LIMIT AT RESONANCES OF LINEARIZATIONS OF SOME COMPLEX ANALYTIC DYNAMICAL SYSTEMS ALBERTO BERRETTI, STEFANO MARMI, AND DAVID SAUZIN Abstract. We consider the behaviour near resonances of linearizations of germs of holomorphic diffeomorphisms of (C, 0) and of the semistandard map. We prove that for each resonance there exists a suitable blow-up of the Taylor series of the linearization under which it converges uniformly to an analytic function as the multiplier, or rotation number, tends nontangentially to the resonance. This limit function is explicitely computed and related to questions of formal classification, both for the case of germs and for the case of the semi-standard map.

Contents 1.

Introduction

2

2.

Germs of holomorphic diffeomorphisms of (C, 0)

3

2.1.

The linearization of a germ

3

2.2.

Existence of non-tangential limits

6

2.3.

Germs with almost resonant linear part

10

2.4.

Rescaling the linearization

13

2.5.

Materialization of resonances

16

2.6.

Two-variable version of Theorem 2.1

18

3.

The semi-standard map

20

3.1.

Introduction to the semi-standard map

20

3.2.

Existence of non-tangential limits

22

3.3.

Explicit formulas for the limits at resonances

24

3.4.

Proof of Theorems 3.2 and 3.3

26

3.5.

Invariance of the limits under formal conjugacy

31

References

32

1

2

ALBERTO BERRETTI, STEFANO MARMI, AND DAVID SAUZIN

1. Introduction It is well known since Poincar´e’s work on normal forms and celestial mechanics, that resonances are responsible for the divergence of the series expansions of quasi-periodic motions in the theory of dynamical systems. In the simplest non-trivial case of one-dimensional holomorphic systems, this difficulty already appears when one tries to conjugate a given system to its linear part z 7→ λz in a neighborhood of a fixed point z = 0.

For a fixed value of the multiplier λ, the linearization (when it exists)

has a complicated analytic structure with respect to its argument. Almost nothing is known, except for some numerical studies (e.g. [1], figures 6 and 7 of [2] and references quoted therein, [3], [4]). If one considers the dependence of the linearization on λ, resonances appear as formal poles of the Taylor series of the conjugation: it is the famous phenomenon of “small denominators”. In some sense (see sect. 2.2), the unit circle T1 is a natural boundary of analyticity for the conjugation with respect to the multiplier λ. Nevertheless the series has non-tangential limits at almost all points of T1 (see sect. 2.1 for more information and [5], [6] for a detailed study in the case of analytic diffeomorphisms of the circle). The relation between the formal poles (associated to the resonances) of the linearization and its natural boundary of analyticity calls for some enlightenment, especially if one places the problem in the perspective of Borel’s theory of uniform monogenic functions [7]. In this paper we prove some results concerning the behaviour at resonances of the linearizations of local diffeomorphisms of C (see e.g. [8]) and of the so-called “semi-standard map” [10], [1]. These two models are actually tightly related (compare sect. 2.6 and 3.1). In particular, for each of the systems we take into account, we prove that the linearizations have well-defined limits as the multiplier (or the rotation number) tends non-tangentially to a root of unity (or a rational value p/q) provided the Taylor series is suitably blown-up, and we compute these limits. In the case of germs, this generalizes a result of Yoccoz [8, sect. 3.3 and 3.4, p. 73-78], as he computes this limit only for the quadratic family Pλ (z) = λ(z − z 2 ).

These limit functions have a finite radius of convergence and ramification

points. By reversal of the blow-up, they provide an approximation of the linearization when the multiplier is very close to a resonance.

LINEARIZATIONS OF COMPLEX DYNAMICAL SYSTEMS

3

Of course it would be quite interesting to generalize these results to general analytic small divisor problems (Hamiltonian systems, real analytic area–preserving twist maps, etc.). [2] studies the standard map case: by a suitable blow-up the Lindstedt series has an explicitly computable limit at the resonances 0/1 and 1/2. The same is true for all resonances p/q, as recently proved in [9]. We also hope to be able to apply the method presented in sect. 3 in order to improve these studies.

2. Germs of holomorphic diffeomorphisms of (C, 0) 2.1. The linearization of a germ. Let G denote the group of germs of holomorphic diffeomorphisms of (C, 0). An important problem in complex dynamics is to describe the conjugacy classes of this group. Since the origin is a fixed point, one has some natural conjugacy invariants like the multiplier λ = f 0 (0) of the germ f at 0 and the holomorphic index: I dz 1 , i(f, 0) = 2πi z − f (z)

where we integrate on a small loop in the positive direction around 0. If λ 6= 1, the origin is a simple fixed point and one clearly has: 1 i(f, 0) = . 1−λ Let Gλ denote the set of f ∈ G such that f 0 (0) = λ. Such a germ f is

said to be linearizable if it belongs to the conjugacy class of the rotation

Rλ (z) = λz. This is always the case if |λ| 6= 1: by the Poincar´e–K¨ onigs linearization theorem [11], [12], one knows that there exists a germ hf ∈ G1

such that:

f ◦ h f = hf ◦ R λ .

(2.1)

The condition hf ∈ G1 ensures the unicity of the solution of this conjugacy equation. From now on, when speaking of conjugacy classes of G without

other specification, we shall always refer to the adjoint action of the subgroup G1 — not the whole group G. The function hf is called the linearization of f : attracting (|λ| < 1) and repelling (|λ| > 1) fixed points of holomorphic diffeomorphisms are linearizable. If one keeps f − Rλ fixed as λ varies, or more generally if

f depends analytically on parameters, then by uniform convergence the

4

ALBERTO BERRETTI, STEFANO MARMI, AND DAVID SAUZIN

dependence of the linearization hf is also analytic. If f is an entire function and |λ| > 1 then hf is an entire function.

When |λ| = 1 the fixed point at the origin is indifferent and one must

distinguish three different kinds of multiplier:

´ ³ 1. parabolic or resonant point: λ = Λ = exp 2πi pq , where p ∈ N, q ∈ N∗ , (p|q) = 1;

2. Brjuno point: λ = B = exp(2πiω), ω ∈ R\Q and ω is a Brjuno number P log qk+1 < +∞, where {pk /qk } denotes the sequence of [13], [14]: ∞ k=0 qk partial fractions of the continued fraction expansion of ω;

3. Cremer point: λ = exp(2πiω), ω ∈ R\Q and ω is not a Brjuno number.

Actually, Cremer proved [15] that Gλ is not a conjugacy class if: sup n

log qn+1 = ∞; qn

however we think that it is quite fair to give his name to the complement of the Brjuno set. ´ In case (1) Ecalle [16], [17] and Voronin [18] gave a complete classification of the conjugacy classes of G contained in GΛ ; among them the class of the rotation RΛ consists of all elements of order q belonging to GΛ , but there are other conjugacy classes in GΛ . In case (2) above, Gλ is a conjugacy class of G [13] (as in the hyperbolic case |λ| 6= 1): for all f in Gλ there exists a unique analytic linearization

hf . In 1987 Yoccoz [8] proved that in case (3) Gλ is not a conjugacy class and there exists at least one non-linearizable germ f ∈ Gλ . A remarkable example is given by the quadratic polynomial:

Pλ (z) = e2πiω (z − z 2 ) which is linearizable if and only if ω is a Brjuno number. Our main motivation in this study is to understand how the formal poles due to resonances give rise to the complicated analytic structure of the linearization. For that reason we shall let λ vary in a non-tangential cone1 with vertex at any root of unity Λ, and we shall treat it as a bifurcation parameter. Consequently, the germ f itself will vary — we thus denote it by fλ henceforth —, and its linearization hfλ will be studied as a function of λ, singular at the resonance Λ. We shall provide all the details for the situation where the nonlinear part fλ − Rλ is kept fixed as λ varies (this 1

By “non-tangential” we mean: non-tangential to the unit circle; a non-tangential cone

with vertex at Λ is any set V = C(Λ, α) with the notation of formula (2.10) on page 11.

