Materialization of Poincaré resonances and divergence of normalizing series

MATERIALIZATION OF POINCARE RESONANCES AND DIVERGENCE OF

NORMALIZING SERIES

Yu. S. ll'yashenko and A. S. Pyartli UDC 517.9+517.5

INTRODUCTION

There are three parallel and closely related theories: the theory of normal forms of analytic differential equations (autonomous and with periodic coefficients) in a neighborhood of a singular point, the theory of normal forms of holomorphic mappings in a neighborhood of a fixed point, and the theory of normal forms of zero-type neighborhoods of elliptic curves.

A program for the parallel development of these theories was proposed by Arnol'd [3]. In Sec. i we develop this program in a sharpened and generalized version, and we then indicate the connections between the theories mentioned; these connections make it possible to prove parallel theorems in only one of the theories.

In this work we prove all theorems listed below on the materialization of resonances and the so-called geometric theorems on divergence of normalizing series connected with them. For equations with the singular point 0 in the space C 3 these results consist in the following. Let % = (%1, %2, %3)~ C 3 be a resonance collection; the numbers %1, %2, and %3 form a triangle containing zero strictly in its interior, and an integral linear combination of them is equal to zero. Let {~(c)} be an analytic, one-parameter family of differential equations whose linear parts at the singular point 0 for E = 0 pass through the resonance %. If {~(E)} is a family of general position, then this resonance is materialized, i.e., as E passes through zero a two-dimensional analytic manifold M(c) separates from the coordinate planes of some chart; this manifold is homotopically equivalent to a two-dimensional torus and is invariant under the equation a(E). If the spectrum %o of the linear part of the equation ~ at the singular point 0 of the space C 3 is a nonresonance spectrum but is pathologically close to a countable number of resonances %s similar to the resonance % described above, while the nonlinear part of the equation is of "general position," then in any neighborhood of zero there exists a countable number of analytic surfaces invariant under the equation ~. These surfaces arise under materialization of the resonances %s and obstruct the convergence of the formal changes reducing the equation ~ to linear normal form. Equations ~ possessing the properties described form a dense set in the space of analytic differential equations defined in a neighborhood of the space C 3 and having singular point 0.

In Sec. 2 we prove an inclusion theorem which asserts, in particular, that any germ of a holomorphism at a fixed point can be realized as the monodromy transformation after a period for a periodic differential equation. This theorem enables us to reduce mapping problems to problems of equations in a manner similar to the way in which inverse reduction is carried out by means of phase-flow and monodromy transformations.

In Sec. 3 we study the geometry of a manifold which is the "materialization of a resonance." It is proved in Sec. 4 that a linear equation whose spectrum is linearly independent over Z cannot have an analytic invariant manifold which is situated in a special manner relative to the invariant planes of the equation. The main results of the paper -- theorems regarding materialization of resonances and divergence -- are proved in Secs. 5-9.

Below (X, Y) denotes a neighborhood of the set Y~ X in the space X; (X, Y)~, (X, Y)~ .... are distinct neighborhoods; (X, Y) is the germ on Y of the neighborhood (X, Y);f:(X, Y) + (X ~, Y') is a mapping taking Y into Y'; a germ of a mapping or a set and its representer are denoted respectively by the same semiboldface and simple symbol; the j-th component of a vector a is denoted by aj. The numbering of definitions, propositions, and remarks begins with i in each subsection; the number of formulas and lemmas begins with i in each section; the numbering of theorems is continuous. Reference to a subsection of a current section is given without indi-

Translated from Trudy Seminara imeni I. G . Petrovskogo, No. 7, pp. 3-49, 1981. Original

article submitted September 28, 1979.

0090-4104/85/3104-3053509.50 O 1985 Plenum Publishing Corporation 3053

cating the section, while the section is indicated if reference is to a subsection of a different section.

SEC. i. PRELIMINARY CONCEPTS AND FORMULATIONS OF THE MAIN RESULTS

This section begins with a survey of classical results.

I. A-Equations (Autonomous Equations)

i. The Space ~A. A-Resonances. Formal Normal Form. We denote by~ A the space of germs at the point 0 of holomorphic vector fields on (C n, 0) with singular point 0 and zero linear part: ~=>/(0)=0, f.(0)~-0 ; we denote by JA the set

IA={(p, j)]/~(l, .), p~Z+=, ]PI=P,+... +p=>l};

we sometimes write JA(n) in order to indicate the dimension of z-space.

We consider the analytic differential equation

~=Az-}-[(z), f~,~A, A = d i a g ~, L= (~, ..... L~) ( 1 . 1 )

The n o n l i n e a r p a r t f o f Eq. (1) i s u s u a l l y c a l l e d t he e r r o r .

D e f i n i t i o n . A c o l l e c t i o n ~ C n i s c a l l e d an A - r e s o n a n c e t i f f o r some (p, j)~]A (L, p)--X~=O.

Otherwise the collection I is called a nonresonance collection; the linear part of Eq. (l.l) is called a resonance or nonresonance linear part correspondingly�9

�9 sT s Polncare Theorem [13]. Equation (i.i) with nonresonance linear part can be reduced to the normal form

@ = A w

in the class of formal power series by the change

z = H ( w ) = w + h ( w ) , h ( O ) = ~ , h , (O)=O.

Such a change i s u n i q u e l y d e t e r m i n e d and i s h e n c e f o r t h c a l l e d a n o r m a l i z i n g change or a n o r m a l i z i n g s e r i e s .

2. Convergence. A normalizing change converges or diverges depending on the degree of closeness of the collection I to resonance collections. The property of I belonging to a Poincare or Siegel domain plays an essential role here. Of two equivalent definitions of these domains [2], the following is convenient for us.

Definition 1. A Poincar4 domain is a maximal open subset of PA in which A-resonances are distributed in nowhere dense fashion; a Siegel domain C n, is the complement to a Poincar~

domain.

Linear equations are subdivided correspondingly into equations of Poincar~ and Siegel

types.

The theorem on convergence of normalizing series for equations in (C n, 0) with a nonresonance linearpart of Poincar6 type was proved by A. Poincar6 [3]; for almost all I in a Siegel domain it was proved by C. L. Siegel; subsequent strengthenings of this theorem were given by

V. I. Arnol'd and A. D. Bryuno.

Definition 2. A collection %~C n is called incommensurate in the Bryuno sense if there

exist positive C and e such that

l( k, P)- - ~'il > Ce-'pll-8, (P, J) ~ JA.

THEOREM (a somewhat stronger assertion was proved in [5]). If a collection I is incommensurate in the Bryuno sense, then Eq. (i) is analytically equivalent to its linear part.

3. Divergence. The condition of incommensurability in the Bryuno sense is close to ne-

ces sary.

#We are obliged to say A-resonance (additive resonance) and not simply resonance in order to

distinguish resonances in different theories from one another.

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Bryuno's Theorem [5, p. 207]. Suppose a nonresonance collection % is pathologically

close to a countable number of resonances, namely, for any positive C and O there exists a countable set Mc]A, such that

I(~, p ) - - ~ l < C e -~ for (p, j)~M.

Then there exists [~A, such that the normalizing series for the system (i.i) diverges.

A stronger version of this theorem (an analytic theorem on divergence) holds for the "majority" of nonlinearities f~A and is proved in i12] based on ideas of E. M. Landis. The same method is used to prove the assertion contained in the title of the paper of Yu. S. Ii'yashenko, "In the theory of normal forms of analytic differential equations, if the con-. ditions of A. D. Bryuno are violated, divergence is the rule and convergence is the exception ~ (Vestn. Mosk. Univ., No. i, 10-16 (1981)), where nonlinear normal forms are considered.

V. I. Arnol'd proposed a program for obtaining geometric divergence theorems [i, 3]. These theorems are also valid for the "majority" of nonlinearities, but an essential condition is imposed on the linear parts: they must be close to a countable number of so~called Poincar6 resonances. If this condition is given up, apparently the first step of Arnol'ds program cannot be carried through -- proof of the theorem on materialization of resonances.

4. Poincar~ and Siegel Resonances. Definition i. An A-resonance $~C n is called a

Poincar~ A-resonance if 0 is an isolated point of the set

{ (P, ~)--ZJl (P, i) ~ ]A} , otherwise it is called a Siegel A-resonance.

Remark i. Any resonance in a Poincar6 domain is a Poincar6 resonance. If the resonance collection % contains a Siegel pair (two numbers with negative ratio and this ratio is irra- tional) or a Siegel triple (three numbers whose convex hull contains 0 together with a neighborhood) and this triple is incommensurate, then the collection % is a Siegel resonance. This

implies the following

Proposition i. Poincar6 A-resonances decompose into two (intersecting) sets;

Type I. All numbers %~ belong to a discrete subgroup with two generators of the group

C of complex numbers with a~dition.

Type II. The numbers %j decompose into two sets: those belonging to the first set lie on some line and are commensurate; those belonging to the second lie on one side of this line.

Remark 2. The set of A-resonances of Type I is dense in the Siegel domain ~A.

We denote by RA(%) the set

Monomials of the form z r for r~RA(~) are called resonance monomials.

Definition 2. The multiplicity of an A-resonance ~ is the dimension over R of the lin-

ear hull of the set RA(%).

Remark 3. A Poincar~ A-resonance of type I has multiplicity n -- 2.

Definition 3. An A-resonance ~ is called positive if RA(%)~Z+ n

Remark 4. In a Poincar~ domain there are no positive A-resonances; in a Siegel domain

there are both positive and nonpositive Poincar@ A-resonances. Poincar~ A-resonances of mul-

tiplicity one are exhausted by the following list ~8]:

Type i: n = 3, the collection % satisfies the relation (~, r) = 0 r 3

Type 2: n is arbitrary; for a suitable renumbering of the components the fraction %~/~2

is a negative rational; the numbers ~3, "-', %n lie on one side of the line (%~, %=).

Arnol'ds program for resonances of multiplicity one of type 1 was carried out hy Pyartli

[16, 17, 3].

5. Materialization of Resonances. A resonance ~ materializes in a family {e(e)} of Eqs. (i) whose linear parts for e = 0 pass through the resonance %, if when e passes through zero

an invariant manifold of the equation e(e) separates from the coordinate planes of some chart

whose geometry is connected with the arithmetic structure of the resonance. A precise defin~-

ition of this connection looks as follows.

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Definition i. An A-resonance % materializes in a k-dimensional family {a(e) Je.~(C k, 0)}, if the equation a(e) has an invariant manifold M(E), which in some chart w on (C n, 0) not depending on E is given by the equation

(1.2)

where

RA(~

is a series of resonance monomials; E:(C n, O) § (C k, O) is a local epimorphism.

The geometry of the manifolds M(E) is described in Sec. 3.

THEOREM la. Each positive* k-fold Poincar~ resonance materializes in a k-dimensional family of Eqs. (i.i) of general position.

Theorem la can be obtained as a consequence of the general theory of invariant manifolds o~ equations developed by A. D. Bryuno. The main results of this theory are contained in the preprints [6, 7]. This same theory suggests that an analog of Theorem i is false for each Siegel resonance (there are special counterexamples on this account [8]).

Theorem la is proved by a technique which makes it possible to apply the principle of contraction mappings where usually the method of majorants was applied. Apparently, this technique is always useful when convergence of formal series in the absence of small denominators is proved; in particular, it makes it possible to overcome the most laborious step in the proof of the theorem on classification of imbeddings of positive type of elliptic curves (Yu. S. Ul'yashenko, "Imbeddings of positive type of elliptic curves in complex surfaces," in press).

6. Nonlinearizability. Suppose a family {~(E)} of Eqs. (i.i) for ~ = 0 passes through a Poincar6 A-resonance of type I and this resonance materializes. We fix a neighborhood of (C n, 0) and then take E sufficiently small. We suppose that the linear part A(e) of the equation a(E) is a nonresonance part, while the equation itself in a neighborhood of (C n, 0) is equivalent to its linear part. Let h:w ~ z = H(w) be a holomorphism taking the equation a(E)

into a linear equation:

w = A i ~ ) ~

Then this equation has an invariant two-dimensional analytic manifold HM(E). It can be proved that if the holomorphism H does not differ too much from the identity, then this manifold is

a resonance manifold in the following sense.

Definition. An analytic manifold imbedding in the coordinate space C n is called a res-

onance manifold if under the projection (wl ..... wn)-+(w, ..... wl), |~l<n, it covers regularly and in single-sheeted fashion a neighborhood of some torus T={lwil-----Ch ]=I, .... l, ~i=O, i=

t+1 . . . . , n} THEOREM 2. Suppose the equation ~ = hw has a resonance invariant manifold. Then the

components of the vector I are linearly dependent over Z.

For a general E the components of the vector I(E) are linearly independent over Z. For

such e there does not exist a holomorphism close to the identity taking the equation a(e) into linear normal form in a neighborhood of (C n, 0).

7. The Geometric Diversence Theorem. Definition. A holomorphism H: (C ~, 0)-+(C n, 0)'

is called admissible if H,(0) = E.

*We restrict ourselves to the case of positive resonances, because, first of all, all Poin- care resonances of multiplicity one in a Siegel domain are positive, and, secondly, all geometric divergence theorems formulated below follow from Theorem la; they would not gain in generality if in place of positive resonances we considered all resonances. Below a somewhat weaker theorem is proved: it is required that the order of the resonance, i.e., !+ min Irl

r s R A (k)

exceed 2. Theorem la holds without this condition, which is imposed in order to prove the

theorem on materialization simultaneously in all three theories.

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THEOREM 3a. In a Siegel domain ~A of the space C n there exists an open set V A and a thick* subset ~ACVA , and for each % ~ in the space j~A ~ there exists a dense set

such that if ~5~A, r~ ~,

then any neighborhood of zero in the space C n contains a countable number of invariant manifolds M s of Eq. (1.1). For any neighborhood of (C n, 0) and chart w on it all manifolds Ms, beginning with some one are resonance manifolds and obstruct convergence of normalizing series for Eq. (1.1). #

It seems undoubtable that for V A it is possible to take the entire set ~A, and for ~A the set of collections % pathologically close to a countable number of positive Poincare resonances. However, it has been possible to study the geometry of the manifold which is a materialization of a resonance only for "convenient" resonances.

The set V A in Theorem 3a consists of all ~ A such that

a) the collection % = (%~,..., Xn ) is hyperbolic, i.e., %i/Xj ~R for i # j;

b) for a suitable renumbering of the variables of the vectors Xa, "'', Xn lie between

the ~ectors --%~ and --%2.

In conclusion we shall say a few words regarding the proof of density of the set ~. In this proof for each germ ~ a sequence of germs ~8~ x converging to it is constructed. The central feature of the construction is that the difference f -- fs is a vectorial polynomial having degree of lower order terms which tends to infinity as s § ~. Thus, the technique of Sec. 9 does not enable us to prove geometric divergence theorems for systems (l.1) with a polynomial right side; even the theorems on materialization of resonances in such systems do not follow from the general theorems and have so far not been proved. On the other hand, it seems likely that all local effects observed for analytic differential equations must also be observed for algebraic differential equations of sufficiently high degree. It would be interesting to carry over the results of the present paper to equations with polynomial right side~ and polynomial mappings. A first step in this direction was taken in the

work [19] for polynomial mappings (C, 0) § (C, 0)'

II. Germs of Holomorphisms at a Fixed Point and Imbeddings of Elliptic Curves

i. Definitions. An elliptic curve is a Riemann surface of genus i. Below r denotes an

elliptic curve and M n is a complex manifold of dimension n.

Definition i. Two imbeddings f.:F ~ M. n, j = i, 2 are called equivalent if there exists a holomorphism ~ : (M~ n, fiF) + (M~ n, f2F)Jfor any neighborhoods of (Mj n, fir), and ~lh~ ~i----~

Definition 2. A line bundle over an elliptic curve is of zero type if the index of self- intersection of the zero section of this bundle is equal to zero. A vector bundle over an elliptic curve is of zero type if it is the direct sum of line bundles of zero type. A line bundle over an elliptic curve is a resonance bundle if it is trivialized over a finite-sheeted

covering of the base.

Definition 3. An imbedding f:r § M n is of zero type if the corresponding normal bundle

Vf over F is of zero type.

2. Sue's THEOREM. Gluing Mappings. THEOREM [18]. An imbedding of a stein manifold F in a complex manifold is equivalent to the imbedding of the manifold r as the zero section in

the space of the corresponding normal bundle.

Remark. As is known, a vector bundle over a Stein manifold homotopically equivalent to

a torus is analytically trivial.

Sue's theorem makes it possible to reduce the classification of imbeddings of elliptic

curves to the classification of so-called gluing mappings as follows.

*A thick set is the countable intersection of open dense sets. tThe topology in the space of germs of analytic vector-valued functions is defined in Sec. 9.

~Note added in proof. This has recently been done by the authors.

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Let f:F + M n be an imbedding, and let U ~ M n be a neighborhood of fF, homotopically equivalent to F. The curve F admits the following standard representation: there exists a vI~C, [ ~ t ] > l , such that

r = c * / { ~ , ~ l k ~ Z } , c * = c - o .

