Renormalization of Vector Fields
and Diophantine Invariant Tori
Hans Koch
1
and Saša Kocić
1
Abstract. We extend the renormalization group techniques that were developed originally for
Hamiltonian flows to more general vector fields on Td ×R` . Each Diophantine vector ω ∈ Rd
determines an analytic manifold W of infinitely renormalizable vector fields, and each vector
field on W is shown to have an elliptic invariant d-torus with frequencies ω1 , ω2 , . . . , ωd . Analogous manifolds for particular classes of vector fields (Hamiltonian, divergence free, symmetric,
reversible) are obtained simply by restricting W to the corresponding subspace. We also discuss
non-degeneracy conditions, and the resulting reduction in the number of parameters needed in
parametrized families to guarantee the existence of invariant tori.
1. Introduction
Classical KAM theory [2,8,28] shows that for every Diophantine vector ω ∈ Rd , there
exist open sets of d-parameter families of Hamiltonian vector fields on Td × Rd , such
that each family has a member with an invariant torus with frequency vector ω. (This
applies to an individual Hamiltonian satisfying a non-degeneracy condition, if one considers
the family of its translates.) Similar results have been obtained for different classes of
near-linear vector fields, mostly by KAM type methods [3,4,28,29] or resummation of
Lindstedt series [9,12–15]. Our goal is to obtain such results within the framework of
renormalization transformations [6,7,10,11,16,17,19–23,25–27], and more importantly, to
develop appropriate techniques for analyzing quasiperiodic motion in a large class of flows.
This approach aims to classify vector fields according to arithmetic properties of their flows,
with the equivalence classes being stable manifolds under renormalization. It combines in
a natural way the arithmetic and geometric aspects of the problem. The construction of
invariant tori represents a basic first application. Other possible applications include the
description of accumulating periodic orbits [1]. The same techniques also apply to nonperturbative problems that are outside the reach of other known methods [20,21]. Our
analysis is centered around three results that should be of independent interest: a normal
form theorem for vector fields (Section 5), estimates on a multidimensional continued
fractions expansion [16], and a stable manifold theorem for sequences of maps (Section 6).
A renormalization group analysis of Diophantine torus flows and/or Hamiltonian vector fields was carried out in [23,25] for d = 2, and more recently in [16] for d ≥ 2. Earlier
results covered a much smaller set of frequencies [19]. One of our goals is to extend the
methods developed in these papers to a large class of vector fields on M = Td × R` , and
to do this in a way that allows for a unified treatment of Hamiltonian, divergence free,
symmetric, reversible, and other types of vector fields. Some of our results are sufficiently
general to be used e.g. in other problems involving renormalization. Despite the increase
in scope, the analysis has in fact become simpler compared to previous work.
1
Department of Mathematics, University of Texas at Austin, 1 University Station C1200, Austin, TX 78712,
email: koch@math.utexas.edu, kocic@math.utexas.edu
1
2
HANS KOCH and SAŠA KOCIĆ
We note that the tori considered in this paper are elliptic, in the sense that they
have zero Lyapunov exponents. It should be possible to adapt our method to hyperbolic
situations, but we will not pursue this question here.
Denote by t 7→ ΦtX the flow for a vector field X. In this paper, an invariant d-torus for
X, with frequency vector ω ∈ Rd , is a continuous embedding Γ of D0 = Td × {0} into the
domain of X, with the property that Γ ◦ ΦtK = ΦtX ◦ Γ for real times t, where K = (ω, 0).
Here, 0 denotes the zero vector in R` . We assume that ω satisfies a Diophantine condition
|ω · ν| ≥ ζkνk1−d−β ,
ν ∈ Z \ {0} ,
(1.1)
for some constants β, ζ > 0.
Our renormalization analysis (Sections 2,3,4) applies to vector fields that are close
to K, after a change of variables, if necessary. We assume analyticity on a complex
neighborhood Dρ of D0 , characterized by the conditions |Im xi | < ρ and |yj | < ρ. We
will also consider certain subclasses of vector fields, including Hamiltonian, divergence
free, symmetric, and reversible vector fields. If G is a linear map on M that leaves Dρ
invariant, we call a vector field X on Dρ symmetric with respect to G if G∗ X = X, where
G∗ X = G−1 X ◦ G is the pullback of X under G. If G ◦ G is the identity, a vector field
is called reversible with respect to G if G∗ X = −X. Notice that G∗ X = ±X implies
In what follows, we will call a vector field symmetric if it is
that G ◦ ΦtX = Φ±t
X ◦ G.
symmetric with respect to G(x, y) = (x, −y), or reversible if it is reversible with respect to
G(x, y) = (−x, y).
Denote by Au the space of all vector fields Y (x, y) = (u, M y + v), with (u, v) a vector
in Cd × C` and M a complex ` × ` matrix. In Section 2, we will introduce Banach spaces
Aρ of vector fields that are analytic on Dρ , and a projection operator P from Aρ onto
the subspace Au . The subspace of functions in Aρ that do not depend on the coordinate
y ∈ C` will be denoted by A0ρ . A function will be called “real” if it takes real values for
real arguments.
The following theorem describes an application of our renormalization group (RG)
approach. It establishes the existence of invariant tori with a given rotation vector ω ∈ Rd ,
for all vector fields on a finite codimension manifold W.
Theorem 1.1. Let K = (ω, 0) with ω ∈ Rd Diophantine. Given ρ > δ > 0, there exists an
open neighborhood B of K in Aρ , and a real analytic map W : (I − P)B → PB, satisfying
W (0) = K and DW (0) = 0, such that the following holds. Let W be the graph of W .
Then every vector field X ∈ W has an elliptic invariant torus ΓX ∈ A0δ with frequency
vector ω. The map X 7→ ΓX is real analytic on W. The restriction of W to symmetric
vector fields takes values in the subspace of symmetric vector fields, and similar statement
holds for reversible, Hamiltonian, and divergence free vector fields.
This theorem, as well as the lemma below on parametrized families, will be proved
in Section 4. The size of the neighborhood B is independent of ω, given the Diophantine
constants and a lower bound on the norm of ω. We note that, for any fixed β > 0, the
measure of the set of vectors ω that violate (1.1) approaches zero as ζ tends to zero [5].
In what follows, Hρ denotes either Aρ , or the subspace of Aρ consisting of all vector fields in a given class (Hamiltonian, divergence free, symmetric, or reversible). The
intersection of Au with Hρ will be denoted by H u .
Renormalization and Diophantine Invariant Tori
3
Theorem 1.1 has an obvious corollary concerning the existence of vector fields with
invariant tori in N –parameter families, where N is the dimension of H u . In particular, any
analytic family f : B ∩ H u → Hρ sufficiently close to the family f0 (s) = K + s intersects
the manifold W ∩ Hρ transversally, and Theorem 1.1 yields an invariant torus ΓX for the
vector field X = f (s) in the intersection.
