti and Marmi
= (A).
(1996) it is proved that :
It follows that ah” = c0 and a\” = a(?), = c,/2, so that, applying once more (3.4). we get :
may be found in Berretti and Marmi (1996) on the majorant series method, plus a careful complexified small denominators near reswe have then the following
Corollary 3.1. Let q 4 0 non-tangentially plex upper half plane ; then the following U’Pyl, E) = l$
zr(d,up,
corollary.
Taking
the limit q-0
from the comlimit exists :
27cp/q+ r/),
upil
=
analytic
_
:
we obtain (‘0 _
2 ‘q-
IgY
erqw+Ir’~~‘J,
2q’ By letting
: +)
and is jointly
to D&(0, E: w)
= iqu(p!qe,
&),
(3.7a)
in the region {1Im (0)l < c,, 1~1< c?).
To compute exactly such function is of course quite a different matter. In Berretti and Marmi (1996) we gave an heuristic argument for an exact form of u@,~); we sketch now some of the basic ideas of the argument, leaving the details to Berretti and Marmi (I 996). We have :
(3.7b) we obtain
easily : (_-II/‘)’= e$ ;
since -_ = 0 means E = 0, i.e. the linear map, we must have
1495
A. Berretti : Analyticity. scaling and renormalization $(O) = 0 ; a unique of 0 is given by :
analytic
solution
in a neighbourhood
log R(i.) + II(to) < (const.). Vcc> 0 : log R(i)+(
$(z)= -2log
1-i (
_)
From (3.7a) and (3.7b) it is easily seen that &‘,*’ has y branch points of infinite order symmetrically located around the origin-as the numerical calculations using Pade approximants also show. To turn this argument into a proof. one would have of course to control the error introduced by the linearizations iteratively applied. Even though it is not a rigorous proof. we believe that the method we used is interesting in providing a bridge between perturbative expansions and the renormalization methods in which the iterated map is studied.
3.3. Sityel’s prnhletti A more complete analysis may be carried over for Siegel’s problem. Let P, be a complex function, analytic in a neighbourhood of the origin. such that : P,,(O) = 0,
P;(o) = i:
Thorettt
3.3. Let : I R,(Z) = ----H,((A-i)’ (A-i)’ ”
“Z);
(I ) f?,(z) is the conjugating F,(I) = ,7-l;,.;
function
for:
P,((A -;.)I
“z).
= H;.(k),
with H; analytic near 0 and such that H,(O) = O&(O) = 1. Poincare ( 1928-l 956) proved that this is always possible if JE.1# I. If /1 if a root of unity, then Hj. does not exist in the class of formal power series, so aforriori as an analytic function. When 1E.I= 1 but i. is not a root of unity, H,(z) exists as a formal power series so the problem consists in proving that it is actually convergent. If: H;(z) = i
where R(j&) is the radius of convergence of (3.8). Here we choose P,(Z) = i.:( 1 --_). for the sake of simplicity and definiteness. though some of the results could be easily generalized. In Berretti and hlarmi (1996) we prove that. as i. tends to a primitive @h root of unity A from inside the unit disk- so that R(j_) -+ O--then. by resealing : suitably, it is possible to obtain a well defined limit R”‘(Z) for the conjugating function H,(z). We also proved that this limit satisfies a differential equation, and we computed it explicitly and determined its singularities. While these results are essentially contained in Yoccoz (1995). the method we use- roughly speaking, iterating y times P,. to reduce to the case A = I -is basically the same as the one we apply to the more difficult case of complex area-preserving maps, like the standard or semi-standard maps. Let then A = e2r’i”‘. (ply) = I, a primitive r/th root ot unity. We have the following theorem.
then :
the problem here consists in determining for which values of j, it is possible to linearize P,, i.e. to analytically conjugate P, to its linear part i.: : P,(H,(:))
1 + r)B(co) 3 (const.),
H,,(A):“.
(3.8)
then in H,,(i.) appear terms containing factors (E.“-i)) ‘. which are arbitrarily large making it difficult, again, to prove the convergence of the series above. Let therefore :
and let (pk:qr) be the sequence of rational approximants to u obtained by truncating the continued fraction expansion of UJ to the kth place. Then Cremer (1938) proved that H;(Z) diverges if i. is such that the sequence log qA+ ,/q/. is unbounded. Siegel (1941) proved the first fundamental result on the topic: if w satisfies the usual diophantine condition then the series in (3.8) converges. It was much later ( 1965) that Brjuno (1971) proved that the condition I&( log yI, + ,; qr ) < ~j is sufficient for the convergence of the series (3.8). Finally, Yoccozproved in 1988 (Yoccoz, 1995) that Brjuno’s condition is actually necessary and sufficient for the convergence of H,(z), and a weak form of Brjuno’s interpolation with /I = 1, i.e. that :
(3) as i. --t A non-tangentially. tion :
H,(Z) tends
to the func-
Notice the singularity structure for E!‘,“(Z): (f branch points of order q symmetrically located around the origin.
4. Conclusions Poincare series at resonances have been studied also in the context of the problem of linearization of vector fields around a singular point. In particular we would like to cite the work of Piartly (1972). who shows how the divergence of those series is related to the accumulation of resonant invariant manifolds at the singular point. See also Arnold (1980) Section 36 for a review and further references. Our analysis is quite severely limited. as it stands, to a very simple model (the standard map) and a few other. Even for generalized standard maps it is not clear if there is some form of Brjuno’s interpolation, and the study of the limit at singularities of conjugating functions is hard to make at the level of rigorous proofs. We believe that, once the suitable resealing is done of the perturbative parameters, analogous results hold. with hyperelliptic functions used to represent 24 at the resonance. Some work is in progress for the simplest such map, i.e. T,,, withf’(s) = sin s+cz sin 2s. This may help to clarify the reason for the large discrepancy between the critical
1496
threshold e, and the radius of convergence of the Lindstedt series p. Even more interesting would be to generalize the above to higher dimensions. Here one of the main problems is that regularizing the Lindstedt series using complex frequencies does not eliminate the small denominators. So an important problem is to properly formulate the dynamics of higher dimensional symplectic maps in the complex domain. Finally, one may ask whether is it feasible to prove the existence of the numerically observed natural boundaries for diophantine rotation numbers. This could be the case for the semi-standard map, which, for real and diophantine rotation numbers, has a Lindstedt series with positive coefficients, and so a “free singularity” on the intersection of the circle of convergence with the real axis; a similar pattern of proof can easily show the existence of a natural boundary for the solutions of linear functional equations of a similar kind. AckttoIllec!qrtttettt.s. The author wishes to thank Alessandra Celletti and the organizing committee of the Second Italian National Meeting of Celestial Mechanics, held in L’Aquila, 2 l23 April 1997. for inviting him and giving him the chance to write this review. Noic at/&d in pruqf: The heuristic method introduced in Berretti and Marmi (1996) to compute the limit function L&‘~’ for the semi-standard map has been turned into a rigorous one in Berrctti c’t al. (1998). Thcorw~~3.1. has been extended in Berretti and Gentile (submitted) to all r E 69 by using a version of the tree expansion of Gentile and Mastropietro (1994). essentially proving Conjecture 1.3 for the standard map.
