TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 317. Number I. January 1990 INVARIANT TORI FOR THE BILLIARD BALL MAP VALERY KOVACHEV AND GEORGI POPOV ABSTRACT. For an n-dimensional domain Q (n ~ 3) with a smooth boundary which is strictly convex in a neighborhood of an elliptic closed geodesic &' , the existence of a family of invariant tori for the billiard ball map with a positive measure is proved under the assumptions of nondegeneracy and Nelementarity, N ~ 5, of the corresponding to &' Poincare map. Moreover, the conjugating diffeomorphism constructed is symplectic. An analogous result is obtained in the case n = 2. It is shown that the lengths of the periodic geodesics determine uniquely the invariant curves near the boundary and the billiard ball map on them up to a symplectic diffeomorphism. 1, INTRODUCTION Let n be a strictly convex and compaCT domain in IRn , n :2: 2, with a boundary an of class Coo, In this paper, we investigate the so-called billiard ball map B near the boundary S* an of the coball bundle L = B* an = {(x,';) E T* an; 1';1 ~ I}, The billiard ball map is an exact symplectic map in the interior of L [6] which is singular on the boundary aL, The local behaviour of B near aL was described by Melrose. Using the equivalence theorem for glancing hypersurfaces proved by Melrose [15] one can introduce local symplectic coordinates (x,';) in T* an near any point P E s* an so that B* an = gn-I :2: O} and the billiard ball map B assumes the form , where x = (XI' ... ,xn_ 2 ) • This result is of particular importance for the construction of a local parametrix in n for the mixed problem for the wave equation. Our goal in the present paper is to construct global symplectic coordinates near a closed curve lying in aL in which B assumes a "normal form" similar to the one described above. First we assume that n :2: 3. Let (# be a closed bicharacteristic in S* an, i.e. a closed trajectory of the Hamiltonian vector field X H with Hamiltonian H(x,,;) = 1';1 for (x,';) E T* an. Denote by &' c an the projection of (# on a n which turns out to be a closed geodesic on an. Received by the editors April 7, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 58G25. The authors were partially supported by the Bulgarian Ministry of Culture. Science, and Education under Contract 52/1988. 45 © 1990 American Mathematical Society 0002-9947/90 $1.00 + $.25 per page 46 VALERY KOVACHEV AND GEORGI POPOV We suppose that &' is elliptic and the corresponding to &' Poincare map P is nondegenerate and N-elementary, N;::: 5. Our aim is to prove the existence of a family of invariant with respect to B submanifolds AI C L near Ii, diffeomorphic to the (n -1) -dimensional torus T n-I = IRn-1/(2nl)n-1 , which are enumerated by I belonging to a Cantor set E with a positive Lebesgue measure in IRn-1 • We shall construct a smooth function K in IRn-1 , K(O) = 0, gradK(O) =I-0 and an exact symplectic transformation U from a neighbourhood of Ii into T* (Tn-I) mapping AI into T n-I x {I} for lEE such that B assumes the following "normal form" B(rp,I) = (rp -grad(~(K(I))3/2),I) for any rp E T n-I and lEE. The existence of a family of invariant tori for the billiard ball map in the case n = 3 was announced by Svanidze in [20]. In contrast to [20] we construct the conjugating diffeomorphism U symplectic and describe precisely the singularity of B near aL. The motivation for the symplecticity of U comes from the microlocal analysis, it arises naturally when one tries to construct quasi modes for the Laplace operator near an using Fourier integral operators. Indeed, the billiard ball map B can be considered as a boundary map l5 + for the pair of glancing hypersurfaces F = T;nlRn, G = S*lRn = {(x,~) E T*lRn; I~I = I}. Using Theorem 1 of the present paper, one of the authors proved in [18] that F and G can be put together into the "normal form" F = {xn = O}, via an exact symplectic transformation x: T*lRn ~ T* (T n-I x IRI) near Ii C T;nlRn where p = 0 whenever lEE. This is the crucial point used in [18] for the construction of quasi modes for the Laplace operator with Dirichlet (Neumann) boundary conditions whose "frequency set" is just the union of the broken bicharacteristics passing over the invariant tori of the billiard ball map. In the case n = 2 Lazutkin [12] proved the existence of invariant curves A()) for the billiard ball map B near any of the two connected components of a L = S: a nus: an, S: a n = a n x {± 1} , with rotation numbers w belonging to a Cantor set R with a positive measure. Moreover, he constructed a diffeomorphism U: T I X (l51 ' l52 ) ~ L, 0 < l51 < l52 ' mapping T I X {w}, w E R, into A(V and such that Bo = U-I B U is given by I (y,w)~(y+w(mod2n),w) onT x{w}. In the present paper we give a more precise symplectic version of Lazutkin's result (see Theorem 2) which provides a symplectic "normal form" of B on the invariant curves in a neighbourhood of aL. As a consequence we show that the lengths of the periodic broken geodesics in n C 1R2 determine uniquely the invariant curves Aw and BII\,,, up to a symplectic map. INVARIANT TORI 47 Let US expose this result in more details. With any periodic point p E 1: \ a1: of the billiard ball map one can associate two integers n, mEN where n is the period and m is the winding number normalized by 2m S n. Denote by r(m, n) the set of periodic orbits 9 = {gl ' ... , gn}' Bgj = gj+i (mod n) with a winding number m. Let L(m, n) be the set of the lengths of the periodic broken geodesics in Q corresponding to the periodic orbits 9 E r( m , n) and denote by £(0) ~ Q(~ ll L(m ,n) U {nlo}) the length spectrum of ° where 10 is the length of ao. Let 01 and 02 be two strictly convex domains in IR 2 . Denote by BI and B2 the corresponding billiard ball maps acting in 1:1 and 1:2 respectively. Guillemin and Melrose conjectured in [6] that if ,C(01) = ,C(02) and the eigenvalues of the linear parts of the Poincare maps corresponding to broken geodesics in 01 and 02 with one and the same length coincide, then BI and B2 can be conjugated by a symplectic map. In the present paper we give a partial answer to this question. We prove that if L1(m,n) = L 2(m,n) for min < J, J > 0, there exists an exact symplectic map X: 1:2 ---+ 1:1 and some sets 1:~ C 1:i , i = 1,2, of positive measure (see conditions (i), (ii) in §2) and consisting of invariant curves for BI and B2 respectively so that X(1:2R ) = 1:R1 and X * (B1Ir,) = B21r2. R R We turn now to an outline of the paper. In §2 the main results are formulated. §3 has a preliminary character. Here we give some facts about the so-called approximate interpolating Hamiltonian introduced by Marvizi and Melrose [14]. This is a COO function C defined in a neighborhood of & in T* ao which defines S* ao near & as {C = O} and such that B(p) = exp(-C I /2X,)(p) + O(Coo ). §4 is devoted to the construction of a completely integrable Hamiltonian Co close to C and of some "action-angle" coordinates for Co. More precisely, making use of the normal form of Birkhoff for P, we find some symplectic coordinates (rp, /) E Tn-I X IRn -1 in which the interpolating Hamiltonian C( rp, /) can be regarded as a perturbation of a polynomial Co (/). The coefficients of Co(/) depend only on the normal form of Birkhoff and the length of the closed trajectori~s of X, on the orbit cylinder associated with the nondegenerate trajectory (9. Now the billiard ball map can be regarded as a perturbation of Bo = exp(-XIr3/2)· ) "0 In §5 we apply the Kolmogorov-Arnold-Moser (KAM) theory to the map Bo close to B. For this purpose, using some ideas of [4, 5], we reduce the problem 48 VALERY KOVACHEV AND GEORGI POPOV to finding invariant tori for a suitable Hamiltonian system close to a completely integrable one. The respective Hamiltonian is non degenerate in the interior of l: but it has a singularity of the form (g/2 on al:. To overcome this difficulty we first consider the corresponding Hamiltonian systems in some compacts a way from aE and apply a refined version of P6schel's theorem [19] following the dependence on the various constants (see the Appendix). Next we glue the symplectic maps obtained together using some uniqueness results about the invariant tori. In §6 we consider the case n = 2. 