If we are just looking for families containing a vector field with frequency vector parallel
(but not necessarily equal) to ω, then the number of necessary parameters is reduced
by one. A further reduction is possible for vector fields that satisfy a non-degeneracy
condition, so that some directions in Au can be generated via y-translations. To be more
precise, let V be some proper linear subspace of C` . Let r > ρ > 0, and let Z = Z(x, y)
be a real vector field in Hr that does not depend on the coordinate x, and that satisfies
PZ = 0.
Given ε > 0, define
gε (z, v) = zK + εPJv∗ Z ,
z ∈ C,
v∈V ,
(1.2)
where Jv (x, y) = (x, y + v). We assume that K + εZ is non-degenerate with respect to
V -translations, in the sense that the derivative Dgε (0) is one-to-one. Let H0u be a linear
subspace of H u that is transversal to the range of Dgε (0), and define
fε (s) = K + εZ + s ,
s ∈ H0u .
(1.3)
We will see later that fε (0) belongs to W, for small ε > 0.
Given an open neighborhood b in some complex Banach space, denote by F(b) the
space of all bounded analytic functions f : b → Hr , equipped with the sup-norm.
Lemma 1.2. If ε > 0 is chosen sufficiently small, and if the vector field K + εZ is nondegenerate with respect to V -translations, then, given an open neighborhood b2 of the
origin in H0u , there exists an open neighborhood B2 of fε in F(b2 ), such that the following
holds. For every family f ∈ B2 we can find a parameter value sf ∈ b2 , and a nonzero
complex number cf , such that X = cf f (sf ) belongs to W and thus has an invariant torus
ΓX ∈ A0δ with rotation vector ω. The maps f 7→ (cf , sf ) and f 7→ ΓX are real analytic on
B2 .
This lemma includes cases where Dgε (0) is onto and thus H0u trivial. In such a case,
every vector field near K + εZ has an invariant torus with frequency vector ω. Consider
e.g. the case of Hamiltonian vector fields, or symmetric vector fields with ` = d. Then
H u is of dimension ` = d. Taking for V some (` − 1)–dimensional subspace of C` not
containing ω, it is easy to write down examples (see e.g. below) of vector fields Z ∈ PHρ
for which Dgε (0) is invertible and thus H0u = {0}. Hamiltonian vector fields of this type
are also called isoenergetically non-degenerate.
The following example covers several classes of vector fields.
Example. Consider a basis {w1 , w2 , . . . , wd } for Rd , with wd = ω. Let k be the minimum
of d − 1 and `. Define Xj (x, y) = (yj wj , 0) for 1 ≤ j ≤ k, and if k < `, define
2
Xj (x, y) = 0, (yj − y` )2 (ej + e` ) ,
X` (x, y) = 0, (y12 + . . . + y`−1
)e` ,
4
HANS KOCH and SAŠA KOCIĆ
for k < j < `, where {e1 , e2 , . . . , e` } denotes the standard basis for R` . Consider now a
real vector field Z = c1 X1 + . . . c` X` , with cj 6= 0 if and only if Xj belongs to Hρ .
In the case of general vector fields, cj 6= 0 for all j, and we can choose V = C` . The
resulting vector field K + εZ is non-degenerate with respect to V -translations, and the
parameter space H0u used in Lemma 1.2 is of dimension d + `2 − 1. The same choice of Z
and V can also be used for divergence free vector fields. In this case, H0u has dimension
d + `2 − 2. For reversible or Hamiltonian vector fields, cj = 0 for j > k. Taking V to be
the span of {e1 , . . . , ek }, we get again non-degeneracy with respect to V -translations, and
the parameter space H0u is of dimension d − 1 − k. In particular, if k = d − 1, then we are
in the situation described above, where every vector field near K + εZ has an invariant
torus with frequency vector ω.
Our proof of Theorem 1.1 and Lemma 1.2 is based on renormalization group techniques. The general idea in this approach is to take a continued fractions algorithm, acting
on frequency vectors, and to “lift” it to a space of vector fields in some appropriate way.
We choose a multidimensional continued fractions expansion [16] which, starting from a
Diophantine vector ω0 ∈ Rd , produces a sequence of vectors ωn = ηn−1 Tn−1 ωn−1 , where Tn
is a matrix in SL(d, Z) and ηn an appropriate normalization constant. The matrices Tn can
be used e.g. to construct successive rational approximants to ω0 . Our RG transformation
Rn that corresponds to the matrix Tn−1 has the property that it maps Kn−1 = (ωn−1 , 0)
to Kn = (ωn , 0). Other properties will be given below.
We start by describing a single RG step. It involves a “scaling” of the torus variable
x by a matrix in SL(d, Z), whose transpose is strongly contracting on the orthogonal
complement of some unit vector ω ∈ Rd . Given such a matrix T , and a nonzero real
number µ, define
Sµ (x, y) = (x, µy) ,
T (x, y) = T x, Tby ,
x ∈ Td , y ∈ R` .
(1.4)
Here, Tb is either the ` × ` identity matrix, or if desired for the renormalization of Hamiltonian vector fields (where ` = d), the inverse of the transpose of T . The scaling of a vector
field X on M is then given by (Sµ T )∗ X, the pullback of X under Sµ T . Recall that the
pullback of a vector field X, defined on the range of a differentiable map U , is given by
U ∗ X = (DU )−1 (X ◦ U ).
Notice that scaling by T ∗ is a singular operation on spaces of analytic vector fields,
since it shrinks the domain of analyticity in the expanding direction of T . Although the
domain loss is of order one (not small), it is possible to associate with X ∈ A% a change
of variables UX , which is close to the identity for X close to K = (ω, 0), such that the
renormalized vector field
R(X) = η −1 T ∗ Sµ∗ UX∗ X
(1.5)
belongs again to A% .
To be more specific, we will identify (in Section 2) a subspace of “resonant” vector
fields, containing K, such that the restriction of T ∗ Sµ∗ to this subspace is compact, and in
fact analyticity improving, for small µ > 0. Then, using a general result from Section 5,
we show that there exists an analytic map X 7→ UX , defined near K, which makes UX∗ X
resonant. In other words, the resonant vector fields, which behave well under scaling, can
Renormalization and Diophantine Invariant Tori
5
be regarded as a local normal form for vector fields. We note that UK is the identity, so
e = (e
the transformation R maps K to K
ω , 0), where ω
e = η −1 T −1 ω.
Theorem 1.3. Let % > 0. Given a Diophantine unit vector ω0 ∈ Rd , there exists a
sequence of matrices Tn ∈ SL(d, Z), and a corresponding sequence of transformations Rn
of the form (1.5), such that the following holds. For n = 1, 2, . . . define ωn = ηn−1 Tn ωn−1 ,
with ηn > 0 chosen in such a way that ωn is a unit vector. Then Rn is well defined
and analytic in some open neighborhood Dn−1 of Kn−1 = (ωn−1 , 0) in A% . The set W
of infinitely renormalizable vector fields X0 in D0 , characterized by the property that
Xn = Rn (Xn−1 ) belongs to Dn for n = 1, 2, . . ., is the graph of an analytic function W
with the properties described in Theorem 1.1 (if % + δ < ρ), where ω = ω0 and B = D0 .
