1. Introduction The resonant structures for hamiltonian systems have been intensively investigated: in fact, they are the bridge between the ordered q

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1 NON RESONANT LIMIT FOR SEQUENCES OF RESONANT ORBITS: THE CASE OF HOLOMORPHIC MAPS by E. Todesco, G. Turchetti Dipartimento di Fisica della Universita di Bologna, ITALY INFN Sezione di Bologna, ITALY The approximation of a nonresonant orbit with a sequence of resonant orbits is considered for the holomorphic maps of the complex plane. The problem is motivated by hamiltonian dynamics (Greene's conjecture) and we consider a complexied hamiltonian map in the region (far from the section of real dynamics) where it can be reduced to an holomorphic map of a single complex variable. For a sequence of maps in normal form with linear resonant frequencies, the limit to a linear map with nonresonant diophantine frequency has a simple interpretation: the ower-like resonant orbits become circles due to the increase of the number of petals and the freezing of radial motion. A similar nontrivial result is proved for small perturbation of the normal forms by investigating the behaviour of the conjugation functions. 1

2 1. Introduction The resonant structures for hamiltonian systems have been intensively investigated: in fact, they are the bridge between the ordered quasi periodic orbits (i.e. KAM tori), and the chaotic motion. However, a complete theory is still lacking. The perturbative approach based on Birkho normal forms gives rise to asymptotic series whose truncation conveniently describes the geometry of resonant families [1]. Phenomenological analysis has been carried out concerning the way a family of resonant orbits approaches a nonresonant one. This limit is of a great importance, since it allows one to detect the critical value of a KAM torus. According to the well-known Greene conjecture [2], a KAM torus can be approximated by a family of resonant orbits (chains of islands) of increasing period; moreover, before the break up any resonant orbit has an equal number of elliptic and hyperbolic xed points, whilst after the break up they are all hyperbolic, and the torus becomes an Aubry-Mather Cantor set. This conjecture provides a very accurate computational tool to determine the critical values of invariant curves for area preserving maps. The above-described statements have not been proved, and even the possibility of approximating a KAM torus using a sequence of resonant orbits before the break up has not been demonstrated. In view of the diculty of this problem, we consider a related one which can be investigated using the Borel transform techniques. After describing the link which exists between a complexied area preserving map and an holomorphic map of a single complex variable [3;4], we show how the invariant curves of a map with a nonresonant linear frequency (Siegel problem [5] ) can be approximated by the orbits of a sequence of maps with resonant linear frequencies [6]. The corresponding normal forms illustrate the nature of the limit in the `integrable' case y : the sequences of ower-like orbits of the resonant normal forms converge to the circular orbits of the nonresonant map via the freezing of the radial motion. A similar picture holds in the `nonintegrable' case, where we consider a sequence of small perturbations of the resonant normal forms whose limit is a linear nonresonant map with a small perturbation. A rigorous proof of the convergence of nonresonant orbits to resonant ones is based on the convergence of the sequence of the conjugation functions that transform the `nonintegrable' maps into their normal forms. This result has been proved at y In the following, we will use the word `integrable' and `nonintegrable' for holomorphic maps to denote the normal form and perturbations of the normal forms respectively; this terminology is borrowed from hamiltonian dynamics. 2

3 the rst order in the small perturbation parameter; in this simpler case the conjugation functions satisfy a homological equation rather than a nonlinear functional equation. The conjugation function and its corresponding limit are considered here at the level of the formal series, while the complete proof is given in [7]. The plan of the paper is the following: in section 2 we give the connection between the hamiltonian and the holomorphic models; in section 3 we analyse the limit in the `integrable' case and discuss the change in the geometry of the orbits. In section 4 the statement for the `nonintegrable' case is given, and a sketch of the proof is outlined. 2. Hamiltonian models and holomorphic mappings The complexication of a hamiltonian map has been considered for the standard map [3;4] where, for an invariant curve with nonresonant frequency, the images of the analyticity strips in the angle were shown to be domains with fractal boundaries. For vanishing perturbation! 0 the width of the strips tends to 1, while it decreases to zero at the critical break-up value of the invariant curve. For! 0 and in the neighborhood of the analyticity boundaries the map can be approximated in the z = e i plane [12] with an holomorphic map of z. Nonresonant case In order to extend the correspondence to the resonant case we consider complexied area-preserving maps in the neighborhood of an elliptic xed point. The limit is easily understood if we consider the nonresonant normal form of the map, namely an integrable mapping, which reads 8 < 0 = e i() U : ; 2 C (2:1) : 0 = e?i() where is a real analytic function. The real area-preserving map is recovered by setting = (where the denotes the complex conjugate): one obtains an amplitude-dependent rotation. One can choose and = [or equivalently (; )] as independent variables and the map (2.1) reads U : 8 < : 0 = e i() 0 = ; 2 C; (2:2) xing we obtain a map of the complex plane with constant frequency. 3