LINEARIZATIONS OF COMPLEX DYNAMICAL SYSTEMS

5

assumption is quite natural: see [5], [6] and [19, chap. III] for the related problem of the dependence of the linearization in one-parameter families of analytic circle diffeomorphisms), but we shall also give (at the end of the next section) the corresponding statements for a more general dependence of fλ on the parameter λ. Note that if, instead of a sectorial neighbourhood of a resonance, one considers a non-tangential cone V with vertex at a Brjuno point B = e2πiω , an easy2 but interesting property of the linearization is its continuity with respect to λ (i.e. hfλ tends to hfB as λ tends to B inside V ). More generally if one assumes that fλ0 is linearizable for a certain λ0 satisfying |λ0 | = 1, when λ tends to λ0 non-tangentially the linearizing

maps hλ are univalent on a small uniform disk3 (they thus form a compact

family) and any limit of these maps when λ tends to λ0 is a linearizing map for fλ0 . These limits clearly coincide if λ0 is not resonant, otherwise this is a consequence of Theorem 2.1 below. In what follows, putting the vertex of V at a resonance Λ, we shall prove the existence of a suitable scaling under which hfλ has a non-tangential limit. These scalings are a slight generalization of the notion of blow-up of a formal power series which is standard in algebraic geometry and which has already been applied to the study of complex analytic differential equations [20, section III]. We shall also compute these limits and study their relationship to the classification of formal conjugacy classes of GΛ . 2

Let us consider the intersection Vρ of V with a disk of center B and of small ra-

dius ρ, i.e. V = C(B, α, ρ) with the notation of formula (2.10). It is immediate to P hn (λ)z n , check from the relation (2.1) which defines the linearization, that hfλ = z +

where each coefficient hn (n ≥ 2) is a finite sum of rational functions of the form

(λj1 − λ)−1 . . . (λjn−1 − λ)−1 with 2 ≤ j1 , . . . , jn−1 ≤ n. Clearly hn is analytic in Vρ and continuous on its closure V¯ρ ; the uniform convergence follows from the standard Brjuno’s

argument, since there exists a positive constant C such that ∀λ ∈ V¯ρ , ∀j ≥ 2, |λj − λ|−1 ≤ C|B j − B|−1 (because there exists c > 0 such that any point of V¯ρ can be written e2πiθ with ∀q ∈ N∗ , dist(qθ, Z) ≥ c dist(qω, Z), and there exist c0 , c00 > 0 such that ∀x ∈ C, |e2πix − 1| ≥ c0 dist(x, Z) and ∀x ∈ R, |e2πix − 1| ≤ c00 dist(x, Z), thus 0 ∀q ∈ N∗ , ∀λ ∈ V¯ρ , |λq − 1| ≥ c 00c |B q − 1|). 3

c

This is very easy to see, as an anonymous referee pointed out to us. Denote by hλ0 a

0 linearizing map for fλ0 . Then gλ (z) = (h−1 λ0 ◦ fλ ◦ hλ0 )(z) satisfies gλ (0) = 0, gλ (0) = λ,

gλ0 (z) = λ0 z. Thus z −1 gλ (z) − λ0 = (λ − λ0 )(1 + zS(λ, z)) (for some analytic function S),

and for any non-tangential neighborhood Vρ of λ0 there exists r0 > 0 such that for any λ ∈ Vρ \ {λ0 } and for any z, |z| < r0 , one has |λ| > 1 ⇒ |gλ (z)| > |z| and |λ| < 1 ⇒

|gλ (z)| < |z|. One deduces from this that, as λ varies in Vρ , the radius of injectivity of hλ is uniformly bounded from below by a strictly positive constant (independent of λ).

6

ALBERTO BERRETTI, STEFANO MARMI, AND DAVID SAUZIN

2.2. Existence of non-tangential limits. Let us consider the one-parameter family of germs of Gλ : fλ (z) = λz +

∞ X

fn z n ,

n=2

where we keep fixed the coefficients {fn }∞ n=2 , i.e. the nonlinear part fλ − Rλ

is constant. We denote simply by hλ the corresponding linearization. We have the following theorem (compare with sections 3.3 and 3.4 of [8]): Theorem 2.1. Let us fix a resonance Λ = exp(2πip/q) where p ∈ N,

q ∈ N∗ , (p|q) = 1. There are two possibilities, according to the q-th iter-

ate fΛ◦q of fΛ :

• If fΛ◦q is the identity, the formulas ³R ¡ ◦q ¢ z ∂ fλ (z) |λ=Λ , U (z) = z exp 0 ( T (z1 1 ) − T (z) = qΛ1q−1 ∂λ

1 z1 )dz1

´

(2.2)

define a germ U ∈ G1 whose reciprocal hΛ linearizes fΛ and has the

following property: for all non-tangential cone V of C with vertex Λ (non-tangential to the unit circle), the linearization hλ tends uniformly on some small open disk around 0 to hΛ as λ tends to Λ inside V .

• If fΛ◦q is the not the identity, there exist a positive integer k and a nonzero complex number A such that:

fΛ◦q (z) = z + Az kq+1 + O(z kq+2 ), and for any non-tangential cone V of C with vertex Λ, the rescaled linearization: ˜ λ (z) = h

¡ ¢ 1 1/kq (λ − Λ) h z λ (λ − Λ)1/kq

(2.3)

tends uniformly on some small open disk around 0 to the function: µ ¶−1/kq A kq ˜ hΛ (z) = z 1 − z (2.4) qΛq−1 as λ tends to Λ inside V . (One can choose any of the kq determinations of (λ − Λ)1/kq .) Remark 2.1. Note that in the case fΛ◦q = Id the linearization of fΛ is not unique. What Theorem 2.1 asserts is that, among all the linearizations of fΛ , one of them is the limit of hλ as λ tends to Λ non-tangentially. The formulas (2.2) make sense and define U ∈ G1 as claimed in the theorem,

since T (z) = z(1 + O(z)); in fact U is the unique solution in G1 of the

ordinary differential equation U 0 = U /T .

LINEARIZATIONS OF COMPLEX DYNAMICAL SYSTEMS

7

Remark 2.2. In the next section, we give some details about the numbers k and A which appear in the theorem. They are classically introduced as formal invariants of fΛ : given two germs fΛ and gΛ in GΛ , the existence of a formal series ϕ(z) = z + O(z 2 ) such that: gΛ = ϕ−1 ◦ fΛ ◦ ϕ, implies: k(fΛ ) = k(gΛ )

and

A(fΛ ) = A(gΛ ),

but the converse is not true: a third formal invariant is necessary in order to ˜ Λ depends only describe all the formal conjugacy classes. Thus, the limit h on the formal conjugacy class of fΛ but does not determine it completely. Remark 2.3. Note that: ¶ 1 µ AΛ kq − kq ˜ , z hΛ = z 1 − q µ ¶ 1 AΛ kq − kq −1 ˜ hΛ = z 1 + z , q

and these maps commute with RΛ . ˜ Λ with the inverse scaling we find that, when λ If we now conjugate h approaches Λ non-tangentially and z approaches the origin in such a way that the quantity z(λ − Λ)−1/kq remains constant, then: 1 ¶− kq µ z kq AΛ · . hλ (z) ≈ z 1 − q λ−Λ

Therefore, for λ very close to a resonance, the linearization behaves as an analytic function with respect to z with kq ramification points that collapse at the origin when λ tends to Λ. We note that this has also been numerically observed (compare with figures 4 and 5 of [4]), with a very good quantitative agreement (compare with the discussion in the last section of [2] for a related problem concerning the standard map). Remark 2.4. Note that one may use any scaling z → s(λ)1/kq z instead of

z → (λ − Λ)1/kq z, provided s is analytic, s(Λ) = 0 and s0 (Λ) 6= 0. In this case the limit is:

µ ¶ 1 As0 (Λ) kq − kq ˜ . hΛ = z 1 − z qΛq−1

Remark 2.5. Let us envisage briefly a more general dependence of the germ f on the parameter, and indicate the corresponding statements. Changing the notations, we consider now a family fσ of local analytic diffeomorphisms

8

ALBERTO BERRETTI, STEFANO MARMI, AND DAVID SAUZIN

which depends analytically on a complex parameter σ. The multiplier of fσ will be denoted by λ(σ). The linearization of fσ , if it exists as an analytic germ of G1 , will be denoted by hσ . Let σ∗ be a point of the parameter-space such that: 1. the multiplier λ(σ∗ ) is not a Cremer point; 2. its derivative λ0 (σ∗ ) is nonzero.

If |λ(σ∗ )| 6= 1, the linearization hσ is analytic at σ∗ (i.e. hσ (z) is analytic for σ close enough to σ∗ and z close enough to the origin).

If |λ(σ∗ )| = 1, we call “non-tangential” a cone in the parameter-space of

the form:

¢ ª ¡ λ0 (σ∗ ) (σ − σ∗ ) ∈] − α, α[ σ | arg ± λ(σ∗ ) for some α ∈]0, π/2[. So, when σ “tends to σ∗ non-tangentially”, this means ©

that the multiplier λ(σ) tends to λ(σ∗ ) transversally to the unit circle. There

are only two possibilities for the asymptotic behaviour of hσ :

• fσ∗ is linearizable; λ(σ∗ ) is a Brjuno point, or a resonant point of order

q but then fσ◦q∗ = Id (where by Id, here and elsewhere, we denote the identity in whatever category we are dealing with): then the linearization hσ tends to some hσ∗ ∈ G1 uniformly on some small open disk

around 0 as σ tends to σ∗ non-tangentially, and the limit germ hσ∗ is

a linearization of fσ∗ (which is thus uniquely determined by fσ∗ in the Brjuno case, but which is determined by case4 ).