The set W can be regarded as the (local) stable manifold for the transformations
R1 , R2 , . . . . A stable manifold theorem that applies to such sequences of maps will be
proved in Section 6. Section 2 deals with a single renormalization group transformation R,
using a normal form theorem proved in Section 5. The composition of such transformations
Rn , according to a multidimensional continued fractions expansion [16], will be described
in Section 3. Section 4 is devoted to the construction of invariant tori.
2. A single renormalization step
In this section we give a precise definition RG transformation R and describe some of its
properties. A matrix T ∈ SL(d, Z) is assumed to be given, subject to certain conditions
that will be specified below.
2.1. Spaces and basic estimates
Unless specified otherwise,
our norm on Cn is kvk = supj |vj |. Another norm that will
P
be used is |v| = j |vj |. For linear operators between normed linear spaces, including
matrices, we will always use the operator norm, unless stated otherwise. Let C be some
finite dimensional complex Banach space. Denote by Dρ the set of all vectors (x, y) in
Cd × C` , characterized by kIm xk < ρ and kyk < ρ. We consider functions on Td × C` with
values in C, that extend analytically to Dρ and continuously to the boundary of Dρ . Our
norm on the space Aρ (C) of such functions f is given in terms of the Fourier-Taylor series
of f as follows:
X
X
kf kρ =
kfν,α keρ|ν| ρ|α| ,
f (x, y) =
fν,α eiν·x y α ,
(2.1)
ν,α
ν,α
P
Q α
where ν · x = j νj xj and y α = j yj j . The sum in this equation ranges over all ν ∈ Zd
and α ∈ N` . If it is clear what space C is being considered, or irrelevant, we will simply
write Aρ in place of Aρ (C). The operator norm of a continuous linear map L on Aρ will
be denoted by kLkρ .
Later on, for the construction of invariant tori, we will also use non-analytic functions,
with real domain D0 = Td × {0}. Denote byP
A0 the Banach space of continuous functions
f : D0 → Cd , for which the norm kf k0 = ν kfν k is finite, where {fν } are the Fourier
coefficients of f . This space can be viewed as a ρ → 0 limit of the spaces Aρ defined above.
6
HANS KOCH and SAŠA KOCIĆ
Proposition 2.1. Let X ∈ Aρ (C) and Z ∈ Aρ0 (Cd+` ), with 0 ≤ ρ0 ≤ ρ. Then
(a) kX(x, y)k ≤ kXkρ for all (x, y) ∈ Dρ .
(b) (DX)Z ∈ Aρ0 (C) and k(DX)Zkρ0 ≤ (ρ − ρ0 )−1 kXkρ kZkρ0 , if ρ0 < ρ.
(c) X ◦ (I + Z) ∈ Aρ0 (C) and kX ◦ (I + Z)kρ0 ≤ kXkρ , if ρ0 + kZkρ0 ≤ ρ.
The proof of these estimates is straightforward and will be omitted. In what follows,
we always assume that ρ > 0, unless specified otherwise.
2.2. Resonant and nonresonant modes
Let 0 < ρ0 < % be given. These domain parameters are now considered fixed for the entire
RG analysis. Choose γ ≥ 1 and χ ≥ kTbk. Let µ and τ be positive real numbers, satisfying
0
eρ /2 µ < µ̂ ≡
ρ0
,
χ%
τ≤
ρ0
,
2%
τ ln µ̂/µ ≤
ρ0
.
2(γ + 1)
(2.2)
Consider the matrix norm |M | = sup|v|=1 |M v|.
Definition 2.2. Denote by S the generator of the one-parameter group of scalings µ 7→ Sµ∗ .
Given any subset J of I = Zd × {−1, 0, 1, 2, . . .}, define P (J) to be the joint spectral
projection in Aρ for the operators (−i∇x , S), associated with the eigenvalues (ν, k) in J.
+
Let now I be the set of all pairs (ν, k) ∈ I satisfying |T ∗ ν| ≤ τ |ν| or |T ∗ ν| ≤ γ −1 τ k,
−
and let I be its complement in I. The projection
onto the “resonant” and “nonresonant”
+
+
+
−
subspace of Aρ is defined
respectively. In addition, we
as I = P I and I = P I ,P
define Ek = P {(0, k)} , for every integer k ≥ −1, and E = k Ek . Notice that EX is the
torus average of X.
The following proposition shows that, unlike in KAM theory [2,8,28], resonant modes
are easy to deal with in the RG approach.
Lemma 2.3. Consider the two linear transformations Sµ and T defined in (1.4). If the
+
condition (2.2) holds, then T ∗ Sµ∗ defines a bounded linear operator from I Aρ0 to A% ,
satisfying
∗ ∗
T Sµ Ek Xk% ≤ N (T ) µ/µ̂ k kEk Xkρ0
(2.3)
∗ ∗ +
T Sµ I (I − E)Xk% ≤ N (T ) bµ/µ̂ γ kI+(I − E)Xkρ0 ,
where N (T ) = T −1 + Tb−1 Tb.
Proof. By our choice of norm (2.1), it suffices to verify the given bounds for vector fields
X = P (J)Y , with J containing a single point. Let
J = {(ν, k)} ,
A = %|T ∗ ν| − ρ0 |ν| + k ln µ/µ̂ .
(2.4)
Then it follows essentially from the definitions that
kT ∗ Sµ∗ P (J)Y k% ≤ N (T )eA kP (J)Y kρ0 .
Setting ν = 0 yields the first bound in (2.3).
(2.5)
7
Renormalization and Diophantine Invariant Tori
+
In order to prove the second bound, assume that (ν, k) belongs to I , and that ν 6= 0.
Consider first the case |T ∗ ν| ≤ τ |ν|. Then |ν| ≥ τ −1 , and we obtain
A ≤ (%τ − ρ0 )|ν| + k ln µ/µ̂ ≤ (% − ρ0 /τ ) − ln µ/µ̂
(2.6)
≤ −ρ0 /(2τ ) − ln µ/µ̂ ≤ γ ln µ/µ̂ .
In the last inequality we have used condition (2.2).
ρ0
= ln(c), and k > γ,
Now consider the case τ |ν| < |T ∗ ν| ≤ γτ k. By using that % γτ ≤ 2γ
we find that
A ≤ %τ γ −1 k + k ln µ/µ̂ ≤ k ln bµ/µ̂ ≤ γ ln bµ/µ̂ .
(2.7)
The second bound in (2.3) now follows from (2.6) and (2.7).