4 ? Similarly, a generic polynomial map F = f(z; w); g(z; w) (where z; w 2 C 2 ) can be expanded in a Laurent series of z with coecients depending on zw according to 8 < P z 0 N? = f(z; w) = z f 0 (zw) + z k=1 z k f k (zw) + z?k f?k (zw) : P w 0 N? = g(z; w) = w g 0 (zw) + w k=1 w k g k (zw) + w?k g?k (zw) (2:3) where? f 0 = e i! 0 + O(zw); f k = O(1) and f?1 = O(zw), f?k = O (zw) k?1. Letting in this case z = p re i# ; w = p re?i# we see that for the real section w = z of the map f k ; f?k are the Fourier components. We can transform the map into a normal form plus a remainder term with a polynomial transformation of order N according to F = (U + E) (2:4) where E = O(jj N=2 ) and the conjugation function, which maps the normal coordinates ; to the initial coordinates z; w, has an expansion similar to (2.3). Letting (; ) be dened by = p e i and = p e?i, we consider the following two limits i)! 0 and Im!?1 keeping nite, which implies! 0 ii)! 0 and Im! +1 keeping nite, which implies! 0. The conjugating function has a Laurent expansion analogous to F [see Eq. (2.3)], but it can be reduced to a Taylor series in for the rst component in the limit i), or to a Taylor series in for the second component in the limit ii). Indeed in the rst case one has k+1 k () = O( k+1 ) are nite whereas?k+1?k () = O( k?1 ) vanish. For the initial map F the limit i) corresponds to freeze zw to a small value keeping z nite and its rst component f becomes a Taylor series in z. The function is then given by a Taylor series in, and conjugates F with the linear map 0 = exp(i()). The analysis can be made rigorous by applying the KAM procedure [13] to the map U + E for any value of such that () is diophantine. Then, for very large negative or positive imaginary part of the angle U + E, one denes two analytic maps of C which can be conjugated with their linear parts. Resonant case When the linear frequency of the map! 0 is close to a resonant value 2 p=q (p and q integers with no common divisor) then the resonant 4

5 normal form is given by a map U which commutes with the discrete group of rotations of an angle 2=q: U : 8 >< >: 0 = e i() + 0 = e i() + N=q? `q A`q () +?`q A?`q () `=1 (2:5) N=q? `q B`q () +?`q B?`q () `=1 where A`q = O(1) and A?`q = O? ()`q?1. If! 0 = (0) = 2 p=q + and () = 2 p=q has a real solution, then on the plane = where a real area-preserving map is recovered, one has a chain of q islands as suggested by the Birkho theorem. The interpolating hamiltonian, whose ow agrees at integer values of time t = n with the n-th iterate of the normal form U, has a pendulum-like structure H(; ) = h() +? `q C`q () +?`q C?`q () [N=q] `=1 h() = i + h 4 () 2 + : : : (2:6) where the C k are related to the A k ; B k and C k = O(1), C?k = O(() k ). At the lowest approximation H = h() + q C q (0) + q C?q (0) (2:7) and in the plane =, one has q symmetric islands of area q=4 whose centers have a distance from the origin. In the exactly resonant case q = e 2ip=q, if we freeze = to a very small value then we obtain from (2.5) a map of C in the form of a Taylor series in, whose linear part is q. The corresponding resonant normal form in C is a map which commutes with the linear part, and can be written according to 0 = q (1 + q ) 1=q ; q = e i2 p=q : (2:8) In the most general case q is replaced by kq in the denominator of (2.8), but for simplicity we consider k = 1. Let us consider the orbits of an area-preserving map in the neighbourhood of a chain of q islands, and more precisely only the orbits which 5