We consider a covering U over U defined by the following condition: the covering r over r naturally imbedded in 0 is biholomorphically equivalent to C*; the holomorphism h:F + C* conjugates the group of superposition transformations of the covering F with the group of transformations C* + C* generated by multiplication by ~:. Let C n be the numerical space. We identify C* with the punctured z: axis of the space C n. Let f~:F + 0 and f2:C* + C n be the natural imbeddings. By Sue's theorem and the remark following it the imbeddings f~ and f2 are equivalent; let ~': (0, F) +(C", C*), and let a~lF =h be the corresponding holomorphism. The group of superposition transformations of the covering U has one generator; the holomorphism ~ conjugates this generator (or the element inverse to it) with the mapping

f : ( C n, C*)-,-(C", C*)',

Fie. : z,---,.~,z,, I v, I ~ 1.

This mapping is called a gluing mapping. Identifying the points z and Fz on the intersection (C n, C*)~(C ,~, C*) ~ , we obtain a manifold biholomorphically equivalent to some neighborhood

(U, fr).

Definition 4. Two gluings are called equivalent if they are conjugated by a homomorphism (c~, c*),- ,-(c~, c*)~

We have proved the

THEOREM. Imbeddings of an elliptic curve are equivalent if and only if the corresponding

gluings are equivalent.

Remark i. Parallel results are contained in the papers [3, 10], but there the gluing mapping is defined not in a neighborhood of (C n, C*) but in a neighborhood of (C n, y~), where u is the circle [z~ I = i. This is sufficient to reduce gluings to normal form but is insufficient for the reduction from mappings to equations by means of which the main theorems of this work are proved.

Remark 2. Let K~ Ce be the annulus {ztl [zt[~[l, vt]} It suffices to find a holomorphism H conjugating the gluings F ~ and F 2, only in a neighborhood of (C n, K) and then extend it

to (C n, C*)x on the basis of the equality

H = F 2k.HoF l-k, k ~ Z .

3. Normal Gluing and Holomorphisms with a Fixed Point. Let f be an imbedding of zero type of an elliptic curve, and let z = (zl, ~), F be the corresponding gluing. The structure of the normal bundle is given by the Dart of the mapping F linear in 2. By definition the

�9 n-1 C* bundle Vf is equivalent to the bundle obtained from C • by the gluing

z-~(diag v)z, ~.= (v, ..... x'~) ~ C *~

In the corresponding chart the gluing F has the form

F (z) = (diag v) z + F (z), F = ( E l , F*),

F ] o = 0 ; oF" [ = 0 . dz C" I

(1.3)

(1.4)

Such gluings are called normal; F is the error of the gluing F.

It is convenient to expound the theory of normal gluings and holomorphisms with fixed

point in parallel.

Let ~i and JA be the same as in subsection I.i. We denote by ~ the space of germs on C* of holomorphic mappings (C n, C*)-+{C n, 0), satisfying condition (1.4). If F~ , then

F-(z)= E F~zpei , where p=(p, , #), JE={(p, j) lpl~Z, ~ Z + n-I, ]~ (1 , hi, IP] >~1 f o r j = 1; ]D[ > (P.DEJE

2 for j # i}#.

#We sometimes write JE(n) in order to indicate the dimension of z-space.

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4. M- and E-Resonances. Definition. A collection v~C*" is called an M-resonance def

(E-resonance) if there exists (p, j)~IM=JA (respectively, (p, j) ~JE) , such that

VP--vj=O.

Otherwise the collection v is called an M-nonresonance (E-nonresonance) collection.

THEOREM.* If ~ is an E-nonresonance collection, then the normal gluing (3) in the class of changes formal in ~ and convergent in zl, can be reduced to the linear normal form diag v.

An analogous theorem for germs of holomorphisms has been known for a long time. The changes mentioned in these theorems are called normalizing series.

THEOREM [i0]. If there exist positive C and ~ such that for all (p,j)~J~ ,

I : - - ~ j l > C l p l -~ ,

then any normal gluing (1.3) is equivalent to its linear part.

An analogous theorem holds for germs of holomorphisms (C n, 0) § (C n, 0) [4, i0].

COROLLARY. If r is an elliptic curve and f:r § M n is an imbedding of zero type to which there corresponds the normal gluing (1.3) and the collection v satisfies the condition of the preceding theorem, then the imbedding f is equivalent to the imbedding of F as the zero section in the corresponding norn~l bundle.

5. Poincar~ M- and E-Resonances. Definition i. A collection ~C *n, [vil=~l, is called

a Poincare E-resonance if 0 is an isolated point of the set

{:--~/I (P, i) ~lE}, and a Siegel resonance o therw ise .

Poincar~ and Siegel M-resonances are defined similarly with Je replaced by JM; the condi-

tion Iv1[ # 1 is dropped.

Proposition i. A collection v~C ~n is a Poincar~ E-resonance if and only if all the

numbers ~. belong to a discrete subgroup of the group C*. J

Proof. Otherwise for some j the set of numbers

{v,P' vjp' [p,~Z, p2~Z+, p2~2}

would be dense in C*.

Proposition 2. to one of two types.

A collection v ~ C ~:n is a Poincar~ M-resonance if and only if it belongs

Type i: all numbers ~j belong to a discrete subgroup of the group C*.

Type 2: the numbers vj decompose into two sets: those belonging to the first lie on the unit circle and their arguments are commensurate; those belonging to the second lie on one

side of the unit circle.

Remark i- The set of Poincar~ E-resonances is dense in the space C n. The set of Poin- car6 M-resonances of type i is dense in the Siegel domain 3* of the space C *n (by definition 2" is the domain consisting of collections ~, whose components lie on different sides of the

unit circle).

We set

The set RM(V) is defined in the same way with JE replaced by JM"

Definition 2. An E-resonance v is called positive if all components except the first of the vectors r~RE(v) are nonnegative; an M-resonance v is called positive if RM(w)cZ+ =

Definition 3. The dimension over R of the linear hull of the set RM(~) (or RE(~)) is

called the multiplicity of the M-resonance (E-resonance) ~.

Remark 2. All Poincar~ E-resonances have multiplicity n -- i.

*This theorem is proved in [3] for n = 2; the case of arbitrary n can be handled similarly.

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Remark 3. Poincar~ E-resonances admit a simple geometric interpretation: the corresponding vector bundle is trivial over a finite-sheeted covering of the base.

Definition 4. The order of a positive E- (M-) resonance ~ is min I~ + ! (respective- rERE(v)

l y , rain I ~ + I ) , where r : ( r t , ~)- rERMCv)

b l a t e r i a l i z a t i o n o f M- o r E - r e s o n a n c e in a f a m i l y of m a p p i n g s {F e} i s d e f i n e d in t h e same way as f o r A - r e s o n a n c e s ; i n t h e d e f i n i t i o n o f s u b s e c t i o n 1 .5 i t i s o n l y n e c e s s a r y to r e p l a c e RA(~) by K~(v) o r RE(~) , and " e q u a t i o n s " by " m a p p i n g s " ; i n p l a c e o f (C n, 0) i n t h e d e f i n i t i o n o f m a t e r i a l i z a t i o n o f E - r e s o n a n c e s t h e r e mus t be (C- , K) whe re K i s t he a n n u l u s d e f i n e d in Remark 2 o f s u b s e c t i o n 3.

THEOREM lb. Each positive Poincar6 M-resonance of multiplicity k of order greater than two materializes in a k-dimensional family of general position of holomorphisms with fixed zero.

THEOREM ic. Each positive Poincar6 E-resonance of order greater than two materialzes in an (n -- l)-dimensional family of normal gluings of general position.

For resonances of multiplicity one these theorems were first proved by A. S. Pyartli without the condition on the order.

A theorem on nonmaterialization in the general case of Siegel M- and E-resonances is

apparently true but has not been proved.

6. Nonlinearizability. It is shown in Sec. 3 that materialization of a Poincar~ E-resonance implies the splitting off from the curve fF of an elliptic curve M(E) lying in a tubular neighborhood U of the curve fF and covering the initial curve fF in a finite-sheeted fashion. The restriction to M(e) of the natural projection U + f is smooth but, generally speaking, not holomorphic. The simplest analog of the theorem on nonlinearizability for elliptic curves is the remark of [3]: a nonresonance linear bundle of zero type over an elliptic curve has no holomorphic sections over any finite-sheeted covering of the base. Thus, a neighborhood U containing the curve M(E) cannot be mapped biholomorphically onto any neighborhood of

the curve F in the space of the normal bundle Vf.

7. A Countable Number of Invariant Manifolds and Convergence of Normal Series. THEOREM 3b. I. In the space C L* there exists an open set V M and a set ~M, thick in V M and for each ~ M there exists a set ~M, dense in the space of errors ~A and such that if v~M,

p ~ M , then the mapping

F : (C", O)---*(C", 0) ' , z--,-(diagv)z+F(z)

has a countable number of invariant curves.

part.

II. set ~0/, s

The mapping F is not equivalent to its linear

In the space C n there exists a thick set ~E, and for each v~E there exists a

dense in the space ~ and such that if v~E, ~ E , then the gluing

F : (C n, C*) -~(C" , C*)' , z-+(diag~)z+~(z)

has a countable number of invariant curves and is not equivalent to the linear gluing z +

(diag ~)z.

It can be proved that the invariant curves mentioned in the theorem obstruct the equiva-

lence of the mapping F to its linear part. This is done in the same way as for equations. However, we shall deduce the assertions on a countable number of invariant curves and on nonlinearizability independently from one another from the analogous theorem formulated below

for periodic differential equations on (C n, 0) and (C n, C*).

III. Imbedding Theorems

It is well known that an arbitrary diffeomorphism of a manifold M 4 M is, generally speaking, not included in a phase flow on M. On the other hand, it is not hard to construct

a nonautonomous periodic differential equation x = v(x, t) on M, v(x, t + 2~) = v(x, t), for which a given diffeomorphism f:M + M is the monodromy transformation after a period. The situation changes abruptly if M is a complex manifold and f is a holomorphism. Until recently,

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theorems of the imbedding of a holomorphism in a periodic system were unknown even for the case of germs of holomorphisms at a fixed point. The following theorem was communicated to one of the authors by V. I. Arnol'd.

THEOREM 4a. The germ of a holomorphism F:(C n, 0) § (C n, 0) can be realized as the monodromy transformations of an analytic equation with periodic right side and singular point 0.

This theorem admits a considerable generalization.

THEOREM 4b. Let M~C n be a Stein manifold such that any analytic vector bundle over US~ M is trivial (Usx is a tubular neighborhood of the oriented circle S x on C), and let F:(C n, M) § (C n, M) be an arbitrary germ of a holomorphism on M. Then there exists an analytic, periodic differential equation ~ = v(t, z) on (C n, M)' for which the manifold M is invariant, and F is the germ of the monodromy transformation after a period.

The manifold M = C* satisfies the condition of Theorem 4b, since the product U ~ • M is S

a Stein manifold homotopically equivalent to the torus T 2 (see the remark of subsection 11.2).

COROLLARY. The germ on C* of a gluing mapping can be realized as the monodromy transformation after a period of a periodic differential equation defined on some neighborhood of (C n, C*) for which the manifold C* is invariant.

The proofs of these theorems and possible generalizations are contained in Sec. 2.

IV. P- and PE-Equations

We call PE-equations periodic differential equations on (C n, C*) whose monodromy transformations are normal gluings (PE comes from periodic elliptic -- periodic equations arising in the theory of elliptic curves) and also equations of a broader class defined below. The exposition of the next subsection is such that if the notation of this subsection is given a different meaning part of what has been written becomes valid for periodic equations on (C n,

0).

i. PE-Equations. The following consideration motivates the general definition of PE-

equations.

A periodic equation a on (C n-~, C*) with invariant manifold C* can be considered an autonomous equation on the product ~ = US~ x (C n-~, C*), where USI is a neighborhood of the unit

circle on C. Let zl be a chart on USI, let z2 be a chart on C*, and let z = (zl, z2, z3, ..., z n) = (z~, z2, z") be a chart on The equation a is equivalent to the equation

Zt=2niz, , ~2=Z2(zJ, z2) .q-72(z), ~"=A(zi , z2)z"--kt"(z), (1.5)

where %[z-=0=0, ?[El=0----0, 0T'l =0 Oz" Iz".=o

The monodromy transformation of the equation a after a period coincides with the restriction to the plane z: = 1 of the phase-flow transformation of Eq. (1.5 after time 1 which for brevity we call the monodromy transformation of Eq. (1.5). Suppose now that this transformation is a normal gluing (C n-t, C*)-+(C n-i, C*), which in the notation of this subsection has the

form

F : (z2, z ) (d[agv(z2, z " ) + F ( z z , z"),

OF" z'=o P=(F , , F'),FI,--_o=0, -6P- =0, v=( '% . . . . . v,,),l%l=/:l. (1.6)

We consider the "part of Eq. (1.5) linear in zx and z":

or, I zl = 2~ izx, z2 = ~', (za, z~) + -~- ,.=o z', ~" = A (zl, z~) z'.

The monodromy transformation of this equation coincides with the linear part of the normal

gluing (z~, z")-+(di'ag v) (zz, z"),

i.e., is linear in all variables. Considerations used in Floquet theory show that there es- ists a holomorphic change Us, X (Cn--i," C*)-+Us, X (C n-l, C')', which is cylindrical in zx, pre-

serves Us, )<C* and reduces Eq. (1.5) to the form

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2 = (d~ag ~.)z+[(z), ~.= (2hi, ~ ) , "v=emp~,

t = (o, h , f') , I I~'=o = 0, af" = 0. OZ" zm=O

(*)

Remark. Since v:exp ~ and Iv2[ =/= 1 hence gt/A.~'l(

Notation. z=(z', z")~C n, z'=(zl, z2),

, we obtain Re~._,=/=O Further, X t : 2 ~ i , and

z " = (z~ . . . . . z . ) , f = if, , f~, V ) , A'=/X' .~ : : {z~C" I 0~<• Iz, I <• oo,

Definition i. The equation

is called a PE-equation.

~-~- (diag ~.)z+f, z ~ ( C " , A') ,

II,,.=o; IJ -I =o. az la"

(1.7)

(l.8)

(1.9)

The space of germs of holomorphic mappings f:(C n, A') + (C n, 0) satisfying condition (1.8) is denoted by 5r (we sometimes write ~v~ or ~r ~(A'), in order to emphasize the dependence on A'). If [~r , then

f(z)--= E [~ zpei' (p,i)EJ

w h e r e J = { ( p , J ) l J ~ (1 , n ) , p = ( P l , P2, P " ) ,

p ~ Z , p2~Z , p " ~ Z + "-'~, ]p"[>~l for j~<2, Ip"l>~2 npH i>2}.

D e f i n i t i o n 2 . A c o l l e c t i o n L ~ C n i s c a l l e d a P E - r e s o n a n c e i f (~ , p) = ~. f o r some (P, ]) ~ J 3

R ( X ) = { r ~ Z " ] ( ~ , r ) = 0 , ~ ] : (r-l-ej, ])~J}. ( 1 . 1 0 )

The s e t R(~) t h u s d e f i n e d i s s o m e t i m e s d e n o t e d by RpE(~)

D e f i n i t i o n 3 . As a l w a y s , t h e m u l t i p l i c i t y o f a r e s o n a n c e ~ i s t h e d i m e n s i o n o v e r R o f t h e l i n e a r h u l l o f t h e s e t R(~)

D e f i n i t i o n 4 . A r e s o n a n c e ~ i s p o s i t i v e i f f o r a l l r~R(~,), r~(r t , rm r"), a l l com- p o n e n t s o f t h e v e c t o r r " a r e n o n n e g a t i v e .

D e f i n i t i o n 5. The o r d e r o f a p o s i t i v e r e s o n a n c e ~ i s minlr"l-t-I erR(M

Definition 6. ~ is a Poincard resonance if 0 is an isolated point of the set {(~,p)--

~l (P, i) ~J}

The definition of materialization of a PE-resonance in the family of equations (1.7)- (1.9) repeats the definition of materialization of an A-resonance with the following changes:

in place of A-resonance we have PE-resonance;

in place of (C n, 0) we have (C n, A').

THEOREM id. Each positive k-fold Poincar4 PE-resonance of order greater than 2 materializes in a k-dimensional family of Eqs. (1.7)-(1.9) of general position.

2. P-Equations. We shall give the following meaning to the notation of subsection i: z = ( z l , z " ) = ( z l . . . . . Zn) , z ' - - z l .

A ' = {z~cn 1 0 < - , < Iz, I < • oo, z " : 0 } .

The set A' thus defined we sometimes denote by A'p;

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Oz a,

sometimes in place of ~ we write ~q~P ;

]=Z•

r ~ - (rl, r")----- ( r l ..... rn).

With this interpretation of &', J and r" Definition 2 gives a P-resonance, and Definitions 3-6 pertain to P-resonances; equality (i.i0) gives a set denoted by R(%) or Rp(%).