References Arnold, V. I. (1980) Chapitres Suppl~mrntuires de la Thkorie drs _l?quutionsDiffkrentielles Ordinuires. Ed. Mir. Moscow. Baker, G, (1975) Essentiuls qf Pude Approsimunts, Academic Press, New York. Baker. G. and Graves-Morris, M. (198 1) Pudt Approximants. Addison-Wesley. Berretti, A. and Chierchia, L. (1990) On the complex analytic structure of the golden invariant curve for the standard map. Nonlinearity 3, 39. Berretti, A. and Gentile, G. Scaling properties for the radius of convergence of a Lindstedt series : the standard map. J. Math. P; et App., submitted. Berretti. A. and Marmi. S. (1992) Standard map at complex rotation numbers : creation of natural boundaries. Ph~,s. Rw. Lett. 68, 1443. Berretti, A. and Marmi, S. (1994) Scaling near resonances and complex rotation numbers for the standard map. Nonlinearity 7, 603. Berretti, A. and Marmi, S. (1995) Perturbative renormalization and analyticity for the standard map and some generalizations. CIIUOS.Solutiotts and Fractals 5, 251. Berretti, A. and Marmi, S. (1996) Limit at Resonances @‘L&teurizutions of’ Some Comples Analytic Dynumicul Systems. preprint. Dip. di Matematica, Univ. di Firenze. Berretti, A., Marmi, S. and Sauzin, D. (1998) Limit at resonances of linearizations of some complex analytic dynamical systems. Ev),odic Th. and Dw. Syst. Berretti. A.. Celetli. A., Chierchia. L. and Falcolini. C. (1992) Natural boundaries for area-preserving twist maps. Journal @‘Stat. Ph_v.s.66, 1613.
A. Berretti : Analyticity,
scaling and renormalization
Brjuno, A. D. (1971) Analytic form of differential equations. Trans. Mosco~r M&h. SOL..25, 131-288. Buric. N. and Percival, I. C. (1991) Modular smoothing and finite perturbation theory. Nonlinearity 4, 98 I. Buric. N., Percival, I. C. and Vivaldi, F. (1991) Critical function and modular smoothing. Nonlitzearit~~ 4, 21. Chierchia, L. and Falcolini, C. (1994) A direct proof of a theorem by Kolmogorov in Hamiltonian systems. Ann. Scuoku Norm. Sup. di Pisu, Serie IV 21, 541. Chandrasekharan, .!$. (1985) Elliptic Functions. Springer. Cremer, H. (1938) Uber die Haufigkeit der Nichtzentren. Muth. Am. 115, 573, Davie, A. M. (1994) The critical function for the semi-standard map. Nonlineurit_v7, 219. Davie, A. M. (1995) Renormuiisution jar Anu!,,tic Arru-presewing Maps. University of Edinburgh. preprint. de la Llave. R. (1992) A renormalization group explanation of numerical observations of analyticity domains. J. Stuf. Phys. 66, 1631. de la Llave. R. and Falcolini, C. (1992) Numerical calculation of domains of analyticity for perturbation theories in the presence of small divisors. J. Stat. P/I,I~s.67, 645. Eliasson. L. H. (1988) Absolute!,, Convergent Series E.~pan.sions ,fiv Quasiperiodic Motions. University of Stockholm. preprint. Eliasson, L. H. (1989) Hamiltonian systems with linear normal forms near an invariant torus. In Noti~ineurD_vnumic.s, ed. G. Turchetti. World Scientific, Singapore. Escande. D. F. (1985) Stochasticity in classical Hamiltonian systems : universal aspects. Pkys. Rep. 121, 165. Feldman. J. and Trubowitz, E. (1992) Renormalization in classical mechanics and many body quantum field theory. J. A&. Math. 58, 2 13. Gallavotti. G. (1994) Twistless KAM tori. Cor~~rlz.Muth. Phys. 164, 145. Gentile. G. and Mastropietro. V. (1994) KAM theorem revisited. P/r.~,.ric.rrD 90, 225. Greene, J. M. (1979) A method for determining a stochastic transition. J. Math. PhJx. 20, 1183. Greene, J. M. and Percival. I. C. (1981) Hamiltonian maps in the complex plane. PI~ysica D 3, 530. Hardy. G. and Wright, E. M. (1979) An Itltroductiott to the Thcor~~of Nutnbrrs, 5th edn. Oxford University Press. Ketoja, J. A. and MacKay. R. S. (1989) Fractal boundary for the existence of invariant circles for area preserving maps : observations and a renormalisation explanation. Plz~~sica D 7,318. Lindstedt, A. (1882) K. .4&i. Ll/iss. St. Pr/rrsburg 31, 4. MacKay, R. S. (1983) A renormalization approach to invariant circles in area-preserving maps. Ph~~sico7D, 283. MacKay, R. S. (1985) Transition to Chaos for Area-preserving Maps. Springer Lecture Notes in Pi7~~.sic~.c, Vol. 247, p. 390. Marmi, S. and Stark. J. (1992) On the btandard map critical function. N~tttl;tteur~~~, 5, 743. Marmi. S.. Moussa. P. and Yoccoz, J. C. (1995) The Brjuno Ft’2rnction.s trnd their Re~yuIurit~’Propcrtkr. preprint. Saclay 95/0X Nuttal, J. N. (1977) The convergence of Pad& approximants to functions with branch points. In Pad&and Rutionul Approsimutions, ed. E. G. Saff and R. H. Varga, p. 101. Academic Press, New York. Piartly. A. (1972) Birth of complex invariant manifolds near a singular point of a vector field depending on parameters. Fzrnct. At&. Appl. 6, 95. Poincart-, H. ( 1893) Les MPthodes Nourelkr de lu Mckhuniyur CJeste. Vol. 11. Gauthier-Villars. Paris. PoincarC, H. (1928-1956) Owrrs. t. 1. Gauthier-Villars, Paris. Siegel, C. L. (1941) Iteration of analytic functions. .4nn. Marh. 43, 607. Vittot. M. (1991) Lindsredt Perturhution Series in Hamiltot~iutt Mechunic.~: E.@cit Formulution viu u Mlrltitlin~c~ttsionol Bur-
.A. Berretti : Analyticity,
scaling and renormalization
mUntZ--lU~il/n~~yr Forrd~, Marseille. preprint CPT911P.2630. Whittaker. E. T. and Watson. G. N. (1927) A Course in Modern Anrr~~.si.s.Cambridge University Press.
1491 Willbrink. J. (1987) Erratic behavior of invariant circles in standard-like mappings. Physicu D 26, 358. Yoccoz, J. C. (1995) Thkoreme de Siegel. nombres de Brjuno et polinomes quadratiques. Asririsyur 231, 3 38.