QED
This lemma shows that the scaling T ∗ Sµ∗ is a contraction (for small µ) on the resonant
subspace of A% , except for a small number of non-contracting directions. In order to exploit
this property, given a vector field X that is not necessarily resonant, we first perform a
change of variables UX , such that
−
I UX∗ X = 0 .
(2.8)
The corresponding linearized equation, which needs to be solved in the process, is of the
−
form I (X − [X, Z]) = 0, where [X, Z] = (DZ)X − (DX)Z. In order to guarantee the
existence of a solution, we need to make the following assumptions.
Assume that there exists a d − 1 dimensional subspace
of Rd where the transpose T ∗
√
of T contracts distances by a factor of at least τ /2 d. Let ω be a unit vector in Rd that
is perpendicular to this subspace, and set K = (ω, 0).
√
−
Proposition 2.4. Choose σ > 0 such that 2 d σkT k ≤ τ . If Z belongs to I A0r then
k[K, Z]kr ≥ σkZkr ,
k[K, Z]kr ≥
σr
kDZkr .
r+γ+1
(2.9)
−
Proof. Assume that (ν, k) belongs to I . In other words, |T ∗ ν| > τ |ν| and |T ∗ ν| > γτ k.
Consider the decomposition ν = νk + ν⊥ √into a vector νk parallel to ω and a vector ν⊥
perpendicular to ω. By using that |ν⊥ | ≤ d|ν|, we obtain
√
√
τ
d σ|ν| < kT k−1 |ν| ≤ kT k−1 |T ∗ ν| − |T ∗ ν⊥ | ≤ kT k−1 |T ∗ νk | ≤ |νk | ≤ d |ω · ν| ,
2
and in particular, σ < |ω · ν|. Similarly, we have
√ σ
√
τ
1
d k ≤ kT k−1 k ≤ kT k−1 |T ∗ ν| ≤ kT k−1 |T ∗ ν| − |T ∗ ν⊥ | ≤ d |ω · ν| .
γ
2γ
2
−
This shows that if Z ∈ I A0r and Y = [K, Z] = (ω · ∇x )Z, then kZkr ≤ σ −1 kY kr and
d X
∂ 1
∂xj Z ≤ σ kY kr ,
r
j=1
` X
∂ γ+1
∂yj Z ≤ σr kY kr .
r
j=1
(2.10)
8
HANS KOCH and SAŠA KOCIĆ
These bounds imply (2.9).
QED
This proposition allows us now to apply the results from Section 5, which describe a
solution of equation (2.8). For convenience later on, let us first restate the assumptions of
Proposition 2.4 in a slightly stronger form:
χ
kT k, kT −1 k ≤ √ ,
2 d
χ=
τ
,
σ
τ
|T ∗ ξ| ≤ √ |ξ| ,
2 d
ξ ∈ ω⊥ .
(2.11)
Lemma 2.5. There exist positive constants C and C 0 , such that the following holds,
whenever (2.2) and (2.11) are satisfied. Denote by D the open ball in A% of radius ε =
C(σ/γ)2 , centered at K. Then for every X ∈ D, there exists an analytic change of
coordinates UX : Dρ0 → Dρ , such that UX∗ X belongs to Aρ0 and satisfies equation (2.8).
The map X 7→ UX is analytic from D to the affine space I + Aρ0 and satisfies the bounds
in Theorem 5.2, with κ = C 0 σ/γ.
−
Proof. The image of I A0r under Z 7→ P [Z, K] = P (ω · ∇x )Z contains all nonresonant
−
Fourier-Taylor polynomials, and thus it is dense in I Ar , for any r > 0. Assumption 5.1
regarding the spaces Ar is satisfied by Proposition 2.1, and the condition (5.3) on K
holds by Proposition 2.4, with κ = ρ0 (1 + ρ0 )σ/(2γ). The hypotheses (5.4) and (5.20) of
Theorem 5.2 are clearly satisfied on D, with ε = C 00 κ2 and C 00 some constant depending
only on ρ and ρ0 . The claims now follow from Theorem 5.2.
QED
2.3. The transformation R
Given T ∈ SL(d, R), a unit vector ω ∈ Rd , and a real number γ ≥ 1, assume that there
exists positive constants µ, σ, τ < 1 satisfying (2.2) and (2.11). In what follows, a quantity
will be called universal if it is independent of the choice of T , ω, γ, µ, σ, and τ .
On the domain D described in Lemma 2.5, we can now define our RG transformation
R according to equation (1.5). The normalization constant η is defined as η = kT −1 ωk, so
that ω
e = η −1 T −1 ω is again a unit vector. Notice that, by construction, UX = I whenever
+
X is resonant. Thus, R ◦ I is linear, and so is R ◦ E.
Let P = E−1 +E0 . The subspace PA% is spanned by vector fields of the form Y (x, y) =
(u, M y +v) and is invariant under R. The restriction of R to this subspace, which is linear,
will be denoted by L.
In the following theorem, Hρ denotes either Aρ , or the subspace of Hamiltonian vector
fields in Aρ , provided that ` = d and we choose for Tb the inverse of the transpose of T .
Theorem 2.6. There exist universal constants R, C0 > 0, such that the following holds.
Let D be the open ball in H% of radius 2R(σ/γ)2 , centered at K. Then R is bounded and
analytic on D, satisfying
k(I − E)R(X)k% ≤ C0 η −1 (γ/σ)(C0 τ /σ)γ+2 µγ k(I − E)Xk% ,
k(I − P)R(X)k% ≤ C0 η −1 (γ/σ)(τ /σ)3 µk(I − P)Xk% ,
kER(X) − R(EX)k% ≤ C0 η −1 (γ/σ)3 (τ /σ)µ−1 k(I − E)Xk2% ,
kL−1 k ≤ C0 η(τ /σ) .
(2.12)
Renormalization and Diophantine Invariant Tori
9
Proof. Let R be half the constant C from Lemma 2.5, so that we can apply the estimates
from Theorem 5.2. Let X be some vector field in D. By Lemma 2.3 we have
k(I − E)R(X)k% = η −1 kT ∗ Sµ∗ (I − E)UX∗ Xk%
≤ C1 η −1 (cτ /σ)γ+2 µγ k(I − E)Xkρ0 + kUX∗ X − Xkρ0 ,
(2.13)
for c = exp(ρ0 /2)%/ρ0 and some constant C1 > 0. Here, and in what follows, C1 , C2 , . . .
denote positive universal constants. Using the bound (5.5) on the norm of UX∗ X − X,
together with the fact that P = P (I − E), we obtain the first inequality in (2.12).
Similarly, Lemma 2.3 implies that
kEk R(X)k% ≤ C2 η −1 (τ /σ)3 µ kEk Xkρ0 + kEk (UX∗ X − X)kρ0 ,
(2.14)
for all k ≥ 1. Summing over k ≥ 1 to get a bound on k(E − P)R(X)k% , and then adding
(2.13), yields a bound analogous to (2.14), but with Ek replaced by I − P. Applying again
the estimate (5.5) on the norm of UX∗ X − X, we obtain the second inequality in (2.12).