6 join the q hyperbolic xed points (i.e. the separatrices); if we let all the hyperbolic xed points collapse on the origin, we obtain a mapping on the separatrices whose dynamics and phase space is analogous to the resonant normal form of period q of the holomorphic mapping (2.8). In the following section we will illustrate the dynamics of this map, showing how in the limit q! 1 it is possible to recover a circular motion, which in the hamiltonian case is analogous to approximating a KAM curve with a sequence of separatrices issued from hyperbolic xed points of increasing order. 3. The `integrable' case and the geometry of the orbits We consider here the maps of C and describe how a sequence of normal forms with resonant frequencies can approximate a linear map with a nonresonant diophantine frequency; this corresponds to the integrable case of hamiltonian dynamics since the maps exhibit explicit iterations and invariants of motion. The normal form [8;9], given by the standard shift (2.8), commutes with the symmetry group 0 = q generated by the linear part of the map, where q is the q-th root of the unity. The map has a trivial iteration (t) = t q (1 + t q ) 1=q (3:1) for t integer, which can be extended to t 2 R in order to build an interpolating ow, and has explicit invariant I() = exp(2i= q ). The dynamics of the map follows the well-known ower structure [6] : in Fig. 1 the orbits of the interpolating ow (solid line) and the rst 1000 iterates of the map (dots) are shown in the case k = 1, q = 3 (left) and k = 1, q = 5 (right). The normal form dynamics can be decomposed in two components: the rst one, responsible of the motion along the petal, which is a map tangent to the identity whose interpolating ow is (t)! ; (3:2) (1 + t q ) 1=q and the second one, responsible for the angular motion, is determined by the linear map whose interpolation is t q. The linear map 0 = q determines a jump from one petal to another skipping p? 1 of them. When p; q! 1 and q! = exp(i!), the normal form tends uniformly in jj R < 1 to the linear part, independently of the arithmetic properties of the frequency!. From the geometrical point of view, the transformation of the petals into closed circles is caused by the freezing of the radial motion on the petals. In fact, if we compute 6

7 the radial velocity of the interpolating ow (3.2) we nd _(t) = d =? [(t)]q+1! 0; (3:3) dt (1 + t kq ) 1=kq q in the limit q! 1 the motion along the petal is frozen, and only the angular motion from one petal to another due to the linear map remains active. When q is increased, the petals become thinner and thinner, and in the limit q! 1 they disappear, giving rise to periodic or dense orbits on the circle if!=(2) is rational or irrational. This is shown in Fig. 1 and Fig. 2, which plot for increasing values of q = 3; 5; 8; 13 the orbits of the interpolating ow (solid lines) and of 1000 iterates of the map. In the case of the q = 8; 13 one already observes the freezing of the radial motion along the petals (see Fig. 2). 4. The `nonintegrable' case We now consider a sequence of mappings which are small perturbations of the normal form (2.8), corresponding to the nonintegrable case of hamiltonian dynamics. q z 1 F q (z) = u q (z) + f(z); u q (z) = (1 + z q ) ; f(z) = f 1=q n z n ; n=2 (4:1) where f(z) is taken independent of q and analytic on the disc D R = fjzj Rg with R < 1. Contrary to the previous case, we have to restrict the frequency of the limit mapping to be sure that there exists invariant closed curves in the neighbourhood of the origin, according to the theory developed by Siegel, Brjuno and Yoccoz [5;11;14]. Therefore we assume that the eigenvalue q = exp(2ip=q) tends to = exp(i!), and! to satisfy the Brjuno condition for the homological equation log q r+1 q r < +1 8r 2 N; (4:2) where q r are the denominators of the approximations of! given by the continued fraction [5]. If! is a diophantine number, the condition (4.2) is always satised. When q! 1, the resonant map F q tends uniformly in D R to the mapping F (z) = z + f(z): (4:3) The orbits of F (z) in the neighbourhood of the origin are deformed closed circles, and one can build a convergent transformation which 7