Definition i' A P-equation is an equation of the form (1.7), (1.8) on (C n, A'p),%1=/=0

Materialization of a P-resonance is defined in the same way as for a PE-resonance only in place of PE-resonance we read P-resonance.

THEOREM le. Each positive Poincar~ P-resonance of multiplicity k and order greater than 2 materializes in a k-dimensional family of general positive of Eqs. (1.7), (1.8).

def In considering A-equations A' is written A' A = {0}.

3. Reductions. Let F be a normal gluing (1.6), let F:(C n-~, C*)-+ (C n-l, C*)', let ~F be a PE-equation for which F is the monodromy transformation, and let ~ = 2hi, f~ = 0. Such an equation exists by the imbedding theorem. Then ~ = exp %. There arises the isomorphism

RE(v)'-~-Rp~(~), r--~(--(r, ~)/2~i, r). Because o f t h i s c o r r e s p o n d e n c e t h e E - r e s o n a n c e v and P E - r e s o n a n c e ~ h a v e t h e same m u l t i -

p l i c i t y and a common order; they are simultaneously positive and are simultaneously Poincar4

or Siegel resonances.

After these remarks Theorem ic follows from Theorem Id. Reduction of Theorem ib to Theorem le is carried out in the same way.

Definition. A holomorphism (C n, A')-+(C, A')" is called admissible if it differs from the identity by an element of the space ~ (for A-equations this definition coincides with the definition of subsection 1.7). A chart w in a region (C n, A') of coordinate z-space is called admissible if the transition function from the chart z to the chart w is an admissible

holomorphism.

THEOREM 3c (on a countable number of invariant manifolds for P- and PE-equations). In the space C n there exists an open cone Vp and sets ~p and ~PE, thick in Vp.and C L~, respectively. To each ~P(~PE) there corresponds a set ~;P(~PE), dense in the space ~P(~') (respectively ~P(A') ) (A' is arbitrary) such that if ~p, ~/~P (or ~PE, f~PE ), then any neighborhood of (C n, A') contains a countable number of two-dimensional invariant analytic manifolds of the equation z= (diag %)z + f(z). For any admissible chart on (C n, A') these manifolds, beginning with some one of them, are resonance manifolds and obstruct the equivalence of the equation to its linear part.

Supplement. The intersection of the sets ~p and ~Pm with the plane %~ = 2~i is thick

in this plane.

Just this theorem is the main result proved in Secs. 5-9. Theorem 3a is proved simultaneously with 3c. Theorem 3b is deduced from Theorem 3c by means of the imbedding theorem.

This reduction is carried out at the end of Sec. 9.

SEC. 2. IMBEDDING THEOREMS

In this section we prove Theorem 4a) Theorem 4b is proved in the same way, and we shall

not consider this. Possible generalizations are discussed.

i. Proof of Theorem 4a. Let F: (C n, 0)-~(C n, 0)', ~-+F(~) be a holomorphism. We consider a neighborhood of the real axis RcC in the product CXCn=C "+i, the vector field el = (i, 0, ..., 0) on C n+1 and the~gluing mapping ~ P: (C n+i, R)-+(C n+i, R)', (z0, ~)--~(z0-~l, F(~)). Identifying the points (zo, z) and F(zo, z), we obtain a manifold M with the annulus Us1 imbedded in it; the projection cn+1--~ C along the second factor becomes a holomorphic retraction w:M + US~ with fibers ~-It=(Cn, 0)t, t~U=, ; the holomorphy type of a fiber, generally speaking, depends on t. The gluing F respects the field el; it therefore goes over into a holo-

3063

morphic field v on M transversal to fibers of the retraction ~ with invariant manifold US,. The monodromy transformation corresponding to the field v on passing around the circle S ~ coincides with F. It remains for us to introduce the chart z by means of the mapping H: (A4, Us,)-~(C "+I, Um) so that on transversals of ~-ip. where p is the initial point of the loop S* ^ ~ its restriction coincides with the chart z on (C , 0), and the retraction ~ coincides with the projection (~, ~)--+z0 Then the field v = H,v will define an equation ~ = v(z) on (C n+*, US,) with monodromy transformation F on passing around S I

By Sue's theorem there exists a biholomorphic mapping H,:(M, Us,) § (C n+*, US,). The fibers H*~ -- *t go over into the planes zo = const by a holomorphism which changes only the first coordinate and is the identity on US, ; we denote it by H a �9 Finally, the required chart on the plane zo = 1 is provided by a change of the form H 3 (zo, z) § (Zo, ha(z)). The mapping H-----H s,H2oH i is the desired one.

2. Generalizations. This construction carries over to the real-analytic case and makes it possible, in particular, to prove the following theorem.

THEOREM. An analytic diffeomorphism of the circle onto itself preserving orientation can-be realized as the monodromy transformation of an analytic differential equation on the torus.

SEC. 3. THE GEOMETRY OF MATERIALIZED RESONANCES

In this section "resonance" means an A-, P-, or PE-resonance. Here we study the geometry of a "materialized resonance" -- a manifold M(e) defined by the equation of subsection

1.1.5:

= [ (w), w ~ (C ~, A'), s ~ (C *, 0),

~(w)= E ~'~/' (3.1) rER(;.)

~:(C n, A') -~ (C k, 0) is a local epimorphism.

Here R(1) and A' denote one of the sets RA(1) , Rp(1), RpE(1), A' A, A'p, A'pE.

We call the manifold M(e) defined by Eq. (3.1) a materialization of the resonance I.

i. Basic Definitions. The lemmas of this subsection follow easily from the theory of

integral modules [9, Sec. 85].

LEMMA 3.1. Let I be a resonance, and let M(I) be the Z-module generated by the set R(1).

Then the module M(I) has a basis belonging to R(1).

Definition 1 The basis r* r k mentioned in Lemma 1 is called a basis of the res- �9 , , , , ,

onance I. The matrix R with rows r*, ..., r k is called a basis matrix of the resonance.

Definition 2. We call an integral matrix simple if its invariant factors are equal to

i. We call a resonance basis simple if the corresponding basis matrix is simple.

LEMMA 3.2. Any resonance basis is simple.

Definition 3. A resonance l, having a basis generating R(I) over Z+ is called primitive.

The corresponding basis and basis matrix area also called primitive.

Definition 4. A positive primitive Poincare resonance is called convenient.

R wrk) ... r k then by definition w = (w rl, . . . . If R is a matrix with rows r*, , ,

2. THEOREM 5. Let % be a convenient resonance, and suppose its materialization M(E) is given by the mapping (3. i) of general position. Then M(E) is given by the equation

where ~ is the germ of a holomorphism (C k, 0)-+(C h, 0)

If H~j(E) # 0, then the manifold M(E) is connected�9

Proof�9 1 ~ We set u = w R. By Definition 3 ~(w)=9(u), ~: (C h, 0)-+(C ~, 0)' is holomorphic;

the requirement of general position on n can now be formulated as follows:

det o~ (0)~:0. (3�9

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The mapping ~ is invertible in a neighborhood of zero; let ~ = n-X. Obviously, the mapping is nondegenerate at zero.

2 ~ . We shall now prove that the manifold M(e) is connected. We set ~ = In___~_~ Con-

nectedness of the manifold M(e) is equivalent to the following condition. All planes Rm = + ~, where ~ is a fixed value of In~(~)/2~i (~(e)=f=0). and l~Z ~ is arbitrary, are obtained

from the plane Rm ffi a by translation by integral vectors or, equivalently, RZn~-Z ~

Now these equalities are equivalent to the simplicity of the matrix R.

Remark. Theorem 5 clarifies the naturality of the conditions of the theorem of Sec. 4 on nonlinearizability.

3. Description of Convenient Poincare Resonances. The material of this subsection is used in the proof of Theorems 3a-3e. A collection g~C m is called hyperbolic if for

LEMMA 3.3. If a hyperbolic A-resonance belonging to a Siegel domain is convenient, then for a suitable renumbering of the components all vectors %s ..... gn~C lie between the vectors --%~ and --%=. The primitive basis matrix R has the form

/r~ r,, r's 0""0 1 R=~rrx . .rr ' . . ? ~'t':'?. J (3.3)

\rn-=l rn-~ 0 0 . . . ~ _ 2 n /

Proof. 1 ~ Since % is a Poincar~ resonance, any Siegel triple of the collection (%:, ..., %n ) is commensurate. Let %i, %j, and %k be an arbitrary Siegel triple of the collection %, and let

be the corresponding resonance relation, where ~, b, and c~Z are positive and relatively prime. By hyperbolicity and positivity of the resonance the vector r = ae i + bej + ce k belongs to a primitive basis of the resonance.

2 ~ . Obviously, a hyperbolic collection ~ C ~ contains precisely two Siegel triples. We renumber the components of the vector % so that the common elements of these triples are %~ and %=. Obviously, the vectors %~ and %~ lie between --%~ and --%=, since the triples (%x, %~, %~) and (%:, %=, %,) are Siegel triples. The first assertion of the lemma for n = 4 is

proved.

Suppose now that n > 5 and % = (%: ..... %5 .... ), (%x ..... %,) ~A ~C4 We renumber the components %~, .~., %4 so that %3 and %4 lie between --%: and --%2, and we suppose that

%5 does not lie between --%x and --%2 (for b~evity we write a~(~, ~), if a vector ~C lies

between the vectors ~, ~C) Suppose, for example, ~(--L1,--~a) Then either ~.

(--~i,--~4), or ~5~(--~4,--~2). Hence, one of the triples (~i, ~,, ~5) or (~4, ~2, ~5) is a Sie-

gel triple. Moreover, the triples (%:, %2, %3), (%x, %2, %4), and (%x, %3, %s) or (%2, %3, %5) are Siegel triples. By part 1 ~ a resonance basis of % contains four vectors corresponding to these triples. Hence, the numbers %:, ..., %~ are connected by four independent integral linear relations and are hence linearly dependent over R, which contradicts the hyper-

bolicity of the resonance.

3 ~ The second assertion of the lemma follows from the first and the assertion of part

1 ~ The proof of the lemma is complete.

LEMMA 3.4. If a hyperbolic Poincar~ P-resonance is convenient and lies strictly within the Siegel domain, then all numbers %., except exactly one lie on one side of the line passing through the vector %~. After a suitable renumbering of variables the primitive basis matrix of a convenient hyperbolic P- or PE-resonance has the form (3.3); for P-resonances the ele -~ ments of all columns except the first and for PE-resonance the elements of all columns except

the first two are nonnegative.

Proof. We shall consider the case of P-resonances. In a manner similar to what was done

above it can be proved that if the collection (--%~, %j, %k ) or the collection (%x, %j, %k )

3065

forms a Siegel triple, then to it there corresponds a vector r = ae~ + be= + ce~ of a primitive basis such that al,+blj+cAh=0, a~Z, O~Z+, c~Z+ �9 The rest of the p~oof i~ the same as

in the preceding lemma.

LEMMA 3.5: If a matrix R has the form (3.3), the numbers rl ~= are pairwise relatively prime and greater than i, the numbers rji and rj,j+ 2 are relatively-prime, and II/%~ER, I)=/=0, R~=0, then ~ is a hyperbolic convenient resonance of multiplicity n -- 2 and R is its primitive basis matrix; ~ is a A-, P-, or PE-resonance if the elements of all columns of the matrix R, beginning respectively from the first, second, or third column, are nonnegative.

Proof. 1 ~ R is a simple matrix�9 Indeed, suppose a prime number p is an invariant fac- n--2

tot of the matrix. Then p divides the minor ~ O.i+~ , and hence p divides one of the num- !

bers rj,j+=, and only one because they are pairwise relatively prime. Suppose, for example,

this is the number r~,~. By hypothesis, p divides the minors r~.! ri.i+~ and r~.~ ri.~+~, 2

Hence, p divides r~,: and r~,= which contradicts the hypothesis.

2 ~ . The multiplicity of the resonance ~ is equal to n -- 2. Otherwise, this multiplicity would be equal to n -- i, and all the numbers ~. would lie on one line which contradicts

the hypothesis. 3

3 ~ The vector r = Z~.rJ is integral only for integral ~. (r j is the j-th column of the

matrix R). We suppose otherwise. Suppose, for example, that ~ is an irreducible fraction.

We consider the matrix

T = i a I a 2 ... fin--2 1 0 l 0 .

�9 . . . . , .

0 O...l

The matrix TR is integral. Its minors of order n -- 2 are equal to the product of det T with the minors of the matrix R standing at the same site. But det T = ~i, and all minors of the matrix TR are integral; therefore, the denominator of the fraction el divides all minors of

order n -- 2 of the matrix R which contradicts its simplicity.

4 ~ . If r = Za.r 3 and at least one of the coefficients ~ is negative, then r~R(l) Indeed, if at leastJone of the coefficients ~j is an irreducible fraction, then r~R(1) If all coefficients are integral and at least one of them is negative (suppose, for example,

this is am) , then rm+= = ~mrm,m+2 < -i, which is impossible for a vector in R(%).

The arguments of parts I~ ~ prove that R is a primitive basis matrix of the resonance

%, and this resonance is positive.

5 ~ . ~ is a hyperbolic Polncare resonance. Indeed, the vectors ~ and ~2 are not collin-

ear by hypothesis. The vectors l~, ~ and ~2, ~ are not collinear, since r~ ir~ 2r~ ~ + 2 # J J . J' J, -,J

0. Finally, the vectors h i and ~ for i >2, j >2 are not collinear. Indeed, it follows from the

simplicity of the matrix R that tie elements of each of its rows are relatively prime. Non-

collinearity of h i and lj now follows from the fact that ri,i+2 and rj,j+= are relatively prime. Finally, from the hyperbolicity of the collection ~ and the equality R% = 0, it follows that ~ is a Polncare resonance of type I. The proof of the lemma is complete.

This lemma is used in Sec. 8 in proving geometric divergence theorems.

4. The Structure of Poincar6 P- and PE-Resonances. The material of this subsection is

usedin a single place: in the proof of the theorem on materialization of resonances which

are not convenient.

LEMMA 3.6. Poincar4 P-resonances decompose into two (intersecting) sets:

Type I. All numbers %~ lie in a discrete subgroup with two generators of the group C �9 J

with addition (A-resonances of type I were described in exactly the same way).

Type II. The numbers ~. decompose into two sets: those belonging to the first lie on the line generated by the vector ~ and are commensurate; those belonging to the second lie

on one side of this line.

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Poincar4 PE-resonances always belong to type I.

Proof. Let X be a Poincar~ PE-resonance. The set {k~%i+kz%z-~kj%j[kl, kz~Z, kj~Z+} does not accumulate at zero (under the additional condition %i/X2~R ) if and only if X. can be expanded in the basis %~, %2 with rational coefficients. This proves the lemma for3PE-reso -

nances,

Suppose ~ is a Poincar~ P-resonance, and let ~/~,~ Then the set {ki,~i-~k~[ki~Z, k~Z+} does not accumulate at zero if and only if %~ and %j are commensurate. If on one side of the line generated by the vector ~ there are no vectors %j, then the P-resonance is of type II. Suppose the vectors ~2 and Xj lie on different sides of the line generated by ~. The set {ki~i~-k~-[-k~Ik~Z , k~Z+, k~Z+} does not accumulate at zero if and only if Xj can be expanded in the basis ~, ~= with rational coefficients. Hence, the resonance ~ is of Type I. The proof of the lemma is complete.

SEC. 4. NONLINEARIZABILITY

The theorems of this section show that analytic invariant manifolds of nonlinear equations and mappings arising in the materialization of resonances obstruct the reduction of the corresponding equation or mapping to linear normal form in a neighborhood containing a 'barge piece"

of the manifold.

Aside from the very elementary Theorem 2, in this section it is proved that the manifold arising in the materialization of a resonance of type I is analytic and is not a topological obstruction to the reduction of the equation to linear normal form. At the end of the section Theorem 2a is formulated which shows that if M is an analytic invariant manifold of a linear equation situated in a special manner in the complement to its invariant planes, then this equation is a resonance equation, and on the basis of the situation of the manifold M it is possible to recover a resonance basis. Only the material of subsections 4.1 and 4.2 is needed

for what follows.

i. Proof of THEOREM 2 of Subsection 1.1.6. We shall prove the theorem for I = 2 (the general case is investigated in the same way, and~the theorem for I = 2 is applied). Let be the projection w = (wl ..... w n) * (wl, w2) = w, w = (w, w), let M be a resonance invariant manifold, let T be the torus, {lwil=ci, [~zl-~-c2. @=0} , let U be a neighborhood of T on the plane w = 0, and suppose the covering ~:M + U is regular and finite-sheeted. Let d be the number of sheets of this covering. We consider the mapping ~:z-+w(z)=z D, where D = diag(d, d, i, .... i). The manifold ~ -:M consists of a finite number of connected components

each of which covers a neighborhood of the torus ~-ZT under the projection z § (z~, z=) in a regular and single-sheeted fashion. Let M" be one such component. It is given as the graph of the mapping ~:~- 'U-+{zI~=O } This graph is invariant relative to the equation

z~_D-IAz,

induced in z -space by the equa t ion ~ = hw under the mapping. Therefore (~--~(2))'~---0 ~-~-~(~), or

(diag f) , (~)=d-'~. (~)(diag ~ .