By Lemma 2.3, we also have
kER(X) − R(EX)k% = η −1 kT ∗ Sµ∗ E(UX∗ X − X)k%
≤ C3 η −1 (τ /σ)µ−1 kE(UX∗ X − X)kρ0 .
(2.15)
Using the bounds (5.5), the norm on the right hand side of this inequality can be estimated
as follows:
kE(UX∗ X − X)kρ0 ≤ C4 (γ/σ)3 k(I − E)Xk2ρ + kE[Z, X]kρ0 ,
(2.16)
−
where Z = I Z is the vector field described in Theorem 5.2. The fact that EZ = 0 implies
E[Z, EX] = 0. Thus,
kE[Z, X]kρ0 = kE[Z, (I − E)X]kρ0
≤ C5 kZkρ k(I − E)Xkρ ≤ C6 (γ/σ)k(I − E)Xk2ρ .
(2.17)
In the last step, we have used the bound on kZkρ from Theorem 5.2. Combining the last
three equations yields the third inequality in (2.12).
The analyticity and boundedness of R on D follows from Lemma 2.5.
In order to bound the inverse of L, let Y be a vector field in PHρ . Then Y can be
written as Y (x, y) = (u, M y +v), and the last inequality in (2.12) now follows from the fact
that (L−1 Y )(x, y) = η(T u, M y + µv). Here, we have used that Tb = I, except (optionally)
in the Hamiltonian case where M and v are zero.
QED
Due to the potentially large factor µ−1 in the third inequality of (2.12), we will choose
the domain of R to be of the form
kP(X − K)k% < r ,
k(I − P)Xk% < r ,
k(I − E)Xk% < rδ ,
with 0 < r ≤ R(σ/γ)2 , and with δ > 0 small (to be determined later).
(2.18)
10
HANS KOCH and SAŠA KOCIĆ
Definition 2.7. Given γ ≥ 1, we will call (µ, σ, τ, r, δ) proper RG parameters if r ≤
R(σ/γ)2 , and if (2.2) holds with χ = τ /σ. The parameters are also assumed to be positive,
and µ, σ, τ < 1. We say that the pair (T, ω) is compatible with these parameters if the
condition (2.11) is satisfied as well. The open subset D of A% defined by equation (2.18)
will be referred to as the domain of R.
3. Infinitely renormalizable vector fields
Our goal now is to compose RG transformations of the type described above.
Let λ0 = 1. Given a sequence of matrices P0 , P1 , P2 , . . . in SL(d, Z), with P0 the
identity, and a unit vector ω in Rd , we define ω0 = ω and
Tn = Pn−1 Pn−1 ,
λn = kPn ω0 k ,
ωn = λ−1
n Pn ω0 ,
(3.1)
for all n ≥ 1. We also define λ0 = 1. Assuming that each of the pairs (Tn , ωn−1 ) is
compatible with some proper set of RG parameters, we can define the corresponding RG
transformation Rn : Dn−1 → A% . Notice that the normalization constant ηn for Rn is
given by ηn = λn /λn−1 .
e n = Rn ◦Rn−1 ◦. . .◦R1 . The domain D
en of the combined RG transformation
Let now R
e
e n that are
Rn+1 is defined inductively as the set of all vector fields in the domain of R
e n into the domain Dn of Rn+1 . By Theorem 2.6, these domains are open
mapped under R
e n are analytic.
and non-empty, and the transformations R
Theorem 3.1. Let α > β, m > 2α + 7, and γ ≥ 2α + 2 be given. Then there exist real
numbers b, C > 0, a decreasing sequence of proper RG parameters (µn , σn , τn , rn−1 , δn−1 )
satisfying
1+α
σn+2 = σn+1
,
µn = σnm ,
2
rn−1 ,
rn = 51 σn+1
n = 1, 2, . . . ,
(3.2)
and for every every Diophantine vector ω ∈ Ω a sequence of matrices Pn ∈ SL(d, Z) yielding
pairs (Tn , ωn−1 ) that are compatible with the RG parameters, and an open neighborhood
B of K = (ω, 0) in A% , such that the following holds. B contains a ball of radius b, centered
en is the graph of an analytic function W : (I − P)B → PB,
at K. The set W = B ∩n D
satisfying W (0) = K and DW (0) = 0. For each X ∈ W and n ≥ 1,
e n (X) − Kn ≤ Cσnm−2α−7 rn k(I − P)Xk% ,
R
%
e n (X) − Kn ] ≤ Cσn2(m−2α−7) rn2 k(I − P)Xk2% ,
P[R
%
e n (X) ≤ Cσn(m−1)γ−2α−6 rn k(I − E)Xkρ .
(I − E)R
%
(3.3)
A proof of this theorem will be given below. It uses a continued fractions expansion
developed in [16,18,24], which we will now describe very briefly, and a stable manifold
theorem given in Section 6. We note that the second bound in (3.3) is not strictly needed
Renormalization and Diophantine Invariant Tori
11
for our subsequent construction of invariant tori. The first bound, with a larger value of
m, could be used instead.
Let F be a fundamental domain for the left action of Γ = SL(d, Z) on G = SL(d, R).
Consider the one-parameter subgroup of G, generated by the matrices
E t = diag e−t , . . . , e−t , e(d−1)t ,
t ∈ R.
(3.4)
Given a Diophantine vector ω ∈ Rd , define W ∈ G to be the matrix obtained from the
d × d identity matrix by replacing its last column vector by a constant multiple of ω whose
last component is 1. Then, for every t ∈ R, there exists a unique matrix P (t) ∈ Γ such
that P (t)W E t belongs to F . To a given sequence of “stopping times” 0 < t1 < t2 < . . .,
we can now associate a sequence of matrices Pn = P (tn ). The corresponding matrices Tn
and vectors ωn are defined as in (3.1).
Let t0 = 0, and define t0n = tn − tn−1 for all positive integers n. Let θ = β/(d + β).
Theorem 3.2. ([16]) There exists c0 > 0, depending only on the Diophantine constants β
and ζ, such that for all n > 0, and for all vectors ξ ∈ Rd that are perpendicular to ωn−1 ,
kTn k ≤ c0 exp{(d − 1)(1 − θ)t0n + d θ tn },
kTn−1 k ≤ c0 exp{(1 − θ)t0n + d θ tn },
|Tn∗ ξ| ≤ c0 exp{−(1 − θ)t0n + d θtn−1 }|ξ| .
(3.5)
n
Proof of Theorem 3.1. Let α > β be fixed. We choose
√ tn = c(1 + α) for each positive
integer n, with c > 0 to be determined. Define c1 = 2c0 d and
σn = exp{−dt0n } ,
τn = c1 exp{−(1 − θ)t0n + d θtn−1 } .