2. MAIN RESULTS First let us recall the definition of the billiard ball map B: l: -+ l: (cf. [6]). Denote by v(x) the exterior normal vector at x E an normed by Iv(x)1 = 1 . If ~ E T; an and I~I < 1, then there exists a unique vector e(x,~) E IRn such that le(x ,~)I = 1, (v(x), e(x ,~)) < 0 and (v ,~) = (v ,e(x ,~)) for any v E Txan. Here (, ) denotes the scalar product in IRn and T; an and TxaQ are identified with IRn -1 via the Euclidean metric. Denote by y the first point of intersection of the ray {x + te(x ,~), t > O} with an. This point is unique if the hypersurface is strictly convex. Let '1 E ~~ an be such that I'll ~ 1 and (v, e(x ,~)) = (v, '1) for any v E Tvan. For (x ,~) E l:n {I~I < I} we define B(x ,~) = (y, '1) and extend B to 8l: by B(x ,~) = (x ,~) for I~I = I. Let us give a precise formulation of the assumptions imposed on the closed geodesic & on the COO hypersurface an. Let p~ be an arbitrary point of & and let Weal: be a local transversal section of & at p of dimension 2n -4. Denote by P the Poincare map associated with the Hamiltonian flow of X H ' H(x,~)=I~I, (X,~)ET*an and by dP the differential of Pat p. First we assume that & is elliptic, i.e. all the eigenvalues Ak ' A; 1 , k = I ,2, ... ,n -2, of dP are on the unit circle {A E C; IAI = I} and Ak i-±I . Next we suppose that the Poincare map P is 2N + I-elementary for some half-integer N ~ 3/2. This means that the eigenvalues of d P are distinct and n-2 n-2 (2.1 ) IT A~k i-I if I ~ L likl ~ 2N + 1. k=l k=l Let w be the canonic symplectic form on T* an. There exist local coordinates (z, () mapping a neighbourhood of p in W into a neighbourhood of 0 E 1R2n -4 such that wlw = L~::12 dZj 1\ d(j and if we introduce polar coordinates (rp, r) by Z j = ~ cos rp j , (j = -[is sin rp j , i = I , 2, ... ,n -2 , then we can represent the Poincare map P: W -+ W as N N+l (2.2) P(rp,r)=(rp+Bo-t-B1r+· .. +O(lrl ),r+O(lrl )). Form (2.2) is called normal form of Birkhoff (cf. [9]). We make the nondegeneracy assumption (2.3) det Bl i-o. INVARIANT TORI 49 In order to formulate our main results we shall need some additional notions. A C l diffeomorphism from T* Minto T* N where M and N are C'X) manifolds is said to be exact symplectic if it preserves the integrals of the fundamental i-form over the fundamental cycles. A function S(y, c;) is said to generate an exact symplectic transformation T: T* M --> T* M , where M is either IRn or Tn if graph T= {(x,C;;T(x,C;));(x,C;) E T*M} has the form graph T = {(y -S~(y ,C;), C;;y, C; -Sy(Y, C;))} and I detSy~(Y, c;)1 < 1 for (y, c;) E T* M. Denote by r c IRn -1 a set of the form { n-I . (2.4) r = IE IR ; CI II ~ Ij ~ C2/1 ,J = 2, ... ,n -2, 2b 2b C3/1 ~ to -In_1 ~ C4 /1 ,0 < II < ao} whe~e Cj ' ao and b < 1/2 are some positive constants and 211:to is the period of &'. F or any pom. t (p , q, qJn-I' I n-I ) E IDn\ n-2 X IDn\ n-2 X TI X 1In0\ 1 we m• trod uce symplectic polar coordinates ('I' ,I) E Tn -I x IRn -1 by p j = J2i; cos 'I' j , q j = J2i; sin 'I' j , j = 1 , ... ,n -2. Denote by V the set of points (p, q , 'I' n-I ,In_I) with symplectic polar coordinates ('1',1) E An -I = Tn -I x r and let &0 = {(O,O,qJn_1 ,to);qJn-1 E TI} c V, V being the closure of V. Our main result is Theorem 1. Let n be strictly convex in a neighbourhood of a closed elliptic geodesic &' c an such that the corresponding Poincare map P satisfies (2.1) with N ~ 2 and (2.3). Then there exists an exact symplectic diffeomorphism V: V --> V(V) c 1:, V(&o) = & = V(V) n a1:, a Cantor set E c f with a positive Lebesgue measure, (0, to) E E, some smooth functions K and g in f and An -I respectively, K(O,to) = 0, gradK(O,to) f:. 0, so that in polar coordinates ('1',1) E An -I the exact symplectic map Bo(qJ,I) = V-I BV(qJ,I) is generated by the function -t(K(I))3/2 + g(qJ, I) and (2.5) n-I g(qJ,I)=O foranYqJET ,lEE. Here K E Coo(r) n CK(f), D;g E Coo(An-l) n CK(An-l) for any multi-index a and V E COO (V) n CK -I (V) where K is the entire part of N -1 . Moreover, if N ~ 3, the set E can be chosen so that b (2.6) mes(E n Ba)/mes(r n Ba) = 1 -O(a ) where B a is the ball of radius a ~ ao . 50 VALERY KOVACHEV AND GEORGI POPOV Remark 2.1. Equality (2.5) yields D;Df g(rp,l) = 0 for any rp E -rn -I , lEE since E has no isolated points. Thus (2.5) is equivalent to n-I Bo(rp, I) = (rp + grad r(1) (mod2n),I) for (rp,l) E T x E where r(1) = -i(K(I))3/2 . Moreover, Df g(rp, 0) = 0 for any P, IPI::; K. Let us turn now to the case n = 2 . Suppose that 0 is a strictly convex domain in R2 with a c'~o smooth boundary ao of length 10 = 2n(. Denote r = [I -°0 , I] C IRI, A = TI x r for some positive constant °0 . Then the following theorem is valid which is a counterpart of Theorem 1 for n = 2 . Theorem 2. Let 0 be a strictly convex planar domain with a smooth boundary of length 10 . Then there exists an exact symplectic diffeomorphism V: A --+ ~, V (T I x {I}) = S: a 0, a Cantor set E c r with a positive Lebesgue measure, lEE, and some COO functions K (I) and g( rp , I) in r and A respectively, K (I) = 0, K' (I) < 0, such that the exact symplectic map Bo = V-I B V is generated in A by the function -i(K(I))3/2 + g(rp,l) and (2.5) is satisfied for any rp E T 1 , lEE. Moreover, the set E can be chosen so that for any ° E (0,°0 ] (2.7) 0-mes(E n [I -°, I n ::; Cno N Remark 2.2. Equality (2.5) for n = 2 yields again Bo( rp ,I) = (rp + r' (I) ,I) + Q( rp ,I) where r(l) = _i(K(I))3/2 and Q(rp,l) = 0 for any (rp, I) E -rl x E. Remark 2.3. The invariant tori given in Theorem 2 can be enumerated by their rotation numbers which in the two-dimensional case determine the invariant tori uniquely. The set R of rotation numbers is defined by the small denominator condition as follows. First for any j E N consider (2.8) R) = {w E (4-) ,4-)+1); Iwkl -k21 ~ ,u2-l-2)lkll-a 2 for any (kl ,k2 ) E 1. \ {O}} where a> 1 and ,u > 0 do not depend on j. Now, we write R = (U~I R) n (0,°0) for some 00 > O. Then the set E can be given by E = {I E r; r' (I) E R} as we shall see in the proof of Theorem 2. Denote ~R = V(-rl x E). Obviously, ~R is a subset of ~ such that (i) ~R is the union of invariant curves of B with rotation numbers in R. (ii) mes(~,)) -mes(~,) n ~R) ::; CNO N for any ° E (0,°0 ], N> 0 and some C N > 0 where ~,) = a0 x [I -°, 1 ] . Let 0 1 and O2 be two strictly convex domains in 1R2. Denote by BI and B2 the corresponding billiard ball maps acting in ~I and ~2 respectively and INVARIANT TORI 51 by Li(m,n) the sets L(m,n) definedasin§1 for Qi' i= 1,2. Let R be given as in Remark 2.3, and ~~, i = 1 ,2, be the union of the invariant curves of Bi , i = 1,2, with rotation numbers in R. Theorem 3. Suppose that LI (m, n) = L 2(m, n) if min < <5 for some <5 > O. Then there exists an exact symplectic map x: ~2 -t ~ I such that X (~~) = ~ ~ and X*(BIII;I) = B21I;2. Moreover, the set E and the restriction of K on E R R are determined uniquely by the set of rotation numbers R and by L(m, n) for mln<<5. Theorem 1 was announced in a slightly weaker form in [11] and Theorems 2 and 3 in [17] where an idea of the proof was also given. 3. INTERPOLATING HAMILTONIAN The equivalence theorem of Melrose [15] for nondegenerate glancing points (cf. also [8]) can be applied to the transversally intersecting hypersurfaces S*lRn = {(x, 0 since the change is nonsingular and in U nl: both 11n-1 and c;n-I are nonnegative. Thus c;n-I = INVARIANT TORI 53 1('1n_ I)+O('1':-I) where I is a smooth function such that 1(0) = 0,/(0) > 0 and 1('1n-l ) >0 for '1n-1 >0. From the last equality of (3.2) we have (3.4) whence we obtain &xn_ 1 &xn_ 1 I 1/2 00 a-y (y, '1) = a-y (y 'Yn-I -'1n-I' '1) + O('1n_I)' n-I n-I Now Lemma 3.2 yields hence Xn_l(y, '1) = a(y', '1)Yn-1 + b(y', '1) + 0('1':-1) where a(y', '1) and b(y', '1) are COO functions. Then from (3.4) we obtain a(y', '1) = (f('1n-1 )/'1n-l) 1/2 + 0('1':-I) for '1n-1 > O. Now, as in [14] we have 1/2 I 00 1 = {C;n-I ,xn_ l } = {/('1n-I) ' (f('1n-I)/'1n-l ) Yn-I + bey ,'1)} + O('1n_ l ) = /('1n_ I)(f('1n_I)/'1n_I)1/2 + 0('1':-1) for '1n-1 > 0, hence 13/2 312 00 ('1n-l) = '1n-1 + C + O('1n_ I). Since 1(0) = 0, then C = 0, i.e. C;n-I = '1n-1 + 0('1':-1) and Proposition 3.1 is proved. Let {qJ j} f= I be a smooth partition of unity in a neighbourhood U of Ii in T* &n and let Cj be local interpolating Hamiltonians for the billiard ball map defined respectively in a neighbourhood of supp qJ j ' Now define C = 2:f= I qJ jCj • From Proposition 3.1 it follows that C -Cj vanishes to infinite order on &r.. ::row we have B(p) = exp( _C 1/2 X,)(p) + O(Coo(p» in a neigh~ bourhood of &. Moreover, C = ° defines &r. in a neighbourhood of & and the bicharacteristic lines passing through the points p E &r. coincide with I {exp(tX,)(p);C=O, tER}. The function C is called an approximate interpolating Hamiltonian. In the case n = 2 its construction is carried out directly in [14] and its uniqueness in the sense of Proposition 3.1 is proved. 