Since ~ 0 ( the mani fo ld M' does not belong to the p lane

for

(4.1)

Let ~(~= E s F~z' i=3

(z~, z=)) at least one of the components ~ is nonzero. Then by (4.1)

This is the desired relation for the components of the vector ~.

2. Admissible Changes. We call a holomorphic change (C n, A') + (C n, A') a normalizing change for A-, P-, or PE-equations if it reduces the equation to linear normal form. In Sec. 8 we shall prove that there is no admissible normalizing change for A-, P-, and PE-equations whose linear part is pathologically close to a countable number of resonances, while the non- linearity is sufficiently " strong." Here we wish to mention that the lack of an admissible normalizing change for these equations implies the lack of any normalizing change. For A- equations this is a well known fact (noted, for example, in [5]). The proof i3 the same for A-, P-, and PE-equations and goes as follows. Suppose % is a strongly nonresonant collection, i.e., the vectors %. are independent over Z, and suppose there exists a convergent normalizing

J

3067

change H I. There always exists a formal admissible normalizing change H ~ (the definition of a formal admissible change and the theorem on its existence are contained in subsection 5.3). The formal change g~=H1oH ~ preserves the linear normal form of an equation whose spectrum is strongly nonresonant by hypothesis. Thus the change ~ must be linear. Hence, the formal admissible change Ho~-~-I.H I converges -- contradiction.

3. Materialization of a Resonance of Type I Is Analytic and Is Not a Topological Ob- struc'tion to Linearizability. The proof of Theorem 2 used the analyticity of the manifold M in an essential way. It turns out that a topological and even smooth invariant manifold covering a neighborhood of the torus T under the projection N in a regular and finite-sheeted manner exists for any nondegenerate linear differential equation whose spectrum lies strictly within the Siegel domain (this observation is due to N. N. Ladis). Indeed, any equation ~ = Az, satisfying, for example, the relation %1 + %2 + k3 has the invariant manifold zlz2z3 = i. By a theorem of Ladis (N. N. Ladis, Differential'nye Uravneniya, 1__3, 2 (1977); Yu~ S. ll'yashenko, Funkts. Anal. Prilozh., i__ii, 2 (1977)) any equation ~ = (diag ~)z, for which the col-

--I --I) _ �9 �9 �9 -- lection (~i ' "''' ~n is R-linearly equivalent to the collection (%x ~, , kn ~), is topologically equivalent to the equation ~ = Az; the conjugating ~omeomorphism is infinitely smooth in the complement to the invariant planes of the equation z = Az. For any collection of numbers (~i -~, ~2 ~3 i), whose convex hull contains 0 strictly in its interior it is not hard to find a R-linearly equivalent collection (kj-t, kz-t %3-t), so that k1+k2q-ks=0, which proves Ladis' remark.

4. Recovery of the Basis of a Resonance on the Basis of the Invariant Manifold. THEOREM 2a. Suppose the equation ~ = (diag %)z has an analytic invariant 'manifold situated in the complement C *n to the coordinate planes zj = 0, and there exists a mapping of this manifold

into the manifold z R = i, homotopic to the identity in C *n. Let R be a simple matrix, and suppose the vectors k. are linearly independent over Z. Then R% = 0.

3 This theorem is proved in the samw way as Theorem 2 of [3]. It is not needed below, and

we shall not present the proof.

SEC. 5. THE FORMAL INVARIANT MANIFOLD

In Secs. 5-7 we prove Theorems la, id, le on materialization of resonances for A-, P-, and PE-equations. In Sec. 5 the equation of the formal invariant manifold is derived. In Sec. 6 the method of proof of convergence is illustrated for the example of the Poincar~ theorem. Finally, in Sec. 7 we prove convergence of the formal series found from the equations of Sec. 5. Below any assertion regarding "equations" pertains in equal measure to A-, P-, and PE- equations unless otherwise specified.

i. Formal A-, P-, and PE-Equations. The material of this subsection is a direct continu- z - - (zs ... . . z . ) , ation of subsection l.lV. Recall that in considering PE-equations ~=(z,, z2), "--

z~(z', z")~C", A'=A'pE={z~C"[xt<Izil<• x3<[z2]<x4, z'=0} ; in considering P-equations

z'=z, , z"= (z~ ..... z~) , z = (z', z") ~ C - , A ' = A ' ; = { z ~ C - I • z, I < x z , z"----0}; 0 < x , < x z < o o , 0 < • oo ; here all inequalities are strict in contrast to the definitions of subsection I.IV.

In considering A-equations we shall assume that z' = 0, z = z", A' = A' = {0}. A

Definition. A formal A-, P-, or PE-equation is an equation

~=~(e)z+T(Z , 8),A=diag ~, z~C n, (*)

where f is a formal Taylor series in z" with coefficients depending on z' and holomorphic in

a ' ; fl*' = o , OV'lOz"lv = 0

An analytic k-dimensional family of formal equations (more briefly, a formal family) is

a family of equations (*)

~=&(e)z+?.(z, e), e~B----(C h, 0) (5.1)

with base B, whereby f can be expanded in a formal Taylor series in z" with coefficients which are holomorphic functions of z', E in the domain A' • B, and

~ v • = 0, o 7 " I = 0 . 0z + la 'xB

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If f(x) and F(x, y) are formal series, then the restriction of F to the "formal manifold" y = f(x) is by definition the formal series F(x, f(x)).

Suppose (5.1) is a formal family. We convert it into a single equation adjoining ~ = 0:

~=A(~)z+T(z, ~), ~=0. (5.2)

The formal manifold e = ~(z) is invariant relative to Eq.(5.2) if ~ = 0, where ~(z)= $,(z) (Az+~z, ~(z))) ; we have in mind equality of formal series.

A mixed series is the formal series

a (z, e) = E a~q eq ~ e l ,

qEZ~. ~,J)E d

the coefficients of which for powers of z" are holomorphic functions on ~' • i.e., the ser-

ies E 4'P"q eq(z')/ converges on A' x B. q,p' We denote by W the space of mixed series, and by V the space of mixed series not depend-

ing on e.

2. The Operator adA. In this subsection in place of the variable z = (zx, ..., z n) we

write w = (wl .... , Wn).

Let A be a linear vector field on cn:A(w) = w. We denote by adA:V + V the operator of

commutation with the field A.

LEMMA 5.1. The operator adA has a complete set of eigenvectors of the form wPe. with eigenvalues (p, ~) -- %j, (j, p) ~ j. 3

The ~roof follows immediately from the formula

a d A :.h-+--Ah+h.A.

We d e n o t e by Vo t h e k e r n e l o f t h e o p e r a t o r a d A , and by V • t h e i n v a r i a n t s u b s p a c e o f t h e o p e r a t o r a d A c o m p l e m e n t a r y to Vo; we d e n o t e by ~o and n i t t ,e p r o j e c t i o n s V ~ Vo, h + ho , and V-+V 1, h-+h • along V• and Vo, respectively. We denote by ~F the restriction adAIV•

and Jo(%) and J• the sets

Jo(,Z) = {(j, p)~][ (p, t ) - -x j=O} ,

]• = ] \ ] 0 ( x ) We e x t e n d t h e o p e r a t o r s ad A , a o and ~• t o t h e e n t i r e s p a c e W: f o r e a c h f i x e d e t h e s e o p e r a - t o r s a c t a s d e f i n e d a b o v e . The n o t a t i o n Wo and W• i s i n t r o d u c e d i n t h e same way a s Yo and

VZ o n l y i n t h e s p a c e W.

D e f i n i t i o n . The m o n o m i a l s awPe i f o r (j, p)~Jo a r e c a l l e d r e s o n a n c e t e r m s ; m o n o m i a l s i n w p" f o r p~R(%) a r e c a l l e d r e s o n a n c e m o n o m i a l s .

LEMMA 5 . 2 . The o p e r a t o r ~ : a d A I V • i s i n v e r t i b l e .

Proof: Suppose

j i

~- , h = ~ ~ ~k e;,~en ~L = hU[(~, k ) - ~d" j •

Since the spectrum of the operator ~ is bounded away from zero, the coefficients of the series ~-lh for powers of w" converge in the same domain as the coefficients for powers of

w" of the series h.

3. A Special Normalizing Change: Definition. A formal change of the form id+h, h~V

is called admissible.

The next theorem shows, in particular, that for equations with nonresonant linear part

a normalizing formal admissible change always exists.

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THEOREM 7. There exists a change formal in w"

(5.3)

reducing Eq. (5.2) to the normal form

w-----,lw+~(w, ~), (5.4)

~=0.

whereby the series ~ contains only resonance terms: ~o~ = ~, while the series h contains only nonresonance terms: ~oh = 0. The coefficients of the series h, ~ for powers of w" converge in the domain A' • B.

A similar theorem for n = 2 without the condition ~oh = 0 was proved in [3] by the method of successive approximations. The general case can be handled similarly, and we shall not give the details.

Remark i. By means of a change which is polynomial in w" it is possible to "kill" the nonresonance terms of Eq. (5.2) to any prescribed order.

Remark 2. The right side of the j-th equation of the system (5.4) is divisible by w.. Indeed--~-~--the region of variation of wj is an annulus (h = 1 in P-equations and j = i, 2~ in PE-equations, then the series on the right side are permitted to contain negative powers of wj. If the region of variation of w. is a neighborhood of zero, then the resonance terms must contain a positive power of wj, sinc~ k is a positive resonance. Hence,

~=w#, where W = diag(wl, ..., Wn) and

~(~,~)= ~ g,(~)~, { o } u R(~.~

where gr(e) are power series converging on B.

4. Existence of a Formal Invariant Manifold. LEMMA 5.3. For transversal passage of

the family {~(s)} through a resonance k the corresponding equation (5.2) has a formal invariant manifold of the form e = ((w), and the expansion of ~ in a series formal in w" whose coefficients are convergent series in w' in the domain A' contains only resonance monomials.

Proof. Let R be a basis matrix of the resonance k. Considering Remark 2, the system

(5.3) can be rewritten in the form

~,=A~,+W~(w,~), ~=0 for we=0, ~=0.

Transversal passage of the family {a(e)} through the resonance k means that det 0 R g (w, 0~

~)I~-= ~ In order that a formal invariant manifold of the form e = ~(w), where ~ is a ser-

ies in resonance monomials, be invariant relative to the system (5.4) it suffices that for any r~R(X) the equality (wr) " = 0 hold. For this it suffices that

R#(w, ~)=0 for ~=,~(w).

The assertion of Lemma 5.3 now follows from the implicit function theorem. Convergence of the series ~ will be proved below. In the definition of materialization of resonances it is required that the mapping ~:w-+~(w). (On, A'~-+(C h, 0) be a local epimorphism. For this it suffices to impose conditions on a finite number of the Taylor coefficients of the series ~. We shall now write out these conditions and show that they are satisfied for nonlinearities

f of general position.

Let R be a basis matrix of the resonance k, which is primitive if the resonance k is convenient. We set u = w R. If k is a convenient resonance, then ~(w) = n(u) and g(w, E) = g'(u, E), where q and g' are formal Taylor series.

0q I (sometimes we write A(f) or A (5) in place of A). If det A # 0 We set A=-'~u u=o and the series ~ converges, then the mapping w § ~(w) is a local epimorphism. Differentiating the relation Rg'(u, q(u)) = 0 with respect to u at u -- 0, we obtain

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Remark. The inequality detA (f) # 0 is satisfied for errors f of general position (we recall that above the resonance E is convenient).

Suppose now that E is an arbitrary s resonance. Suppose the collection of vectors {rJ, qS} generates R(%) over Z+, uj = w rj, v s = wq s. Then E(w) = q(u, v), and q is a formal

Taylor series. If det =#0 and the series ~ converges, then the mapping ~ is a lo-

cal epimorphism. It is not hard to show that the last inequality is also satisfied for errors of general position.

5. Restriction to the Formal Invariant Manifold. According to Bryuno's idea, although a normalizing change ~ diverges its restriction to the formal invariant manifold may converge.

Let v and Vo be the right sides of Eqs. (5.2) and (5.4), respectively. We set

(v, f, H, h, Vo, g) = (v, f, H, h, Vo, g ) I ~ = ~ .

Since multiplication by resonance monomials commutes with the projections ~o and ~• and the series ~(w) consists only of resonance monomials, restriction to the formal invariant manifold takes Wo into Vo and W• into V• ; in particular,

W g ~ V o , h ~ V -L.

6. The Functional Equation. By definition of the change H

fl,5o:~oFl.

Because of the invariance of the manifold ~ = ~(W)

H.un~uoH o r

h A - - A h = f o t t - - W g - - h . W g . (*)

Since the fo rma l i n v a r i a n t man i f o l d s a t i s f i e s the equa t i on Rg(m, E) = O, i t f o l l o w s t h a t

Rg=O. From t h e c o n d i t i o n ~oh = 0 i t f o l l o w s t h a t

~0h=O. (5.6)

7. Transformation of the Functional Equation. We write the equation (*) in the form

ad Ah~-f,H--Wg'--h.Wg (5.7)

and apply to both sides of it the operator ~o. Considering Eq. (5.6), we obtain

no (foil) = Wg. From this we find g in terms of h and ~:

g = W-~no (y .H) . (5.8)

We recall that f./4(w):f(w~-h(w), ~(;e))

The equation Rg = 0 makes it possible to write down a relation connecting ~ and h:

RW-lu0 (f, H) =0. (5.9)

Finally, applying the projection ~• to both sides of Eq. (5.7) and noting that ~lh=h, we obtain

ad Ah = ~-1- (f.It) --h,WtL ( 5 . 1 0 )

Theorem 8 of S e c . 7 shows t h a t t h e f o r m a l s o l u t i o n s ~, h , and g o f Eqs . ( 5 . 8 ) - ( 5 . 1 0 ) c o n -

v e r g e ,

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SEC. 6. THE METHOD OF STRONGLY CONTRACTIVE OPERATORS

The concept ef a strongly contractive operator is introduced in this section. The majority of applications of the method of majorants known to us can be replaced by the principle of contraction mappings with the use of strongly contractive operators; estimates of the domain of the solution are hereby obtained automatically. Converence of the series ~ defining the invariant manifold will be proved in the next section using strongly contractive operators. The basic ideas of this proof will be demonstrated in this section for the example of Poincarff's theorem.

The Majorant Norm. We denote by ~ any of the spaces ~A (subsection i.I.i), ~P (subsection I.IV.2) or ~PE (subsection I.IV.I). Let the set A' be the same as in subsection 5.1, and let A O be a o-neighborhood of the set 8' in the norm [lwII=maxlwjl (w is to be

!

inserted in the definition of &' in place of z). We denote by ~p the space of germs f~, whose representers can be expanded in power series in w which converge absolutely on Ap. Let

be the operator ~,-+~p ,

,ett: ~. h~w"e, ,----~_,lhilw~'ei. ,I d

The operator J/ is defined on the space of matrices whose columns belong to ~r in a similar way. On the space ~r there is defined the usual norm

11 h 16 = max [ h (w) I

(the Euclidean modulus of the vector is intended) and the majorant norm

Ihl.=ll~h{6. we define For h~o, o>p

I h 6 = II(~h) . 16 = max II ( ~ h ) . II,

w h e r e I1"11 i s t h e norm o f a m a t r i x i n E u c l i d e a n s p a c e ,

2. A Stron$1y Contractive Operator. We denote by Qe the open ball in the space ~r with center at zero os rad-i-us p (the norm is the majorant norm). Let ~) :~-+~r be an arbitrary operator defined on all balls Q_ for sufficiently small p. We denote by ~pO) the Lipschitz constant of the operator @ on t~e ball Qp.

Definition. An operator ~):~r is called strongly contractive with constant q > 0 if for all sufficiently small p the following inequalities hold:

{cD0l,<qp2 and LPp(D< qp.

LEMMA 6 . 1 . A s t r o n g l y c o n t r a c t i v e o p e r a t o r has a f i x e d p o i n t a in any b a l l Qp f o r p <_ 1/2q and la[p < 2qp 2.

P r o o f . For p <__ 1/2q t h e o p e r a t o r @ t a k e s the b a l l QO i n t o i t s e l f , s i n c e f o r h~Qp.

Iq~h Io--< ImO I p+'l @h--@O I .<qpZ+qp.p~<p.

Since the operator ~IQp is contractive with coefficient qp <__ 0.5, the fixed point a can be

defined as follows:

whence

a = lim ~"0 = (DO + ~ ((D~ O-- (Ds-'O), tl---,-iiD

o o

E q p= [a6<qp= + (qp)Sqp== 1--qp s = l

---.< 2qp ~.

3. An Example of the Application of Strongly Contractive Operators. We consider the

A-equation

z - - - -Az+f (z ) , f ~ t A (n) ( 6 . 1 )

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with a nonresonance linear part of Poincar@ type.

Poincar@'s THEOREM. Equation (6.1) is analytically equivalent to its linear part.