(3.6)
Then Theorem 3.2 guarantees that the conditions (2.11) are satisfied. By using that t01 = t1
α
and t0n = 1+α
tn for n > 1, we obtain the bounds
α
tn } ,
σn ≤ exp{−d 1+α
τn ≤ c1 exp{−tn } ,
(3.7)
1−θ
with = 1+α
(α − β) > 0. Let now µn = σnm with m > 1 fixed. Then it is clear that the
conditions (2.2) are satisfied as well, for any γ > 0, provided that c is chosen sufficiently
large. Here, and in what follows, any condition that is said to hold for large values of c is
implicitly being satisfied by choosing c as large as necessary.
Next, let r0 = R(σ1 /γ)2 , and define r1 , r2 , . . . as in (3.2). Then rn−1 ≤ R(σn /γ)2 , for
all n ≥ 1. Thus, we have shown that (µn , σn , τn , rn−1 , δn−1 ) are proper RG parameters, in
the sense of Definition 2.7, and that (Tn , ωn−1 ) is compatible with these parameters. This
is independent of the choice of δn−1 > 0, which we will describe below.
Consider now the rescaled RG transformation Rn , defined by the equation
Rn (Z) = rn−1 Rn (Kn−1 + rn−1 Z) − Kn .
(3.8)
The domain of Rn is given by (2.18), with r = 1 and δ = δn−1 , and with K replaced by
the zero vector field. The restriction of Rn to PAρ , which is linear, will be denoted by Ln .
12
HANS KOCH and SAŠA KOCIĆ
−1
By using the bound on L−1
n from Theorem 2.6, we obtain kLn k ≤ 1/5, for large c > 0.
Here, we have used also that σn < kTn k−1 ≤ ηn ≤ kTn−1 k < σn−1 . The same inequalities,
and Theorem 2.6, also imply that
k(I − E)Rn (Z)k% ≤ εn k(I − E)Zk% ,
εn = 5C0 γσn(m−1)γ−2α−6 ,
k(I − P)Rn (Z)k% ≤ ϑn k(I − P)Zk% ,
ϑn = 5C0 γσnm−2α−7 ,
kPRn (Z) − Rn (PZ)k% ≤ ϕn δn−1 k(I − E)Zk% ,
(3.9)
ϕn = C0 γ 3 σn−m−2α−5 ,
for all Z in the domain of Rn . Assume now that m > 2α + 7 and γ ≥ 2α + 2. Then
εn ≤ ϑn ≤ 1/5, if c is sufficiently large. Furthermore, by setting δn−1 = (5ϕn )−1 , it is easy
to check that εn δn−1 ≤ δn , provided again that c has been chosen sufficiently large.
At this point we have verified the hypotheses of Theorem 6.1 with ε = ϑ = 1/5.
This includes the condition (6.4), since the third inequality in (3.9) remains true if δn−1
is replaced by k(I − E)Zk% . The assertions of Theorem 3.1 now follow from Theorem 6.1.
QED
The above also proves Theorem 1.3, except for the reference to the statements in
Theorem 1.1 that concern the symmetry properties of W , and invariant tori. The latter
will be proved in the next section.
The fact that W preserves the type of a vector field (Hamiltonian, divergence free,
symmetric, or reversible) can be seen as follows. The elimination step X 7→ UX is typepreserving by design; see the discussion at the end of Section 5. In the case of divergence
free, symmetric, or reversible vector fields, the same is true for the scaling T ∗ Sµ∗ , so the
subspace of vector fields with the given symmetry property is invariant under renormalization. In these cases, W is clearly type-preserving. The same applies to Hamiltonian
vector fields, if we choose for Tbn = I the inverse transpose of Tn .
Consider now the renormalization of Hamiltonian vector fields with Tbn = I. In this
e
case, X
with respect to the standard symplectic
Pn = Rn (X0 ) is in general not Hamiltonian
P
form j dxj ∧ dyj , but with respect to j dxj ∧ (Cdy)j , where C is the inverse of the
transpose of Pn . Still, the second (C` ) component of Xn has a zero torus average. Thus,
our RG analysis could be restricted to vector fields with this property, replacing e.g. Rn
by R0n = E ◦ Rn , where E is the canonical projection onto vector fields whose second
component has a zero torus average. Since E does not affect Hamiltonian vector fields,
the resulting stable manifold W 0 coincides with the corresponding restriction of W. This
shows that W preserves all of the types considered.
4. Construction of invariant tori
Following [21,23], our construction of invariant tori is based on the relation between an
invariant torus of a vector field X and the corresponding torus of the renormalized vector field R(X). We start with an informal discussion of this relation. Then we prove
Theorem 1.1 and Lemma 1.2.
Renormalization and Diophantine Invariant Tori
13
4.1. Preliminaries
Let X ∈ A% . Notice that R(X) is obtained from X by a change of coordinates (that
depends on X), combined with a rescaling of time. Thus, the flow for R(X) is related to
the flow for X by the equation
−1
ΛX ◦ ΦtR(X) = ΦηX
t
◦ ΛX ,
ΛX = UX ◦ Sµ ◦ T .
(4.1)
−1
In particular, T ◦ ΦtR(K) = ΦηK t ◦ T on D0 . The identity (4.1) can also be used to relate
an invariant torus for X to an invariant torus for R(X). To this end, if F is any map from
D0 into the domain of ΛX , define
MX (F ) = ΛX ◦ F ◦ T −1 .
(4.2)
e with frequency vector ω
Assume that R(X) has an invariant torus Γ
e = η −1 T −1 ω, taking
e Then, by using (4.1), together with
values in the domain of ΛX , and define Γ = MX (Γ).
the fact that R(K) = (e
ω , 0), we obtain
−1
e ◦ T −1 ◦ ΦtK = ΛX ◦ Γ
e ◦ Φηt
Γ ◦ ΦtK = ΛX ◦ Γ
R(K) ◦ T
−1
e
e ◦ T −1 = ΦtX ◦ Γ .
= ΛX ◦ Φηt
= ΦtX ◦ ΛX ◦ Γ
R(X) ◦ Γ ◦ T
This shows that Γ is an invariant torus for X with frequency vector ω.
In order to make these identities more precise, we need to estimate the difference
Y (t) = ΦtX − ΦtK between the flow for a vector field X and the flow for K = (ω, 0). This
can be done by solving the integral equation
Z t
Y (t) =
[(X − K) ◦ ΦsK ] ◦ [I + Y (s)] ds .
(4.3)
0
t
Notice that ΦtK = I + tK and I + Y (t) = Φ−t
K ◦ ΦX .
Proposition 4.1. Let τ be a positive real number and X a vector field in Aρ , such
that τ kX − Kkρ < r < ρ. Then the equation (4.3) has a unique continuous solution
t 7→ Y (t) ∈ Aρ−r on the interval |t| ≤ τ , and
kΦtX − ΦtK kρ−r ≤ kt(X − K)kρ .