◦q ∂ ∂σ (fσ )|σ=σ∗

in the resonant

• λ(σ∗ ) is resonant of order q and fσ◦q∗ 6= Id: then there exist a positive integer k and a nonzero complex number A such that: fσ◦q∗ (z) = z + Az kq+1 + O(z kq+2 ), and the rescaled linearization: ¡ ¢ 1 1 ˜ σ (z) = (σ − σ∗ )− kq hσ (σ − σ∗ ) kq z h

tends uniformly on some small open disk around 0 to the function: µ ¶− 1 kq A kq ˜ σ (z) = z 1 − h z ∗ 0 q−1 qλ (σ∗ )Λ as σ tends to σ∗ non-tangentially.

4

In the resonant case, any linearization of fσ∗ can be reached at the limit: one checks

that, if a germ F of order q is given with a particular linearization H ∈ G1 , the formula fλ = F + (λ − Λ)

TF0 Λ

with T = H −1 .(H 0 ◦ H −1 )

defines a family fλ for which the linearization hλ tends to H.

LINEARIZATIONS OF COMPLEX DYNAMICAL SYSTEMS

9

All this is proved by adapting the proof of Theorem 2.1, to which the next two sections are devoted. The easier case where fΛ◦q is the identity is examined at the end of sect. 2.4 only. Formalizing a little more, we could say that we are studying the regularity properties of the “linearization” mapping L: L : f ∈ G 7→ hf ∈ G1 , with a particular attention to the singular set: ˆ = {f ∈ G | |f 0 (0)| = 1}, G through its restrictions to some analytic paths. We call path a map from some open connected subset U of C in G: σ 7→ fσ , and we call it analytic if for all σ∗ ∈ U , there exists r > 0 such that the

path induces an analytic function (σ, z) 7→ fσ (z) on the polydisk { (σ, z) ∈ C × C | |σ − σ∗ | < r and |z| < r }. If an analytic path satisfies

∂ 0 ∂σ fσ (0)

6= 0 for some parameter σ∗ , we can

change (at least locally) the parametrization in order to obtain a straightened path: an analytic path λ 7→ fλ such that fλ0 (0) = λ. ˆ the composition with L When the image of an analytic path lies outside G, is well defined and yields an analytic path in G1 . But if the path is straight-

ened and defined in a neighbourhood U of λ∗ ∈ T1 , we get an analytic

path hfλ on each side of the unit circle and generally no “analytic continuation” to a neighbourhood of λ∗ . It follows indeed from the corollary stated

in sect. 2.5 that in the case where the germs fλ or their inverses extend to entire functions of z, the resonant points5 of U are true singularities of hfλ , unlike the Brjuno points, where continuous extension in non-tangential sectors is always possible. This situation is quite reminiscent of a monogenic uniform function according to Borel’s definition [7], even if we cannot go much farther than a simple analogy for the moment. This work is indeed a first step in the understanding of the behaviour of the linearizations at these singular points, the idea being that the interesting information is localized there. 5

At least those Λ = exp (2πip/q) such that fΛ is not of order q

10

ALBERTO BERRETTI, STEFANO MARMI, AND DAVID SAUZIN

2.3. Germs with almost resonant linear part. We now begin the proof of Theorem 2.1 and consider the one-parameter family of germs fλ ∈ Gλ , with λ close to a fixed resonance Λ = exp(2πip/q). We denote by fλ◦q the

composition of fλ with itself q times, and we suppose that fΛ◦q is not the identity. Let us first focus on the germ at the resonance. Lemma 2.1. There exists a positive integer k and a nonzero complex number A such that: fΛ◦q (z) = z + Az kq+1 + O(z kq+2 ).

(2.5)

Proof. This is quite easily proved by comparing the Taylor expansions of both sides in the equation fΛ ◦ fΛ◦q = fΛ◦q ◦ fΛ . Remark 2.6. If f2 = . . . = fn = 0, then kq +1 ≥ n. But if fλ is a polynomial

of degree d, then 1 ≤ k ≤ d − 1 (see [21, Prop. 6, p. 9]).

Remark 2.7. The numbers k and A are invariant under formal conjugacy of fΛ . But there also exists a unique B ∈ C such that fΛ◦q belongs to the

analytic conjugacy class of a germ g of the form:

g(z) = z + Az kq+1 + A2 Bz 2kq+1 + O(z 2kq+2 ); the coefficient B = i(fΛ◦q , 0) is invariant under formal conjugacy of fΛ , and

fΛ◦q is formally and topologically conjugate to the polynomial z + Az kq+1 + A2 Bz 2kq+1 . Conversely, if two germs of GΛ have the same formal invariants (k, A, B), they must belong to the same formal conjugacy class [16], [17], [18]. We now let λ vary in a disk around the resonance: |λ − Λ| < ρ where ρ is some small positive constant, but we always impose |λ| 6= 1. The

power series expansion:

hλ (z) = z +

∞ X

hj (λ)z j

(2.6)

j=2

for the linearization hλ of fλ can be recursively determined by means of (2.1). However the equation fλ◦q ◦ hλ = hλ ◦ Rλ◦q

(2.7)

LINEARIZATIONS OF COMPLEX DYNAMICAL SYSTEMS

can be used instead, and one then gets:    h1 (λ) = 1, j X X 1 ◦q  f (λ) h (λ) =  j i  λjq − λq

j1 +...+ji =j

i=2

hj1 (λ) · · · hji (λ),

11

for j ≥ 2, (2.8)

denoting {fj◦q (λ)} the Taylor coefficients of fλ◦q : fλ◦q (z)

q

=λ z+

∞ X

fj◦q (λ)z j .

j=2

Let us choose a positive constant r (independent of λ) such that the Taylor coefficients of fλ◦q satisfy: ∀j ≥ 2,

|fj◦q (λ)| ≤ r1−j

(in order to do this, we use the Cauchy inequalities and then choose r small enough). Each coefficient fj◦q (λ) is a polynomial in λ, with a zero at Λ if 2 ≤ j ≤ kq, thus there exists c1 > 0 such that for all |λ − Λ| < ρ: For j = 2, . . . , kq,

|fj◦q (λ)| ≤ c1 |λ − Λ|r1−j .

(2.9)

Let α ∈]0, π/2[. We introduce notations for a cone of vertex Λ and aper-

ture 2α and for its intersection with a disk around Λ: ª © ¡ λ − Λ¢ ∈] − α, α[ , C(Λ, α) = | arg ± Λ © ª C(Λ, α, ρ) = C(Λ, α) ∩ |λ − Λ| < ρ .

(2.10)

Lemma 2.2. For all α ∈]0, π/2[ and for all sufficiently small ρ > 0, there

exists a positive constant c2 < c1 such that: ∀j ≥ 2, ∀λ ∈ C(Λ, α, ρ) :

|λjq − λq | ≥ c2 |λ − Λ|.

(2.11)

Proof. For all λ ∈ C(Λ, α, ρ) and j ≥ 2 we can write:

where A(λ) = λq ·

λq −Λq λ−Λ

λjq − λq = A(λ)Bj (λ), λ−Λ

λ(j−1)q −1 λq −1 . written e2πiθ

is bounded from below, and Bj (λ) =

There exist θ0 , c > 0 such that, for all λ in C(λ, α, ρ), λq can be

with 0 < |θ| < θ0 and ∀t ∈ R, |θ − t| ≥ c|t|; in particular, ¡ ¢ ¡ ¢ ∀j ≥ 2, dist (j − 1)θ, Z ≥ min (j − 1)|θ|, c .

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ALBERTO BERRETTI, STEFANO MARMI, AND DAVID SAUZIN

But there also exist c0 , c00 > 0 such that for all θ in C:  |θ| ≤ θ0 ⇒ |e2πiθ − 1| ≤ c0 |θ| ≤ c0 θ0 , ¡ ¢  ∀j ≥ 2, |e2πi(j−1)θ − 1| ≥ c00 dist (j − 1)θ, Z .

Therefore, ∀λ ∈ C(Λ, α, ρ), ∀j ≥ 2:

c c00 min(j − 1, ) ≥ c000 , c0 θ0

|Bj (λ)| ≥

where c000 > 0 does not depend on j. Now let:

   σ1 = 1, j X    σj =

X

σj 1 . . . σj i ,

i=2 j1 +...+ji =j

for j ≥ 2.

We have the following lemmas:

Lemma 2.3. If ρ > 0 is sufficiently small, α ∈]0, π/2[ and j ∈ N∗ , then,

∀λ ∈ C(Λ, α, ρ), we have:

|hj (λ)| ≤ σj

µ

c1 c2 r

¶j−1 µ

where bxc denotes the integer part of x.