Proof. Let

{zl IzJl <p}; let A be the vector field A (z) = z. In this subsection only we denote by L: ~A_+~A the operator inverse to commutation with the field A, and by v the right side of Eq. (6.1). The desired change H = id + h satisfies the equation H,A=wH of adAh=fo(id+h), whence

h = L f . ( i d + h ) .

The operator ad A in the space ~ has a full set of eigenvectors zPej with eigenvalues(~, p)~

%J, P~Z+ ~, IPI>| �9 By the hypothesis of the theorem its spectrum is bounded away from zero. Suppose all its eigenvalues exceed 6 in modulus. Then the norm of the operator L on all spaces ~q~p does not exceed 6 -~.

LEMMA 0. The operator ~:h-~f~(id~-h) is strongly contractive.

Remark. If I.IR in the definition of a strongly contractive operator is replaced by the assertion of the l~ma becomes a standard fact of analysis (see, for example, [17]). II" llp Proof of LEMMA 0. Let F=. /K f Since the germ f and hence also F have zero linear

part, there exist positive q and P, such that

Therefore,

a) I 010= Ifl0=ItFllp<qp b) Suppose a,b~Qp

I IF l l ,<qp 2, I lF. l lo<qp.

Th en

!

I tDa - - tDb [o = I , ! f . ( id + a + ~ (b - - a)) (b - - a) dt Io~< 0

max II(dt f). (id q- J~. a + t oJ~ (b - -a ) ) dt (b--a) lk,~< I F. la, l b - -a lo . tE[o,I]

Hence,

.~'~ ~ < [ F. [3o < 3q p.

Lemma 0 and with it Poincar~'s theorem have been proved.

SEC. 7. CONVERGENCE

In this section by means of strongly contractive operators we prove that Eqs. (5.8)- (5.10) have convergent solutions, and we deduce from this the theorem on materialization of resonances. We formulate the convergence theorem in subsection III together with the est&- mates needed for the theorems on a countable number of invariant manifolds.

We henceforth consider only Poincar~ resonances.

I. Preliminary Normal Form of a Family

If % is a resonance of order not lower than two and the family of equations {~(E)} for e = 0 passes through this resonance, then by a holomorphic change of variables it is possible to kill terms of low degree in e and z" in this family. Namely, we introduce the following

definition.

Definition. The order of the monomial eqzPej, q~Z+ k, (p, j) = J, is the quantity

lil + I I for j = 1 q + p" --i for j > i, if J = Jp;

+ p _ 1 for j > 2 if J = JPE" + for j < 2 and q + p"

Proposition i. Suppose {a(e)} is an analytic family (5.1) of equations on (C n, A') with base B = (C k, 0) whose linear parts for E = 0 pass through a positive resonance % of order greater than 2. Then in some neighborhood of (C n x B, &' x {0}) there exists a holomorphic

change reducing the family (5.1) to the following "preliminary normal form":

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i~=Az+ZI'~q-a(~, z), (7.1)

where Z = diag(z~, ..., Zn) , F is a constant n • k matrix, and the mixed series for a does not contain terms of order lower than 2.

Remark. It follows from this proposition that the error a satisfies assertion 1 ~ of Lem-

ma 7.1 below.

Proof of Propositon i. We shall consider the case of A-equations. Since the resonance

I is positive and its order is greater than two, all resonance terms at E = 0 have degree in z no lower than three. Terms of degree 0 in e and no greater than 3 in z can be annihilated by a change of variables. Terms of first degree in z and E are separated out into the second term; therefore, the error a contains only terms of order no lower than 3. This proves the

proposition for A-equations.

The definition of the order of monomials is specially coordinated with the definition of the order of a P- and PE-resonance: for positive P- and PE-resonances of order greater than two the resonance terms zPe. for any j have order no lower than three. The rest of the proof

is the same as for A-equatiOns.

The space of germs of mappings a(C n x B, A' x {0}) § (C n, 0), whose expansions in power

series in (z, E) do not contain terms of order lower than 2 we denote by ~ . We set

B . = { e l te j l<P} , ~.=& XBp,

and ~ is the space of germs a~E~ whose=Tepresenters can be expanded in power se r ies in the va r i ab les z, r converging absoZutely on Ap. The majorant norm [ 'Jp on the space ~p i s def ined by the e q u a l i t y

I z ~ ~ z~ ej I~ = max I Z i a~ I ~ z~ e~ I-

Further, we denote by 0 the space of germs of analytic mappings (C", A')-+(C k, O) ; we denote b__y_ Op the subspace of germs ~O, expandable in power series in z, absolutely converg-

ent on Ap with the naturally defined majorant norm l' "iO'~l~ We set gg=~XO, ~0=~pX �9 if y~p, y=(h, ~), h~d~p, ~Gp , then [y[p-----rnax([h[p, (the spaces ~r are defined in subsection 6.1). The projections ~A~-~Z~, ~A~--~ along the factors we denote by ~ and ~,

respectively.

II. Estimates Uniform with Respect to

Let ~ be a compact set in C n such that %j[~ # 0, Im(%i/% j) # 0, for i # j for any %~.

Agreement. For resonances of type I positive constants depending on the norm laIo and

the dimensions of the domain &o and common to all resonances %~ , we denote by C, C:, C2, C', C" .... In different assertions the same symbol may denote different constants. Since for resonances of type II the existence of a countable number of invariant manifolds will not

be proved, for these resonances the constants C, CI, C', ... are allowed to depend on %. If in inequalities containing ['[p it is not indicated for which p they are valid, then we assume

that they are satisfied for all sufficiently small p.

In analogy with the partition z = (z', z") for any error of the family (7.1) we set

a' = 0, a" = a, if (7.1) is a family of A-equations;

a' = (al, 0 ..... 0), a" = (0, a2, a3, ..., an) , if (7.1) is a family of P-equations;

a' = (a~, a2, 0 .... , 0), a" = (0, 0, a3, ..., an) , if (7.1) is a family of PE-equations.

For brevity we set v=(z, ~)~, v~ O),y=(h, ~)~

LEMMA 7.1. For each a~o there exists a C > 0, such that the following assertions

hold:

1 o. la'lo<Cp~, la"10<cp~ fo~ p < o ;

2 ~ la' .~10<Cp, la" .~ l~<Cp ~ fo~ p < o / 2 ;

3 o. [w(w+y,)-w(o0+y~)I~<Cply ' -y*lo,

la"(vO+y,)-a"(v~+yD.lo<Cp~ly'-y*l~ fo~ p<a/6,

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Remark. If the assertions of the lemma hold with different constants C. in place of C, then they are also valid with a common constant C.

Proof. The first two assertions follow from the definition of the space ~ and the Cauchy estimate; we shall prove assertion 3 ~ .

Proposition 2. Let ~ , ~ , l~l~<P Then

lioCv~ Ip<4 I/:1.,,,. We carry out the proof for ~ In the other cases the proof is similar.

Let f = fo + f~ + f2 + fa where fo is a Taylor series in all variables; for fJ, j -- i 2, the series contains w. only in negative powers while the remaining variables are contained in nonnegative powers; 3fa is the series containing w: and w= only in negative powers. We set v -- (wa, ..., w n, 0 .... , 0), y = (ha, ..., hn, ~i, ..., ~k ), i.e., v ~ = (w~, w2, v), y = (y~, y=, y). Let

> = (~ f~ (~ + ~ y),

T, = (~ f~). (=~ - ~ y,, =~ + ~ ~, ~ + ~ ~),

p=(~p).(w~ +~y~, ~-~y~, ~+ ~),

It can be easily verified that the series fJ majorizes the series JKp o (u0+9) Further, l~ilp=lltilIp (for series with positive terms the norms l'Ip and ll'llp coincide);

I17 ill~< llJCf~ll2,= I fJ i 2,,

but IfJl < Ifl p, since by deleting terms of the series we do not increase the majorant form. Th s7 10<l l 0, which proves the proposition.

The third assertion of Lemma 7.1 now follows from the first two by means of the computa- tions used in the proof of Lenlna 0.

LEMMA 7.2. For all ~ the inequality I~2/~iI > C holds.

This follows from the compactness of ~.

LEMMA 7.3. Let ~, g~C n and suppose g = ~ + B~, ~, ~C Then I~I < Clg I,

Isl < clgl.

Proof. Let T(~)= ~ ~. Since Im(~a/X~) # 0 in ~, it follows that det T(X) # 0.

Further (~)= T-l(~)(gl).~ g~ By the compactness of ~ we have the inequality HT-~(~)Ii<C,

whence the assertion of the lermna follows.

LEMMA 7.4. Suppose R is a primitive basis matrix (3.3) of a hyperbolic convenient res-

onance ~, ~ Then C'r i i+2 < [rijl < Cri i+ ~, i ~ (i, k), j = i, 2.

Proof. The vector ri-- -- (ri~, ri2, r i i+2) is orthogonal to the plane spanned by the v_ec- tors % = (%~, %2, %i ) and ~, and is hence proportional to their vector product n(%) = [~, ~]. By hypothesis the collection % is hyperbolic. Hence, the vector n(%) has nonzero components. Because of the compactmess of ~ the assertion of the lemma now follows from this.

III. Formulations

THEOREM 8. Suppose the linear part of the family {~(e)} passes transversally through a

positive resonance % of order greater than two. Then

i) the formal solutions h, ~, g of Eqs. (5.8)-(5.10) converge;

2) the mapping w + w + h(w) is a holomorphism in some neighborhood of the set A', and

the resonance % materializes in the family {~(e)}.

To prove Theorems 3a-3c on the existence of an infinite number of invariant manifolds we need the estimates of the domain of the functions E, h and the ranges of these functions which

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will be obtained during the course of the proof of Theorem 8. In order that it not be neces~ sary to search for these estimates in the proof itself, we present them in the following supplement.

Supplement. Let A be a convenient hyperbolic resonance with primitive basis matrix R of the form (3.3), and let a~.~o, ~ = m a x r j j + z . ,

F =

k

oo...o 0 0 . . . 0 I l o . . . O l 0 1 . . . 0 I

o" o'. "11

n, k = n - - 2 . (7.2)

Then the series 6, h constructed for the family (7.1) converge in the region &po:

po : C ~ - k ,

and for 0 < po

Ih[o< C~p~P ~, [glo < C]p~' pZ. (7.3)

Remark. Theorems la, d, and e on materialization of resonances follow from Theorem 8, the supplement to it, and the remark at the end of subsection 5.4. Indeed, the first of inequalities (7.3) shows that (id + h)-X:z + w(z) is a holomorphism of some neighborhood of (C n, &') § (C n, &')' In a neighborhood of (C n, A') there is an invariant manifold M(e) of the equation ~(e) given by virtue of Lemma 5.3 by the equation

~=~(w);

the series for ~ consists of resonance monomials.~ The local epimorphic property of the mapping ~: (C n, ~')-+(C k, 0) holds for nonlinearities f of general position by the remarks at the end of subsection 5.4.

(7.2')

IV. Proof of THEOREM 8

The proof will be broken down into a number of parts. It will contain some lemmas needed to prove the Supplement. The proofs of these lemmas will be presented at the end of the section.

Below R denotes a primitive basis matrix for a hyperbolic convenient resonance and a simple basis matrix for any other resonance.

Transversal passage of the linear part of the family (7.1) through the resonance means that det RF # 0.

i. Transformation of the Functional Equation. In Sec. 5 the formal series h and ~ were written for the variable w = (wl, ..., Wn). Noting that the family (5.1) is written in the

form (7.1) and setting ~ = diag (hl, ..., hn), we obtain

f (z) = i (z, ~(w(z))) =zr~(w (z) ) +a ( z , ~(w(z) ) ), foil= wr~+~r~+a(w+h, ~).

We o b s e r v e t h a t n ~ r ~ = O , ~ i g ~ r ~ = ~ r ~ , s i n c e ~oh = 0 and Wo$ = ~, and hence , m u l t i p l i c a t i o n of 9~r on the l e f t by ~ p r e s e r v e s a l l monomials in the expans ion in the s e r i e s by non re son - ances.

From (5.9) we obtain

whence

o r

RW-tno(WP~+a(w+h, $ ) ) = 0 ,

RF~=--RW-I~oa (w +h, ~),

~=--(Rr)-,RW-'noa(w+h, ~).

In the term WF~ of Eq. (5.8) we replace ~ by the expression (7.4):

(7.4)

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g = W-~noa(w+h, ~). ( 7 . 5 )

We set L----(adAlvx)-' ; the operator L exists by Lemma 5.2. equality ~• it follows that

h.=L (~F~+n.i-a(~+h, ~)--h.Wg).

From Eq. (5.10) and the

(7.6)

2. Passage to Operators. We seek a solution h, g, g of the functional equation (7.4)- (7.6) whereby (h, g)~r and g is expressed in terms of h and g by formula (7.5); substitut- ing this expression for g into (7.6), we reduce Eqs. (7.4)-(7.6) to equations for h and g. A solution of the equations^obtained will be found as the fixed point of a certain strongly contractive operator 5~5~ . Everywhere below if .Y~-(h, ~)~5~ , then h=~ty, ~=nzy. We define the following operators from 5~ to 5~:

Om~ _" y--+(a(w+h, ~), 0), ": y-+(n.l-a(w+h, ~), 0),

�9 3 : y---+(Lstia(w+h, g), 0), Wo : y-.-~(W-hxoa(w+h, ~), 0), �9 , : W-'noa(~+h, ~), 0), ~ . : y--~(O, --(RF)-iRW-tnoa(w+h, ~)), ~Fs : y--~(9~F~, 0), q~ : y--+(Lo~Fg, 0), E : y-+(--Lh, Wg, 0), Where g~---ni~gty.

Proposition 5. If y = (h, ~) is a fixed point of the operator @ = ~ 3 + ~ 4 + E + ~ 2 : y = @ y , then the collection h, ~, g = ~i~iy is a solution of Eqs. (7.4)-(7.6).

Proof. The equality g = ~t~ly coincides with Eq. (7.5); the equality h = ~xCy coincides

with Eq. (7.6); the equality ~ = ~2r coincides with Eq. (7.4).

To prove convergence of solutions of the system (7.4)-(7.6) it suffices to show that the operator Cs + ~4 + E + ~2 is strongly contractive and use Lemma 6.1.

3. The Operators r r Are Strongly Contractive with Constant C. It follows from Lem- ma 7.1 that l~i01.~<Jal.<Cp2 and .~p~)1<Cp ; therefore r is a strongly contractive operator. The operator ~• is linear and does not exceed l.lp; therefore, r is a strongly con-

tractive operator with constant C.

4. The Operator ~o Is Strongly Contractive with Constant C. For the proof it is conven-

ient to introduce the following notation:

-I _i Wn-1 ) (W-I) ' = diag(wl -I, w= , 0, ..., 0), (W-I) '' = diag (0, 0, ws , ..., for PE-equa-

tions ;

(W-~) ' = diag (wt -I, 0, ..., 0), (W-t) '' = diag (0, w2 -I, ..., Wn -~) for P-equations;

(W-~) ' = 0, (W--~) ''= W -I for A-equations.

We consider two operators ,.~'-~,~: �9 'o : y ~ ( (W-')'~oa'(w+h, ~), 0), ~"o : y--+( (W-t)" noa" (w+h, g), 0).

Since ~o = ~' o + ~" o , it suffices to show that the operators ~'o and ~" are strongly

contractive with constant C.

We shall investigate the operator ~' o. For A-equations ~'o = 0. For P- and PE-equations the operator (a, 0)-+((W-1)'~0a ,, 0) is bounded, since the operator no does not increase the norm, and I I ( W - t ) ' [ & l J < C ' (the variable wl for P-equations and the variables w~, wa for PE-equations in the region Ap are bounded away from zero). From Lemma 7.1 for a' it now fol-

lows that

I V'o (o) I o < C p ~, I~e'o (y , ) -V'o (y~) I~< C ' o l y ' - y Z l �9

We shall investigate the operator ~"o. The norm ll(W-9"l&ll is unbounded; we use the

following modification of Schwarz' lemma.

LEMMA 7.5. If a function f is holomorphic in the disk Izl < P and f(0) = 0, then Iz-'fJd~<p-ll[Ip ([-I0 is the majorant norm in the space of power series converging absolutely on

the closed disk).

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From Lemma 7.1 for a" and Lena 7.5 it follows that

,IV%(0) Io<C'p =, IV"o(~')--V"o(~=)Io<Cplr which was requ i red to prove.

5. The Operators ~ and ~= Are Strongly Contractive with Constants N1, N=. This assertion follows easily from the fact that ~o is a strongly contractive operator, and

N , = Cl l--r (RF) -~R + E l l . N== Cll (RF)-~R II.

To prove the Supplement we obtain an upper bound for the constants N~ and N=.

Noting the form of the matrices R and ~ (see (3.3) and (7.2)), we obtain RF = diag(rla,

r:~, ..., rkn) ,

( r nr]~ ~ r ~ r ] ~ ' 1 0... 0\ (RF) -~/;t= r~r~ ~ r~r~' 0 1 0 ) .