(4.4)
Proof. By using Proposition 2.1 and the contraction mapping principle, the equation
(4.3) is easily seen to have a unique continuous solution t 7→ Y (t) ∈ Aρ−r for t near 0.
The solution can be continued as usual, as long as kY (t)kρ−r < r. But on any interval
containing zero, where kY (t)kρ−r < r, we have by (4.3) the bound
kY (t)kρ−r ≤ kt(X − K)kρ .
(4.5)
Here, we have used also that Z 7→ Z ◦ ΦsK is an isometry on Aρ . Thus, equation (4.3) has
a continuous solution Y for all times t satisfying kt(X − K)kρ < r.
QED
14
HANS KOCH and SAŠA KOCIĆ
4.2. Existence of tori
Consider now a fixed but arbitrary vector field X on the stable manifold W of our RG
transformations Rn . Let X0 = X, and Xn = Rn (Xn−1 ) for n ≥ 1. In order to simplify
notation, we will write Uk and Mk+1 in place of UXk and MXk , respectively. Our goal is
to construct an appropriate sequence of functions Γk : D0 → D% , satisfying
Γn−1 = Mn (Γn ) = Λn ◦ Γn ◦ Tn−1 ,
Λn = Un−1 ◦ Sµn ◦ Tn ,
(4.6)
for all positive integers n. Then we will show that Γk is an invariant torus for Xk , with
frequency vector ωk , for each k ≥ 0.
We assume that α, m, and γ satisfy the conditions given in Theorem 3.1. For every
integer n ≥ 0, define Bn to be the vector space A0 , equipped with the norm
kf k0n = arn−1 kf k0 = arn−1
X
kfν k .
(4.7)
ν
Here, a is some positive real number, to be specified later. Denote by Bn the unit ball in
Bn , centered at the identity function I.
Proposition 4.2. Assume that (m − 1)γ > 3α + 8. If a has been chosen sufficiently large,
then there exists an open neighborhood B of K in A% , and a universal constant C1 > 0,
such that for every X ∈ B ∩ W, and for every n ≥ 1, the map Mn is well defined and
analytic, as a function from Bn to Bn−1 , and it takes values in Bn−1 /2. Furthermore,
kDMn (F )k ≤ C1 σn , for all F ∈ Bn .
Proof. Clearly, Mn is well defined in some open neighborhood of I in Bn , and
Mn (F ) = I + g + (Un−1 − I) ◦ (I + g) ,
g = Sµn ◦ Tn ◦ f ◦ Tn−1 ,
(4.8)
where f = F − I. By Theorem 5.2 and Theorem 3.1, we have for n > 1 the bound
−
(m−1)γ−2α−6
kUn−1 − Ikρ ≤ C2 σn−1 kI Xn−1 k% ≤ C3 σn−1 σn−1
rn−1 k(I − E)Xk%
≤ C4 σn−1 rn−1 k(I − E)Xk% ≤ C5 σn rn−1 ,
(4.9)
with C2 , . . . , C5 universal constants. The first inequality and the final bound in (4.9) also
hold for n = 1, if the neighborhood B of K has been chosen sufficiently small.
Recall that ρ0 < ρ < % have been fixed. The composition with I + g in equation (4.8) is
controlled by Proposition 2.1, using that kgk0 ≤ σn−1 a−1 rn kf k0n < ρ0 independently of n,
if a has been chosen sufficiently large. Here, and in what follows, we assume that F ∈ Bn .
2
By using that rn /rn−1 = σn+1
/5, we obtain kgk0n−1 ≤ σn /5. When combined with
(4.9), this yields the bound kMn (F ) − Ik0n−1 ≤ σn /2, if the neighborhood B of K has been
chosen sufficiently small.
When restricting Un−1 to the domain Dρ0 , we obtain a bound analogous to (4.9)
for the derivative of Un−1 . This, together with the fact that the inclusion map from Bn
2
into Bn−1 is bounded in norm by σn+1
/5, shows that kDMn (F )k ≤ C1 σn for all n ≥ 1,
Renormalization and Diophantine Invariant Tori
15
and for all F ∈ Bn , where C1 is again a universal constant. This completes the proof of
Proposition 4.2.
QED
Denote by Φn and Ψn the flows for the vector fields Xn and Kn , respectively.
Proposition 4.3. Assume that m > 2α+7+p with p > 0. If a has been chosen sufficiently
large, then there exists an open neighborhood B of K in A% , such that the following holds,
for every X ∈ B ∩ W, and for every n ≥ 1. If F ∈ Bn /2 and |s| ≤ σn−p , then Φsn ◦ F ◦ Ψ−s
n
belongs to Bn .
Proof. We will use the identity
s
−s
−s
−s
Φsn ◦ F ◦ Ψ−s
=
I
+
f
◦
Ψ
+
Φ
◦
Ψ
−
I
◦
I
+
f
◦
Ψ
.
n
n
n
n
n
(4.10)
Let ε = m − 2α − 7 − p. By Proposition 4.1 and Theorem 3.1, we have the bound
s
ε
Φn ◦ Ψ−s
n − I ρ0 ≤ ks(Xn − Kn )kρ ≤ Cσn rn k(I − P)Xk% ,
(4.11).
provided e.g. that the right hand side of this inequality is bounded by ρ − ρ0 . This is
certainly the case if ε is positive and kX − Kk% sufficiently small, independently of n. The
composition by I+f ◦Ψ−s
n in equation (4.10) is controlled the same way as the composition
by I + g in the proof of Proposition 4.2, using also that kf ◦ Ψ−s
n k0 = kf k0 . As a result,
the third term on the right hand side of (4.10) belongs to Bn and is bounded in norm by
Caσnε kX − Kk% , which is less than 1/2 for any n ≥ 1, if X is sufficiently close to K. QED
Assume that α, m, γ, and a have been chosen in such a way that the hypotheses of
Theorem 3.1, Proposition 4.2, and Proposition 4.3 are satisfied, with p = 1 + 1/α. Let
F0 , F1 , . . . be a fixed but arbitrary sequence of functions in A0 , such that Fn ∈ Bn for all
n ≥ 0. Then we can define
Γn,m = Mn+1 ◦ . . . ◦ Mm (Fm ) ,
0 ≤ n < m.
(4.12)
Theorem 4.4. Under the above-mentioned assumptions on α, m, and a, there exists an
open neighborhood B of K in A% such that the following holds. For every X ∈ B ∩ W,
the limits Γn = limm→∞ Γn,m exist in Bn , are independent of the choice of F0 , F1 , . . ., and
satisfy the identities (4.6). Furthermore, Γ0 is an elliptic invariant torus for X, and the
map X 7→ Γ0 is analytic and bounded on B ∩ W.