4. CONSTRUCTION OF SUITABLE "ACTION-ANGLE" COORDINATES 4.1. Let S be a transversal to Ii in r. and Sh = {p E S; C(p) = h}, So = W. Since 1 is no!, an eigenvalue of the differential d P of the Poincar~ map P at PESo n & , there exists a smooth family of closed trajectories &h of X, contained respectively ~ the_sets {p E r.; C(p) = h} for Ihl small which form an orbit cylinder and &0 = & . We shall use the normal form of Birkhoff for the Poincare map P,: S, ---+ S, depending smoothly on the parameter C. 54 VALERY KOVACHEV AND GEORGI POPOV Proposition 4.1. Let the Poincare map be 2N + I-elementary, N ~ 3/2. Then there exist symplectic coordinates (p, q , ( , t) in a neighbourhood of S , i.e. w = E;:12 dpj 1\ dqj + d( 1\ dt such that S c {t = o} and if we denote p((p, q) = (p* , q*), then p; = p j cos j (r' , 0 -q j sin j (r' , 0 + f; (p , q , 0 , (4.1) q; = Pj sin j(r' , 0 + qj cos j(r' , 0 + f n-2+j(p, q, 0, j=I, ... ,n-2, ( 4.2) If;(p, q, 01 ::::; c(lpl 2 + Iql 2 )N +lj2 , j = 1 , ... , 2n -4. Moreover, , 8A, j(r ,0= ~(r ,0, j = 1, ... , n -2, ) where A is a polynomial of r' of degree at most N with COO coefficients and P~ (0, 0, h) lie on the cylinder of orbits of X(' We choose A(r', 0 so that A(O,()=O. The proof of Proposition 4.1 is analogous to that given in [9, 16], constructing a generating function of the respective canonical transformation depending smoothly on (. Thus we construct the coordinates p and q in S. Then we extend p and q to a neighbourhood of S so that X(Pj = X(qj = 0, j = 1 , ... , n -2 , and supplement p, q , ( to a symplectic basis in a neighbourhood of S. It is easily seen that in this basis ( 4.3) Let 2nto be a period of &-. Denote by G-: Vo --+ G-( Vo) c IR 2 n-2 the symplectic map G(p) = (p , q , t /to' toO defined in a neighbourhood Vo of Sand set G o = G-ls _' For J sm~ll enough we denote n-2 n-2 I ~={(p,q,<;On-l,ln-l)EIR xlR xT x[to-J,to+J]; pJ + qJ ::::; 2t5, j = 1, ... , n -2} with a symplectic form OJ = dp 1\ dq + dln_ 1 1\ d<;On_l . We shall denote by (<;o,I)=n(p,q,<;On_l,ln_l)En n-l =T n-l x(O,J] n-2 x[to-J,to+J] the respective polar coordinates in ~ determined by Pj = j2i; cos <;OJ' qj = j2i; sin <;OJ' j = 1, ... , n -2. N ote t hat •l or w-0 = "L..,Jn j=-lI dIj 1\ d <;0 j we h ave n * w-0 = w-'1 0 nn-l . INVARIANT TORI 55 4.2. Let 1(,) be the period of periodic trajectory &', from the cylinder of orbits, thus 1(0) = 27rto . Denote by 1°(') and AO(r', ') Taylor's expansions of 1(,) and A(r, ') with respect to , at ,= 0 up to degrees M and M + 1 respectively, M sufficiently large. Let '0 be the solution of the equation (4.4) 27rto -1r0 ' I °( h)dh-A °( r, , 0=27rIn_ l , ,(O,to) =0, for (r',to-In_I)E[-J,Jt-l • Then we have 8'0(0, to) 0 (4.5) 81 <. n-I The following proposition provides "action-angle" coordinates for a completely integrable Hamiltonian '0 which is close to ,. Proposition 4.2. For J small enough there exists a neighbourhood U~ of &' in T* 80. and an exact symplectic transformation G: ~ ----> u~ of class COO such that G*' = , 0 G has the form G*,(p,q,(jJn_1 ,In-I) = '0(/) + e(p,q,(jJn-1 ,In_I) and (4.6) where C does not depend on J, I' = (/1' ... ,In_2 ), Ij = (p~ + q~)/2, j = 1, ... ,n -2, I = (I' ,In-I). Proof. first we shall construct suitable smooth coordinates in a neighbourhood U of &'. Denote by gl the Hamilton flow of rn _ 1 = to' and for PES" P in a neighbourhood of p~ = & n So denote by r(p) the smallest positive time such that gT(P)(p) E S,' r(p~) = 27r. Let s(p) be the smallest nonnegative time for which g -s(p) (p) E S for p in a neighbourhood of &. _ Before constructing a suitable coordinate system near &' it is convenient to represent the map P, in the form P, = po + pI (for the sake of brevity we drop 0 where pO(p, q) = (p, q), Pj = Pj cos j(r' ,') -qj sin j(r' ,'), qj =Pjsinj(r',,), j= 1, ... ,n-2, I P (p,q) = (f1(p,q,O, ... ,izn-4(P,q,,)). Obviously, pi (p) = O(lr' (p)I N+I/2). Let X E Coo(IRI), 0::; X::; 1, x'(t) ~ 0 and X(t) = 0 in (-oo,e), X(t) = 1 in (27r -e ,00) for some e > 0 small enough. Denote by PIO: 1R2n -4 ----> 1R2n -4 the map Plo(p, q) = (p* ,q*) where p; = Pj cos(X (t)j (r' , 0) -qj sin(x(t)j(r', ')), 56 VALERY KOVACHEV AND GEORGI POPOV j = 1, ... ,n -2. Denote by p/the map p/(p) = X(t)pl(p) and set pI = PIO + p/. Now consider the map (4.7) F' (p) = (p(p) ,q(p)) = ps(p) 0 GO 0 g -s(p) (p). It is easy to see that F' is a well-defined COO map in a neighbourhood of &-' since P I = Id for t::; e and P I = P, for t ~ 21l -e . Next we define the smooth function rpn-l as follows. Denote T(t, p) = t + X(t)(21l -r(p)). We set for p E V a neighbourhood of &', rpn-l (p) = T(s(p) , g-S(P\p)). It is easy to see that rp n-l: V ---> 1" I is a well-defined COO function. Moreover, the map V 3 P ---> F(p) = (F' (p), rpn-l (p), rn_1 (p)) is a diffeomorphism for V small enough and Flvo = Glvo • Let (rp, r) = 1l (p , q , rp n-I ' r n _ I) be the respective polar coordinates Pj = J2S cos rpj' qj = J2S sin rpj' j = 1, ... ,n -2. , Denote r = (rl ' ... ,rn_2) . Now in polar coordinates (rp, r) we denote by VI the set VI = {p E V;Clrl(p)::; rj(p)::; C2r l (p),j = 2, ... ,n -2} where 0 < CI < 1 < C2 are some constants. Lemma 4.3. In VI we have I{rj , rk}(p)1 ::; Cjr, (p)1 N+I , j, k ::; n -1, ( 4.8) , N {rj,rpk}(p) = f5jk + O(lr I), j::; n -2,k::; n -1, {rn_ l , rpn-I}(P) = 1 + 0(1r'1). Proof. From (4.1), (4.2), and (4.7) it is easy to see that (4.9) It I ::; 41l. Indeed, (gl) * F' (p) = ps(p)+I 0 GO 0 g -s(p) (p) = ps(p)+I 0 p-s(p) 0 F' (p) = p;(P)+I 0 po-S(P) 0 F' (p) + 0(lr'IN+I/2) and (gt)*rn_1 = rn_1 which imply (4.9). Moreover, from the above equality it follows that for j, k::; n -2 (4.10) t * , N (g) rpj(p) = (X(s(p) + t) -X(s(p)))j(r(p)) + O(lr (p)1 ). INVARIANT TORI 57 Therefore, On the other hand, X'n_1 {s(p) , r) = 0(lr'(+1/2). Thus {s(p) , rj } = 0(lr'I N+I/2) for j ::::; n -2 and } -s(p) , N ~ , N {rj' 'Pk (p) = {rj' 'Pk}(g (p)) + O(lr 1 ) = ujk + O(lr 1 ). Now we shall prove that (4.11 ) T(p) = J(r(p)) + O(lr, 1N +I/2 ), PES, for some smooth function J such that J(O) = 2n. Let l E S, P = (0, r(p)) in polar coordinates. Denote by g;, t E HI , the Hamiltonian flow of rj , j = 1, ... ,n -1. Then PI = g~tt:.;) (l) E Sand r(PI) = r(po) + O(lr'IN+I) . Now taking Tj(l) = ctlj(r'(pO) ,,(po)), j = 1, ... ,n -2, from Proposition 4.1 we have (4.12) glT I(pO) o .. ·ognT_n1-2 (po) gnT(-pIO )( p 0) -_ p° + 0(1 r 'IN+I/2) . On the other hand, and as in [2] it is easy to see that (4.13) gI/ogkIk( P)=gkIk og/I (p)+O(lr(,p )1 N+I/2 ). Let PES, r(p) = r(po). Then g;I(P) 0 ••• 0 g~n~~(P)(po) = p for some tj(p) E HI and in view of (4.12), (4.13) we obtain glT dpo) 0 •.• 0 gnTn--22 (po) 0 g T(pO)p( ) = p + 0(Ir' IN+I/2) , thus gT(PO)(p) = PI + 0(lr'I N+I/2) where PI = g;~2-2(PO) 0 ... 0 g~TI(PO\p) E S or dist(gT(pO)(p) ,S) = 0(1r'I N+I/2). Therefore T(p) = T(pO) + 0(lr'I N+I/2) , po = (O,r(p)). Obviously T(pO) is a smooth function of V'I, ... ,Jrn-2 . Let J(r) be its Taylor's series up to order 2N. Then J(r) satisfies (4.11) and J(O) = 2n. On the other hand, consider 58 VALERY KOVACHEV AND GEORGI POPOV r(p) as a function of p, q and denote by r * (p , q) its Taylor series up to order 2N. Then * 2 2 2 2 r (p,q) =f((PI +ql)/2, ... ,(Pn-I +qn_I)/2) which implies tha,t f is a polynomial of rj = (pJ + qJ)/2, j = 1, ... ,n -2. The second estimate of (4.8) for k = n -1 follows from (4.11) which yields 'Pn-I (p) = T(s(p) , r(p)) + 0(lr'(+1/2). Here T(t, r) = t + X(t)(271 -f(r)). Thus X,'P n_ 1 = O(lr'IN) , j ~ n -2. 