' - ; . . . . - ' i . . . . . .

r~r~ . r ~ r , . 0 0 . . . 1

By Lemma 7.4 II(Rr)-IRI]<C ' , and hence NI = C1, N2 = C2 (see the Agreement of subsection II).

6. The Operator ~a Is Stron$1y Contractive with Constant C. It is obvious that ~aO = 0. Further, if ~,~=diagh~,j=l, 2 , then

I ~ , r t , - - ~ r t ~ l o . < I ( ~ ' - - ~ ) r ~ ' Io+ I ~r ~) I p- It is easy to see that

therefore,

Similarly, we obtain

I1~ ( ( ~ - - ~ ' - ) r~ ~) II ~ l l ~ (~ , ~2)II-I lr l l- I1~11, I1~ (~ ' - -~=) I I ~ l l~ (h ' - -h ~) II,

I (~ ' - -~ ) r~ ' l .< , l ,m- -h ' - l . l l r l l I~' I.~<Cplh'--h=l."

1~2F(~' -~ =) I. < Cpl~'-~21. - Hence, I ,~""""""~ir~'--,:~)~'2r~ 2 I p < Cp l y , - - y = I~

7. The Operators @a and ~, Are Strongly Contractive. We set N=max l(q,~)-Ll Since (q.~)~O (q~ej,j)EJ

the action of the operator L reduces to division of the coefficients of wP+ejej by the quantity (p, %) # 0, it follows that IILI]~N Since the operators @= and ~3 are strongly contractive, the operators Ca and ~4 are also strongly contractive with consant CN.

To prove the Supplement we must estimate N.

LEMMA 7.6. If X is a convenient hyperbolic resonance with primitive basis matrix R, ~ , then N < C~ k (~ is defined in the formulation of the Supplement).

Thus, under the conditions of the Supplement it may be assumed that the operators @3 and ~4 are strongly contractive with constant C~ k.

8. The Operator Z is Strongly Contractive. It is obvious that Z0 = 0. Further,

Lb. We = ~ ((p, ~) - - .~A- ' h~ (p, g) w ~ ej. jl(x) (7.7)

LE~VIA 7.7. Suppose % is a resonance with basis matrix R, g~C", RE~0, q~Z", ~ j : (q~-ej, j)~], (q. ~):~ . For a resonance of type I we additionally assume that ~ Then l(q,

g) l < C I$I IgI,where Igl is the Euclidean modulus of the vector.

To estimate the quantity ((p, %) -- X.)-I (p, g) we apply this lemma, setting q = p -- e.. Since (p, ~) -- %j = (q, %), (p, g) = (q, ~) + gj, it follows that 3

I((P, L)--M)-'(P, e)I <Clel +#, le l < C , N l e l .

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From (7.7) we obtain

Let V~yi= (gol) Then

ILh.~gl,, < C,Nlgl,lhl0.

Ey~- -Zy~ = Lb. ~lVg ~ - - Lh,~IVg2= L (h'--hZ), IVg'-~Lh.alV (g~--g~).

A c c o r d i n g t o t h e p r e c e d i n g e s t i m a t e

IZU'--ZY I~ le'l~176 le,--e lo.

Since the operator ~ is strongly contractive with constant N~, it follows that

Ie'--e Io<N, Since for small p the operator ~ maps the ball Qp into itself, it follows that Igllp Therefore,

< p.

IEU'--EUzI.<C,NoIh'--hZI.+C,NpN, 19'--9~[~ C,Np(I-I-N,) Iv,-u~l,. Thus, the operator Z is strongly contractive. Under the conditions of the Supplement its constant has the form Cm k.

9. Completion of the Proof of the Theorem and the Supplement. The operator ~ = ~ +

~ + E + ~2 is strongly contractive and by Lemma 6.1 it has a fixed point in the ball Q^ for sufficiently small p. The latter means that the equation y = ~y and together with it t~e system (7.4)-(7.6) have as solution the series h, ~, g converging in the region A 0 for sufficiently small p. The proof of Theorem 8 is complete.

To prove the Supplement we not~ that the operator ~= ~a + ~4 + E + ~2 is strongly contractive with constant q = CI + C2m < Cam k. By Lemma 6.1 this operator has a fixed point in any ball Qp for

P < P o = 0 . 5 0 ; - ' ( o - ~ = Co~ - k < I , 2q 2 and the norm of this fixed point in the ball Qp does not exceed 0o-~p , i.e., the solutions

' �9 < P o - J~ l < P ~ ~ h ~ of Eqs (7.4)-(7.6) converge in the region ~p for p < Po and lhlp P The proof of the Supplement is complete.

V. Proofs of the LEMMAS

Lemma 7.5 is proved in the same way as Schwarz' Lemma.

Proof of LEMMA 7.6. Since R is a primitive basis matrix for a resonance of the form (3.3), it follows that %i,j~(8, n) are linear combinations of X I and X2 with rational coefficients, whereby the denominators of the coefficients are equal to rj,j+2 j~(l, k) Hence, all vectors %j, j~(l, n) as well as the vectors (p, ~), p~Z n lie at the nodes of the lattice generated by the vectors ~-IXi and ~-~X:, wher~ ~ is the smallest common multiple of the numbers rj,j+2, j~{l, k) . It is obvious that ~ < m . The assertion of the lemma follows from the fact that I(P, X)l is not less than the distance from the nearest node of the lattice to zero.

Proof of LEMMA 7.7. Suppose X is a resonance of type I. We consider the plane N in C n of dimension 2 o~thogonal to the linear hull of R(X). It is obvious that X, ~ g~H and the vectors X, X form a basis in H. Therefore, g = a~X + ~2X.

By lemma 7.3 lail<Ctlg[, lazl<Ctlgl , and hence

(q, g) ~-ai(q, 2~) -I-o.2 (q, ~) ~-a16-l-a26-- According to the estimate obtained,

I(q, g)l<C, lel 181+C, lel la]<Cl61 lel. Suppose X is a P-resonance of type II (see Lemma 3.6; for A-resonances of type II the

proof is similar). Let X2, "'', Xl lie on the line generated by the vector X~; let X~+I, ...,

Xn lie on one side of this line. For each vector a~C n we set d=at, .... at; d=a~+i,..., an We recall that if (q+ej, j)~Jp , then all components of the vector q = ej except the first

3 0 7 9

are nonnegative. Therefore, there exists C such that for any vector q satisfying the hypothesis of the lemma either (~, ~):0, or ](#, ~)[>Ci~ [ We further note that all vectors Xi ..... ~l are pairwise commensurate and Rg = 0; therefore, ~=gi%i-i~

a. Let (#,~)-----0 . Then (#, ~) :0 , and

I(q, e ) l= l ($ , g)l = 181<C, lel 181. b. Suppose' (#, ~) ~=0 Then

and I#1 < C - ' l (#, E) I <c-'161

I'(q, e)I-<1 (#, )I+1 (#, (c,+c-,)I 1181. The proof o f the 1emma i s complete.

SEC. 8. A COUNTABLE NUMBER OF INVARIANT MANIFOLDS

The central concern of this section is a lower bound for the dimensions of a basis of the family of invariant manifolds arising in materialization of a resonance X in the family of equations

~:Az+Zrs+f(z). (8.1)

Below A = diag X, F is a matrix of the form (7.2), and fEA, z~(C n, 8') The desired estimate must be uniform with respect to all resonances in some region ~cC" and with respect to all errors in some thick set (which, in particular, was the motivation for the Agreement of subsection 7.2). The error f must make the family (8.1) essentially nonlinear; this means that some nonlinear terms in the formal normal form of the family (8.1) must be sufficiently large. In order to prove thickness of the set of such errors we must learn how to estimate the change of several terms of the normal form at once as the error f varies. It is possible

to do this if the elements of the basis matrix of the resonance % are not too far from one another, and this is the motivation for considering 'b-regular matrices" in subsection 8.1.

At the beginning of the section a set of exceptional points %~C" is constructed --

points pathologically close to a countable number of resonances with m-regular basis matrices, and their thickness is proved. At the end of the section we prove the theorem on a countable number of invariant manifolds.

i. Exceptional Points. Everywhere below the family of manifolds arising in the materialization of a resonance is called simply a '~aterialization of the resonance." The word "resonance" means only a hyperbolic, convenient resonance; the multiplicity of such a resonance is equal to n -- 2 by Lemmas 3.3 and 3.4. A materialization of resonances is a family of two-dimensional manifolds. The primitive basis matrix (henceforth called simply a basis matrix for simplicity) of the resonance has the form (see subsection 3.3)

R =

t r~l r~2 r~8 0 ... 0 \ ,,; r. o o ) rkl rk~ 0 0 . r~

(8 .2)

where k = n -- 2. We set X = min r i j+2, m -- max rj j+2. If R is a basis matrix of the reson-

ance X, then we write R($), m(%), -X(%); to the ma-ttix R s there correspond constants Xs, m s �9

Definition i. An integral matrix R is called m-regular, if

1 ~ R is the basis matrix of some resonance, and the numbers rj j+2 are pairwise rela-

tively prime.

2 ~ . m--x<m.

We proceed to the definition of exceptional points.

We first define the sets VA, Vp and VpE. They consist of all %EG n satisfying the

conditions :

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1 ~ ~i/%j ~ ~ for i # j; the points %~VpE are otherwise arbitrary.

2 ~ for ~VA the vectors ~3, "'', %n lie between --~ and --~:; for ~Vp the vectors --%2, %3, ..., %n lie on one side of the line passing through 0 and %1.

The set V denotes VA, Vp, and VpE , respectively, in considering A-, P-, and PE-equations. It is obvious that V is open.

Let ~V be a domain with compact closure. By Lemma 7.4 the basis matrix (8.2) of the resonance %~ satisfies the conditions

~' rH +2 < I ri~ I < ~ r ji.+2, (8 .3 )

where the constants ~, ~' depend only on ~: ~ = E(~), ~' = ~'(~) (in order to distinguish these constants from other, less important constants we denote them by Gothic rather than La- tin letters).

Definition 2. Let ~V be an arbitrary domain with compact closure, and let ~{~) be the corresponding constant. A point ~V is called (~, m)-exceptional if there exists a sequence of resonances ~s~, converging to %, whereby

i ~ the basis matrices R S = R(~ S) are m-regular;

2 ~ for some T

( 8 .4 ) ~>2(2g+l)k

and for each matrix R~{Rs}

IR~i < ~ - ~ ; ( 8 . 5 )

3 ~ the collection ~ is strongly nonresonant, i.e., (~, p) # 0 for any p~Zn--{0}.

A point % is called exceptional if it is (~, m)-exceptional for some ~ and mo

THEOREM 9. The set of exceptional points is thick in V.

Theorem 9 follows easily from the following lemma.

LEMMA 8.1. There exists m(k) such that for each s the set of resonances whose basis ma-

trices are m-regular and for which X > s is dense in V.

To prove Lemma 8.1 we need two more lemmas.

LEMMA 8.2. For any k there exists m such that for any X there are k pairwise relatively

prime numbers rj:x < rl < ... < rk, such that r k -- rl < m.

Proof. Let p~, "''' Pk-- k be the first k prime numbers. The numbers rj = XPk ! + pj

are the desired ones, m = Pk"

LEMMA 8.3. Let p/q be an arbitrary fraction. If q is sufficiently large then the numer- ator p can be changed by a quantity not exceeding 2/qq in such a way that the fraction obtained

becomes irreducible.

Proof, It suffices to consider the case p < q. As is known [14], on the intervals (p, p + fqq) and (/qq, 2/qq) for sufficiently large q there are no more than three prime numbers. If p > /qq, then at least one of them does not divide q; we denote it by pl. The frac-

tion Pl/q is the desired one.

If p < ~q, then the desired fraction is el/q, where pl~(~q , 2Vq) is a prime number

not dividing q.*

We shall prove Lemma 8.1 for A-resonances.

Let %~VA be an arbitrary point, and let U be a neighborhood of it; let ~>s. By

definition of V A, for j > 2

where ~. > 0, B. > 0. By Lemma 8.2 there exist pairwise relatively prime numbers r., such o J that ~<rl<... <r~ and r k -- r~ < m. By Lemma 8.3 the numbers ~j, B~. can be appr]oximated

o

*The authors are grateful to N. M. Korobov for providing them with this proof.

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by irreducible fractions of the form p_-./r. PJa/r'' respectively, with error no greater than i/s + 2/r We define the matrix R ofJthe 3' J form (3.3) and the resonance ~ by the equalities

ri~ = Pi l , ri~ = Pi~, r i l + 2 : r i ; (8.6)

rii+= r i l ,~

For sufficiently large g )/~-U By Lemma 3.5 %' is a hyperbolic convenient resonance with basis matrix R which was required to prove.

The case of P- and PE-resonances can be handled similarly.

Remark. If the initial point % lies on the plane %1 = 2~i, then the resonance %' close to % constructed in the proof of Lemma 8.1 lies in the same plane.

COROLLARY. There exists m such that the set of resonances with an m-regular basis matrix is dense in the intersection V~{~l~-2~i}. The set of exceptional points is thick in this same intersection.

2. (d, %~-Nondegenerate Germs and the Theorem on Materialization of Resonances with Es- timates. Let ~ be a resonance, let R be its basis matrix, and let u = w ~. By Theorem 8 a materialization of the resonance % in the family (8.1) is given in a chart w on (C n, A') by

the equation ~(w) = E.

As in subsection 5.4, ~(w) = n(u), A= a~du lu=0 (we sometimes write A~(f)).

Definition 3. Let d > 0 and let ~ be a resonance. A germ f~A is called (d, %)-nondegenerate if

IIA~ -~ (f) il <d'(~L ( 8 . 7 )

The following theorem is the central feature of the section. It refines Theorem 8 and for the case of PE-equations shows that materialization of a resonance in the family (8.1) is

a family of two-dimensional manifolds M(E) which on passage of g through 0 separate from the plane (wl, w=) and for small E cannot pass too far from this plane. For A- and P-equations the analogous assertion is false. In Theorem i0 we shall trace the 'piece" of the invariant manifold arising on materialization of A- and P-resonances and remaining close to the plane

(wl, wf).

Let N be the projection w § (w~, wf), let the domain A 0 be the same as in subsection 6.2,

and let A ~ be the plane w3 = ... = w = 0. We set n

3

Op = Aps/4 N {w I I w, I > p ~, i = l, 2; I w~l < o ~, ] > 2}. ( 8 . 8 )

Let C be the same constant as in inequality (7.2').

THEOREM i0. Let ~ V be a domain with compact closure, and let ~=~(Q), k=n--2, d>0, T>2(2~q-l)k. Suppose that the resonance ~ belongs to ~ and the germ f is (d, ~)-nondegenerate. Then, if m = ~(%) is sufficiently large, then the equation a(E),

z=Az+ZFe+f(z )

for IE] < Eo = ~-x~ has an invariant manifold M(E), which in some chart w on (C n, 5') is giv-

en by the equation

whereby

1 ~ .

2 ~ .

o.

wR=r

~is a biholomorphic mapping defined in the ball IE[ < go; ~(0) = 0;

if p(03) = C~ -k, then

II (M (e) ND~(~)) = A~

the transition function H:z = H(w) is holomorphic in Dp(~).

For the proof of this theorem a lemmais required which estimates the size of the region

in which the inverse mapping is defined.

(8.9)

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LEMMA 8.4. Suppose the mapping ~:x-+y, x,y~Cn is defined in the region Ix[<8,

~(0)----0, B= O~'lHx ~=0' detB~=0, l~le=M Then the inverse mapping ~-~ is defined in the re-

gion [yI<~/KMHB-'[[ ~ and [~-'(y) l<211B-~lllY[ , where K depends only on n.

Proof. Suppose ~(x)----Bxq-~(x) ; then the series ~(x) does not contain linear and free terms. Obviously, IBIQ < M. Let Bx + B(x) = y. We consider the mapping ~):x-+B-'(y--~(~)) We shall show that this mapping is contractive for [YI<0=/K~IlB-'[[

If ix[ < B/2, then by the Cauchy inequality

OXm 8xl O ~

Hence, for ]xi < p < 012, ll3s/ xll < =) , where K depends only on n. Further, forlxll<p, Ix=I<P, IO( xl)-~)(x=)l<l B-~(~(x~)--~(x'))l<llB-~lllfM0-='glx=-xll" If p<p==0=/2/fMIIB-~Jl , then ~) is a contractive mapping with Lipschitz constant not exceeding 0.5.

If Jgl<p/21iB-'ll , then ~takes the ball Ixl <__ p into itself, since

l B-~ l ~ C x ) l = l C ~ C x ) - ~ C O ) ) + ~ ( O ) l < - T - I x l i I Yl<P-

Hence, for lYl < P/21B-']I , P < P=, the equation

Sx+~(x) =~

has a unique solution x = q0-Z(y), satisfying the condition Ixl < P. Hence, I~-'(y) l<2lJB-'HJyl for lyl<p~/(2llB-'ll) The proof of the lemma is complete.