1 c2 |λ − Λ|

¶

j

j−1 kq

k

,

(2.12)

Proof. We proceed by induction: (2.12) is clearly true if j = 1; assume that it holds at ranks 1, . . . , j − 1 for some j ≥ 2. Let λ ∈ C(Λ, α, ρ); thanks to the subadditivity of the integer part we get: |hj (λ)| ≤ with: Hi,j (λ) =

µ

j X

Hi,j (λ)

i=2

X

j1 +...+ji =j

σj1 · · · σji ,

¯µ ¶ ¯ ¶ 1 c1 j−i ¯¯ fi◦q (λ) ¯¯ ¯ λjq − λq ¯ c2 |λ − Λ| c2 r

j

j−i kq

k

From the inequality (2.9) and Lemma 2.2, it follows that:  1−i ¯ ¯ ◦q ¯ fi (λ) ¯  c2r|λ−Λ| for i = 2, . . . , j, ¯ ¯ ¯ λjq − λq ¯ ≤  c1 1−i if moreover i ≤ kq. c2 r

On the other hand: ¹

j k j−1   kq j−i ≤ j k  kq  j−1 − 1 º

kq

for i = 2, . . . , j, if moreover i ≥ kq + 1.

.

LINEARIZATIONS OF COMPLEX DYNAMICAL SYSTEMS

13

Therefore, using the inequality c2 < c1 , we get in all cases: j k ¶ µ ¶ j−1 µ kq 1 c1 j−1 . Hi,j (λ) ≤ c2 r c2 |λ − Λ| Lemma 2.4. There exists a positive constant c3 such that: √ ∀j ≥ 1, σj ≤ c3 (3 − 2 2)1−j . Proof. The generating series σ(z) = tion:

P∞

i=1 σi z

i

(2.13)

satisfies the functional equa-

σ(z)2 , 1 − σ(z) so that: √ 1 + z − 1 − 6z + z 2 σ(z) = 4 √ is analytic in the disk |z| < 3 − 2 2 and bounded and continuous on its σ(z) = z +

closure; (2.13) then follows by Cauchy’s estimate.

As Lemma 2.3 clearly shows, the radius of convergence of hλ cannot tend to zero faster than |λ−Λ|1/kq as λ tends to Λ non-tangentially. We shall now

perform the rescaling (2.3) which will compensate this possible divergence.

2.4. Rescaling the linearization. Let us fix ρ > 0 sufficiently small and ˜ λ is: α ∈]0, π/2[. The j-th coefficient of the Taylor series of h ˜ j (λ) = (λ − Λ) h

j−1 kq

hj (λ).

According to j(2.8)kthe coefficients hj are rational functions of λ, with a pole ˜ j (λ) at Λ of order j−1 at most according to (2.12). Thus each coefficient h kq

1

˜ j (Λ) and the chosen determination of (λ − Λ) kq does not tends to a limit h ˜ λ converges formally as λ tends to Λ in C(Λ, α, ρ). Moreover, using matter: h (2.12) and (2.13), we see that: j k µ ¶j−1 ¶ j−1 µ kq j−1 1 ˜ j (λ)| ≤ σj c1 ≤ c4 R−j , |h |λ − Λ| kq c2 r c2 |λ − Λ| √ 1/kq where R = (3−2 2) cc21r c2 and c4 is some positive constant. Thus, the for˜ Λ has nonzero radius of convergence and h ˜ λ converges uniformly mal limit h

14

ALBERTO BERRETTI, STEFANO MARMI, AND DAVID SAUZIN

on any disk around 0 of radius less than R, by a standard compactness argument for sets of holomorphic functions 6 . ˜ Λ , we introduce the germ: To compute the limit function h 1

1

− f˜λ (z) = (λ − Λ) kq fλ ((λ − Λ) kq z),

˜ λ , and we note that: whose linearization is just h f˜λ◦q (z) = λq z +

kq X j=2

fj◦q (λ)(λ − Λ)

j−1 kq

◦q z j + fkq+1 (λ)(λ − Λ)z kq+1

+ O((λ − Λ)1+1/kq ).

Because of the inequality (2.9) and by definition of A, we have:  f ◦q (λ) = O(λ − Λ) for j = 2, . . . , kq, j f ◦q (λ) = A + O(λ − Λ), kq+1

hence:

1

f˜λ◦q (z) = z + (λ − Λ)(qΛq−1 z + Az kq+1 + O((λ − Λ) kq )).

(2.14)

˜ λ of f˜λ is the one of f˜◦q as well: But the linearization h λ ˜ λ (z)) = h ˜ λ (λq z), f˜λ◦q (h

(2.15)

and by Taylor formula: ˜ λ (λq z) − h ˜ λ (z) = (λ − Λ)qΛq−1 z h ˜ 0 (z) + O((λ − Λ)2 ), h λ which yields after substitution of (2.14) into (2.15): ˜ 0 (z) = h ˜ λ (z) + zh λ

1 A ˜ kq+1 kq ). ( h (z)) + O((λ − Λ) λ qΛq−1

Taking the uniform limit as λ tends to Λ, we obtain the equation ˜ 0 (z) = h ˜ Λ (z) + zh Λ

A ˜ (hΛ (z))kq+1 , qΛq−1

(2.16)

which can be rewritten as a regular ordinary differential equation ˜ Λ (z), H(z) = z −1 h 6

H 0 (z) =

A z kq−1 (H(z))kq+1 qΛq−1

Indeed, if R0 ∈]0, R[ and ε > 0, we have for any l ≥ 3: ˜ λ (z) − h ˜ Λ (z)| ≤ |z| ≤ R0 ⇒ |h

l−1 X j=2

˜ j (λ) − h ˜ j (Λ)| + 2c4 |h

X R0 j ( ) , R j≥l

and we can fix l big enough so to make the second sum in the right-hand side smaller than ε/2; then, for λ close enough to Λ, we can ensure that: ˜ λ (z) − h ˜ Λ (z)| ≤ ε, |z| ≤ R0 ⇒ |h which is the property of uniform convergence.

LINEARIZATIONS OF COMPLEX DYNAMICAL SYSTEMS

15

whose unique solution with initial condition H(0) = 1 gives rise to the formula (2.4). Lastly, we consider the case where fΛ◦q is the identity. The Taylor expan-

sion of fλ◦q may be written as:

fλ◦q (z) = λq z + (λ − Λ)

X

τj (λ)z j ,

j≥2

where the τj are some analytic functions (in fact polynomials). Let us choose r > 0 small enough so that ∀j ≥ 2,

|τj (λ)| ≤ r1−j .

The induction formulas (2.8) become:    h1 (λ) = 1, j X λ−Λ X  τ (λ) h (λ) =  i j  λjq − λq i=2

j1 +...+ji =j

hj1 (λ) · · · hji (λ)

for j ≥ 2.

This allows to prove inductively the existence of a non-tangential limit for each coefficient. We obtain that, for each j ≥ 1, the coefficient hj (λ) converges to a number hj (Λ) as λ tends non-tangentially to Λ, with the following recursion formulas:    h1 (Λ) = 1, j X Λ  τi (Λ)  hj (Λ) = (j − 1)q i=2

X

j1 +...+ji =j

hj1 (Λ) · · · hji (Λ),

for j ≥ 2. (2.17)

Using (2.11) and the same numbers σj as in sect. 2.3, we find: |hj (λ)| ≤ σj (rc2 )1−j , and we conclude like before that the formal limit hΛ of the linearization hλ is analytic and that hλ converges uniformly to it on a disk around 0 with small enough radius. Moreover, taking the uniform limit of the conjugacy equation for fλ , we obtain fΛ ◦ hΛ = hΛ ◦ RΛ . Finally, we introduce T (z) =

∂ 1 qΛq−1 ∂λ

¡

fλ◦q (z)

and we observe that the equation

¢

|λ=Λ

=z+

zh0Λ (z) = T (hΛ (z))

Λ q

P

j≥2

τj (Λ)z j

16

ALBERTO BERRETTI, STEFANO MARMI, AND DAVID SAUZIN

holds (either directly from the formulas (2.17) or by a chain of reasoning analogous to the one that led to the equation (2.16)). One then easily identifies its unique solution in G1 . 2.5. Materialization of resonances. Using the notations of Theorem 2.1 and considering the second situation which it describes, we can prove the following corollary on the radius of injectivity r(λ) of hλ (i.e. the supremum of the radii of the closed disks on which this function is univalent): Corollary 2.1. For any non-tangential cone C(Λ, α) and for all R > 0: ¯q¯1 1 ¯ ¯ kq ⇒ ∃ρ > 0 : ∀λ ∈ C(Λ, α, ρ), r(λ) < R|λ − Λ| kq . (2.18) R>¯ ¯ A

˜ λ: Proof. Let r˜(λ) be the radius of injectivity of h r˜(λ) = r(λ)|λ − Λ|

1 − kq

.

¯ ¯1/kq ˜ Λ . We Suppose R > ¯ Aq ¯ , i.e. R exceeds the radius of convergence of h

must check that r˜(λ) < R for λ close enough to Λ inside C(Λ, α).

Suppose that this is not true. We could then find a sequence {λn } in ˜ λ converges C(Λ, α) converging to Λ and such that r˜(λn ) ≥ R. But h n

˜ Λ uniformly in a small disk around 0 and the set of the functions towards h ˜ Λ should belong of G1 univalent in {|z| < R} is a compact normal family: h to it, which is not the case.