1 Finally, we have X'n_l 'Pn-I = 1 + O(lr'l) since f(O) = 271. It is easy to see that in polar coordinates ('P, I) we have j~n-l, where 'N+I ,N Icjk(p)1 ~ C1r 1 , Idjk(p)1 ~ C1r I, j ~ n -2, k ~ n -1, (4.14 ) , N+I ICn-I,k(P)1 ~ C1r 1 , k ~ n -1, Idn-I,k(P)1 ~ C, k ~ n -2, Idn-I,n-I (p)1 ~ C1r'l, uniformly in VI. Indeed, we have n-I ( a a ) 7l*F*X,/p) = L {rj' rk}(p) ar + {rj' 'Pk}(P) am k=1 k Tk and by Lemma 4.3 we prove (4.14). Let c E (0,~)n-2 X (-~ ,~) and Me = {r = c}. Then Me is a compact connected (n-2)-dimensional submanifold of Yo. Denote by Ie the embedding Me ~ Yo and by a the canonic symplectic I-form a = e dx in T* an, w = -da. Lemma 4.4. There exists P E 1\ I (V) such that p = 0 in a neighbourhood of S, (4.15) IIPllp ~ C1r'(p)I N+I/2 where II· lip is the norm in T;(T*aQ) andfor a l = a + P we have (4.16) d(I;(F*)-l al ) = 0 for any C E (0,~)n-2 X (-~ ,~). Moreover, ( 4.17) where { , } I is Poisson's bracket induced by WI = -d a I . INVARIANT TORI 59 Proof. We denote o 8 n-I 8 Zj = 8rp" + L djk(p) 8rp , J k=1 k n-I 8 L~ = L cjk(p) 8r' j 5, n -1, k=1 k and Zj = n-;I Z:' L j = n-;I LJ. Then the vectors Zj are tangential to Me and IL)p 5, Clc'IN+I/2 for any p E Me. Here 1·lp is the norm in Tp(T*"8n). Thus we have ( 4.18) * * -I * *-1 -d(le (F) a)(Zj' Zk) = Ie (F) w(Zj' Zk) = w(X'j ,X,)I,=c + O(lc'IN+I) = O(lc'IN+I). We write (F*)-Ia = pdq + td' + dQ + p where P is a I-form with COO coefficients. Moreover, from (4.18) we have (4.19) d(le* P-)(Zj ,Zk) = d(le* (F*) -I a)(Zj ,Zk) + O(lc , IN +I ) = O(lc, IN +I ). On the other hand, where n-2 P = L(ajdPj + bj dq) + aod, + bodt j=1 and aj , bj are COO functions. We expand aj , bj , j = 1, ... , n -2, in Taylor's series up to degree 2N with respect to P and q around the point P = q = ° and bo up to degree 2N + 1. We obtain P = PI + P2 where PI = O(lr'IN+I) is the remainder in Taylor's formula. Then from (4.19) we have ( 4.20) n-I n-I d(I;P2 ) = LLPjk (rp',t,y'C7, ... ,Jcn_ 2 ,cn_ l )drpj" drpk j=1 k=1 where Pjk are polynomials of .jC(, ... ,Jcn -2 of degree at most 2N + 1 . In view of (4.20), the definition of Z: and (4.14) we obtain successively that all coefficients of the polynomials Pjk are equal to zero. Hence n-2 C E (0,l5) x (-l5 ,l5). We denote P = -F* PI. Then WI = -dal , a l = a + p, is closed on Mc for any c. \ Denote by X~ the Hamilton vector field of the function f with respect to the symplectic form WI. Let us fix c E (0,l5)n-2 x (-l5 ,l5), P E Mc. Then I wI(X'j,L) = (drj,L) =0, VLE TpMc. 60 VALERY KOVACHEV AND GEORGI POPOV Since TpMe is Lagrangian with respect to WI' we obtain Xr/p) E TpMe. Hence I I {rj , rk}1 (p) = WI (Xrj ' Xr) = ° which proves the lemma. Let P E V. As in [2], consider the map IRn -1 3 t = (tl ' ... ,tn_I) --> TI(p) = F:1 0··· 0 F:~i (p) E V where Flj is the flow of the Hamilton vector field Xl . Let us note that Flj J rj J commute with one another in view of (4.17). Since Me is compact, the stationary group of t --> FI(p) has n -1 generators el , ... ,en_I which can be found explicitly in our case. In a neighbourhood Vo of S we have WI = w, hence X:j = 8/8qJj where (qJ, r) = n(p, q, qJn-1 ,rn_ l ) are the respective polar coordinates. Then in Vo we have for j ~ n -2 I F/(qJ,r) = (qJl' ... ,qJj_1 ,qJj +tj,qJj+I' ... ,qJn-1 ,r). In particular, Fjj: S --> S for j ~ n -2 and ej = ( 0, ... ,0 ,2n, 0, ... , ° ), ~ ~ j~n-2, j-I times n-j-I times is a stationary point for F' (p). Let tn_I (r) be the smallest positive time such ~n 0 0 that Fn"--11(P )ES for p =(O,r)ES, r=(rl' ... rn_ I ). Then F:~i(r)(/) = (RI(r), ... ,Rn_2(r),0,r) = FIR 1(r) 0···0 FRn-n2-2 (r)( p0). Therefore en_l(r) = R(r) = (RI(r), ... ,Rn_l(r)), Rn_l(r) = tn_l(r), is a stationary point of FI, i.e. Fen-l (r) (p) = p for any p E Mr . Moreover, Rj(r) are smooth functions of (p,q,qJn-1 ,rn_ l ) and since they do not depend on qJ j ' we obtain that R j are smooth functions of r in [0,<5t-2 x [-<5,<5]. Now, for (t, r) E -rn -I x [0, <5t-2 X [-<5, <5] we set GI (t, r) = F:"--iRn-1(r)(t1 + tn_I RI (r), ... ,tn_2 + tn_ IRn_2(r) , 0, r). Thus we obtain a smooth map n-I GIn in the coordinates (p, q , tn_I' r n-I) : -I G ( ) Fln-IRn-l(r)(M(p) ° ) n In p,q,tn_1 ,rn_1 = n-I q , ,rn_1 where n is the respective polar change of the coordinates, M is the blockdiagonal matrix M = diag(MI ' ... ,Mn _ 2 ), M. = (C~S(tn_IR) -Sin(tn_IR)) J sm(tn_IR) cos(tn_I R) , j = 1, ... ,n -2. INVARIANT TORI 61 Since the vectors ej E IRn -1 , j = 1, ... , n-l , are generators of the stationary group of Rn -I 3 t ----FI(p), PES, we obtain that G1 is a diffeomorphism of ~ onto a neighbourhood Vo of iff in T* on for t5 small enough. As in [3], we can construct some polar symplectic coordinates ('P, l) with respect to WI in a neighbourhood of {r = O} starting from (t, r) which will give us the smooth exact symplectic transformation G: ~ ----Vo in Proposition 4.2. Unlike [3], we work in a neighbourhood of {r = O} and not near a fixed torus and are interested in the smoothness of the respective maps near {r = O}. First we shall show that besides the equalities {rj , rkh = 0, j, k :::; n -1, we have {rj ,tk}I=t5jk , j:::;n-2,k:::;n-l, (4.21 ) {rn_l,t)1 =-Rj(r)/Rn_l(r), j:::;n-2, {rn_1 ,tn-I}I = I/Rn_ l (r) and (4.22) Equalities (4.21) follow from F;GI(t,r)=GI(t+sej,r), j:::;n-2, F:_ I GI (t, r) = GI (tl -sR I /Rn_1 ' ... , tn_2 -sRn_2/Rn_1 ' tn_1 + s/Rn_1 ' r). We shall prove equalities (4.22). From Jacobi's identity we have X;;{tj,tk}I=O, j,k:::;n-l,i:::;n-2, XrnI _1{ tj , tkh = cjk(r) where cjk are smooth functions of r E [0, t5)n-2 x (-t5 , t5) . Hence {tj , tk}1 (FnR:,l,(r)(p)) = Rn_1( r)cjk(r) + {tj , tkh (p), pES. On the other hand, {tj' tk}1 (p) = {tj' tkh (Fen-I (p)) = {tj' tkh (FnR:"l,(r)(p)) = Rn_1( r)cjk(r) + {tj' tkh (p), pES. Thus cjk(r) = 0, j ,k:::; n -1. Moreover, in Vo {tj,ln_I}(P) = {'Pj -'Pn_IRRj ,~n-I} n-I n-I ( oRj ORn_l ) 1 ='Pn-I ~-~ T n-I ) n-I and since X;n_1 = %'Pn_1 in Vo' we have {tj' tn_l}(p) = 0 for p E Vo and therefore for any p E V. Analogously we obtain {tj' tk}(p) = 0, P E V, j, k :::; n -2, which proves (4.22). 62 VALERY KOVACHEV AND GEORGI POPOV We put G2(t, r) = ('P,I) where 'Pj = tj' j ~ n -1, Ij = rj , j ~ n -2, rn-1 = In_ 1 + Q(l), Q(O,to) = -to' For the function Q(I) we obtain the equations R .(r) {In-I' 'P)l = -R ] (r) -{Q(I) , t)l n-l R .(r) n-l 8Q = -R ] (r) -L 81 {Ik' tj } n-l k=l k R.(r) 8Q 8Q = -R ] (r) -81 -~{In-l,'P)I' j~n-2, n-l ] n-l 1 8Q {In-l ,'Pn-1}1 = R (r) -~{In-l ,'Pn-l}l' n-l n-l That is why we put 8Q(I)/8Ij = Pj(1) , j = 1, ... ,n -1, where Pj(I) = -Rj(r)/Rn_1(r) for j ~ n -2 and Pn-1(1) = 1/Rn_ 1(r) -1, r = (I' ,In-l + Q(I)) . These equations are solvable in a neighbourhood of {r = O} since Pj are smooth functions and 8Pj /8Ik = 8Pd8Ij' j, k = 1, ... ,n -1 (cf. [3]). Indeed, we have 8Pj 8Pk 8Ik = -{tk' {rn-1, t)l}l = -{tj , {rn_ 1, tk}l}l = 8Ij ' Then the map 7r -1 G 2 G 1 7r is a diffeomorphism in Vo. Lemma 4.5. There exists a diffeomorphism G 3: UJ ---+ U such that G; W 1 = W and G3 = Id +G4 where IG4 (p)1 ~ Clr' (p)IN+l/2 . Proof. Following Moser [16], consider at = a + tP, t E [0,1] and WI = W -tdp. From (4.15) it follows that wt is a nondegenerate 2-form in U if U is small enough. Let ~ be a vector field determined by W t -.l ~ = P and /p) = exp(t~)(p). Here -.l denotes the inner product. Then from (4.15) we obtain lt(p) -pi ~ Clr' (p)I N+1/2 , Moreover, we have O~t~l. dd/cf>t) * wt = (t) * (-dP + d (wt --lI ~)) = 0, hence G 3 = 1 is the diffeomorphism we seek. Denote G = G37r-1G2GI 7r. Then G(Vo) = U is a neighbourhood of & in T*8n and n-2 G* W = L dpj 1\ dqj + dln_ 1 1\ d'Pn_ 1. j=1 It remam. s to prove (4.6). Denote Pj = FjR J (Pj_l)' 1 ~ j ~ n, where Po E S has polar coordinates (0, r(I)), r(I) = (r' ,toO = (I' ,to((I))· Then Pn = Po INVARIANT TORI 63 and l' = U;:-/l'j' where l'j = {Fj(Pj_I) ' 0::; t::; Rj_l(r)} , is a cycle on M,. Since the form a -id rp is closed in Vo' we have (4.23) l a -lldrp = la -2nln_ 1 = c = const. y y y Therefore, for (rp, l) E nn-I (4.24) 2nln_1 = 1a -C = 1a l + 1P -C = 1a l + 0(11 'NI +l/2 ) -c. y y y y On the other hand, Pj=(RI(r), ... ,Rj(r), 0, ... ,0, r) and l'j = {(RI(r), ... ,Rj_l(r),tRj(r),O, ... ,0,r);0::; t::; I} for j = 1 , ... ,n -1 . Denote L j = {(RI(sI' ,to'O, ... ,Rj_l(sI' ,toC), tRisl' ,toO'O, ... ,o,sl' ,toC);O::; s::; 1,0::; t::; I}, j::; n -2, L n_1 = {F:_ I (RI (sl' ,toC), ... ,Rn_ 1 (sl' ,toO, sl' ,toO; 0::; s ::; 1,0::; t ::; Rn_ 1 (r)) and 2n-3 times Note that l', belongs to the cylinder of orbits of X/o,. Now we have Moreover, L n _ 1 is isotropic so that fLn _ 1 WI = 0 while n-2 n-2 n-2 n-2 I f; lj WI = f; t; 1, dlk /\ drpk = f; In IjRisl' , toC) ds n-2 I = f; In I j ;~ (sl' ,C) ds + 0(II'I N +I/2) = A (I' ,C) + 0(II'I N +I/2) since A(O, C) = O. Finally, on the cylinder of orbits L = {FLI (0, ... ,0, h); o ::; h ::; n we have 1 al = 1 a l +/WI = 27rto+/dh/\ dt ;', 1'0 L L r' rRn-1(0.