Proof of THEOREM I0. a) The theorem is proved by simple computation. We first sum- marize the results of Theorem 8 and the Supplement to it.

By Theorem 8 on (Cn, A ') there exists a chart w such that the equation ~(e) for sufficiently small e has an invariant manifold of the form

~=n(u). (8 .1o)

By the Supplement to Theorem 8 the transition function from the chart w to the chart z is equal to w + h(w) and is defined in the domain Apt~) , whereby for P~<p(~o)

I h] ,< C,p-' (~o) p=. (8.11) I~ l .<C,p- ' (,~) V. (8.12)

By the (d, %)-nondegeneracy of the germ f the matrix ~q/3u(0) is invertible; hence there exists an inverse mapping ~ = n-~; therefore, Eq. (8.10) is equivalent to Eq. (8.9).

b) The domain of the mapping q contains the image of the domain A^(, ~ under the mapping �9 k W

w + w R. Because of the esimtates (8.3) this image contains a polydlsk ~uil < e, where e = p(w) (2~+~)m. Applying Lemma 8.4 to the mapplng ~--N with linear part B ~ A and noting that lql8 < M = C~p(m) by (8.12) and [[A-'il<d ~ by (8~4), we find that the mapping ~ = q-~ is defined in the polydisk [Ejl < e', where e' = p(~)2k2 +i d-2W(KC1)-1. Now by (8.4) E' > m -Tm= eo for sufficiently large w. This proves the first assertion of the theorem.

From Lemma 8.5 it also follows that for

I , (~)1 <2d=[8[.

c) In order to prove the second assertion of the theorem, we estimate the restriction of the function w. (j > 2) to the intersection of the manifold M(E) with the region [wi[ >

( i = l , 29 f o r <

Thus, let w R = ~(~), I EI < re_T= and lwil > p3/=(m), i = i, 2. Then by virtue of the

estimates (8.3)

I w~ I '~ < I wj ( .+ ' = I ~ (~) I I w~ I - ~ , I w= I -r# < p_.~o ] ~lO (~)-3~',0 < 2d'~ ~<-'+~'") ) =C-3~'~;

l wj I < 2'/~ d C -3~ m3~ ~-~ < C~ m-~k = p~ (m).

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The last inequality is satisfied for large m, since T>(3~q-2)k by (8.4).

d) Finally, we shall prove the third assertion. By Cauchy's inequality and the inequalities (8.11) there exists a constant C=, depending on 6 and n such that for ~ > 0

For ~ = 1/8 and sufficiently large ~ we obtain (8.13) I

[[ h. ~os/4(~) < C 1 C, p 8 (~) , ( 8 . 1 3 )

t h e m a p p i n g h on A p s / , ( m ) i s c o n t r a c t i v e a n d H [ D p ( ~ ) i s a h o l o m o r p h i s m . T h e o r e m 10 h a s b e e n p r o v e d .

3. ~ - N o r m a l Germs and a C o u n t a b l e Number o f I n v a r i a n t M a n i f o l d s . A s e q u e n c e o f m - r e g u - l a r m a t r i c e s R s i s c a l l e d m o n o t o n e i f X~+l>os

D e f i n i t i o n 4 . A s e q u e n c e o f r e s o n a n c e s ~s w i t h b a s i s m a t r i c e s R s i s s a i d t o a p p r o x i m a t e a n e x c e p t i o n a l p o i n t X, i f t h e m a t r i c e s R s a r e m - r e g u l a r f o r some m, t h e s e q u e n c e {Rs} i s mon- o t o n e , and ~ = ~s + F e s , w h e r e F h a s t h e f o r m ( 7 . 2 ) , w h e r e b y [ r < ~ s - ~ s .

LE}~qA 8 . 5 . I f ~ i s an e x c e p t i o n a l p o i n t , t h e n a s e q u e n c e a p p r o x i m a t i n g i t e x i s t s .

L e t ~ b e a (~ , m ) - e x c e p t i o n a l p o • a n d s u p p o s e t h e s e q u e n c e {R s} i s t h e same a s i n D e - f i n i t i o n 4 . I t may b e a s s u m e d w i t h no l o s s o f g e n e r a l i t y t h a t {R~} i s m o n o t o n e ( f r o m an u n - b o u n d e d s e q u e n c e • U s i t i s a l w a y s p o s s i b l e t o s e l e c t a m o n o t o n e s u b s e q u e n c e ) . We c o n s i d e r t h e s y s t e m o f e q u a t i o n s f o r ~s and Es : ~ = ~s + F E s , Rs~S = 0. I t s s o l u t i o n h a s t h e f o r m e~-(RsF)-lRs~, Xs=~--Fe s Since the first two rows of the matrix F are zero, the first two components of the vector ~s coincide with %~ and %~. Therefore, %~s/h2s~, Rs%S~-O ; by Lem- ma 3.5, %s is a resonance with basis matrix R .

S

Considering the form of the matrices R s and F and dropping for simplicity the index s, we find

( R F ) - ' = d iag(r ,a- ' ..... r ~ - ' ) , II (RD-' i l < 1.

Hence, i~I<i~i<~-=o~ The proof of the lemma is complete.

We proceed to the definition of %-normal germs.

For any mixed series F in the variables z or w we denote by F(~) the segment of the series F containing all terms of this series of degree no higher than ~ in z" (respectively in W II ) .

Suppose now that % is an exceptional point (from this, in particular, it follows that the collection X is a nonresonance collection). For each r~ there is defined a set of germs of admissible holomorphisms H~.~:(C",A')-+(C~,A') killing terms of degree no higher than m + 2 in z" in the error of the vector field A + f (m is the same as in Definition 4). For all such series H%,f the segments Hx,f(m+~) coincide. We define two operators ~-+~ :

~ f = H . ( A + f ) o H -~ - - A,

w h e r e H = ~ x f . By d e f i n i t i o n ( ~ x f ) ( m + 2 ) = 0 .

Definition 5. Let X be an exceptional point, and let {~s} be a sequence approximating it. A germ rE~ is called %-normal if the germ ~%f is (d, %S)-nondegenerate for all sufficiently large s and some positive d.

An analogous concept for an A-resonance of multiplicity one on the plane is defined in [16] in a considerably simpler way, namely, the germ f itself must be (d, xS)-nondegenerate for all sufficiently large s. The more involved Definition 5 is needed to prove thickness of the set of X-normal points; the additional difficulties are connected with the high multiplicity of resonances.

We recall that a holomorphism G: (Cn, A')-+(Cn, A') ' is called admissible if G~id~-h, hE~

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We now formulate the main result of this section.

THEOREM ii. Let ~ be an exceptional point, and suppose that the germ f~ al. Then

1 ~ The equation

~=Az+f(z) (8.14)

in any neighborhood of (C n, A') has a countable number of invariant manifolds

2 ~ Let G: (C",A')-+(Cn, A') ' be an arbitrary admissible holomorphism, and let N be the projection z~-~ (zl, z2) Each of the manifolds GMs, beginning with some one, under the projection ~ cover in a regular and finite-sheeted manner a neighborhood of some torus

T,={z~C"l Iz, I=c,, I==1 =c=, =~ . . . . . z.=O}. 3 ~ Equation (8.14) is inequivalent to its linear part.

Proof. Let {X s} be a sequence of resonances approximating X (it exists by Lemma 8.5),

and let c s be the same as in the definition of an approximating sequence. We consider the family of equations {~(c)}:

z = A s z n L Z F e - b ( ~ f ) (z) . ( 8 . 1 5 )

S F o r ~ = E t h e e q u a t i o n a ( e ) h a s t h e fo rm

~----Az--F (W,f) (z)

is E-norm~

(8.14')

and in some neighborhood of zero it is analytically equivalent to the original equation; the

conjugating holomorphism is equal to (~xf)-1; we denote it by Ho.

We set p_ = p(m ). By Theorem i0 there exist a chart w_ on (C n, A') s, a biholomorphic mapping ~s defined i~ a neighborhood of zero of the space C n~2, containing E s, a domain D s = Dp , given in the chart w s by relations (8.8) for p = p , and an admissible holomorphism H s

extendable biholomorphically to Ds, such that the equation (8.14) has an invariant manifold

of the form HSMs ', where

M ~ ' - - { w ~ O ~ I - (w~) , , = ~ , (~,) }.

The original equation (8.14) has the invariant manifold Ms=HooHSMs ' This proves

the first assertion of the theorem.

We shall prove the second assertion. For this it suffices to prove that for large s the restriction of the mapping H to GM s is regular, and the range of this restriction contains

the torus T constructed below. S

Regularity of HIGMs will be proved if it is established that the tangent planes to GMs

transversal to the planes (zl, z2) = const, for example, lie in the cone

lazJl < Idz, I + I~z=l. (8 .16)

For simplicity we shall write w, R, m in place of w s, R , ms, while retaining the notation H s, Ds, M s . The tangent plane T w to the manifold w R = const at the poznt w~Os is gzven by

the equation

o r

RW-~dw=O,

d ~ v j - - w j r p , d w I _ _ w i r p d w 2 , j = 3 . . . . . n . u~ rii+~ t~ rji+9

Since in the domain D for large s the maximum of the function w. (j > 3) is much less S ~ �9 --~

than the minimum of the functions wl and w2 (relatzon (8.8)) whzle th~ ratzos rji/rjj+2, i = i, 2 are bounded by a constant not depending on s by virtue of (8.3), we find that ~or large

s the plane T w lies in the cone

Id~jl <c(Idw, I + Idw=l), where c << i. By inequality (8.13) the plane H,T w lies in the cone

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Idz~l <9_c(laz, I + I dz~l), (8.17)

if m is sufficiently large. Finally, since GoHo is an admissible holomorphism its derived mapping at any point of the domain HSDs takes the cone (8.17) into the cone (8.16) for sufficiently large s. For example, if we consider A-equations, the domains HSDs converge to zero as s + =, and (GOHo),(0) = E. The case of P- and PE-equations can be investigated similarly�9

This proves regularity of fllHSM s.

The tori T s are chosen differently for A-, P-, and PE-equations. For A-equations

The domain D for A-equations has the form S

D, = {wl I w I; e (pyz, PYg, ,: = I, 2, I w;I < l > 2}�9 By assertion 2 ~ of Theorem i0 the boundary for the intersection Ms'nD~ (we denote it

by Bs) belongs to those parts of the boundary of the domain D on which lwzl or~lw=l assumes one of the values p~/2 or p~/4. By inequalities (8.11) [IMIB~li<Clps3/' Since G = GoHo is an admissible holomorp~ism, it~follows that G = id + g, and there exists a constant k' such that IIgi[0<h'p ~ for any sufficiently small p. Since H s = id + h s, we find that the restriction of the superposition GoH s to B_ differs from the identity mapping by no more than C3p3_/2. Now

�9 ~ " " 0 " 3 2 "~ the distance from the torus To to the zntersectaon az)~nA :,.s reater than C3 / There- ~ g - - . P s " ,

fore, the degree of any point PeF.Ts relative to the mappings [IoGoHsJm" and I-I/Ira" , where I:w + z(w) -- w is the same. Hence, the manifold GM s has over T s as l ~any sheets ~ ~s IM s. Assertion 2 ~ for A-equations has been proved. The case of P- and PE-equations is treated similarly.

The third assertion of the theorem follows easily from the second assertion and the theorem on nonlinearizability. Indeed, if Eq. (8.14) is equivalent to its linear part, then by the assertion of subsection 4.2 there exists an admissible holomorphism taking this e~ua- tion into linear normal form w = Aw. Then the equation w = Aw in any nezghborhood of (C , A') has an analytic invariant manifold satisfying the conditions of Theorem 2. Hence, the collection % is not strongly nonresonant- contradiction.

SEC. 9. THICKNESS OF THE SET OF A-NORMAL GERMS

THEOREM 12. Suppose ~C" is an exceptional point. Then the intersection of the set of ~-normal germs with each of the spaces ~p, p>0, contains a subset thick in the space

The topology in the space of germs is described in subsection 4; from this description it follows that Theorem 12 completes the proof of Theorems 3a and 3c. At the end of the section Theorem 3b is deduced from Theorem 3c.

i. The Set ~ . Let ~C" be an exceptional point, let {A s} be its approximating sequence, and let d be an arbitrary positive number. For each resonance ~ we denote by Gd, ~ the set of (d, A)-nondegenerate germs f~ (subsection 8.2, Definition 3). Let # and T be the same operators as in subsection 8.3. We set P

"~,~ U ~" ' -~- I10d,~. s, s>q

= , q

= U d>O

Proof of THEOREM 12. By definition, ~ is a set of ~-normal germs. We fix an arbitrary p > 0. We shall prove that for sufficiently large d the intersection ~N~0 is thick in ~p ; this intersection is the thick subset mentioned in Theorem 12. We immediately obtain Theorem 12 from the following lemmas.

LEMMA 9.1. The intersection ~J~,-IOd.~N~p is open for any resonance A and positive number d.

LEMMA 9.2. The intersection ~"a,qN.~o is dense in ~p for any q and sufficiently large

d.

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The next two subsections are devoted to the proof of these lemmas.

2. Proof of LEMMA 9.1. 1 ~ We recall the definition of a (d, %)-nondegenerate germ (subsection 9.2). Let ~ be a resonance (the Agreement at the beginning of Sec. 8 remains in force), let R = R(~) be an m-regular matrix, and let X~X(~), (0~(~) be the same as at the beginning of Sec. 8. We shall assume that • > m + 2. Let A', w' and w" be the same as in Sec. 5. We consider the equation

~=Az-Ff(z), A=diag~, f~,.~ (9.1)

and the formally normalizing change z = H(w) = w + h(w) expanded in a mixed series, i.e., in normal Taylor series in w" whose coefficients are functions of w' holomorphic in A'. The change H takes Eq. (9.1) to the formal normal form

whereby the formal series h does not contain resonance terms: ~oh = 0 (we recall that Zo is a projector preserving resonance terms and killing nonresonance terms, x•

Let F be a matrix of the form (7.2). We set

Og I A=A~(f)=--(RF) -~e ~ll~ (9.3)

From formula (5.5) it follows easily that this definition is equivalent to the original definition of subsection 5.4.

The germ f is called (d, %)'-nondegenerate if

tl A~-' (f)II < ,~.

2 ~ . We proceed directly to the proof of Lemma 9.1. For A-equations this lemma is trivial, since the coefficients of the normalizing series and of the normal form are polynomials in the Taylor coefficients of the error. For P- and PE-equations this is not case; in particular, to compute the matrix A%(~uf), the change Cpf and the new error ~f a countable number of operations on the coefficients of the mixed series of f are required, in general.

We consider the case of P- and PE-equations.

~ Proposition i. 1 ~ For each nonresonance ~C n , any p>0 and f~p there exists a p and a neighborhood (~p,f) such that the operators Cp and ~F~: (~p,f)--~ are continuous.

2 ~ The operator ~p-+CW-2) ~, f-~-A~(IF,[) is continuous for any p > 0.

Remark. Lemma 9.1 follows immediately from assertion 2 ~ of Proposition i.

Proof of Proposition i. The segment ~(m+1) of the formal series on the right side of the normal form (9.2) can be found after a finite number of steps of the method of successive approximations. Each step is carried out as described in [4, pp. 189, 190] and involves the solution of a homologous equation whose right side does not contain resonance terms. The solution operator of this equation has been studied in Sec. 7; its boundedness was proved in part 7 ~ of subsection 7.VI. Therefore, if the error f belongs to the space ~p , then the correc- tion h computed on the basis of f at the next step of the~method of successive approximations belongs to the same space; for some p' < p the new error f belongs to the space ~p,, and the transition operator to the next error is continuous in some neighborhood of (~,~)' as an operator (~p,~)'-~p. From this it follows that there exists a constant p < p and a neighborhood of (~p, ~) such that for each ~'~(~p,~) the change ~uf' and the new error ~pf' belong to the space L/~ , and the operators ~)p:(~A~p, f)-~A~ and ~: (~/[p,h--~J[~ are continuous. Similarly, the operator ~p-+C (n-2)', f--~Ax(~,,f) is continuous. The proof of the prop-

osition is complete.

3. Proof of LEMMA 9.2. The fact that all basis matrices of the resonances considered are m-regularplays a decisive role in the arguments of this subsection; just these arguments

made it necessary to consider m-regular matrices.

1 ~ We first prove a proposition which makes it possible to change the matrix A~(f) in

a comparatively strong manner without changing f too strongly if fLm.2) = 0.

Proposition 2. Let ~ be a resonance, let R = R(%) be an m-regular matrix, and let u= ~e X~%(~) ' ~=[0(~), f~ and f(m+2) =0. Suppose that (9.2) is the normal form of Eq. (9.1)

with these ~ and f, and that H = id + h is a normalizing change. Then

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w ( ~ e ) ~=0U=no[ t (w ~ hCx-,~(w))]~+u. (9.4)

Proof. We note that monomials of the form wiwrJ have degree in w" no lower than • and no higher than ~ + i, while the terms nonlinear in u have degree in w" higher than 2X; now 2%q-l>~q-l, since X > m + 2. Therefore,

W ( ~ u g )L=0u=~{~+l)(w ). (9.5)

Equation (9.1), the normal form (9.2), and the change H are connected by the relation

Hence,

n . (w) ( A w + , (w)) = (A+O �9 n (w).