8 conjugates the map to its normal form, i.e. the linear part 0 =. If one expands the transformation in a power series of, = (z) = z + (z)+o( 2 ), the rst order term satises the homological equation (z)? (z) =?f(z) whose solution is (z) = +1 n=2 f n? n zn ; (4:4) using the Brjuno condition one can prove that is analytic in a neighbourhood of the origin. In order to analyse the dynamics of the small perturbation F q of the resonant normal form, we perform a conjugation with the integrable map and discuss the analytic properties of the conjugating function for the homological equation; in order to simplify the analysis, we conjugate the resonant map F q to a preliminary normal form at rst order in, dened according to U q () = u q () + u (1) q (); u (1) q () = through the transformation 1 j=1 u qj+1 qj+1 (4:5) = q(z) = z + q (z) + O( 2 ) (4:6) which contains only nonresonant terms q(z) = 1 j6=kq+1;j2 q;jz j j q;jz j : (4:7) At rst order in, the functional equation q F q = U q q explicitly reads q(u q (z)) + f(z) = u 0 q(z) q (z) + u (1) q (z) (4:8) (where u 0 q(z) is the derivative of u q (z) with respect to z) and can be solved by projecting into the subspace of normal forms y and its complement. Letting be the projector into the space of normal forms, we obtain u (1) q = f q(u q (z))? u 0 q(z) (z) + (1? )f(z) = 0: (4:9) form. y The subspace of normal forms is the set of monomials which are in normal 8

9 The last equation can be explicitly solved if we replace u 0 q with q. To this end we add to the preliminary normal form a remainder term, namely we choose U q () = u q () + u (1) q () + r q () (4:10) where r q () = O( q+2 ) is not in normal form. Then the remainder term appears in the r.h.s. of equation (4.8) which can be written q(u q (z))? q q (z)+f(z) = u (1) q (z)?[ q?u 0 q(z)] q (z)+r q (z) (4:11) and is solved according to 8 < : q(u q (z))? q q (z) = (1? )f(z) P fj z j u (1) q (z) = f(z); r q (z) = [ q? u 0 q(z)] q (z): (4:12) In the following, we will prove that the formal solution of the functional equation (4.12) for q is given by 1 q(z) = 1 f j z j 1 q k(j?1) q q (1 + kz q ) = f j j=q j k=0 j=2 q? j z j +O(z q+2 ) q (4:13) and therefore in the limit q! 1, q!, one formally recovers the conjugation function of the homological Siegel problem (4.4). Similarly, one can verify that the preliminary normal form u (1) q () and the remainder r q () formally vanish in the limit q! 1; as a consequence the preliminary normal form plus the remainder formally converge to the normal form. In order to prove (4.13), we introduce the mapping w = z?q of C to C q, and we dene (w) (w?1=q ) = (z); f(w) (1? )f(w?1=q ) = (1? )f(z); (4:14) which are given by the series expansions 1 q(w) = q;nw?n=q ; f(w) = fn w?n=q : (4:15) n=1 The rst functional equation of Eqs. (4.12) then reads? e?2ip (1 + w)? q (w) + f(w) = 0 (4:16) 9