() r' = 2nto -10 10 dt /\ dh = 2nto -10 /(h) dh. 64 VALERY KOVACHEV AND GEORGI POPOV Note that for I' = 0, In _ 1 = to (4.24) yields C = 0 in (4.23), thus G is exact symplectic and 277:In_ 1 = 277:to -10' I(h) dh -A(/', 0 + 0(II'IN+1/2 ). Now if (0 (/) is a solution of ( 4.4) 277:In_ 1 = 277:to -1ro' 1° ( h)dh-A °( I, , (),((O,to)=O, where 10(() and A°(/', 0 are Taylor's expansions of 1(0 and A(/', () with respect to ( up to degrees M and M + 1 respectively, M large enough, we obtain which proves (4.6). From the representation of the billiard ball map by means of the interpolating Hamiltonian ( and from (4.6) we obtain B(rp,I) = exp( -.JeX'o)((rp,I) + R(rp ,I)) in Va n {( ~ O} where R(rp,I) = (0(II'I N -1/2) + 0((:+2), 0(II'IN+1/2 ) + 0((:+2)) and R( rp ,/) = R 1 (rp , v? ,v'c) where R 1 is a smooth function of its arguments for (rp, I) E Tn-1 x IDn-1 • Here we have denoted IDn -1 = {I E IRn-1 ; 0:::; I j < J, j = 1, ... ,n -2, IIn _ 1 -tol < J, (0(/) > 0,11'1:::; C(o(/)} , where C > 1 is such that "(rp,I) -(0(/)1 :::; t((0(/))M+2 for I E IDn-1 and rp E Tn -1 (here we have used (4.5)). Then we have (4.25) for (rp, I) E Tn-1 x IDn-1 where g(rp ,I) = (0(II'IN -1/2 ) ,0(II'IN+1/2 )) + O( (:+2) and g is symplectic in Tn-1 x IDn -1 • 4.3. Denote (4.26) ID an -l = {I E IR n-l , C1a :::; Ij :::; C2a, ]. = 1 , ... ,n -2, 2b 2b C3a :::; (0(/) :::; C4a } where 0 < C1 < 1 < C2 , 0 < C3 < 1 < C4 , 0 < b < 1/2 small enough. Henceforth Cj will be constants which do not depend on a but they will INVARIANT TORI 65 depend on the concrete circumstances. Note that for a ~ ao' ao small enough, we have 0:-1 cOn-I. We shall denote A:-I = lrn -I x 0:-1 . Denote for 0 c IRm , p > 0 , (4.27) 0 + p = {z E em; dist(z ,0) ~ p} and set H°(l) = -1('0(1))3/2, H(ffJ,I) = -1('(ffJ ,1))3/2. Proposition 4.6. For ao small enough we have (4.28) II HI°l ( I) II ,1I(H°Il (I)) -I II ~ CI, a -b , n-I IEOa +p, (4.29) for each a E (0, ao)' 0 < p ~ a/2, where II· II is the norm of the respective (n -1) x (n -I)-matrix. Moreover, the map n-I ° [) a + p :3 I --> HI (I) is a diffeomorphism. Proof. We have H~(I) = _,~/2 grad '0(1) . But ° 0, -I ° , grad '0(1) = -(/('0) + A,(I ''0)) (All (I ''0), -1), thus HI° = 'r0l0/2([0 ('0r0 ) +A0,(I, ,'0r 0)) -I (Al°l ( I , ,'0r0) ,-1 ). Set K = '0(/°('0) + A~(I', (0))-2. We have 8K(0)/8,0 =I-0 and this allows us to regard the map I --> H~(I) as a composition of the following three maps: , , , ° (I ,In_I) --> (I ,(0) --> (I ,K) --> HI (I). This gives us the representation H O I _ 8H~ 8(I',K) 8(1','0) Il( ) -8(1' ,K) 8(1', '0) 8(1' ,In-I)' The first two maps are diffeomorphisms in a sufficiently small neighbourhood of & and for the respective functional matrices we have 8(1' ,K) _ (En_ 2 On_2) 8(1' ,(0) -KII 8K/8,0 8(1' ''0) _ (En_2 On_2) 8(1' ,In-I) -gradII '0 8'0/8In_ 1 where En-2 is the unit (n-2)-matrixand 0n-2 is the zero (n-2)-dimensional column matrix. Thus it remains to consider the matrix 8H~ /8(1' ,K) for IE 0;-1 + p , i.e. when the variables (I', K) run over a set of the form [C-I a, C-2a] n-2 x [C-3a 2b ,C-4a 2b ] + C-5P. 66 VALERY KOVACHEV AND GEORGI POPOV We have H~ = KI/2(A~,(/' ,(0), -1) where '0 = KAI(I' ,K) and the function A I (I' ,K) satisfies A 1(0,0) i-O. Then for the entries of the matrix oH[° 1 0(1, , K) we have A.k = KI/2 (0 2 A°(l' ''0) + KoA°(l' ''0) . OAI (I' ,K)) j o~o4 o~ 04 for 1 ::; j, k::; n -2 , A = !K-I/20A°(l',,0) KI/202A°(l',,0) (AI(I' K)+KOAI(/',K)) j,n-I 2 01 + 01.0' ' oK j j for j = 1 , ... ,n -2, An_I k = 0 for k = 1, .. , ,n -2, A n_ 1 n-I = _~K-I/2 No~ it is easy to see that the validity of Proposition 4.6 follows from the invertibility of the matrix A~, [' (I' ,C) for I', , small enough which is a consequence of the nondegeneracy assumption (2.3). This completes the proof of Proposition 4.6. 5. MAIN THEOREM Denote Qa = {w = H~ (I); I E O;-I} and let the numbers (J and y be such that (J > n -1 and y > 0 . Then denote (5.1) Q~ = {w E Q a ;I(w,k') -knl ~ ylkl-a for any k = (k' ,kn) E In \ {O}}. The next lemma shows that the family of invariant tori we are going to obtain has a positive Lebesgue measure. Lemma 5.1. Let y < (:6al+1b, (:6> 0, I> O. ThenJor a small enough we have y lb n-2+b mesQa~(l-cla )c2a . Proof. By analytic-geometrical arguments we obtain mes Qa ~ (:7an-2+b . Then by arguments close to those in [1] we find mes(Qa \ Q~) ::; (:sya n -3+b where (:8 depends only on nand (J. This yields the desired inequality. Denote by (Jy(rp,/) the map ('1''/) --> (rp,yI) and by 1I'lIs,p-1xn' s > 0, the respective Holder norms of the functions in Tn-I X Q as well as IIJllsY-lxn;y = IIJ 0 (J)ls,a;:-1(p-1xn) (see [19]). Let Csa l+1b < Yo < C6al+1b , 1 < Cs < C6' I> 0, and denote y = yoa3b . Theorem 1 follows from the following theorem. Theorem 5.2. Let N ~ 2, 0 < a < ao where ao is small enough. Then there exists an exact symplectic diffeomorphism U: A;-I --> A;-I and a Junction K, INVARIANT TORI 67 K (I) > 0 in 0;-1 , of class COO such that (5.2) V-I BV((jJ,I) = ((jJ + r/(I),I) , r(l) = -iK(l)3/2 , for any (jJ E Tn-I, I E O;~~ = {I E 0;-1; r/(l) E n~O}. Moreover, (5.3) 11K -r II _ < C N-I/2-(/+2)b "0 p ,D~ l;y _ pa and the generating function S((), I) of V satisfies (5.4) In order to prove Theorem 5.2 we reduce the problem to finding invariant tori for the flow of a suitable Hamiltonian. For this purpose we use some arguments from [4, 5]. Now we shall use the fact that the map Id + g from (4.25) is exact symplectic. This is a consequence of (4.25) and the following assertion proved in [6]: Proposition 5.3. The billiard ball map B is exact symplectic in ~. Let ", E COO (IR I), ", = 0 in a neighbourhood of 0, ", = 1 in a neighbourhood of 27l' and let (jJt be exact symplectic with a generating function ",(t)S where S generates (jJ21C = Id + g. Denote by Bt the exact symplectic map Bt = exp(tXHO ) 0 (jJt' t E IRI . Let ~t = dBt/dt 0 (Bt)-I be the respective vector field which is well defined for t E TI since ~t = X HO in some neighbourhoods of t = 0 and t = 27l'. Following Douady [4], we obtain wn_ 1 -.J ~t = dht for a suitable COO function ht' t E TI . Moreover, using (4.25) we have (5.5) ht((jJ, I) = H°(l) + Fa L (J')"/2Q" (t, (jJ, ji;, ... ,J I n_2, Fa) j"j=2N+I + ,:+S/2Q, (t, (jJ, ji;, ... ,Jln-2, Fa) for ((jJ, I) E A;-I where Q", Q' are smooth functions of (t, (jJ , ..;r;, ... , Jln-2'~)' Denote y' = (jJ, y = (y' 'Yn)' ",' = I, ", = (",', "'n) where the variable "'n runs over a neighbourhood of 0 in IRI and Yn E TI . Set -0 0 ' -, , (5.6) H (",) = H (", ) + "'n' H(y,,,,) = hYn (y ,,,, ) + "'n' Denote by (F)tt EIRI the fl owo f Xii and let A' ={(y,,,,)EAan; -H(y,,,,)=O, y n = O} where A; = Tn X 0; , 0: = {", E IRn ; C7 a :::; "'j :::; Cga, j = 1 , ... ,n -2, 2b , 2b C9a :::;'o(",):::;ClOa ,1"'nl:::;CII } and Cj are some positive constants such that C7 < 1 < Cg , C9 < 1 < CIO • It is easy to see that A' has the form A, ={(y, ,0,,,,, , -H0 (",, ));(y " ,,,,)EAan -I }. 68 VALERY KOVACHEV AND GEORGI POPOV Denote by 1: Aan -I -+ A the map " " " , , 0 , 1 (y ,r, ) = (y ,0, r, , -ho (y ,r, )) = (y ,0, r, , -H (r, )). It is easy to see that F 2rr : A' -+ A' satisfies (5.7) 1 -I 0 F 2rr 0 1 = B. , ~o ~o 2 ~ ~2 Denote Ho = H + (H) , H' = H + H . Now, from Proposition 4.6 we obtain (5.8) 11 8;;; II, (a;;;f <; Ca -b ;n D:+ p where 0 < a S ao' b > 0 small enough are fixed and p S a12. Moreover, in A; we have (5.9) , , , , 0 , ~ 0, H (y, r,) -Ho(r,) = (hyJY ,r, ) -H (r, ))(1 + H(y, r,) + H (r,)) = ab O(Ir,"(+1/2) where r," = (r,1 ... ,r,n-2) which follows from (5.5) for M large enough. Introduce the sets 0' = {w = grad Ho' (r,) ; r, E [)n} and 0' = {w E 0' ; a YI l(w,k)1 ~ 111kl-a for any k E lLn \ {O}} where a is as above and II = Cia -3b = Cio' 0 < C < I . We reduce the proof of Theorem 5.2 to Theorem 5.4. Let H~ (r,) be an analytic function in [); + p such that (5,10) II ~:;t,+p' (a;~ f ",+p <; R, R 2 I, and let 8 H~I 8 r,: [); + p -+ en be invertible on [); + p . Then for any fixed A > a + 1 > n, a > 1, a i A = {i I A + j; i, j ~ 0 integer} there exists c> S C1R -3 such that if H' E COO (A;) and (5.11) RIIH " -Holls,AZ;/1 S/IC2 >' s=aA+A+a, 0O where J = a4b , R = Ca-b , N ~ 2 and 0 < b < 1/(41 + 8). Thus Theorem 