(ad A) h (w) = f (w-}-h (w) ) -- h, (w) , (w) -- a~ (w). (9.6)

Dropping from both sides of Eq. (1.6) terms of degree greater than m + 2 in w" and noting that X > m + 2, we obtain h (m+2)(w) = 0. Hence,

[h. ( w ) , (w)f'+" = 0. Moreover,

If (w + h~-'~ (w))p+, = ff (w + h (w))f~+~.

Dropping on both s i de s of Eq. (9.6) terms of degree h i g h e r than w + 1 in w" and app ly ing the projector ~o, we obtain

0 = ~0 [f (w + h~x-,, (w))p+'~ -- ,~+'~ (w)

Together with Eq. (9.5) this gives (9.4). Proposition 2 is proved.

0 2 ~ . We set ~.={f~ogplfc~+2~=0} . We shall first prove that the intersection ~a~qN

o~ ~ is dense in the subspace ~r We note that for all [~o, ~,[=id , and hence ~ f = f. Thus, P

~,~ n ~ = U (G~,~, n ~). s>q

Therefore, it suffices to prove

Proposition 3. For each [~r there exists a sequence fs~Ga.xs N~A~ ~ converging to f in ~r .

Proof. Suppose the germ f is not (d, %S)-nondegenerate. We set A s = A s(f). We choose a matrix B ' such that ,'I B~[I< d -us , and the matrix A + B ' is not degenerate. Then Asq- Bs'=~sUs s where ~s is Hermitian and positive definite an~ U is a unitary matrix

B s B s " Then s B s'' = d -~s Us, = ' + Bs "

lIB, H< ~-%

II (As + B,)-' II < a', .

We let be a matrix of order n • (n -- 2), and we set

b' = - - Z B, z Rs, l' = f + b s. From formulas (9 .3) and (9 .4 )*

a~s(p)= A ~ s ( f ) + n s = As+Bs �9

Because of (9.8), the germ fs (d, X s) is nondegenerate.

We set

(9.7)

(9.8)

It remains for us to show that

*Let id + h and id + h s be normalizing changes for the fields A + f and A + fs, respectively. Then h (X-I) = h s(X-z), since (f -- fs) (X) = 0; h (m+2) = 0, since f(m+2) = 0 (X > m + 2).

Further, [b s o (id + h (X-1))](~+z) = be; --(RF)R-zZ-IbS = Bsu.

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I b ' l p ~ O as s ~ . (9.9)

Let K= max [zi[, and let ~ be the same constant as in inequality (8.3). Then for K > i, iE(l,n),zE &p

r i , j > 0

I bs lo < V 'n lf2~ ~ pX, d-~s < p- , , ( Kz~. P ) = ~

Tf d > p ] ( ~ , t hen the q u a n t i t y on the r i g h t tends to zero as s § =, wh ich was r e q u i r e d to p r o v e . The case K <_ 1, r ~ , j < 0 or r = , j < 0 can be t r e a t e d s i m i l a r l y .

3 ~ . I t rema ins f o r us to p rove an ana log o f P r o p o s i t i o n 3 f o r an a r b i t r a r y f ~ p .

- - I Proposition 4. For each ~d~ there exists a sequence fs ~ ' Ga.~s ['}r converging to f in d~p .

f, f = ~ f, A s = A% s (f) B s is the matrix satisfying conditions Proof. We set H = @ P , (o) (9.7) and (9.8) for the matrix As, B = = 0 , b*=ZBsz Rs, indicated now ~s=~+b s . Proposi-

B, tion 4 and with it Lemma 9.2 and Theorem 12 follow from the assertions:

A ~ . p+bs~.-,Gaa~; B ~ IbSlp + as s +

Assertion B ~ was already proved above.

Proof of Assertion A ~ By definition b s(xs)= 0 Xs > m + 2; therefore,

Hence, (f + bS ) (m+2)

We set H,bSoH -z = ~s.

O.(f+bO=tt, z ~ / - / ( z ) = w , 'tr.(/-t-b ') =l+H.b" o H-'.

Assertion A ~ will be proved if it is established that

A~, ( f + ~s) = As + B,.

f (m+2) since

( 9 . 1 0 )

By formulas (9.3) and (9.4) for any resonance ~ and ~a~p

Ax (f) w r = - - (R F ) - ' RW - t a0 [fo (id + h~7~))] (~+') (9 .11)

( i d + h ~ , f i s t h e n o r m a l i z i n g c h a n g e f o r Eq. (9.1)) . T h i s f o r m u l a w i l l b e a p p l i e d i n t h e c a s e

w h e r e ~ = A s , and i n p l a c e o f f , we t a k e f + b S . S i n c e b s ( X s ) = 0 , we o b t a i n

Further, ~ (m+=)

h(xs--l) I (Xs--l) x ~. 7+~ TM = "}3, T

(m+2) = 0. Therefore = 0, and hence, hA,f

(Xs--I) (O)s+l) ~'s((~s +1) h (zs-z))l(%+z) - - [ r , ( i d + hx,,7 )] + [([ + b s ) ~ + x'.T "

P r o p o s i t i o n 5.

(9.12)

x0 b s(~ = bs

The proof makes essential use of the structure of the basis matrix R s of the resonance

For brevity we henceforth drop the index s. A s .

Arguing as at the beginning of the proof of Proposition 2, we find that the vector-valued function ~of (m+z) for any f~o contains only terms of the form w p with complex coefficients

where p = e i + r l , i ~ ( l , n)-----] ] ~ (1, n - - 2), r i=elr i iJre2r i+2-[ - eirij+2, rll+~>O, j ( n - - 2 The l a s t e q u a l i t y follows from formula ~3.3). By definition b is a vectorial polynomial consisting of monomials with exponents e i + rJ, for which i > 2. The expansion of the vector-valued function b in a mixed series contains only monomials whose exponents are obtained from e i + r 3 for some i, j,

i > 2, by means of the following elementary operations.

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A. Addition of a vector of the form --el+p, /~(I, n), p"~Z$ -2 satisfying the condition

]p"[~2 for I>2. (9.13)

Remark. Since the holomorphism H is admissible, the exponent of any monomial of the superposition'boH -x can be obtained from e i + rJ for some i and j by means of a finite number of operations of the form A.*

B. Addition of a vector p whose components, beginning with the third, are nonnegative.

If an operation of the class B does not belong to the class A it is permitted to be applied once; it corresponds to multiplication by H, on the left.

It is obvious that by means of a nonzero number of elementary operations from the exponent e. + rJ, i > 2 it is not possible to obt@in the exponent ei, + rJ'. Indeed, if j # j'~ then

1 to obtain the exponent e i, + rJ' from e i + r 3 it is necessary to apply the operation A for Z = j no fewer than r=.+2 times in order to annihilate the term e.r.=+2 in the exponent e i +

JJ ,, 3 ] rJ. But by (9.13) the degree in w of the monomial with the exponen~ obtained then exceeds 2 X -- 1 > m and can no longer be reduced under further elementary operations -- contradiction.

Suppose now that j = j' The exponent r j + e i, cannot be obtained from r j + el, i > 2, by means of elementary operations: for all operations of the class A the degree in w" of the corresponding monomial increases, while among operations of the class B there is no addition of the vector --e. + e.,, i > 2.

1 1 ~

Thus, the expression ~ob (~+I) contains only monomials with the same exponents as the expression b; they are obtained from the monomials contained in b by means of a zero number of elementary operations on the exponents, i.e., they are the same as if H -I = id, H, = E. The proof of Proposition 5 is complete.

Formula (9.10) and together with it assertion A ~ follows from Proposition 5 and formulas (9.11) and (9.12). Proposition 4, Lemma 9.2, and Theorem 12 have now been proved.

4. The Topology in the Space of Germs is the inductive-limit topology. We consider a sequence PZ ~ 0 and set ~(0~--~% . The topology in the space ~l) is generated by the majorant norm. By definition, convex sets in ~, whose intersection with each of the spaces ~0 is a convex neighborhood of zero in the space ~u~ form a fundamental system of neighborhoods of zero in j~.

It now follows from Theorem 12 that the set ~, is dense in the space ~, which completes the proof of Theorems 3a and 3c.

5. Reduction of THEOREM 3b To THEOREM 3c. We shall deduce the first part of theorem 3b for holomorphisms with a fixed point from Theorem 3c for P-equations. The second part of Theorem 3b is derived from Theorem 3c for PE-equations in a similar way.

To each P-equation

2-~AznUfz, z~ (C", A/) (9.14)

there corresponds a p~riodic differential equation on (C n-l, 0), or, equivalently, a P-equation with first component z~ = 2 iz~. It is obtained from (9.14) by division of the right side by 1/2 izl (llzl + fl(z)). By a change of scale it is possible to arrange that &' contains the circle [z1[ = I. Thus, to each P-equation a there corresponds a monodromy transformation after a period (i.e., as the variable z~ passes around the unit circle) of the corresponding periodic equation. We call this transformation a monodromy transformation of equation ~ for brevity.

Let ~P(n) and ~A(n--l) be the respective spaces of germs of holomorphic mappings (C",A/)-+(C",O) and (Cn-',0)--~(C"-l,0) ' defined in subsections 1.IV.2 and i.I.l; the elements of the space ~P(a) are errors of P-equations; elements of the space ~A(n--|) are errors (nonlinear parts) of holomorphisms with a fixed point. In correspondence with the notation of subsection 1.IV.2 z=(zl, z")~C", ~' is a chart on the plane zl = 1 in a neighborhood of the point (i, 0, ..., 0), and we henceforth identify this neighborhood with (C n-~, 0).

from the definition of the space ~ (for fE~ af--J-~ =0 ) and from the fact *This follows az" A"

that the holomorphism H is admissible.

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To each X~C" there corresponds a mapping A~:~P(~)--~tA(n--[), defined as follows: the monodromy transformation of the equation

z=hz+f(z), [~,~P(n) has t h e fo rm

z" -~Nz"+ (AV) (~'), w h e r e

N = d i a g v , v----exp (2~ik"/X0.

L e t A be an e x c e p t i o n a l p o i n t , and l e t f be a A - n o r m a l ge rm. Then by Theorem 3c Eq. ( 9 . 1 4 ) h a s a c o u n t a b l e n u m b e r o f t w o - d i m e n s i o n a l i n v a r i a n t m a n i f o l d s o b s t r u c t i n g t h e e q u i v a - l e n c e o f t h i s e q u a t i o n to i t s l i n e a r p a r t . The c o r r e s p o n d i n g monodromy t r a n s f o r m a t i o n F h a s a c o u n t a b l e n u m b e r o f i n v a r i a n t c u r v e s a r i s i n g f rom t h e i n t e r s e c t i o n of t h e p l a n e z~ = 1 w i t h the invariant surfaces of Eq. (9.14). The mapping F is inequivalent to its linear part. Oth- erwise Eq. (9.14) could be reduced by a holomorphic change to linear normal form in the same way that in Floquet theory nonautonomous linear equations are converted into autonomous equa-

tions.

We set VM={e ~" ]k~Vp, ~,~2~i}.

A" We f i x a n a r b i t r a r y A f o r w h i c h e = v , A1 = 2 ~ i , a nd we s e t

Obviously, the set ~M={e~Ik~P, k1=2ni} is thick in VM, since the intersection of ~p with the plane AI = 2~i is thick in this plane by the corollary proved in subsection 8.1. It remains for us to prove that the set ~M is dense in the space of germs ~t~(n--l) This

follows from the following facts.

i ~ The set ~P is dense in ~P(n)

2 ~ . The mapping A A is an epimorphism by the inclusion theorem.

3 ~ . The mapping h A is continuous by the theorem on continuous dependence of a solution

of a differential equation on the initial condition.

This completes the reduction of Theorem 3b to Theorem 3c.

LITERATURE CITED

i. V. I. Arnol'd, "Remarks on singularities of finite codimension in complex dynamical systems," Funkts. Anal. Prilozhen., 3, No. i, 1-6 (1969).

2. V. I. Arnol'd, "Lectures on bifurc--ations and versal families," Usp. Mat. Nauk, 27, No.

5, 119-184 (1972). 3. V. I. Arnol'd,"Bifurcationsof invariant manifolds and normal forms of neighborhoods of

elliptic curves," Funkts. Anal. Prilozhen., iO, No. 4, 1-12 (1976). 4. V. I. Arnol'd, Supplementary Chapters of the Theory of Ordinary Differential Equations

[in Russian], Nauka, Moscow (1978). 5. A. D. Bryuno, "The analytic form of differential equations," in: Trudy MMO, Vol. 25, 26,

Moscow State Univ. (1971, 1972). 6. A. D. Bryuno, "Analytic integral sets," Preprint IPM No. 98, Part i, Moscow (1974), 7. A. D. Bryuno, "Analytic integral sets," Preprint IPM No. 98, Part 2, Moscow (1974). 8. A. D. Bryuno, "The normal form of differential equations with a s~all parameter," Mat.

Zametki, 16, No. 8, 407-414 (1974). 9. B. L. van d---er Waerden, Algebra [Russian translation], Mir (1976).

i0. Yu. S. ll'yashenko and A- S. Pyartli, "Zero-type imbeddings of elliptic curves," in: Trudy Seminara im. I. G. Petrovskogo, No. 5, Moscow State Univ. (1979).

ii. Yu. S. Ii'yashenko, "Materialization of Poincar~ resonances and divergence of normal

series," Usp. Mat. Nauk, 33, No. 3, 130-131 (1979). 12. Yu. S. ll'yashenko, "Divergence of series reducing an analytic differential equation to

linear normal form at a singular point," Funkts. Anal. Prilozhen., i_~3, No. 3, 87-88

(1979) o 13. A. Poincar@, Oeuvre, Paris (1916).

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14. K. Prakhar, The Distribution of Prime Numbers [Russian translation], Mir, Moscow (1967). 15. A. Pyartli, "Generation of complex invariant manifolds near a singular point of a vector

field depending on a parameter," Funkts. Anal. Prilozhen., 6, No. 4, 95-96 (1972). 16. A. S. Pyartli, "Cycles of a system of two complex differential equations in a neighbor-

hood of a fixed point," in: Trudy MMO, Vol. 37, Moscow State Univ. (1978). 17. W. Rudin, Foundations of Mathematical Analysis [in Russian], Nauka, Moscow (1976). 18. Y. T. Sue, "Every Stein subvariety admits a Stein neighborhood," Inv. Math., 38, No. i,

89-100 (1976). 19. V. P. Tareev, "On conditions for a countable number of cycles for a complex pursuit

function," in: Methods of the Qualitative Theory of Differential Equations [in Russian], Gor'kii, (1978), pp. 167-175.

SPECTRAL ASYMPTOTICS OF DIFFERENTIAL AND PSEUDODIFFERENTIAL OPERATORS. I

K. Kh. Boimatov UDC 517.9

i. INTRODUCTION

Spectral asymptotics of differential and pseudodifferential operators (DO and PDO) have been studied in many works. The state of investigations up to 1975 is expounded quite com- pletely in the survey works [1-5]. Of the works appearing later we mention [6-36].

The present work is devoted to the investigation of spectral asymptotics of semibounded differential and pseudodifferential operators with operator symbols. The technique applied is based on estimates of Green functions of parabolic equations and application of Tauberian theorems.

The following questions of spectral asymptotics are studied:

a) asymptotics of the distribution function of eigenvalues;

b) the spectral problem with a small parameter;

c) asymptotics of the weighted trace.

The principal achievements of the work are listed below.

i. A New Tauberian Approach to the Investigation of Asymptotics of the Spectrum of Dif- ferential Operators in Nonsmooth Domains (bounded and unbounded). Earlier a Tauberian method was applied for nonsmooth domains only in the works [37, 38, and 8].

We shall explain the essence of the technique we apply for the example of the Laplace operator defined in a bounded domain ~cRn, mesaS)=0

oo

We denote by P the Friederichs extension of the operator Po = (--W), D(Po) = Co (~) (here and below D(~ is the domain).

We consider the auxiliary operator

Qo------A+Mp --l-" (x), D (Qo) ---- Co" (f~),

where M, e > 0 and,the function p is such that p(x) ~ dist(x, Of 2) , and the inequality (x) l < MaPl-a j (x) holds. I Dx a0

Since ~ > 0, the closure Q of the operator Qo is a self-adjoint operator in the space L2(~). This fact facilitates the study by the Levi method of the Green function of the parabolic equation*

Ou = Qu, u It=o = g . ( 1 . 1 ) Ot

*Namely, to construct a parametrix it is possible to avoid applying the procedure of "recti- fying the boundary, changes of variables,..." (see [39]).

Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 7, pp. 50-100, 1981. Orig- inal article submitted July 2, 1979.

3092 0090-4104/85/3104-3092509.50 O 1985 Plenum Publishing Corporation

Documents

Materialization of Poincaré resonances and divergence of normalizing series