Spuw SC;., Vol. 46, No. 1 I ‘12, pp. 1487- 1497. 1998 (“ 1998 Elsevier Science Ltd All rights reserved 0032-0633:‘981$ - see front matter
PII:SOO324633(98)OOO62-2
Analyticity, scaling and renormalization for some complex analytic dynamical systems Albert0 Berretti II Universita di Roma (Tor Vergata), Dipto di Matematica. Via della Ricerca Scientifica, 00133 Rome, Italy Received
I I July 97 : revised 7 April 1998 : accepted
15 April 1998
Abstract. We review some results about the analytic structure of Lindstedt series for some complex analytic dynamical systems: in particular, we consider Hamiltonian maps (like the standard map and its generalizations), the semi-standard map and Siegel’s problem of the linearization of germs of holomorphic diffeomorphisms of (C,O). The analytic structure of those series can be studied numerically using Padt approximants, and one can show the existence of natural boundaries for real, diophantine values of the rotation number; by complexifying the rotation number, we show how these natural boundaries arise from the accumulation of singularities due to resonances, providing a new intuitive insight into the mechanism of the breakdown of invariant KAM curves. Moreover, we study the Lindstedt series at resonances, i.e. for rational values of the rotation number, by suitably resealing to 0 the value of the perturbative parameter, and a simple analytic structure emerges. Finally, we present some proofs for the simplest models and relate these results to renormalization ideas. 0 1998 Elsevier Science Ltd. All rights reserved
1. Introduction The analytic structure of perturbation series in all branches of theoretical physics is a very important object of study for a variety of reasons; besides the issue of their convergence, from the study of their properties we gain valuable information about the analytic structure of the functions they represent : this is well known in many branches of modern theoretical and mathematical physics. from statistical mechanics to quantum field theory. Classical (Hamiltonian) mechanics poses some special challenges since the perturbative series which are here relevant are characterized by the occurrence of the so-
E-mail : berrettiicl mat.uniroma?.it
called small denominators. So the study of their convergence presents some additional difficulties of arithmetic nature, and their analytic properties (i.e. the nature of their singularities and their dependence on the parameters of the problem) reserve many surprises. In this article we review some of the results we obtained so far on some of the simplest model problems in which these difficulties arise: while of only indirect application to physically relevant concrete problems, we believe that an understanding of those models is a prerequisite to the study of anything more complex and realistic. We believe that this point of view is complementary to the one based on “renormalization” techniques (see Escande, 1985 ; MacKay, 1983). in much the same way as, in quantum field theory, perturbative renormalization (based on renormalized perturbation theory. running coupling constants and the Callan-Symanzik equation) completes the other, more “modern”. approach to renormalization (based on coarse-graining of the degrees of freedom and self-similarity). Actually, this would be very much in the spirit of taking the method of “tree expansions” of Eliasson (1988, 1989); [see also Vittot (1991): Feldman and Trubowitz ( 1992) ; Gallavotti ( 1994) : Chierchia and Falcolini (1994) ; Gentile and Mastropietro ( 1996) to name a few] as the analogous, for classical mechanics. of the Feynman graphs of quantum field theory.
1.1
Tk .strrtdmi map und its genernli=atiott.r
The generalized standard map is a discrete time. one dimensional dynamical system generated by the iteration of the area-preserving (symplectic) map T,,, of the cylinder U x R into itself given by : (1.1) where XE U, J’E R andf’(s) is a trigonometric polynomial. To make things simpler. we shall always assume thatJ’(s) is odd. If ,f(.u) = sins. than we have the standard map,
1488
A. Berretti : Analyticity, scaling and renormalization
henceforth denoted simply by T,.We refer the reader to the huge literature on the topic (see e.g. Mackay (1985) as a starting point) for further properties. A map of the above kind, with of replaced by a specific periodic function analytic around the real axis, has been considered in Willbrink (1987) and studied with (heuristic) renormalization methods. It is useful to reformulate the dynamics given by (1.1) in a “Lagrangian” form, by eliminating _t’to obtain a second order finite difference equation for s : .Y,,+, -2X,+X,,_
, = q(x,,).
(1.2)
higher frequency in the trigonometric polynomial ,f is p,then the sum over k in (1.5) turns out to be limited to Ikl
As is well known from KAM theory, the homotopically non-trivial invariant curves %,,,,,with rotation number (tj of the map T,:,, may be determined by finding a change of coordinate on U : .Y =
D,~,(@,w) = sinH,
(1.3)
d+Zl(t),E,O),
x
where obviously 1 + u”(t), E, tu) > 0, such that in the new variable 8 the dynamics reduces to simple rotations by tr) :
We say that the coordinate transformation (1.3) conjugates the dynamics on the invariant curve to a rotation. and refer to the function u as the conjugating function. It is easily seen that it satisfies the functional equation :
D,?,u(-,E,LO) = M,u(',E,w),
(1.4)
where the operators Df,and IV,: act on functions of 6 as follows :
If we impose that the average of u over 8 be 0, then the solutions of (1.4) are formally unique and odd as functions of 8. To each (sufficiently smooth) solution to (1.4) there corresponds an invariant curve %‘*:.,,, given by the parametric equations : .Y =
6+U(&E,W),
.
1..u, .
c I' =
to+
U(H,E,W)-
L4(6-O~,E,CO);
it clearly has the same smoothness properties of u( *, E,oj).The existence problem for smooth curves of the (generalized) standard map is related to the same problem for the solutions of
1.2. Perturbation
as those invariant therefore eqn (1.4).
theory and small divisors
We can try to attack the problem perturbatively,
setting :
+
14,,,(B,w) . . . u,,,,(b).
o,, =
(1.6b)
PI
,,$,Lzu,,.do)sinkQ&“.
on the space of smooth, 2rr-periodic functions of 0. Note that the eigenvalues of D, on the space of 271periodic, zero-mean functions are given by 2isin (nkw), k # 0 ; the spectrum of D,,, therefore contains 0, either as an eigenvalue (in case co/2n is rational) or as an accumulation point of eigenvalues (when 0/27c is irrational). We then clearly run into problems when o/2n is rational, since then the operatorDz, is not invertible. On the other hand, if 0/2rc is irrational,DJ’ is unbounded and the coefficients of the Lindstedt series will be hard to controLtheir growth being determined by the arithmetic properties of 0/27r. This is the small divisors problem, for which the superconvergent methods of KAM theory are effective; these are not really necessary, however, as the recent developments of the methods based on tree expansions mentioned above-e.g. Gallavotti (1994) ; Chierchia and Falcolini (1994) ; Gentile and Mastropietro (1996)-are also effective-implies. The well-known scenario emerging from KAM theory can be summarized as follows. If w satisfies the usual diophantine condition :
This expansion is known (since Poincare, 1893 who cites Lindstedt, 1882) as the Lindstedt series for the problem and it is the main tool for investigating the analytic properties of u and, therefore, %‘>_‘,,,,. Incidentally, if the
(1.7)
then there are constants c,, c2 such that u is jointly analytic in 0, E for 1~1< c,,lIm 01 < c2. We note that the above diophantine condition (1.7) could be restated as follows : let : -_= 2rL
(1.5)
I I
K
Z-J! >--Iqj’+” 2n q
vp,qEz,q#0,35,K>o:
1
0
U(&E,O) = 1 u,z(&w)En n2 I =
c
We could also write (1.4) as a fixed point problem :
0’ = o+o.
v
II , +
(1.6a)
a,+
~ a?+:
= [a,.a:....]
1 1 .
be the continued fraction expansion of w/271,and let pk/qr be the rational approximants-the convergents-to 0)/2x obtained by truncating this expansion to order k: then the diophantine condition takes the form :
A. Berretti : Analyticity, scaling and renormalization 44 i-l
=
1489 Conjecture
0(4X)
with ;’ 3 2 ; see e.g. Hardy and Wright (1979) for details. As c: grows, a “breakdown threshold” is reached, and the invariant curve ceases to exist as a smooth object, leaving place to more complex invariant sets. Quantitatively, we can define the analytic breakdown threshold--also called the critical function-as follows : E,(~O)= sup it:’ 3 0 :
v E” < E’ 3 ‘G’;.,,<,,],
that is the supremum of those real. nonnegative E’ for which there is always an analytic invariant curve with rotation number tr) at lower values of E. We remark that the radius of convergence of the Taylor series in ( 1.5). given by :
is in principle a different object than E,.(W); obviously ~~((0) > p(tu), but the inequality could be strict, depending on the analytic structure of u in the complex E plane. We believe that the relation between these two quantities is an important issue to clarify : as we shall see, the numerical results suggest that the relation depends strongly on the nature of the trigonometric polynomialf: Finally, observe that if ~o/27c is rational then E,(CU)= p(cv) = 0. So they are zero on a dense subset of the interval [0, I] and nonzero on another dense subset of [0, I] (the diophantine irrationals).