5.4 holds for H' and H~ and from (ii) we obtain (5.13) II~SII < C N-l/2-(p+l+l)b C 0 p,A~ ;/1 -pa lor any p > . Note that Sy'l is small enough if ao is small enough and N ~ 2 since in view of (5.13) we have ISY'l1 ~ Ca N -3/2-(2+21)b in A:. Then S generates a canonic map T, i.e. graph T = {(Y -S'I(Y, 11),11; Y, 11-Sy(Y, 11»)}. Now we shall concentrate our efforts on the construction of the function K and the symplectic map U of Theorem 5.2 starting from the nongenerate Hamiltonian K' and the symplectic transformation T. First we shall need some preliminaries. ~ ~ 2 I Let us denote Kl (y, 11) = H(T(y, 11». Then Kl + K( = K and K( does not depend on y for any 11 EO: 'I' Set K2 (11) = Kl (0,11)· The manifold ~o = {(y, 11) E A:; H(y, 11) = O} is contained in {(y, 11) E A: ; H' (y, 11) = O}. Moreover, if A'I n ~o =I-0 for some invariant torus A'I T(Tn x {11}), then A'I C ~o and FII A., = exp(tXH')IA~ where FI(p) = exp(tXH)(p) = (/(p), 111(P». ~ ~ 1 n Denote ~ = T-(~o) = {(y, 11) E Aa ;Kl (y, 11) = O}. Since T is a canonic map, we have (5.14) F I (T(Y,11» = T(Y + tgradK2 (11),11) for (y, 11) E An 'I n f, i.e. when y E Tn , 11 E 0an , and K2 (11) = O. We write a" I ,II down T(y,l1) = (P(Y,l1), Q(y,l1». From (5.6) we have Yn = 8H/811n = 1, thus y~ = y~ + t and we have 1 (5.15) Pn(y, 11) + t = Pn(y + tgradK2 (11), 11) for any t E R . As in [4] we conclude that gradI'Pn(Y' 11) does not depend on y E Tn if 11 E 0:,/1 and K2(11) = O. Therefoie, Pn(y, 11) = Yn + f(l1) for some function f on 0: ';'1 • On the other hand, 70 VALERY KOVACHEV AND GEORGI POPOV thus ~ ~ -S~n (0, rO + !(rt) = Pn ( -S~(o, rt), rt) = ° and !(rt) = 5 (0, rt). From (5.15) we obtain ~n 8K Yn + !(rt) + t = Yn + t~(rt) + !(rt) , rtn thus ( 5.16) Denote g(rt) = 5~(0, rt), rt E [): ; then the map Tl (y, rt) = (y -g(rt) , rt) is symplectic with a generating function 5(0, rt). Moreover, the map To Tl is generated by the function 5(Y, rt) -5(0, rt) which satisfies (ii) and iI(T(TI (y, rt))) = K 2(rt) for rt E [): ,/1 . Thus we can suppose that ( 5.17) for (y, rt) E An " n f and y small enough. a,,1 n Then Theorem 5.4(i), (5.14) and (5.17) yield Pn(y' ,O,rt) = ° and (5.18) 2rr ~ , ~ , F (T(y ,O,rt)) = T(y +grad~,K2(rt),2n,rt) for (y', 0, rt) E f n A:,/I . We turn now to the construction of the function K(I) in Theorem 5.2. First we prove that (5.19) 11K2 -H~oll P'[)~;/2<-CP aN -I /2-(l+I)b f oranyp_> °' Y2=y1ab. Indeed, we have ~O ~ ~ ~o ~ K 2 (rt) -H (rt) = (H(O, rt -Sy(O, rt)) -H (rt -Sy(O, rt))) ~o ~ ~o + (H (rt -Sy(O, rt)) -H (rt))· We evaluate the first addend by (5.12) and (5.l3) using the following estimate for the norm of a composite function FoG: p-l (5.20) IIF 0 Gllp ,II ::; 1IFIlp ,/1 L cp )IDGII~~y~ , P ? 1 , ,,=0 where DG is the matrix of the first derivatives of G and C I = 1 . For the p,psake of simplicity we have dropped out the dependence of the above norms on the domain [):. For the second term we have iIo (rt -5 y (0, rt)) -j{0 (rt) = -( 5 y (0, rt), 10 I (grad j{o) (rt -t5y (0, rt)) d t ) INVARIANT TORI 71 whose (p, yl)-norm can be estimated by CpaP , p = N -1/2 -(p + I + l)b, using again (5.13) and (5.20). This proves (5.19). Now, for N ~ 2, we can use the implicit function theorem to solve the equation K2 (1'1) = ° with respect to l1n' We find a smooth function r( 11') choosing the constants Cj in the definition of 10:-1 and 10: appropriately so { , , that the set LI = 11 ;K2 (11) = o} can be represented as LI = {(11 , -r(l1)); 11' E IO:-I } . Moreover, making use of (5.19) and (5.20), we find (5.21 ) Il r + 13: .,30/211 p ,[)~ _1 ;Y2 <_ Cp a N-I/2-(/+I)b , p ~ 0. Thus r < ° in 10:-1 and the function K(l1') = (-~r(l1'))2j3 is smooth and positive in 10:-1 • We shall prove that K satisfies the requirements of Theorem 5.2. First, in view of (5.21) and the equality K -'0 = '0((1 -~,~3/2 R)2/3 -1), R = r + t,;/2, we obtain ( 5.22) 11K -'IoI p ,[)~ _1 ;/2 _< Cp a N-I/2-(/+2+2p)b which yields (5.3) for Y = y2a2b = y1a3b . It remains to construct the symplectic diffeomorphism U of Theorem 5.2. We shall write down explicitly the generating function of U starting from S. Lemma 5.5. Suppose that 11' E IO:~~ = {11' E 10:-1 ; grad r(l1') E Qyo}' Then (11', -r(l1')) E 10:,/1 and gradK2(l1', -r(l1')) = (grad r(l1'), 1) for ao small enough. Proof. We have (gradK2)(l1', -r(l1')) = (grad r(l1'), 1)(8K21811n)(l1', -r(l1')) which yields (gradK2)(l1', -r(l1')) E Q~I since Y1 < Yo and 18K21811n -11:::; Ca N-3/2-(2/+3)b. Therefore (gradK')(l1' , -r(l1')) = (gradK2)(l1' , -r(l1')) E Q~I and (8K21811n)(l1', -r(l1')) = I in view of (5.16) which proves the claim. Now using (5.17), we obtain T(y' ,0,11', -r(l1')) E A' for any 11' E [):~I, ,/0 y' E Tn-I where A' = Lo n {Yn = o} has the form , , , 0, " n-I A = {(y , °, 1 1 , -H (11 )) ; (y , 11 ) E A a } • Moreover, F 2rr = IBI-1 : A' -+ A' and from (5.18) and Lemma 5.5 we have -I -, , , -, , , , (5.23) IBI (T(y, 0,11 , -r(l1 ))) = T(y + grad r(l1 ), 2n, 11 , -r(l1 )) , n-I d n-I for any 11 E lOa,yO an YET . 72 VALERY KOVACHEV AND GEORGI POPOV Denote by 'I the immersion 'I (y' ,,,') = (y' ,0,,,' , -.(,,'» E Tn X 1:1 ' (y' ,,,') E A:-I , and set S(y', ,,') = S(Y', 0,,,', -.(,,'». This function satisfies (5.4) in view of (5.13) and (5.21). Denote by U the symplectic map generated by S. Then U maps Aan -I m. to a set 0 f t h e same type. M oreover, Uy(,",,) =, -I 0 T-0'1 (Y' ,,,' ) for ,,' E D:~,,,I0 since (oS/o"n)(Y'"'' -.(,,')) = 0 for ,,' E Oan ,Y-oI in view of (5.17) and Lemma 5.5. Now (5.2) follows directly from (5.23). This completes the proof of Theorem 5.2. Remark 5.1. Instead of K 2(,,) we can use any function K2(,,) = K 2(,,) + g(,,) such that g(,,) = 0 on O:,YI and (5.24) II g II p,y <_ Cp a N-I/2-(I+I)b in IlLnJIna • Then the corresponding solution r(,,') of the equation K2 (,,', "n) = 0 with respect to "n equaI s .("' ) l£"o r any " , E Da n ,-YoI and t h e sets 0 an ,-YIo and D-na -,YIo defined by • and r coincide. Indeed, writing down f(,,') = .(,,') + r(,,'), K 2(,,) = ("n + .(,,'»h(rr) with some function h (,,) =F 0 in a neighbourhood of D: and K2(rr) = (rrn + .(,,') + gl (,,»h(,,) , gl (,,) = g(,,)/h(,,) , we have The function g2 satisfies the estimate (5.25) II g2 II p ,Y2 <-Cp aN-I/2-(I+I)b in view of (5.21) and (5.24). Then (,,' ,r(,,'» E D: for rr' E 0:-1 and by Lemma 5.5 we obtain the equality (5.26) r(,,') = g2('" ,r(,,'» -g2(rr' ,0) = r(,,')v(,,') £l"o r any rr ' E Dan ,-YIo ' Here ( ') -1V" -1 aog 2"(' ,sr( "' »d s -_ O( a N-3/2-(2/+3)b) , o "n thus Iv(,,')1 < 1 for 0 < a ::; ao and ao small enough. Now (5.26) implies r(,,') = 0 on D:~~. Since O:~~ has no isolated points, we have grad r(,,') = grad.(,,') for ,,' E D:~~. On the other hand, the map 0:-1 3,,' ----gradr(,,') is injective in view of Proposition 4.6 and (5.21) which proves the relation Dn-I = 5n-' . a,yo a,yo Proof of Theorem 1. Set aj = qj, 0 < q < ao, Dj = D:j-I , Aj = Tn-I X Dj , D., = Dn-,~, yO = yo(a) = C5a J +1h and let S((),I) be the function S((),I) J ,/a) " J J J J INVARIANT TORI 73 defined by Theorem 5.2 for a = aj • Choose r' of the same type as r, r c r' c U~I OJ' r being defined by (2.4) with some constants c2, k = 1, ... ,4, while OJ are defined by (4.26) with some constants Ck ' k = 1 , ... ,4. Denote by O~ the set (4.26) with constants C2 and a = aj • We can suppose that OJ n 0j+v = 0 for any j and v ~ 2 and that O~ n O~+l has a nonempty interior. Proposition 5.6. There exist functions K j , Sj defined by Theorem 5.2 in Aj with yJ = C5a;+/b and such that (5.27) ( 5.28) o for any I E 0j.y n 0j+1 and y ~ Yj . Proof. Denote by Tj the corresponding canonical transformation defined by Theorem 5.4. From Remark 1 after Poschel's Theorem A [19] it follows that -1 -n n Tj-o Tj+l(y,tl) = (y+g(11),11) for 11 E 0aj,r]noaj+l . Suppose that the exact symplectic map T = Tj-I 0 Tj +1 is defined by the generating function R(Y, tl). Then grady R(Y, 11) does not depend on Y E Tn for 11 E O].yno]+1 ' y ~ Y; and since R(Y, 11) is small enough, we obtain R(Y, 11) = R(tl). Thus Tj+1( y, 11) = Tj(Y + grad R~), 11), 11 E 0].1 n 0]+1' _ Moreover, the function Sj (Y , 11) + R( 11) generates the map Tj 0 (Id + grad R , Id) . Let 'P j = 'P j (y , 11) solve the equation 'P-8Sj ('P,11)_8R(11)=Y 811 811 and let 'Pj+1 = 'Pj+1 (y, 11) solve 'P -8Sj+1( 'P, 11)18 11 = y. Then ('Pj+1 (y, 11),11-(8Sj+,18Y)('Pj+, (y, 11),11)) = Tj+ 1 (y , 