Obviously, the asymptotic inequality (2.18) holds for the radius of in◦(−1)

jectivity of hλ too (since these functions, properly rescaled, converge ◦(−1) ˜ uniformly to h on some small disk). Λ

When |λ| < 1 and fλ is an entire function (resp. when |λ| > 1 and the

inverse function f ◦(−1) is entire), the linearization hλ is univalent in its disk of convergence, as is easily checked by means of the functional equation

f ◦ hf = hf ◦ Rλ (resp. the functional equation f ◦(−1) ◦ hf = hf ◦ Rλ−1 )

which is then satisfied in this whole disk, and not only in a neighbourhood of the origin. Thus under these more restrictive assumptions the previous corollary gives an information on the decrease of the radius of convergence of hλ . Arnol’d [22] raised the question of the origin of the divergence of the series of classical perturbation theory: L’id´ee de la mat´erialisation des r´esonances est de trouver des obstacles topologiques ` a la convergence des s´eries de la th´eorie

LINEARIZATIONS OF COMPLEX DYNAMICAL SYSTEMS

17

des perturbations dans le comportement des orbites du syst`eme perturb´e dans l’espace des phases complexe. We shall now give an elementary illustration of such a manifestation of the resonances in the singular behaviour of the linearizations.

According

to Remark 2.2, we may think of the function ¡ AΛ z kq ¢−1/kq . z 1− q Λ−λ ◦(−1)

as a first-order approximation of hλ

for λ tending to Λ in a non-tangential

cone. This function has ramification points at the boundary of its disk of injectivity (which coincides with its disk of convergence), and we can write them e2πim/kq σ,

m ∈ Z/kqZ, ¤1/kq £ . where σ denotes some determination of Aq (1 − Λλ ) ◦(−1)

Now, since fλ = hλ ◦ RΛ ◦ hλ

, we may expect that something occurs

near these points in the dynamic of fλ , explaining the result of Corollary 2.1. Lemma 2.5. Let us denote by σ a determination of

¤ λ 1/kq . A (1 − Λ )

£q

There

exists positive constants ε and ρ such that, for 0 < |λ − Λ| < ρ, the local diffeomorphism fλ admits exactly k orbits of period q inside the pointed disk

Dε∗ = {z ∈ C | 0 < |z| < ε}. Moreover, the kq fixed points of fλ◦q in Dε∗ are

analytic functions of σ which can be written

zm (λ) = e2πim/kq σ(1 + O(σ)),

m ∈ Z/kqZ

(with fλ (zm (λ)) = zm+kp (λ)); we thus get the following upper bound for the ◦(−1)

radius of injectivity r0 (λ) of hλ

:

¯1/kq ¯q . r0 (λ) ≤ sup{ |zm (λ)|, m ∈ Z/kqZ } ∼ ¯ (λ − Λ)¯ A

Note that here the multiplier λ is not required to lie in a non-tangential cone with vertex at Λ. Proof. The relation between the periodic orbits of fλ and the radius of in◦(−1)

jectivity of hλ

is due to the fact that a whole periodic orbit cannot be ◦(−1)

included in the disk of injectivity of hλ

, for in that disk the dynamic is

conjugate to Rλ and |λ| 6= 1. More precisely: whenever z 6= 0 belongs to

the disk of convergence of fλ◦q and both z and fλ◦q (z) belong to the disk of ◦(−1)

injectivity of hλ

, we have:

◦(−1)

hλ

◦(−1)

(fλ◦q (z)) = λq hλ

◦(−1)

(z) 6= hλ

(z),

18

ALBERTO BERRETTI, STEFANO MARMI, AND DAVID SAUZIN

thus fλ◦q (z) 6= z.

We now begin the proof of the first statement by writing the q-th iterate

of fλ as fλ◦q (z) = λq z + Az kq+1 (1 + zB(z)) + (λ − Λ)z 2 C(λ, z) where B and C are analytic for λ close to Λ and z close to the origin. On ¡ ¢ the other hand, 1 − λq = qΛq−1 (Λ − λ) 1 + (Λ − λ)D(λ) with D analytic

at Λ. Thus, the equation of the non-zero fixed points of fλ◦q is equivalent to the equation

λ q (1 − )E(λ, z) A Λ ¡ ¢¡ ¢−1 where E(λ, z) = 1 + (Λ − λ)D(λ) + z Λq C(λ, z) 1 + zB(z) is analytic z kq =

at (Λ, 0) and E(Λ, 0) = 1. This function can be written E = F kq where F has the same properties, and our equation amounts to ∃m ∈ Z/kqZ |

¢ ¡ Aσ kq ), z e2πim/kq z = σF Λ(1 − q

which is equivalent, by the implicit function theorem, to saying that z is the value of one of kq functions zm (λ) analytic in σ, with zm = e2πim/kq σ(1 + O(σ)). Finally, since fλ (zm (λ)) = Λzm (λ) + O(σ 2 ) is also fixed by fλ◦q , we must identify it with zm+kp (λ).

2.6. Two-variable version of Theorem 2.1. In this section we consider λ as a variable — just like z — rather than a parameter, so we pretend we are dealing with mappings of C × C∗ : Φ(z, λ) = (φλ (z), λ) analytic on subsets of C × C∗ and acting trivially on the second argument.

Our purpose now is simply to rephrase the results of sect. 2.2 in this twovariable context, in view of extending them to one case where the second variable is not inert any longer (see the next section). We start with a family of germs fλ ∈ Gλ like before, except that we do not

assume the nonlinear part fλ − Id to be independent of λ, and we translate the results of the end of sect. 2.2 (with λ = σ) for the map F (z, λ) = (fλ (z), λ)

which is assumed to be analytic on Dr × U , where r > 0 and U is some open

subset of C∗ .

LINEARIZATIONS OF COMPLEX DYNAMICAL SYSTEMS

19

The rotations Rλ become the map: R(z, λ) = (λz, λ),

(2.19)

and the linearizations hλ , available at least at the points λ of U which do not lie on the unit circle S 1 , become the map: H(z, λ) = (hλ (z), λ), solution of the conjugacy equation: F ◦ H = H ◦ R, with the normalizing condition H − Id = (O(z 2 ), 0).

As a function of two variables, H is analytic at all the points of the form

(0, λ∗ ), where λ∗ lies in U but not on S 1 .

For λ∗ ∈ U ∩ S 1 , calling V the intersection of U with any non-tangential

cone with vertex at (0, λ∗ ), we can state the following facts about the be-

haviour of H in Dr × V :

1. If λ∗ = B is a Brjuno point, there exists ρ > 0 such that H admits a continuous extension to Dρ × V .

2. If λ∗ = Λ is resonant of order q and if F ◦q (z, Λ) = (z, Λ), the same conclusion holds.

3. If λ∗ = Λ is resonant of order q but not all the points (z, Λ) are qperiodic, we can attach to Λ a positive integer k and a nonzero complex number A defined by: F ◦q (z, Λ) = (z + Az kq+1 + O(z kq+2 ), Λ) and such that the same conclusion holds for the rescaled mapping: ˜ = S −1 ◦ H ◦ S, H ¢ ¡ 1 with S(z, λ) = (λ − Λ) kq z, λ and with: ¡ ¡ ˜ Λ) = z 1 − H(z,

¢− 1 ¢ A z kq kq , Λ . q−1 qΛ

20

ALBERTO BERRETTI, STEFANO MARMI, AND DAVID SAUZIN

3. The semi-standard map 3.1. Introduction to the semi-standard map. We now extend the results of the previous section to the biholomorphic symplectic mapping F of C/2πZ × C defined by: F(x, y) = (x1 , y1 ),

 x1 = x + y + eix , y = y + eix .

(3.1)

1

This is the so-called semi-standard map, which has been studied by many authors [23], [24], [25], [26], [27] as a model-problem of symplectic twist map. In particular it provides a simple model for the study of invariant circles of symplectic twist maps, with power series involved instead of trigonometric series [23, §32, p.173]: indeed, for Im(x) large, we may see F as a perturba-

tion of the rotation R(x, y) = (x + y, y) and ask whether the invariant curves y = constant of R have any counterpart for F; i.e. we fix ω ∈ C and we look for an invariant curve parametrized by θ:  x = θ + ϕ(eiθ ), y = 2πω + ψ(eiθ ),

in such a way that F(x, y) corresponds to θ + 2πω, with ϕ and ψ analytic and vanishing at the origin. If ω ∈ R, the above problem admits a solution (ϕ, ψ) = (ϕ2πω , ψ2πω ) if and only if ω is a Brjuno number [24], [25].