1.3. Brjuno’s imerpolutiow In the standard map case. the radius of convergence p(w), defined in (I .X). has some interesting arithmetic properties which we are going to illustrate ; since most of them hold only for the standard map. in this paragraph we shall assume that .f’(.\-) = sins. In fact, we have that p(w) is related to an arithmetic function. introduced by Yoccoz (1995), and named, after the work of Brjuno ( 197 I), Brjuno’s function : B(t) = -logt+tB(t~‘).
for ~EIR\Q,
0 < I
I. 1. Let p E IR and : C(a) = ~logp(i:,++)~:
(1.10)
then 38 > 0 such that C(to) is bounded, and actually strictly positive. continuous function of C~J.
a
That is, Brjuno’s function captures the “most singular” behaviour of the radius of convergence of the Lindstedt series. The exponent fl is the critical exponent of the radius of convergence.
Brjuno’s interpolation (Cor!jecturr 1. I ) for the standard map was introduced in Marmi and Stark (1992). Davie ( 1994) actually proved that log p(o)) + 2B(&27r) is bounded from above: this implies that Brjuno’s condition is necessary for the convergence of the Lindstedt series. Brjuno’s interpolation relates also to the modular smoothing technique of Buric et ul. (1991): Buric and Percival( 1991), as shown in Marmi and Stark ( 1992). Of course, one could make analogous conjectures for E,(V)): there is II priori just no reason for the critical exponent for the breakdown threshold to be the same as for the radius of convergence. As we shall see in a moment, the numerical results show that for generalized standard maps the situation is different : therefore an important issue would be to find a substitute of conjecture I.1 when the perturbation contains several frequencies- -as is the case in practically any concrete model.
1.4. Sculiny of‘ the c,or?ju~yating,filnc.tio77 Brjuno’s interpolation implies the following formula for the asymptotic behaviour of the radius of convergence of the Lindstedt series as o tends to a rational valuep,‘y (e.g. through a sequence of diophantine (1)‘s): (1.11)
(1.9a) B(r) = B(t+ 1) = B( -t). It is easily seen that B(t) diverges is finite if and only if the series :
if t is rational.
(1.9b) and it
where prlqa is the sequence to convergents to t, is convergent-this is Brjuno’s condition. It is also not too hard to prove that formulas (1.9a), (1.9b) define B(t) uniquely on each L,,. 1 < p < K, by a contraction argument. For more results on the properties of Brjuno’s function see Marmi et N/. (1995) and references quoted therein. The relation between p(to) and B(t) is stated in the following conjectures.
this follows from elementary properties of Brjuno’s function obtained directly from its definition (Marmi and Stark. 1992). One may ask for more, and in particular if the whole conjugating function II scales as ~j.3~ tends to a rational value. We make this statement precise in the following conjecture. Conjecture 1.3. There exists a real. positive number /j such that for every rational number r = p/y in the interval [O. I] there is a function u~‘~’ (8, e), jointly analytic in (U, C) in a strip around the real axis and E in a disk around the origin, such that:
A. Berretti : Analyticity,
1490 The exponent fl is the critical exponent for the radius of convergence, and it is equal to two. In other words, we keep the radius of convergence fixed as w tends to resonant values and see what happens to the resealed conjugating function. One might even wonder whether it is possible to calculate explicitly these functions, and get exact values for their singularities on the complex 8 and E planes: note that a priori this is far from being obvious or simple. 1.5. Complex$cation
of’w
Before we get any further, we must take a fundamental step. As we are looking at analytic properties of the function CI(and of the curves %“,,,,,),we must take all variables to be complex, i.e. we must regard the dynamical system defined by the iteration of the map T,:3fas acting not on the real cylinder U x R but on C’. At this point, there is just no reason to consider only real rotation numbers, so we will take the liberty of considering any complex value of w. For the generalized standard maps there is an obvious symmetry (related to their Hamiltonian origin and, ultimately, to time reversal symmetry) so that changing o into its complex conjugate w* preserves the form of the conjugating function II, so we can take w to be in the complex upper half plane X = .(m E @I Im (0) > O)-or rather in its closure, to include also real o’s Note that if Q E X’, then the small denominators disappear as the spectrum of D,,, does not contain 0 so that the convergence of the Lindstedt series may be proved directly by simple estimates on its coefficients and using the smallness of 1~1. Taking the limits 0/2rr -p/q now is much simpler, as we may take any continuous path in X to a rational point on the real axis, with some “regularity”condition (e.g. non-tangentiality). On the other hand, the definition of limit in (1.12) and similar equations is plagued by the fact that we must take a sequence of diophantine values of to/2n converging to a rational value, and one is left with the task of proving that such a limit is independent on the choice of the sequence. One may also think of the complexification of w as a new way to approach the problem of existence of KAM invariant curves for the standard map and its generalizations. In fact, if w E 2 we have one complex analytic invariant curve parameterized by w-and this is easy to prove, since small denominators do not exist. We may therefore regard the imaginary part added to Edas a “regularization”, and consider the problem of obtaining a real analytic curve in the limit o + co,,, 0,/27r being real and satisfying a diophantine (or more generally Brjuno’s) condition, as a problem of “removing the regularization”, dealt with by uniform estimates for ~0 in a cone in fl over w,, (see Carletti and Marmi (in preparation) for further details). Finally, we remark that it would be interesting to prove some regularity properties for p(a) when w E X-or at least to provide a strong numerical analysis. For example, to prove that p is harmonic in X. 2. Numerical results In this section we review the main numerical results we obtained on the problem of the singularity structure of u.
scaling and renormalization
We shall comment briefly on these results in a later section, after a discussion on the main analytical results obtained so far. First of all, we note that the most trivial thing one could imagine to do, that is : (1) compute numerically the coefficients u,,(e, tn) of the Lindstedt expansion (1.5) for some “random” values of 0 # 0, rr and diophantine (I) (e.g. the golden mean) (2) use the root criterion to obtain numerical estimates of P(to)
is not effective. In fact, due to the presence of small denominators, the coefficients u,(~,o) vary in a very erratic way, so that the root criterion fails to give a decent approximation to p(w) even when high orders in the Lindstedt series are used. Take a look at Fig. 1 to understand the scale of the phenomenon (and see Berretti and Chierchia (1990) for more details); here o is the golden mean, that is (0, = (& I)/2 = [I’], and up to order 440 in F the root criterion still fails to relax to the correct value (computed in this case with Greene’s (1979) criterion, represented for comparison by the horizontal line in the figure. On the other hand, the use of Greene’s criterion to extensively compute p(w) for several values of o and several kind of maps (beyond just the standard map) is computationally too hard to be feasible : in fact, one has to go quite close to the resonance to be able to see the asymptotic behaviour conjectured, and this proves to be just too hard to do (see Marmi and Stark (1992) for an attempt to do exactly this). Moreover one would have to define and implement a Green’s criterion for complex rotation numbers : and it is not quite clear how to do it. In de la Llave and Falcolini (1992) nevertheless Greene’s criterion was indeed used to compute breakdown thresholds for complex values of E, though only for real values of (rj. As we are interested in the more general case in which everything is complex, we stick to the method of Pade approximants, introduced in this context in Berretti and Chierchia (1990) and Berretti et ul. ( 1992). We shall not present here the theory of PadC approximants, as this would be completely off topic in the context of this review, and many fine monographs exist on the matter (see e.g. Baker, 1975; Baker and Graves-Morris, 1981). It will be enough to recall that, given a function F(Z), analytic near the origin, its Pade approximant [M/M is a rational function P(;)/Q(=), with A4 and N being respectively the degrees of the P and Q, such that Taylor expansion coincides with the Taylor expansion of F up to order M + N; formal uniqueness is obtained by imposing e.g. that Q(0) = 1. Under suitable non-degeneracy conditions on F such approximants are shown to exist, are unique and can be recursively computed from the first M+ Nf 1 coefficients of the Taylor expansion of F. For meromorphic functions F, their Pade approximants may be proved to converge to F as their order increases. uniformly on compact sets not containing poles, and the singularities of the Pade approximants may be used to approximate the singularities of the function F. For functions F not just meromorphic, but presenting a more complicated pattern of singularities (e.g. branch point or essential singularities) much less is proved. but
A. Berretti : Analyticity, scaling and renormalization
1491
nevertheless one can use the location of the singularities of the Padl approximant as an estimate for the location of the singularities of F (Nuttal, 1977). Of course, many spurious pole/zero pairs will be created in the process, and these pairs can be detected computing the residue of each pole found and eliminating those which have an extraordinary low residue (how low is low enough depends on the scales of the problem). See Baker (1975) more information. for references and for the practical formulas used to compute numerically the coefficients of the polynomials P and Q. See also Berretti er al. (1992) for details on the techniques used in this context.