11) = ('P j (y , 11) , 11 -(8 Sj 18 Y) ('P iY , 11) , 11)) and we have --(5.29) 'Pj+1 = 'P j , grady (Sj+ 1 (Y, 11) -S(Y, 11)) = ° for 11 E O]'f n 0]+ 1 • As in the proof of Theorem 5.2 we have --KI(Y' 11) = H(Y, 11-(8Sj /8Y)(Y, 11)) .j and --K2 ·(11) = H(O,11-(8Sj /8Y)(O,11)) . • j 74 VALERY KOVACHEV AND GEORGI POPOV In view of (5.29) we have K2 ,), (11) = K2 ,)' +1 (11), 11 E On) ,r n On) + l . Now instead of K2 )'+1 (11) we consider K2 '+1 (11) = K2 '+1 (11) + g(11) where g(11) = , ,j ,j K 2)11) -K2,j+1 (11) in O~ n D~+I ' g(11) = 0 on D~+I ,J' U (O~+I n D~+2) and II II < C aN -1/2-(/+I)b. g p,Dj+1,r -p )+1 According to Remark 5.1 we can replace K2 '+1 by K2 '+1 + g. Thus we can ,) ,) suppose that K2 ,) '+1 = K2 ,)' on On) n) On+ 1 . Then.) ' +1 (I) = •) , (I) in [)), ,r n 10 )'+ 1 and so in IDj n IDj +1 . Therefore, Sj+1 (0, I) = 8j+1( 0,0, /, -'j+1 (I)) = 8)0,0,1, -'j(l)) = Sj(O,l) for /E 10 j ,y n [) j+ I' Y ~ YJ and any 0 E Tn-I . This completes the proof of Proposition 5.6. Remark 5.2. Making use of the representation (5.5) and approximating the smooth functions Q", Q' with real-analytic ones, we can choose the sequence of analytic functions approximating the smooth Hamiltonian in the proof of Poschel's Theorem B [19] independent of the domain. Thus equalities (5.27) and (5.28) may be assumed fulfilled by construction for any /E OJ nOj +1 and OETn -l . Now we shall prove Theorem 1. We can find smooth functions 'Pj E C:(O) such that 2:':1 'Pj(I) = 1 for /E r' and 'Pj = 1 on O~, (5.30) IDP'P) ~ Cpa;IPI. Denote S(O,l) = 2:':1 'Pj(,])Sj(O,l) and by K(I) the smooth function K such that K(I) = Kj(I) in IDj' .(I) = _~(K(I))3/2. Then S(O,l) = Sj(O,l) in 0), ,,0 • For J.I.. > 0 and a > n -1 denote by r the set "J p r k I+lb -(J p = {/E r; I (grad '(I), } -knl ~ J.l..11 Ikl for any k = (k' ,kn ) E In \ {O}}, E = fp = rp U {(O, to)}' We choose C5 in the definition of Yj so that rp n 10 j C 10 j,y &l'o r any Y ~ Yj0_ l . Th"I S IS satl.s f i e d'1f J.I.. (Cl aj )I+lb ~ C 5aI+j_lbI ' 1. .e. C5 ~ J.I..(C1q)l+lb. Then we have (5.30) and choosing b < 1/(4/+ 8) in an appropriate way, we can prove the desired smoothness of K, g and U. Now suppose that N ~ 3. From estimate (5.3) we obtain 11K -'II ' < C aN -5/2-(31+7)b ) 0 2,DJ -2 and the right-hand side can be estimated by C2ab if b < 1/(6/+ 16). In particular, (5.31 ) INVARIANT TORI 75 Let /E r \ r,u. Then /E OJ for some j and 'leI) E naj \n~~ where Yj = J.l(C2a/+1b . Then by Lemma 5.1 we obtain mes (na j \ naYjj ) <_ C yjajn -3+b <_ C' a jn -2+U+I)b • Then in view of (4.29) and (5.31). Here and below we denote by C, C' various constants. Thus we obtain 00 00 mes(r \ r,u) :::; L mes(1D j \ o?) :::; C L a;-2+ U+4-n)b j=1 j=1 00 = C L(qj)n-2+U+4-n)b < C' qn-2+U+4-n)b. j=1 On the other hand, 00 mes r >_ C~'"a jn -2+2b > C q n-2+2b , j=1 hence mes(r\ r ,u)/mes r :::; c' qU+2-n)b :::; C' qb provided that 1+2 -n ~ 1 , i.e. I ~ n -1. This estimate immediately implies (2.6) and Theorem 1 is proved. 6. THE TWO-DIMENSIONAL CASE Proof of Theorem 2. Let , be an approximate interpolating Hamiltonian for the billiard ball map B [14], i.e. B(p) = exp(-,1/2X,)(p) + 0(,00) in a neighbourhood of s:an. Denote by M, the closed curve M, = g = r} for r E (-0,0), 0 small enough. For any p E M, consider the map RI :3 t --+ exp(tX,)(p) E M, and denote by I(r) its period. Let S be a transversal to Mo. For p E M, denote by tep) the smallest positive time t such that exp( -tX,)(p) E S. Then (t(p) , '(p)), 0 :::; t(p) :::; I(C(p)) are symplectic coordinates in a neighbourhood of s:an. We define rp(p) = 21lt(p)/I(C(p)) and complement it to exact symplectic coordinates (rp, I) E T* (T I). For this purpose we seek for a function /= g(e) so that the change x: T* an --+ T* (T, I) transforms the symplectic I-form (J = i; ds on ~ into /drp. Then /and C satisfy the relation (6.1 ) r'(p) 21l/(p) = -10 I(r) dr + 10 , 76 VALERY KOVACHEV AND GEORGI POPOV 10 = J../+ (J = 1(0) = 2n[, where y+ is s:an with the positive orientation. Denote by '(I) the solution of (6.1) with respect to , with initial condition '(0 = 0 and by 'M(I) its Taylor expansion up to degree M which will be chosen large enough. Now, in the coordinates (rp, I) the exact symplectic map Bo = XBX-' can be written as Bo( rp ,I) = (rp + r~(I) ,I) + 0(([ -I)M+3/2) , r M = -1,;;2 (I) in a neighbourhood of T' x {[}. As above, we first consider Bo in the sets , , A j =1" x [[-C,a j ,[-C2a)=1" xrj , j=I,2, ... , where 0 < C2 < C,' aj = 4-j and A = U~, Aj . We choose M j = N + 3j/2, N large enough, and consider Bo in Aj as the perturbation of the map (rp, I) ----> (rp + r~ (I) ,I). By arguments similar to those in the proof of Theorem 5.2, I replacing I, band N by j, 1/2 and N + (3 j + 1) /2 respectively, we construct exact symplectic transformations Vj: Aj ----> Aj such that Bj = V j-' B Vj are generated by the functions rj(1) + H)rp, I), j = 1,2, ... , Hj(rp, I) = 0 for (rp,I) E 1"' x E j where E j are the preimages of the sets R j (see (2.8)) under <. Here for brevity we write rj instead of r Mj • Now we choose E = U~, Ej . We have Ej C {l E r j ; I" 0, (J > 1 and repeating some details of the proof of (2.6), we obtain (2.7). We can glue K j , Hj and Vj together using again Remark 1 after Poschel's Theorem A [16] which in the case n = 2 is quite obvious. Moreover, la: (K -'M)I :::; Cpl/-qN-'-5P/2+j('-P/2) , /E rj , thus the function K is COO in r. This completes the proof of Theorem 2. Denote R = {w(l) = r' (I); /E E} and by w ----> /(w) the inverse map of the diffeomorphism [[-t50 '[) 3/----> w(l). Set Aw = V(T' x {/(w)}). Then r R = V(1"' x E) is the union of the invariant curves Aw of B with rotation numbers p(AJ = w E R. It obviously satisfies the inequality mesr6 -mes(rR n r 6 ) :::; CN t5 N for any N> 0, t5 > 0 where r 6 = an x [l -t5, 1]. Corollary 6.1. There exists an approximate interpolating Hamiltonian , for the billiard ball map B in a neighbourhood of s:an such that B(p) = exp( _,'/2 X()(p) for any pEr R . INVARIANT TORI 77 Proof of Theorem 3. Let 9 E f'(m, n), 9 = {gl ' ... ,gn}, i.e. Bj(gl) = gl+j' j = 1, ... ,n -1 , Bn(gl) = gl and 9 has a winding number m. Denote by Xj the reflection points on 8n corresponding to gj and set gj = (Sj' 0) where Sj is the value of the natural parameter on 8n at Xj and uj is the cosine of the angle between the oriented line segment linking Xj with xj +1 and the oriented tangent line at xj . Let d be the maximal length of the arcs linking Xj with Xj+1 ' j = 1, ... ,n, and then we have d ::; mlojn < 010. Thus we obtain o ::; 1 -uj ::; C(o) , C(o) > 0 as 0 > 0 which means that for 0 > 0 small enough and min < 0 we have gj E U(A) , 1 ::; j ::; n, if 9 E f'(m, n). From Birkhotrs theorem it follows that f'(m, n) = 0 for n ~ no' no large enough. We shall use the following lemma. Lemma6.2. Let (mk,n k) E ~i, k = 1, 2, ... , and 2nmkjnk ~ was k ~ 00. Let 9 k -{glk ' ... ,gkn ) E f'(mk , nk) an d U -I (gkj) = (rpkj ,1kk) whe re the map U is given by Theorem 2. Then the sequence {I;, 1::; j ::; nk , k = 1,2, ... } is convergent and tends to I (w) . Proof. Denote by QI (rp ,I) the first component of the function Q( rp ,I) in Remark 2.2 and p(rp,l) = r'(1) + QI(rp,l). Then PI> O· for 0 < [-1< e, e > 0 small enough. Let 9 E f'(m,n) and U-I(g) = (rpj,lj)' gj E g, j = 1, ... ,n, and let I(w l ) < II < I(w2 ) , I(w) E E, i = 1,2. Then I(w l ) < I j < I(w2 ) for j = 1, ... ,n and Wi E R, WI < w2 • Moreover, for the rotation numbers of Aw and 9 we have p(Aw) = r'(I(w) = Wi' p(g) = 2nmjn and since P is monotonely increasing with respect to I for rp fixed, we easily obtain WI < 2n m j n < w2 • Note that the set R has no isolated points, thus if there exists a sequence 9k E f'(mk' nk), k = 1,2, ... such that Ijkk fI. (I(w) -e, I(w) + e), then 11 fI. (I(w l ), I(w2» for any k, j and some I(wi ) E E appropriately chosen, I(w l ) < I(w) < I(w2 ). Hence 2nmklnk fI. (WI' w 2 ) which contradicts the condition of the lemma. By Theorem 2 Bo is an exact symplectic map with a generating function r(1) + G(rp,l). It is easy to see that for aD = Idrp we have B~ao -aD = df where f(rp,I) = Ir'(1) -r(l) + h(rp,l) and h(rp,l) = IGI(rp,l) -G(rp,I) vanishes on E. Since the L-spectrum of B is a symplectic invariant which coincides with the length spectrum of n (see [6]), we have n n L(g) = "Lf(B~(rpl ,II» = "Lf(rpj,l) j=1 j=1 where