We shall see that for ω ∈ C \ R, there exists always a solution, which we

still denote by (ϕ2πω , ψ2πω ), and we shall study its behaviour as ω tends to a resonance, i.e. a rational number. For the very same reason as the one

mentioned in the note 2, the solution will depend continuously on ω in any cone non-tangential with respect to the real axis with vertex at a Brjuno number. Since we consider now ω as a variable rather than a parameter, we can state the problem as a conjugacy problem: find H of the form H(x, y) = ¡ ¢ x + ϕy (eix ), y + ψy (eix ) such that F ◦ H = H ◦ R. The relationship with the previous section is best seen using the following

variables: z = eix ,

λ = eiy ,

LINEARIZATIONS OF COMPLEX DYNAMICAL SYSTEMS

which give to F the form7 : F (z, λ) = (z1 , λ1 ),

21

 z1 = λzeiz , λ = λeiz . 1

The points (0, λ) are fixed by F , and our problem consists in finding a map H fixing these points and satisfying the conjugacy equation: F ◦ H = H ◦ R,

(3.2)

where R corresponds to R and coincides with the previously defined rotation (2.19). Thanks to the relation z1 = λ1 z, it is easy to check that H can be written as: H(z, λ) = We choose the normalization:

µ

h(z, λ) h(z, λ), h(λ−1 z, λ)

h(z, λ) = izeφλ (z) ,

¶

.

φλ (0) = 0,

so that ϕy = π/2 − iφeiy . This leads us to the equation: φλ (λz) − 2φλ (z) + φλ (λ−1 z) = −zeφλ (z) ,

(3.3)

and to the formula: ¡ ¢ −1 H(z, λ) = iz eφλ (z) , λei(φλ (z)−φλ (λ z)) .

(3.4)

The existence and the analyticity of H for λ ∈ C∗ \S 1 and |z| small enough

is guaranteed by the stable manifold theorem applied in the intermediate set of variables (z, y). Indeed, in these variables the semi-standard map takes the form:

 z1 = zei(y+z) , y = y + z; 1

the points (0, y) in these variables are fixed, and the jacobian at such points has eigenvalues 1 and λ = eiy . We thus get, for each y of positive imaginary part (0 < |λ| < 1), a local stable manifold, and for each y of negative imaginary part (|λ| > 1), a local unstable manifold; these are analytic imbedded

disks tangent to the eigenvectors (λ − 1, 1), which we can parametrize by z, so that the dynamic on them is conjugate to the rotation z 7→ λz. By glueing

these parametrizations and using the appropriate set of variables, we get the

maps H or H. Note that when y = 2πω, where ω is Brjuno number (i.e. when 7

Instead of the semi-standard map F itself, we are thus considering the quotient map

from C/2πZ × C/2πZ in itself, but this does not change anything for the kind of invariant

circles which we are interested in; observe that ϕy and ψy must be 2π-periodic in y.

22

ALBERTO BERRETTI, STEFANO MARMI, AND DAVID SAUZIN

¡ ¢ λ is a Brjuno point), we may consider the image of z 7→ z eiϕy (z) , y + ψy (z)

as a center manifold [23], [28].

3.2. Existence of non-tangential limits. The functions under study are the unique solution φ of (3.3) such that φλ (0) = 0 and the normalized solution H of (3.2) which is determined by φ through (3.4). With reference to sect. 2.6, we (improperly) call them linearizations. Theorem 3.1. Let us fix a resonance Λ = exp(2πip/q) where p ∈ N,

q ∈ N∗ , (p|q) = 1, and let us define the maps: sλ (z) = (λ − Λ)2/q z,

S(z, λ) = (sλ (z), λ)

(how we choose the determination of (λ − Λ)2/q does not matter), and the rescaled linearizations:

φ˜λ = φλ ◦ sλ ,

˜ = S −1 ◦ H ◦ S. H

(3.5)

For any non-tangential cone V with vertex Λ, the function φ˜λ converges to some analytic function φ˜Λ , uniformly on some small open disk Dr around 0, as λ tends to Λ in V ; and for ρ > 0 small enough, if we denote by U the ˜ λ) extends continuously in disk around Λ of radius ρ, the function H(z, Dr × (V¯ ∩ U ) by:

˜ Λ) = (iz eφ˜Λ (z) , Λ). H(z,

Proof. Expanding φλ into powers of z: φλ (z) =

∞ X

φj (λ)z j

j=1

and substituting into (3.3) one finds:  1   φ1 (λ) = D1 (λ) , j−1 X X 1 1  φ (λ) =  j  Dj (λ) k! k=1

j1 +...+jk =j−1

φj1 (λ) · · · φjk (λ)

for j ≥ 2,

where Dj (λ) = −(λj/2 − λ−j/2 )2 .

Since λ tends to Λ in V , there exists a positive constant c5 < 1 such that:  c5 if j 6= 0 (mod q), |Dj (λ)| ≥ c |λ − Λ|2 if j = 0 (mod q). 5

We now check by induction that, for all j ≥ 1, ¶b j c µ q 1 −j ∗ |φj (λ)| ≤ c5 σj , 2 c5 |λ − Λ|

(3.6)

LINEARIZATIONS OF COMPLEX DYNAMICAL SYSTEMS

23

where the numbers σj∗ are defined according to the formulas σ1∗

= 1,

σj∗

=

j−1 X

X

k=1 j1 +...+jk =j−1

σj∗1 · · · σj∗k

(j ≥ 2).

Indeed, the inequality (3.6) is satisfied for j = 1 (whether q = 1 or q ≥ 2).

Let us suppose that it holds at ranks 1, . . . , j − 1 for some j ≥ 2: thanks to the subadditivity of the integer part we get |φj (λ)| ≤

c−j+1 σj∗ Aj , 5

1 | with Aj = | Dj (λ)

µ

1 c5 |λ − Λ|2

¶b j−1 c q

.

j j−1 j But we have always b j−1 q c ≤ b q c, and also b q c ≤ b q c − 1 in the case where

q divides j, thus

µ

1 Aj ≤ c5 |λ − Λ|2 in all cases, which proves (3.6) at rank j. c−1 5

¶b j c q

The generating series for the numbers σj∗ is easily computed: 1 1/2 ], 2 [1 − (1 − 4z)

P

∗ j j≥1 σj z

=

it defines a holomorphic function which extends continu-

ously to the closure of the disk of radius 1/4 with center at the origin, thus σj∗ ≤ const 4j and the first statement of the theorem clearly follows by the arguments already used in sect. 2.4. We note that φ˜Λ is in fact a function ˜ of z q , and this gives the second statement relative to H. Remark 3.1. If λ tends to 1 non-tangentially, one can easily compute the limit φ˜1 . Note that φ˜λ is the solution of: ˜ φ˜λ (λz) − 2φ˜λ (z) + φ˜λ (λ−1 z) = −(λ − 1)2 zeφλ (z) ;

since: 1 φ˜λ (λz) = φ˜λ (z) + (λ − 1)z φ˜0λ (z) + (λ − 1)2 z 2 φ˜00λ (z) + . . . , 2 1 (1 − λ)2 2 ˜00 1 − λ ˜0 φ˜λ (λ−1 z) = φ˜λ (z) + z φλ (z) + z φλ (z) + . . . , λ 2 λ2 one finds at the limit: ˜

z φ˜01 (z) + z 2 φ˜001 (z) = −zeφ1 (z) , with the initial condition φ˜1 (0) = 0. Thus: ³ z´ . φ˜1 (z) = −2 log 1 + 2

(3.7)

(3.8)

24

ALBERTO BERRETTI, STEFANO MARMI, AND DAVID SAUZIN

Remark 3.2. If the term eix in (3.1) is replaced by γ(eix ) where γ is any analytic function vanishing at the origin (even with finite radius of convergence), the conjugacy equation (3.3) becomes: φλ (λz) − 2φλ (z) + φλ (λ−1 z) = iγ(izeφλ (z) ) and Theorem 3.1 can be proved in this more general case too. But the differential equation for the non-tangential limit at Λ = 1 is then: ˜

z φ˜01 (z) + z 2 φ˜001 (z) = iγ(izeφ1 (z) ); the solution with initial condition φ˜1 (0) = 0 might be very different, and its analytic continuation much more savage in that case.

3.3. Explicit formulas for ³the limits at resonances. We now compute ´ p ˜ with (p|q) = 1. the limits φΛ for all Λ = exp 2πi q

We attach to such a resonance Λ the q − 1 positive numbers: rp Dr (Λ) = 2 − Λr − Λ−r = 4 sin2 ( π), r = 1, . . . , q − 1 q

whose product is q 2 (note that Dq−r (Λ) = Dr (Λ)). Theorem 3.2. The limit at the resonance Λ is given by the formulas: µ ¶ Λ2 C(Λ) q 2 ˜ z , φΛ (z) = − log 1 + q 2q à µ ¶− 2q ! 2 C(Λ) Λ ˜ Λ) = iz 1 + H(z, zq ,Λ , 2q where the number C(Λ) is obtained from auxiliary coefficients C1 , . . . , Cq−1 according to the formulas: 1 , C1 = D1 (Λ) r−1

1 X 1 Cr = Dr (Λ) n! n=1

q−1 X 1 C(Λ) = n! n=1

X

r1 +...+rn =r−1

X

r1 +...+rn =q−1

Cr1 · · · Crn for r = 2, . . . , q − 1,

C r1 · · · C rn .