10 A solution can be found introducing the Borel transforms q B (t) and f B (t) of (w) and f(w) respectively.according to q(w) = w?1 Z 1 0 e? q B w d = Z w?1 1 0 e?tw q B(t) dt; (4:17) where in the second integral the integration path on t = =w is a straight line issued from the origin with the phase of w?1, and q B is the Borel transform dened by the analytic continuation of the series q B(t) = 1 n=1 q;n t n=q?1 nq? 1! : (4:18) In the second integral of (4.17) the integration path can be changed to straight lines with a phase dierent from that of w?1 to perform the analytic continuation in w [10]. Using the relation g B (t) = e?t af B (at) where g(w) = f(a?1 (w + 1)), the Borel transform of (4.16) reads e?t q B(e 2ip t)? q q B (t) + f B (t) = 0: (4:19) Replacing t! e 2ip t in (4.18) and multiplying by e?t?1 q one obtains a new equation. Repeating the procedure on the new equation and so on q times one obtains q equations; taking the sum, all the terms with q B(e 2ikp t) with k = 1; : : : ; q? 1 cancel and one has q B(t) = 1 q (1? e?qt ) q?1 k=0?k q e?kt f B (e 2ikp t) (4:20) Replacing in (4.20) the expression of f B (t) we obtain q B(t) = = 1 q (1? e?qt ) ` f` `q? 1! q?1 k=0 e?kt k q t`=q?1 q? `qe?t ` f` t`=q?1 e `q? 1 2ikp(`=q?1) =! (4:21) Taking the Laplace transform after some algebraic manipulation the result (4.12) is obtained. It can be proved that both U q and q are given by a collection of 2q functions analytic on sectors of aperture smaller than 2=q. In the limit 10

11 q! 1, q!, one can easily verify that U q formally converges to U = and q formally tends to the solution of the homological equation (4.4). Moreover, using the subsequence of q r ; p r given by the continued fraction expansion of!=2, one can prove that the convergence is not only formal and that analyticity in a close neighbourhood of the origin is recovered in the limit. More precisely, the following result holds. Proposition. If! satises the Brjuno condition and the subsequence q r ; p r approximates! according to the algorithm of the continued fraction, then for r! 1, qr uniformly converges to on a neighbourhood of the origin. The proof is based on the Cauchy estimates of the remainders of the series. Using the Borel and the Laplace transform, one can prove that the formal series dening the conjugation function can be resummed to 2q functions, analytic in sectors of aperture smaller than 2=q. The Borel transform has poles whose residues dierent from zero are responsible of the sectorial analiticity, and therefore of the divergence of the Taylor series. In the limit q! 1 a function analytic in a disc is recovered, since these residues vanish, provided that! q!! where!=(2) is a Brjuno number [11]. A Cauchy estimate of these 2q functions show that they all uniformly converge to the same function, analytic in a disc. Acknowledgements We wish to acknowledge Professors J. P. Ramis and Dr. F. Fauvet for helpful suggestions and stimulating remarks; we also want to thank Prof. J. C. Canille Martins and Prof. C. Camacho for helpful discussions, and IMPA for hospitality in summer 1992 when this work was started. References [1] A. Bazzani, M. Giovannozzi, G. Servizi, E. Todesco and G. Turchetti 1993 Physica D [2] J. M. Greene 1979 J. Math. Phys [3] I. C. Percival and J. M. Greene 1981 Physica D [4] I. C. Percival 1982 Physica D 6 67 [5] C. L. Siegel 1942 Ann. Math [6] C. Camacho 1978 Asterisque

12 [7] E. Todesco and G. Turchetti, `Approximation of linearizable mappings of (C; 0) through resonant dieomorphisms', submitted to Ann. Inst. Fourier [8] J. Ecalle 1982 Les fonctions resurgentes et leurs applications (Orsay, Publication Math. d'orsay) [9] J. Martinet and J. P. Ramis 1983 Ann. scient. Ec. Norm. Sup [10] J. P. Eckmann and P. Witter 1985 Computer methods and Borel summability applied to Feigenbaum's equation (Berlin, Lecture Notes in Physics Vol. 227, Springer Verlag) [11] A. D. Brjuno 1990 IHES M 36 [12] L. Billi, M. Malavasi and G. Turchetti 1994 Integrability and chaotic behaviour ed T. Bountis (Berlin, Springer Verlag) to appear [13] A. Bazzani and G. Turchetti 1992 J. Phys. A 25 L427 [14] J. C. Yoccoz 1988 C. R. Acad. Sci. Paris Ser. I

13 Figure captions Fig. 1 Phase portrait of the interpolating ow (solid line) and of the rst 1000 iterates (dots) relative to four dierent initial conditions of the standard shift with q = 3 (left) and q = 5 (right). Fig. 2 Phase portrait of the interpolating ow (solid line) and of the rst 1000 iterates (dots) relative to four dierent initial conditions of the standard shift with q = 8 (left) and q = 13 (right). 13