L(g) is the length of the periodic geodesic in n corresponding to ~ = {gl' ... ,gn}Ef'(m,n), (rpj,l)=U-I(g),providedthat mjn 0 where e = e(w) is chosen so that 0 < e < 78 VALERY KOVACHEV AND GEORGI POPOV II(w) -tl/2. Then r(I) = _j(K(I))3/2 is a function of class COO in the interval II -I(w)1 < e and k k IL(g )Ink -(wI(w) -r(l(w)))1 ~ C(w) max IIj -I(w)l. Hence the sequence L(l)lnk , k = 1, 2, ... , is convergent and for any choice of 9k E r(mk ,nk ) we have (6.2) where S(w) = wI(w) -r(l(w)) is just the Legendre transform of r(l). Thus L( m , n), min ~ a , recover uniquely the Legendre transform of r(l) in R. Since E has no isolated points, one can recover from (6.2) the function r(l) for any lEE. Now let n l and 02 be two strictly convex domains in JR2, r.i = B* ani; let Bi be the respective billiard ball maps, i = 1, 2, and let the assumptions of Theorem 3 be satisfied. Let Ei be defined by E; = {I E ([i -a, [i]; <(I) E R} , [i = 1~/21C, i = 1, 2. It is easily seen that l~ = l~ since l~ = limn---+oo L(gn,i) , gn,i E ri (1 , n) . Now for all w E R we have i = 1,2. Since R has no isolated points, we can differentiate with respect to w which yields II (w) = 12(w). Hence EI = E2 and r I (I) = r 2 (1) for I E EI ' thus Bo,1 (rp, I) = BO,2(rp, I) for any (rp, I) E TI X EI . Conjugating with Vi' i = 1, 2, we obtain the assertion of Theorem 3. 7. CONCLUDING REMARKS 7.1. A question may arise whether there exist strictly convex hypersurfaces with closed elliptic geodesics satisfying conditions (2.1) and (2.3). We can answer this question positively at least in the case n = 3, perturbing arbitrarily little the metric on an ellipsoid with three different axes in an arbitrarily small neighbourhood of any point of the shortest or the longest ellipse and applying a classical result of Nirenberg (see [7]) on the isometrical embeddings of compact oriented surfaces with Riemannian metric with strictly positive curvature as smooth convex surfaces in R3 . 7.2. Theorem 1 and a theorem of Birkhoff-Lewis type (cf. 116]) show that the closure of the periodic points of B in a neighbourhood of &' has a positive measure. 7.3. The results obtained hold also for any generalised billiard ball map arising as a boundary map for a pair of glancing hypersurfaces (see [15]). Indeed, INVARIANT TORI 79 we used in the proof only some properties of the approximated interpolating Hamiltonian but not the specific structure of ~. ApPENDIX As noted in §5, Theorem 5.4 can be derived from Poschel's Theorem A [19]. The constants participating in the estimates throughout [19] depend only on n, a, A, {J, Rand Q, but not on the domain or the parameter y, i.e. their dependence on R, and, in Theorem A, on {J, is not stated explicitly. In our case we have R = Ca-b , 0 < {J :::; a12, the parameters a and b participating in the definitions of the domain 0: and the parameter y as well, thus we have to follow the dependence of the·various constants in [19] on R and, in Theorem A, on {J as well. The Main Lemma in [19] holds for J > 0 small enough depending only on n, a, A, Rand Q, but not on the domain, y or {J. We put J:::; CI R-3 and follow the dependence of the constants c2 ' c3 ' cg , CII in estimates (i), (ii), (iii), (v) on R. In the proof of the Main Lemma a version of the implicit function theorem is used to solve for a local diffeomorphism rp and a small perturbation rjJ the equation (A.l) rp(n(')) = rp(') + rjJ(O for a map n close to the identity I: Lemma A.I [19, Lemma 4.2]. Let rp, rjJ be real analytic on the {J-neighbourhood A + {J of a domain A in en , with IDrp I, IDrp -II :::; S there. If (A.2) with a constant c = c( n , S), then there exists a unique real analytic map n:A---.A+{J, such that (A. 1 ) holds. In addition, analytic dependence on parameters carries over from rp, rjJ to n. Also, n -I is 2n-periodic in " if rp and rjJ are. Following the proof of the above lemma, we see that we can choose the constant c = c(n,S) in (A.2) in the form c = c(n)(1 +S3). Further on, for any J :::; CI R-3 , R large enough, the Main Lemma holds and the constants c2 ' c3 ' CII can be chosen independent of R while cg depends linearly on R. From the Main Lemma we can derive Theorem B for the normalized value of y = 1 :::; {J. We find that the constant c12 in [19, estimate (4.20)] depends linearly on R while the constant c13 in [19, (4.23)] can be chosen independent of R. This provides an additional factor R-I in the left-hand side of the first estimate (3.24) in [19] for y = 1 while the second estimate (3.24) and (3.25) in [19] for y = 1 remain unchanged, with constant c p independent of R. Now we pass from the normalized value of y = 1 :::; {J to the general case o < y :::; {J. We replace the functions F and G in the formulation of Theorem 80 VALERY KOVACHEV AND GEORGI POPOV Bin[19]by FO=y-2pOOfIy' G=y-2 GofIy respectively and apply Theorem B for y = 1 and the same R to the functions FO and G. This gives us functions (j), F, and r satisfying equalities (3.22), (3.23) in [19] for FO , G instead of pO, G as well as estimates (3.24), (3.25) in [19] for y = 1, the left-hand side of the first inequality of (3.24) being multiplied by R-1 and the constant cp independent of R. Now, the functions = fIy 0 (j) 0 fIy-l , P = l F 0 fIy-1 and r = yr 0 fIy-1 satisfy (3.22), (3.23) in [19] with the functions pO, G. Moreover, estimates (3.24) in [19] hold for any y :$ p , the left-hand side of the first estimate (3.24) being multiplied by R-1 • On the other hand, we see that the exponent of y in the right-hand side of (3.25) in [19] must be -1 and not -2. Now let the assumptions of Theorem A be satisfied, the left-hand side of inequality (3.3) in [19] being multiplied by Rand 0:$ C1R-3 , C1 independent of Rand p, i.e. the assumptions of Theorem 5.4 hold. Then from Theorem B [19] we find, on account of the correction in estimate (3.25), that instead of (3.7) in [19] the generating function S satisfies the estimate < -I P+I ° IISlIpJ.,p;y _ Cpy R IIH -H IIPJ.+.l.+a;y Cp independent of Rand p, which corresponds to estimate (ii) in Theorem 5.4. Moreover, Theorem A provides a diffeomorphism T: Tn X n --+ Tn X I transforming the Hamiltonian vector field X H on Tn X I into the vector field T* X H on Tn X Q so that T* XHhnxn = (w,8/8()}. Now, in order to obtain Theorem 5.4, we replace S, y, H, Ho and I by S, YI' H', H~ and D:, respectively, and the canonical map T generated by S satisfies for (y,r/) E A:'/I the equality T(y,r/) = T(y,(8K'/811)(I1)) where K'(I1) is the nondegenerate Hamiltonian in (i) of Theorem 5.4 (K(P) in [19, (3.5)]). REFERENCES I. V. I. Arnold, Small denominators and problems of stability of motion in classical and celestial mechanics, Russian Math. Surveys 18 (1963), 85-193. 2. _ , Mathematical methods of classical mechanics, Springer-Verlag, Berlin and New York, 1978. 3. V. I. Arnold, V. V. Kozlov and A. I. Neustadt, Mathematical aspects of classical and celestial mechanics, Current Problems in Math., Fundamental Directions 3, Moscow, 1985. (Russian) 4. R. Douady, Une demonstration directe de ['equivalence des theoremes de tores invariants pour difRomorphismes et champs de vecteurs, C. R. Acad. Sci. Paris Ser. 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Magnuson, Symplectic singularities, periodic orbits of the billiard ball map, and the obstacle problem, Thesis, M.LT., Cambridge, Mass., 1984. 14. Sh. Marvizi and R. Melrose, Spectral invariants of convex planar regions, J. Differential Geom. 17 (1982), 475-502. 15. R. Melrose, Equivalence of glancing hypersurfaces, Invent. Math. 37 (1976),165-191. 16. J. Moser, Proofofa generalizedform ofafixed point theorem due to G. D. BirkhofJ, Lecture Notes in Math., vol. 597, Springer-Verlag, Berlin and New York, 1977, pp. 464-494. 17. G. Popov, Invariant circles and length spectrum of the billiard ball map, Preprint. 18. __ , Quasimodesfor the Laplace operator (in preparation). 19. J. Poschel, Integrability of Hamiltonian systems on Cantor sets, Comm. Pure Appl. Math. 35 (1982), 653-696. 20. N. V. Svanidze, Existence of invariant tori for a three-dimensional billiard, which are concentrated in the vicinity of a "closed geodesic on the boundary region,~ Uspekhi Mat. Nauk 33 (1978), 225-226. (Russian) INSTITUTE OF MATHEMATICS, BULGARIAN ACADEMY OF SCIENCES, ACAD. G. BONCHEV STREET 8, SOFIA 1113, BULGARIA