The constant C(Λ) is always an algebraic positive number. Obviously C(Λ−1 ) = C(Λ) (since Dr (Λ−1 ) = Dr (Λ)); here are the first few values of C(Λ):

LINEARIZATIONS OF COMPLEX DYNAMICAL SYSTEMS

25

p q C(e2πip/q ) 0 1 1 1 2 1/4 1 3 1/6 1 4 5/24 1 5 2 5

√ 63−11√5 120(3−√5) 63+11√5 120(3+ 5)

1 6 99/80

This result agrees with the one computed for Λ = 1 in the previous section, and in the proof below we shall suppose q ≥ 2. We shall use a

slightly different scaling and new variables s and η: ϕ(s, η) = φΛeη (η 2/q es ), so that:

lim ϕ(s, η) = ϕΛ (s) = φ˜Λ (Λ−2/q es ), n.t. η− −→0 n.t.

where the notation η −−→ 0 means that the limit exists for η tending to

zero in any non-tangential cone of the half-plane {Re η < 0} with vertex 0, uniformly in some half-plane {Res ≤ s0 }. It is sufficient to show that:

µ ¶ C(Λ) qs 2 e , ϕΛ (s) = − log 1 + q 2q

and, just like at the end of the proof of Theorem 2.1, this formula will derive from a differential equation; we need only to show that: ϕ00Λ (s) = −C(Λ)eqs+qϕΛ (s) .

(3.9)

We shall also prove the following. Theorem 3.3. The following expansion holds for ϕ(s, η): ϕ(s, η) = ϕΛ (s) +

X

1≤k< 2q

η 2k/q Ck eks (1 +

C(Λ) qs 2k 2q e )

+ O(η)

(3.10)

where the Ck are the ones defined in Theorem 3.2. We believe that eq. (3.10) provides the beginning of an infinite asymptotic expansion in powers of η 2/q ; moreover the method that we use leads to a system of equations involving the same operators as in [29], where these operators are shown to produce resurgence in the variable η.

26

ALBERTO BERRETTI, STEFANO MARMI, AND DAVID SAUZIN

3.4. Proof of Theorems 3.2 and 3.3. In this section we shall prove the two theorems stated above. We shall prove them through a series of lemmas. Proof of theorem 3.2. In the variables (s, η), the conjugacy equation (3.3) becomes: ϕ(s + Ω + η, η) − 2ϕ(s, η) + ϕ(s − Ω − η, η) = −η 2/q es+ϕ(s,η) ,

(3.11)

where Ω = 2πip/q. We introduce the following linear combinations of the Ω-translations of ϕ: σr (s, η) =

q−1 −kr X Λ k=0

q

ϕ(s + kΩ, η), for r = 0, 1, . . . , q − 1.

We also introduce the Kronecker symbol on Z/qZ:  1 if a = b (mod q), ∀a, b ∈ Z, δ˜a,b = 0 otherwise.

The following identity:

∀a, b ∈ Z,

δ˜a,b =

r=0

allows us to write the inverse formulas: ϕ(s + kΩ, η) =

q−1 X r=0

q−1 X (Λa−b )r

q

(3.12)

Λkr σr (s, η) for k = 0, 1, . . . , q − 1,

and by combining the Ω-translations of equation (3.11), we obtain the system of equations: (∆r σr )(s, η) = −η 2/q es+σ0 (s,η) Sr for r = 0, 1, . . . , q − 1,

(3.13)

where the operator ∆r is acting on a function ψ(s, η) according to: (∆r ψ)(s, η) = Λr ψ(s + η, η) − 2ψ(s, η) + Λ−r ψ(s − η, η) and: Sr =

q−1 −k(r−1) X Λ k=0

q

exp(

q−1 X

0

Λkr σr0 (s, η)).

r 0 =0

Developing the exponentials and using again the identity (3.12), we find: X X 1 (3.14) δ˜r1 +...+rn ,r−1 σr1 · · · σrn . Sr = δ˜r−1,0 + n! n≥1

1≤r1 ,... ,rn ≤q−1

We know that the functions σr and Sr tend to some analytic 2πi-periodic n.t.

functions of s as η −−→ 0. We may add, thanks to the uniformness statement and because these are analytic functions of z = es , that the same is true for

the partial derivatives with respect to s of the functions σr ; this allows us to

LINEARIZATIONS OF COMPLEX DYNAMICAL SYSTEMS

27

give the following approximation for the operator appearing in the left-hand side of equation (3.13):  η 2 (∂ 2 σ0 (s, η) + O(η 2 )) s (∆r σr )(s, η) = −D (Λ)σ (s, η) + O(η) r

r

if r = 0,

(3.15)

if r = 1, . . . , q − 1,

thanks to the Taylor formula. This was indeed the purpose of using these functions σr : to get rid of the difference operator in the approximation of our equation, replacing it by a differential operator (one might consider that the operator ∆r is a “differential operator of infinite order”; some series to which the inversion of a closely related operator gives rise are studied in [29]). This implies immediately that σr = O(η 2/q ) for r = 1, . . . , q − 1. But we

shall see better estimates in the following lemma. Lemma 3.1.

σr (s, η) = O(η 2r/q ) for r = 0, 1, . . . , q − 1. Before proving Lemma 3.1, we introduce a notation for the intervals of integers: for a, b ∈ Z such that a ≤ b,

[[a, b]] = {r ∈ Z | a ≤ r ≤ b} ,

and we state an easy combinatorial lemma: Lemma 3.2.

a) If n ∈ N∗ and r, r1 , . . . , rn ∈ [[1, q − 1]], δ˜r1 +...+rn ,r−1 6= 0 ⇒ r1 + . . . + rn ≥ r − 1.

b) Suppose that q ≥ 3 and r0 ∈ [[1, q − 2]]. Define Nr for r ∈ [[1, q − 1]] as

Nr = min(r, r0 ). Then, if n ∈ N∗ , r ∈ [[r0 + 1, q − 1]] and r1 , . . . , rn ∈

[[1, q − 1]],

δ˜r1 +...+rn ,r−1 6= 0 ⇒ Nr1 + . . . + Nrn ≥ r0 . Proof of Lemma 3.2.

a) Suppose that δ˜r1 +...+rn ,r−1 6= 0. There exists

m ∈ Z such that r1 + . . . + rn = r − 1 + mq. The positiveness of all rj implies that mq ≥ n + 1 − r > −q, so m > −1, thus m ≥ 0.

b) Suppose that we are given q, r0 , r and r1 , . . . , rn like in the second part of the lemma. Since r − 1 ≥ r0 and because of a), it is sufficient to prove that:

r1 + . . . + rn ≥ r0 ⇒ N r 1 + . . . + N r n ≥ r0 . But if we suppose Nr1 + . . . + Nrn ≤ r0 − 1, the positiveness of all

Nrj implies that each one satisfies Nrj ≤ r0 − 1, and thus Nrj = rj by

28

ALBERTO BERRETTI, STEFANO MARMI, AND DAVID SAUZIN

definition of N . Therefore r1 + . . . + rn = Nr1 + . . . + Nrn ≤ r0 − 1 and we are done.

Proof of Lemma 3.1. The property to be proved was already checked for r = 0 and r = 1; this settles the case of q = 2. We shall suppose q ≥ 3 and

prove by induction that for all r0 in [[1, q − 1]], ∀r ∈ [[1, q − 1]],

2

σr = O(η q

min(r,r0 )

).

This property at rank r0 = q − 1 is nothing but the desired estimate. It

was already checked for r0 = 1, so let us suppose it to be true for some

r0 ∈ [[1, q − 2]]: in order to establish it at order r0 + 1 we only need to show

that:

∀r ∈ [[r0 + 1, q − 1]],

σr = O(η 2(r0 +1)/q ).

Now, if r0 + 1 ≤ r ≤ q − 1, the property at rank r0 and Lemma 3.2 imply

that:

∀r1 , . . . , rn ∈ [[1, q − 1]],

δ˜r1 +...+rn ,r−1 σr1 · · · σrn = O(η 2r0 /q ),

thus Sr = O(η 2r0 /q ); because of the equation (3.13), ∆r σr = O(η 2(r0 +1)/q ) and the estimate (3.15) implies that σr = O(η min(1,2(r0 +1)/q) ), hence the result in the case where 2(r0 + 1)/q ≤ 1.

If 2(r0 + 1)/q > 1, we have σr = O(η) and this allows for a refinement of

the approximation (3.15): ∆r σr = −Dr σr + O(η 2 ), thus σr = O(η 2(r0 +1)/q ) in that case too. n.t.

We shall now study the behaviour of η −2r/q σr (s, η) as η −−→ 0, again by

induction on r. As for σ0 (s, η), its non-tangential limit is nothing but ϕΛ (s) P since ϕ = q−1 r=0 σr and σr tends to zero for 1 ≤ r ≤ q − 1. (We also see in

this way that all the Ω-translations of ϕ coincide at the limit: ϕΛ must be 2πi -periodic, and φ˜Λ must be function of z q .) q

Lemma 3.3. For all r in [[1, q − 1]], ¡ ¢ σr (s, η) = η 2r/q Cr er(s+σ0 (s,η)) + O(η) .