is the golden mean: we observe a power law behaviour, with an exponent r approximately one. The exact value of this critical exponent should be related (de la Llave, 1992) to the exponent 6 of MacKay’s fixed point of the renormalization group for the standard map. The exponent 6 is computed in Ketoja and MacKay (1989) to be (5 2 1.6280, and according to de la Llave (1992) c( = - logto/log6, which for UJ equal to the golden mean gives x 2 0.9874. which is well within numerical errors.
We consider now the standard map, and look numerically, using Padl approximants, to the analytic structure of the conjugating function U. We compute numerically the coefficients u,,(H, Q) for selected values of 0 and for some real, diophantine values of OJ, and then we compute Padl approximants; by looking at the roots of their denominators. we can locate the singularities of 11. We find (Berretti and Chierchia, 1990) that, independently of the values of 0 # 0,~ there exists on the complex t: plane a natural boundary, i.e. a dense set of singularities which precludes analytic continuation. The size and shape of the natural boundary does not seem to depend on 0 (except, of course, in the trivial cases t3 = 0,7c in which u = 0 identically!). The shape of the natural boundary, at least for diophantine (o/271 “far” from a rational number, appears to be circular within numerical errors. so that this would suggest that E,.= p-but the situation could be different as we move close to resonances : in fact Davie (1995) claims that as the rotation number tends to 0 (staying real. so through a sequence of diophantine values of to127r)E,.and p tend to zero with a power law characterized by different critical exponents. As we shall see, this is coherent with our findings, both numeric and analytic. about the scaling of the conjugating function 11. In Fig. 2 we see the poles of the Padtt approximants [35/35], [50’50]. [60/60] and [85/85] for ~(H,E,o), when 0 = 1, and (I) is the golden mean : we clearly see the coalescence of poles of the Padl approximant to form the natural boundary. All the poles corresponding to spurious pole/zero pairs have been deleted from the plot. It is interesting to look also to the singularities in the complex 0 plane : here any structure must be symmetricin respect to the real 0 axis. It is convenient to use the variable <=e “I; on the complex [ plane we see a pair of circular natural boundaries, of radius respectively 1 +a and I/( I +o), with CJ+ 0 as i-:+ c,((o): we took the golden mean as a rotation number. In the complex 6, plane therefore the analyticity region is a strip, symmetric around the real axis. bounded by a natural boundary; the width of the strip tends to 0 as E tends to the breakdown threshold. In Fig. 3 we plot the poles of the Padt: approximant [SO/SO]in the complex < plane for the function U, at E = 0.8 to the outer and (0 = (II~: only the poles corresponding boundary have been plotted. In Fig. 4 we plot CTvs F, for the standard map, when o)
It is natural to look numerically at what happens to the singularities of II as OJ becomes complex, to try to understand the mechanisms leading to the formation of the natural boundary (Berretti and Marmi, 1992). As small denominators disappear, so does the natural boundary. The PadC approximants show radial lines of poles alternating with zeros-typically representing the presence of a branch point for u at the end of the line closer to the origin. and as Im (TV)+ 0. while Re (to)/h tends to a diophantine, the number of branch points grows. building up the natural boundary as OJ becomes real ; see Fig. 5 where we plot the poles of the Padt approximants [40/40]. [40/40] and [30/30], at H = 1, respectively for OJ&T = (0, +0.3i,cu, +o.o%, UJ, +o.oii. where OJ, is the golden mean. Taking into account (1.11) and Co+c.tur.r 1.3. we can get a better insight to the mechanism of breakdown of the invariant curves by looking at the structure of singularities of 11as ~/2x tends, from the complex upper half plane ,Y?‘. to rational values: if the imaginary part of (1~ is chosen small enough, so as that only the terms “in resonance” with the given rational value play a significant role in the Lindstedt series, and OJ -+ 2zppiy, then we find that the Pad6 approximants have exactly 2q radial lines of alternating poles and zeros, accumulating on the circle of convergence of the series. These lines of singularities can therefore be interpreted as branch points for 14. and those which appear when Re ((c))/27( is diophantine can be interpreted as the contributions of the leading resonances corresponding to the best rational approximations given by truncated continued fraction expansions of Re (01);271: as Im (0) tends to 0,resonances of higher and higher order contribute to U. adding singularities on the circle of convergence and creating the natural boundary. An analogous mechanism happens if we look at singularities on the complex (1plane. In Fig. 6 we see poles of the Pad& approximants [X$/28]. [40/40] and [48/48], at 0 = I. respectively for CJJ/?~~ =
I /2
+
0.0 Ii, 215 + O.OOOOi.3% + 0.00005i.
We can use Pad& approximants also to check the conjectures stated in the preceeding section. In particular, we can compute the radius of convergence p(to) as the distance from the origin of the closest singularity of II, whose location is estimated using PadC approximants. While crude in terms of numerical errors (which can be roughly estimated of the order of magnitude of about 1%). we can
A. Berretti : Analyticity, scaling and renormalization
1492 get quite deeply into the asymptotic region, i.e. quite close to the resonance. In the following table, taken from Berretti and Marmi (1994), we show the results of the least squares fit of the empirical data for p(o) to the scaling law ~(01) = K(o)lcc, - 2xp/qlP1yas Q + 27cp/q. P/q
B
O/l l/2
2.01 2.04 2.03 2.00 2.02 2.00 2.00 2.00 2.00 2.00
l/3 l/5 215 l/3 l/3 O/l l/2 O/l
exponent is no longer Z/q-actually it is no longer of the form /?/q for some p E [w! We present here the results of some calculations done for the same map as the example of Fig. 8, that is for f(s) = sin x+ l/20 sin 2.x. In this case the following scaling law seems to hold :
Comments 34.9 26.3 14.0 5.9 7.43 13.5 13.4 32.2 23.0 32.4
In rows (a) o tends to the resonance through a vertical straight line in the complex plane; in row (b) cu tends to the resonance through an arc of parabola ; in row (c) cc) tends to the resonance through a curve with an infinite order of contact to the real axes; in rows (d), (e) and (f) o tends to the resonance through the real, diophantine sequences cl,, = [n, l”], o, = [2, n, l”] and w,, = [nr], respectively. The conjectured scaling law for the radius of convergence is therefore verified quite well. Figure 7 instead show the plots of p(27cp/q+ iAw) vs Aol, with p/q = O/l, l/2, l/3, l/S, 2/5 ; we can easily verify here the power law behaviour characterized by an exponent 2/q.
where r.yl denotes the lowest integer greater or equal to x and again p seems to be two, as the data from the following table suggests. Pirl
B
O/l
2.001 2.000 1.998 2.008 2.015
l/2 l/3 l/5 215
Clearly Brjuno’s interpolation does not hold, at least with the same Brjuno’s function we use for the standard map. Work is in progress to strengthen the numerical analysis and to extend it to more maps.
3. Analytical
results
Here we summarize some of the (few) analytic results we have obtained about the analytic structure of the conjugating functions for the generalized standard map. We also present some results obtained for simpler models, like the so-called semi-standard map (Green and Percival, 1981) or Siegel’s problem (see Yoccoz (1995) and the references therein).
2.3. Numerical results_for the generalized standard maps We also performed a similar numerical study on the more general maps of the type T,,( defined in (1.1). In Berretti et al. (1992) we looked at the analytic structure for u for real, diophantine rotation numbers. We found a number of surprising results. First of all, we still have a natural boundary, but in general its shape this implies that in some cases is not circular: E,.(Q) > p(tu), even significantly; in fact, E,.is given by the intersection of the natural boundary with the real, positive E axis, and in some cases the natural boundary elongates in that direction, so that it distance from the origin is strictly smaller. An example is given in Fig. 8, where we plot the poles (after removal of ghost pole/zero pairs) of the PadC approximant [70/70] in E for the map with ,f(s) = sins+ 1/20sin2x, rotation number equal to the golden mean and 0 = 1 : here F, and p differ by about 30%. More examples of this kind have been given in de la Llave and Falcolini (1992) where the boundary of the analyticity region is computed by a complex version of Greene’s criterion, at diophantine rotation numbers, and compared with Pade approximants. The situation is more intriguing if we look at the scaling of p(w) as (1) tends to resonant values. Indeed, we found (Berretti and Marmi, 1995) that a scaling law of the type we have for the standard map still holds, while the
3.1. Scaling near resonances.for the standard map : O/l and 112 In Berretti and Marmi (1994) we have been able to prove indeed that eqn (1.12) holds for the resonances O/l and l/2. Theorem 3.1. Let x2 be the space of 2rr-periodic functions f(0) analytic and bounded in a neighbourhood of the complex strip :
where];. are the Fourier coefficients closed complex cone defined by : %;. = {~E@Jlrn~
> 0.
off, and let g:. be the
[Rev] d IjIrny),
7 > 0.
Then if v]E g; and p/q = O/1 or l/2 the limit :
A. Berretti : Analyticity. scaling and renormalization
I403
exists in 11- II1 norm, 0 < x’ < z, uniformly in 1~1< I’ = const. em I. Moreover, they satisfy the differential equations : I$, ’ ‘(8, c) = c sin (H + zr’“’“(0, E)), d” I’( 0. E) = 11”’“(27L E) = 0,
kK(k) = 7~~:i..
(3.2)
See e.g. Whittaker and Watson (1927) and Chandrasekharan (1985) for details and for an elementary introduction to the theory of elliptic functions from the complex-analytic point of view. It is then immediately found that the singularities of .Y in t closer to the real axis are branch points of infinite order at :
II” “(0. F) = II” y27c. E) = 0. Notice that : where K’ = K(k’) lished notation :
The proof comes in two parts: first one shows the existence of the limit, then we compute it explicitly by showing that it satisfies the differential equation. Full details may be found in Berretti and Marmi (1994). While the case (0 -+ 0 is very simple, the case (u + II (p/q = 11’2)is slightly tricky as it uses some special symmetries of the standard map, specific to the resonance l/3 case. The basic idea is to prove that the 112 resonance corresponds, loosely speaking, to a O/l resonance for the map iterated twice. We believe that the full formalism of trees is necessary to extend the proof to all resonances. We therefore make the following conjecture. Conjrcturr
3.1. u”’ ‘I’ satisfies : 2& “’ = C, /
for some real constant
sin q(8 + dp "I).
C, ‘i. with boundary
11”’“‘(0, 8) = u”‘qh,
:
conditions
e) = 0.
Note that, for p/q = O/1 or l/2 and conjecturally for all differential equations permit us to compute explicitly the singularities in 8, E or u@“I),since they can be solved exactly in terms of Jacobian elliptic functions. In fact. let :
p/q, these
x(t) = q(o+Lf'"Y'(&E)). f = q&
i"= qCryeY.
and
t=ip.
k”+k’
k” K
= 0. lJsing
_
z-c
r,i
a well-estab-
(3.3)
we see that the singularities are at I = rc(I Is) or. in the variable [ = e”, at --_“. - 1;:. Using (3.1) we read off the singularities for 11(““’ in 8, and explain-also in a quantitative way (Berretti and Marmi, 1994)-the pattern of singularities found for the Pad6 approximants when (!I is close to a resonance. Concerning the singularities in I:. it is useful to expand .Yin Fourier series :
which converges and is analytic for /:I < 1 (i.e. Im (r) > 0). Notice that as s(t) depends on ELonly through z (because of the boundary conditions) this means that (1) the singularities of s(t) in /. do not depend on t. except for the trivial cases t = 0. TTand (2) they are found as the singularities of z as a function of Jb. As one can write i. in terms of S-functions of z- -actually, using the notations of Whittaker and Watson ( 1927). 1. = 3:(0, z)/4-we can locate the singularities closest to 0 to r as a function of i. as two branch points on the imaginary i. axis : i. = k ii,. Again using (3.1) we found a complete qualitative explanation of the numerical observations, and also a quantitative agreement with /)((I,) as crj -+ 2xpP;q (Berretti and Marmi. 1994).
(3.1) so that .Ysatisfies : X” = i. sin s,
s(0) = 0,
s( 271) = 27l;
Before attacking the problem of proving Conja-tzrrc 3.1 for the standard map, one may first try to study the simplest semi-standard map :
then : .y(t) = n-?am(F(t-n).k)
s, : where am( -, k) is the Jacobian amplitude K(k) is the quarted-period function :
of module
k,
K(k) = and k is determined boundary conditions tion) :
I’ = .v+ 1.’ i J.’ =
j’ +
this map can be put in Lagrangian
i:
e”
form as :
s,,_ , - 2s,, + .v,~ , = i: e”“. by the condition (coming from the imposed to the differential equa-
Note that is an intrinsically complex dynamical system : s’, J.’ are not real for real .Y.~3. The function ld which conjugates the dynamics to rotations by (1)satisfies :
1494
A. Berretti : Analyticity, scaling and renormalization D,;,u
=
Ee”“+“’
(3.4)
It is easily seen that u(~,E,w) is actually a function of sein and w only ; this suggests to use the following variables :
where
: uk = u(eA, L co), e, = e + kw, p = “,p,
&Z, 3”) = iU(0, E, 0) ;
(0 = 27tp/q+rj,
then 4 satisfies :
a:“’ are given by :
and the coefficients a),“’= 1,
a?’ = - 2,
a?’ = 0
otherwise.