(3.16)

For all r in [[1, q]],

¡ ¢ Sr = η 2(r−1)/q Er e(r−1)(s+σ0 (s,η)) + O(η) ,

(3.17)

LINEARIZATIONS OF COMPLEX DYNAMICAL SYSTEMS

29

with Sq = S0 and: E1 = 1, X 1 Er = n! n≥1

X

r1 +...+rn =r−1

C r1 · · · C r n

for 2 ≤ r ≤ q.

Proof. If 1 ≤ r ≤ q −1 the estimate for σr in (3.16) follows from the estimate of Sr in (3.17), by virtue of equation (3.13) which yields: ∆r σr = −η 2r/q (Er er(s+σ0 ) + O(η)) and of the refinement of the approximation (3.15): ∆r σr = −Dr σr + O(η 1+2r/q ), which is made possible by the fact that σr = O(η 2r/q ). Indeed, Cr = Er /Dr . If 1 ≤ r ≤ q, the estimate of Sr in (3.17) will be deduced from the

estimates (3.16) at ranks 1, . . . , r − 1, i.e. we proceed by induction on r.

For r = 1, we apply the first part of Lemma 3.2 inside equation (3.14)

and get: S1 = 1 +

X 1 n! r

n≥1 m≥0

X

σ r 1 . . . σ rn .

1 +...+rn =mq

But in each term of the sum, since the rj must be positive, the integer m must be positive too and the corresponding product σr1 · · · σrn which is

O(η 2(r1 +...+rn )/q ), is also O(η 2 ). Therefore S1 = 1 + O(η 2 ).

For 2 ≤ r ≤ q, supposing (3.16) to be true at ranks 1, . . . , r−1, we rewrite

(3.14) taking into account the first part of Lemma 3.2: X X X 1 X 1 σr1 · · · σrn + Sr = n! n! n≥1

r1 +...+rn =r−1

n,m≥1

r1 +...+rn =r−1+mq

σ r1 · · · σ r n .

The second sum is O(η 2+2(r−1)/q ) because of Lemma 3.1. The induction hypothesis applies to each term of the first one, for this sum involves only values of the indices r1 , . . . , rn ranging from 1 to r − 1. Thus: Sr =

X 1 n!

n≥1

X

r1 +...+rn =r−1

η 2(r−1)/q ·

· Cr1 · · · Crn e(r−1)(s+σ0 ) (1 + O(η)) + O(η 2+2(r−1)/q ),

hence the desired estimate. We can now finish the proof of the theorem. On one hand the estimate (3.17), when specialized to the case r = q, introduces the coefficient

30

ALBERTO BERRETTI, STEFANO MARMI, AND DAVID SAUZIN

C(Λ) = Eq : Sq = S0 = η 2(q−1)/q (C(Λ)e(q−1)(s+σ0 ) + O(η)). On the other hand the equation (3.13), together with the estimate (3.15), yields: η 2 (∂s2 σ0 + O(η 2 )) = −η 2/q es+σ0 S0 . Therefore: ∂s2 σ0 (s, η) = −C(Λ)eqs+qσ0 (s,η) + O(η),

(3.18)

and the uniform limit ϕΛ of σ0 must satisfy the limit equation (3.9), which characterizes it since all our functions are 2πi-periodic and tend to zero as Re(s) tends to −∞. Proof of theorem 3.3. A kind of analytic Gronwall lemma may be applied to the estimate (3.18) in order to prove that: σ0 (s, η) = ϕΛ (s) + O(η).

(3.19)

This provides estimates of σ0 , σ1 , . . . , σq−1 (with the help of (3.16)) which prove (3.10). More precisely, let us explain how (3.19) may be derived from (3.18). Let χ(s, η) = σ0 (s, η) − ϕΛ (s). We can write the difference between (3.18)

and (3.9):

∂s2 χ = −C(Λ)eqs+qϕΛ (eqχ − 1) + O(η) as: ∂s2 χ = f (s, η)χ + g(s, η), where f = −C(Λ) e

qχ −1

χ

eqs+qϕΛ (s) = O(1) (because χ is known to be o(1))

and g = O(η).

Coming back to the variable z = es , we get three functions χ∗ (z, η) = χ, f ∗ (z, η) = f and g ∗ (z, η) = g, which are analytic for z in some fixed disc Dρ centered at the origin and η in some non-tangential cone V0 with vertex 0. The three of them vanish when z = 0, so that their Taylor expansions are: X X X χ∗ = χn (η)z n , f ∗ = fn (η)z n , g ∗ = gn (η)z n . n≥1

n≥1

n≥1

Because of the uniformity of our estimates and Cauchy inequalities, there P exist a convergent series α = n≥1 αn z n with constant positive coefficients and a positive constant η0 such that: ∀η ∈ V0 , ∀n ≥ 1,

|η| ≤ η0 ⇒ |fn (η)| ≤ αn and |gn (η)| ≤ |η|αn .

LINEARIZATIONS OF COMPLEX DYNAMICAL SYSTEMS

31

When expanding into powers of z the differential equation: (z∂z )2 χ∗ = g ∗ + f ∗ χ∗ , we see that the coefficients of χ∗ can be bounded by |χn | ≤ |η|βn provided

that:

    β1 ≥ α1 , ¡ 1  αn + β ≥ n 2  n 

X

n1 +n2 =n

¢ α n1 βn2 .

Such a requirement may be fulfilled by a convergent series β = e.g. β = α/(1 − α), which is sufficient to conclude.

P

n≥1 βn z

n,

3.5. Invariance of the limits under formal conjugacy. We have associated to the semi-standard map a set of numbers {C(Λ)} which determine

the non-tangential limit of the rescaled linearization at any resonance Λ, and

we mentioned at the end of sect. 3.2 the existence of non-tangential limits for the more general case of the map: Fγ (z, λ) = (z1 , λ1 ),

 z1 = λzeiγ(z) , λ = λeiγ(z) , 1

for any function γ analytic and vanishing at the origin. We shall denote ˜ γ = S −1 ◦ Hγ ◦ S the corresponding linearization and its rescalby Hγ and H ing.

We now prove the invariance of these limits under a suitable notion of formal conjugacy. All the mappings from C × C∗ in itself that we consider

leave the points (0, λ) fixed: these points are the null section of the bundle C × C∗ 7→ C∗ .

Let us fix a resonance Λ. The formal conjugating diffeomorphisms that

we shall use are of the form: µX ¶ X n n ξ(z, λ) = αn (λ)z , λ + βn (λ)z , n≥1

n≥1

where the coefficients αn and βn are continuous functions defined in a neighbourhood of Λ. We may think of such a ξ as a local diffeomorphism of ¡ 2 ¢ C , (0, Λ) , formal in z, continuous in λ, leaving the null section fixed. ¡ ¢ Let F ∗ be some analytic local diffeomorphism of C2 , (0, Λ) leaving the

null section fixed, defined in Dr × U , where r is a positive number and U is

some open subset of C∗ . If we assume F ∗ to be formally conjugate to Fγ by

32

ALBERTO BERRETTI, STEFANO MARMI, AND DAVID SAUZIN

some ξ, we find immediately the relation between its linearization and the one of Fγ : F ∗ ◦ H ∗ = H ∗ ◦ R,

with H ∗ = ξ ◦ Hγ .

Thus, applying the same rescaling S to the linearizations of Fγ and F ∗ , we obtain: ˜ ∗ = S −1 ◦ H ∗ ◦ S = S −1 ◦ ξ ◦ S ◦ H ˜γ. H

But (S −1 ◦ ξ ◦ S)(z, λ) tends to (z, Λ) as λ tends non-tangentially to Λ, thus

we have proved that for any non-tangential cone V with vertex Λ, and for ˜ ∗ extends continuously in ρ > 0 small enough, the rescaled linearization H ˜ γ at the points (z, Λ). Dρ × (V¯ ∩ U ) with the same value as H

Acknowledgements This work was supported by the CEE contract ERB-CHRX-CT94-0460.

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34

ALBERTO BERRETTI, STEFANO MARMI, AND DAVID SAUZIN

` di Roma (Tor Alberto Berretti, Dipartimento di Matematica, II Universita Vergata), Via della Ricerca Scientifica, 00133 Roma, Italy and INFN, Sez. Tor Vergata E-mail address: [email protected] ` di FirenStefano Marmi, Dipartimento di Matematica “U. Dini”, Universita ze, Viale Morgagni 57a, 50134 Firenze, Italy and INFN, Sez. Firenze E-mail address: [email protected] `mes dynamiques”, CNRS – Bureau des David Sauzin, “Astronomie et syste longitudes, 77, avenue Denfert-Rochereau, 75014 Paris, France E-mail address: [email protected]