We shall apply q- 1 times a transformation three steps :
and if:
defined in
0 write everything in terms of r&+, - hk + u/,~, apply the functional eqn (3.4) 0 linearize : l
(3.5) determines a recursion relation for the coefficients c$,, of the Taylor expansion of 4. It is easy to see that if o is real, then (3.6) is a power series with positive coefficients. This map has been introduced by Green and Percival (1981). Davie (1994) was able to prove a weak form of Brjuno‘s interpolation; in fact, denoting with pSS(i) the radius of convergence of (3.6) he proved that :
is bounded. Since the Lindstedt series (3.6) for the semistandard map is so much simpler, we could actually prove the equivalent of conjecture 1.1 in the following theorem. Theorem 3.2. Let $(z,l) be a solution of the functional eqn (3.5) satisfying &O, 1) = 0, and let 111< 1 ; let 1 be a primitive qth root of unity (i.e. x4 = 1 but Xk # 1 for all 0 < k < q). Then the following limit exists :
and is analytic in a neighbourhood of the origin, provided the path taken by 3. is non-tangential to the boundary of the unit disk. The proof and is based estimate of onances. Of course
Applying this transformation iteratively we obtain a sequence of expansions :
2:
(Se ‘(“~~+“~~))y~‘(a/“~, +ab”u,+a(A’,u_,)
In Berretti and Marmi
= (A).
(1996) it is proved that :
It follows that ah” = c0 and a\” = a(?), = c,/2, so that, applying once more (3.4). we get :
may be found in Berretti and Marmi (1996) on the majorant series method, plus a careful complexified small denominators near reswe have then the following
Corollary 3.1. Let q 4 0 non-tangentially plex upper half plane ; then the following U’Pyl, E) = l$
zr(d,up,
corollary.
Taking
the limit q-0
from the comlimit exists :
27cp/q+ r/),
upil
=
analytic
_
:
we obtain (‘0 _
2 ‘q-
IgY
erqw+Ir’~~‘J,
2q’ By letting
: +)
and is jointly
to D&(0, E: w)
= iqu(p!qe,
&),
(3.7a)
in the region {1Im (0)l < c,, 1~1< c?).
To compute exactly such function is of course quite a different matter. In Berretti and Marmi (1996) we gave an heuristic argument for an exact form of u@,~); we sketch now some of the basic ideas of the argument, leaving the details to Berretti and Marmi (I 996). We have :
(3.7b) we obtain
easily : (_-II/‘)’= e$ ;
since -_ = 0 means E = 0, i.e. the linear map, we must have
1495
A. Berretti : Analyticity. scaling and renormalization $(O) = 0 ; a unique of 0 is given by :
analytic
solution
in a neighbourhood
log R(i.) + II(to) < (const.). Vcc> 0 : log R(i)+(
$(z)= -2log
1-i (
_)
From (3.7a) and (3.7b) it is easily seen that &‘,*’ has y branch points of infinite order symmetrically located around the origin-as the numerical calculations using Pade approximants also show. To turn this argument into a proof. one would have of course to control the error introduced by the linearizations iteratively applied. Even though it is not a rigorous proof. we believe that the method we used is interesting in providing a bridge between perturbative expansions and the renormalization methods in which the iterated map is studied.
3.3. Sityel’s prnhletti A more complete analysis may be carried over for Siegel’s problem. Let P, be a complex function, analytic in a neighbourhood of the origin. such that : P,,(O) = 0,
P;(o) = i:
Thorettt
3.3. Let : I R,(Z) = ----H,((A-i)’ (A-i)’ ”
“Z);
(I ) f?,(z) is the conjugating F,(I) = ,7-l;,.;
function
for:
P,((A -;.)I
“z).
= H;.(k),
with H; analytic near 0 and such that H,(O) = O&(O) = 1. Poincare ( 1928-l 956) proved that this is always possible if JE.1# I. If /1 if a root of unity, then Hj. does not exist in the class of formal power series, so aforriori as an analytic function. When 1E.I= 1 but i. is not a root of unity, H,(z) exists as a formal power series so the problem consists in proving that it is actually convergent. If: H;(z) = i
where R(j&) is the radius of convergence of (3.8). Here we choose P,(Z) = i.:( 1 --_). for the sake of simplicity and definiteness. though some of the results could be easily generalized. In Berretti and hlarmi (1996) we prove that. as i. tends to a primitive @h root of unity A from inside the unit disk- so that R(j_) -+ O--then. by resealing : suitably, it is possible to obtain a well defined limit R”‘(Z) for the conjugating function H,(z). We also proved that this limit satisfies a differential equation, and we computed it explicitly and determined its singularities. While these results are essentially contained in Yoccoz (1995). the method we use- roughly speaking, iterating y times P,. to reduce to the case A = I -is basically the same as the one we apply to the more difficult case of complex area-preserving maps, like the standard or semi-standard maps. Let then A = e2r’i”‘. (ply) = I, a primitive r/th root ot unity. We have the following theorem.
then :
the problem here consists in determining for which values of j, it is possible to linearize P,, i.e. to analytically conjugate P, to its linear part i.: : P,(H,(:))
1 + r)B(co) 3 (const.),
H,,(A):“.
(3.8)
then in H,,(i.) appear terms containing factors (E.“-i)) ‘. which are arbitrarily large making it difficult, again, to prove the convergence of the series above. Let therefore :
and let (pk:qr) be the sequence of rational approximants to u obtained by truncating the continued fraction expansion of UJ to the kth place. Then Cremer (1938) proved that H;(Z) diverges if i. is such that the sequence log qA+ ,/q/. is unbounded. Siegel (1941) proved the first fundamental result on the topic: if w satisfies the usual diophantine condition then the series in (3.8) converges. It was much later ( 1965) that Brjuno (1971) proved that the condition I&( log yI, + ,; qr ) < ~j is sufficient for the convergence of the series (3.8). Finally, Yoccozproved in 1988 (Yoccoz, 1995) that Brjuno’s condition is actually necessary and sufficient for the convergence of H,(z), and a weak form of Brjuno’s interpolation with /I = 1, i.e. that :
(3) as i. --t A non-tangentially. tion :
H,(Z) tends
to the func-
Notice the singularity structure for E!‘,“(Z): (f branch points of order q symmetrically located around the origin.
4. Conclusions Poincare series at resonances have been studied also in the context of the problem of linearization of vector fields around a singular point. In particular we would like to cite the work of Piartly (1972). who shows how the divergence of those series is related to the accumulation of resonant invariant manifolds at the singular point. See also Arnold (1980) Section 36 for a review and further references. Our analysis is quite severely limited. as it stands, to a very simple model (the standard map) and a few other. Even for generalized standard maps it is not clear if there is some form of Brjuno’s interpolation, and the study of the limit at singularities of conjugating functions is hard to make at the level of rigorous proofs. We believe that, once the suitable resealing is done of the perturbative parameters, analogous results hold. with hyperelliptic functions used to represent 24 at the resonance. Some work is in progress for the simplest such map, i.e. T,,, withf’(s) = sin s+cz sin 2s. This may help to clarify the reason for the large discrepancy between the critical
1496
threshold e, and the radius of convergence of the Lindstedt series p. Even more interesting would be to generalize the above to higher dimensions. Here one of the main problems is that regularizing the Lindstedt series using complex frequencies does not eliminate the small denominators. So an important problem is to properly formulate the dynamics of higher dimensional symplectic maps in the complex domain. Finally, one may ask whether is it feasible to prove the existence of the numerically observed natural boundaries for diophantine rotation numbers. This could be the case for the semi-standard map, which, for real and diophantine rotation numbers, has a Lindstedt series with positive coefficients, and so a “free singularity” on the intersection of the circle of convergence with the real axis; a similar pattern of proof can easily show the existence of a natural boundary for the solutions of linear functional equations of a similar kind. AckttoIllec!qrtttettt.s. The author wishes to thank Alessandra Celletti and the organizing committee of the Second Italian National Meeting of Celestial Mechanics, held in L’Aquila, 2 l23 April 1997. for inviting him and giving him the chance to write this review. Noic at/&d in pruqf: The heuristic method introduced in Berretti and Marmi (1996) to compute the limit function L&‘~’ for the semi-standard map has been turned into a rigorous one in Berrctti c’t al. (1998). Thcorw~~3.1. has been extended in Berretti and Gentile (submitted) to all r E 69 by using a version of the tree expansion of Gentile and Mastropietro (1994). essentially proving Conjecture 1.3 for the standard map.
References Arnold, V. I. (1980) Chapitres Suppl~mrntuires de la Thkorie drs _l?quutionsDiffkrentielles Ordinuires. Ed. Mir. Moscow. Baker, G, (1975) Essentiuls qf Pude Approsimunts, Academic Press, New York. Baker. G. and Graves-Morris, M. (198 1) Pudt Approximants. Addison-Wesley. Berretti, A. and Chierchia, L. (1990) On the complex analytic structure of the golden invariant curve for the standard map. Nonlinearity 3, 39. Berretti, A. and Gentile, G. Scaling properties for the radius of convergence of a Lindstedt series : the standard map. J. Math. P; et App., submitted. Berretti. A. and Marmi. S. (1992) Standard map at complex rotation numbers : creation of natural boundaries. Ph~,s. Rw. Lett. 68, 1443. Berretti, A. and Marmi, S. (1994) Scaling near resonances and complex rotation numbers for the standard map. Nonlinearity 7, 603. Berretti, A. and Marmi, S. (1995) Perturbative renormalization and analyticity for the standard map and some generalizations. CIIUOS.Solutiotts and Fractals 5, 251. Berretti, A. and Marmi, S. (1996) Limit at Resonances @‘L&teurizutions of’ Some Comples Analytic Dynumicul Systems. preprint. Dip. di Matematica, Univ. di Firenze. Berretti, A., Marmi, S. and Sauzin, D. (1998) Limit at resonances of linearizations of some complex analytic dynamical systems. Ev),odic Th. and Dw. Syst. Berretti. A.. Celetli. A., Chierchia. L. and Falcolini. C. (1992) Natural boundaries for area-preserving twist maps. Journal @‘Stat. Ph_v.s.66, 1613.
A. Berretti : Analyticity,
scaling and renormalization
Brjuno, A. D. (1971) Analytic form of differential equations. Trans. Mosco~r M&h. SOL..25, 131-288. Buric. N. and Percival, I. C. (1991) Modular smoothing and finite perturbation theory. Nonlinearity 4, 98 I. Buric. N., Percival, I. C. and Vivaldi, F. (1991) Critical function and modular smoothing. Nonlitzearit~~ 4, 21. Chierchia, L. and Falcolini, C. (1994) A direct proof of a theorem by Kolmogorov in Hamiltonian systems. Ann. Scuoku Norm. Sup. di Pisu, Serie IV 21, 541. Chandrasekharan, .!$. (1985) Elliptic Functions. Springer. Cremer, H. (1938) Uber die Haufigkeit der Nichtzentren. Muth. Am. 115, 573, Davie, A. M. (1994) The critical function for the semi-standard map. Nonlineurit_v7, 219. Davie, A. M. (1995) Renormuiisution jar Anu!,,tic Arru-presewing Maps. University of Edinburgh. preprint. de la Llave. R. (1992) A renormalization group explanation of numerical observations of analyticity domains. J. Stuf. Phys. 66, 1631. de la Llave. R. and Falcolini, C. (1992) Numerical calculation of domains of analyticity for perturbation theories in the presence of small divisors. J. Stat. P/I,I~s.67, 645. Eliasson. L. H. (1988) Absolute!,, Convergent Series E.~pan.sions ,fiv Quasiperiodic Motions. University of Stockholm. preprint. Eliasson, L. H. (1989) Hamiltonian systems with linear normal forms near an invariant torus. In Noti~ineurD_vnumic.s, ed. G. Turchetti. World Scientific, Singapore. Escande. D. F. (1985) Stochasticity in classical Hamiltonian systems : universal aspects. Pkys. Rep. 121, 165. Feldman. J. and Trubowitz, E. (1992) Renormalization in classical mechanics and many body quantum field theory. J. A&. Math. 58, 2 13. Gallavotti. G. (1994) Twistless KAM tori. Cor~~rlz.Muth. Phys. 164, 145. Gentile. G. and Mastropietro. V. (1994) KAM theorem revisited. P/r.~,.ric.rrD 90, 225. Greene, J. M. (1979) A method for determining a stochastic transition. J. Math. PhJx. 20, 1183. Greene, J. M. and Percival. I. C. (1981) Hamiltonian maps in the complex plane. PI~ysica D 3, 530. Hardy. G. and Wright, E. M. (1979) An Itltroductiott to the Thcor~~of Nutnbrrs, 5th edn. Oxford University Press. Ketoja, J. A. and MacKay. R. S. (1989) Fractal boundary for the existence of invariant circles for area preserving maps : observations and a renormalisation explanation. Plz~~sica D 7,318. Lindstedt, A. (1882) K. .4&i. Ll/iss. St. Pr/rrsburg 31, 4. MacKay, R. S. (1983) A renormalization approach to invariant circles in area-preserving maps. Ph~~sico7D, 283. MacKay, R. S. (1985) Transition to Chaos for Area-preserving Maps. Springer Lecture Notes in Pi7~~.sic~.c, Vol. 247, p. 390. Marmi, S. and Stark. J. (1992) On the btandard map critical function. N~tttl;tteur~~~, 5, 743. Marmi. S.. Moussa. P. and Yoccoz, J. C. (1995) The Brjuno Ft’2rnction.s trnd their Re~yuIurit~’Propcrtkr. preprint. Saclay 95/0X Nuttal, J. N. (1977) The convergence of Pad& approximants to functions with branch points. In Pad&and Rutionul Approsimutions, ed. E. G. Saff and R. H. Varga, p. 101. Academic Press, New York. Piartly. A. (1972) Birth of complex invariant manifolds near a singular point of a vector field depending on parameters. Fzrnct. At&. Appl. 6, 95. Poincart-, H. ( 1893) Les MPthodes Nourelkr de lu Mckhuniyur CJeste. Vol. 11. Gauthier-Villars. Paris. PoincarC, H. (1928-1956) Owrrs. t. 1. Gauthier-Villars, Paris. Siegel, C. L. (1941) Iteration of analytic functions. .4nn. Marh. 43, 607. Vittot. M. (1991) Lindsredt Perturhution Series in Hamiltot~iutt Mechunic.~: E.@cit Formulution viu u Mlrltitlin~c~ttsionol Bur-
.A. Berretti : Analyticity,
scaling and renormalization
mUntZ--lU~il/n~~yr Forrd~, Marseille. preprint CPT911P.2630. Whittaker. E. T. and Watson. G. N. (1927) A Course in Modern Anrr~~.si.s.Cambridge University Press.
1491 Willbrink. J. (1987) Erratic behavior of invariant circles in standard-like mappings. Physicu D 26, 358. Yoccoz, J. C. (1995) Thkoreme de Siegel. nombres de Brjuno et polinomes quadratiques. Asririsyur 231, 3 38.