Proof. By Proposition 4.2 there exists N > 0 such that Mn : Bn → Bn−1 /2 contracts
distances by a factor of at least 1/2, if n ≥ N . Thus, if N ≤ n < m < k, then the
difference Γn,k − Γn,m is bounded in norm by 2n−m+1 . This shows that the sequence
m 7→ Γn,m converges in Bn to a limit Γn , and that the limit is independent of the choice
of the functions Fm . By choosing Fm = Γm for all m, we obtain the identities (4.6). The
analyticity of X 7→ Γ0 follows via chain rule from the analyticity of the maps used in our
construction, and from uniform convergence.
16
HANS KOCH and SAŠA KOCIĆ
In order to prove that Γ0 is an invariant torus for X, let t ∈ R, and define tn = λn t
for all n ≥ 0. Our goal is to apply the identity
ηn s
ns
Φsn−1 ◦ Mn (F ) ◦ Ψ−s
◦ F ◦ Ψ−η
,
(4.13)
n
n−1 = Mn Φn
which follows from (4.1). To be more precise, given t ∈ R, define tn = λn t for all n ≥ 0.
By using that λn = η1 η2 · . . . · ηn , together with the bound ηn ≤ kTn−1 k < σn−1 , and the
recursion relation (3.2) satisfied by σ2 , σ3 , . . ., we obtain a bound λn ≤ C2−1 σn−p , for some
universal constant C2 > 0. Thus, if |t| ≤ C2 , then |tn | ≤ σn−p for all n ≥ 1. Proposition 4.3
now allows us to iterate (4.13), to get the identity
tm
−tm
,
(4.14)
Φt0 ◦ Γ0,m ◦ Ψ−t
0 = M1 ◦ . . . ◦ Mm Φm ◦ Ψm
for all m > 0. As was shown above, the right hand side of this equation converges in A0 to
Γ0 , and thus the left hand side converges to Γ0 as well. In addition, Γ0,m → Γm in A0 , and
since convergence in A0 implies pointwise convergence (see part (a) of Proposition 2.1),
and the flow Φt0 is continuous, we have Φt0 ◦ Γ0 ◦ Ψ−t
0 = Γ0 . This identity now extends
to arbitrary t ∈ R by using the group property of the flow, together with the fact that
composition with Ψs0 is an isometry on A0 .
Finally, notice that by Theorem 3.1,
λn kDXn kρ ≤ C3 σnε rn k(I − P)X0 k% ,
(4.15)
where C3 is some universal constant and ε = m − 2α − 7 − p. The left (and thus right)
hand side of this equation is an upper bound on the modulus of the Lyapunov exponents
for the flow of λn Xn on the the range of Γn . Since X0 is obtained from λn Xn by a change
of coordinates, and Γ0 is the corresponding invariant torus for X0 , the same upper bound
applies to the flow for X0 on the torus Γ0 . Taking n → ∞ shows that this torus is elliptic.
QED
4.3. Analyticity and families
In what follows, the torus Γ0 associated with X ∈ B ∩ W will be denoted by ΓX . The
domain parameter ρ used in the introduction is renamed to %0 , to avoid notational conflicts.
The following theorem together with Theorem 3.1, and the discussion at the end of
Section 3 (concerning the restriction of W to specific types), implies Theorem 1.1.
Theorem 4.5. Let %0 > % + δ with δ > 0. Under the same assumptions as in Theorem 4.4,
the map X 7→ ΓX defines (via extension) a bounded analytic map from B 0 to A0δ , where
B 0 is some open neighborhood of K in A%0 .
Proof. For every u ∈ Rd , define a translation Ru on Cd ×C` by setting Ru (x, y) = (x+u, y).
If X is a vector field on one of the domains Dr , then Ru∗ X denotes the pullback of X
under Ru . And for functions F : D0 → Dr we define Ru∗ F = Ru−1 ◦ F ◦ Ru . An explicit
computation shows that the RG transformation R, and the maps MX defined in (4.2)
satisfy
R ◦ Ru∗ = RT∗ −1 u ◦ R ,
MRu∗ X = Ru∗ ◦ MX ◦ (RT∗ −1 u )−1 .
(4.16)
Renormalization and Diophantine Invariant Tori
17
Here, we have used that the translations Ru∗ are isometries on the spaces Ar , and that the
domain of R is translation invariant; see Definition 2.7. This also implies that the manifold
W is invariant under translations Ru∗ , which is used in the second identity in (4.16).
It is convenient to extend the function X 7→ ΓX to an open neighborhood of K in A%
by projecting X onto a point X 0 ∈ W and defining ΓX = ΓX 0 . More specifically, we take
X 0 = (I + W )((I − P)X), where W is the map defining W, as described in Theorem 3.1.
If restricted to a sufficiently small open ball B ⊂ A% centered at K, the map X 7→ ΓX is
now analytic and bounded on all of B.
The construction of Γ0 in the proof of Theorem 4.4, together with the identities (4.16),
and the invariance property W = W ◦ Ru∗ , shows that ΓRu∗ X = Ru∗ ΓX , for all X ∈ B.
Thus, if u ∈ Rd then
ΓX (u, 0) = Ru ◦ ΓRu∗ X (0, 0) ,
X ∈ B.
(4.17)
The idea now is to extend the right hand side of (4.17) analytically to complex u, by using
the analyticity of X 7→ ΓX . To this end, choose an open neighborhood B 0 of K in A%0 ,
such that Ru∗ B 0 ⊂ B, for all u ∈ Cd of norm r = %0 − % or less. Then the right hand side
of (4.17), regarded as a function of (X, u), is analytic and bounded on the product of B 0
with the strip kIm uk < r. Denoting this function by G, we clearly have G(X, .) ∈ A0δ for
all X ∈ B 0 . The analyticity of X 7→ G(X, .) is obtained e.g. by using a contour integral
formula for (g(t) − g(0) − tg 0 (0))/t2 with g(t) = G(X + tZ, .).
QED
Proof of Lemma 1.2. Let % + δ < %0 < r. (Recall that ρ has been renamed to %0 .) By
Theorem 3.1 and Theorem 4.5, there exists an open ball B in H%0 , centered at K, such
that U = P − W ◦ (I − P) defines an analytic map from B to H u , and such that X ∈ B has
an invariant torus ΓX ∈ A0δ with frequency vector ω, whenever U (X) = 0. Furthermore,
X 7→ ΓX is analytic on B ∩ W.
By our non-degeneracy assumptions, the function Gε (z, v, s) = zK + (1 + z)Jv∗ (εZ + s)
defines a diffeomorphism φ = P ◦ Gε between open neighborhoods of the origin in the
spaces C ⊕ V ⊕ H0u and H u . Let b be an open ball in H u of radius < R/2, centered at
the origin, where R is the radius of B. If ε > 0 and b are chosen sufficiently small, then
G0ε = (I − P)(Gε ◦ φ−1 ) is a family in F(b) of norm < R/2, and the equation F (φ(z, v, s)) =
(1 + z)Jv∗ f (s) defines an analytic map f 7→ F from some open neighborhood B2 of fε
in F(b2 ), to F(b). The image of fε under this map is the family Fε given by Fε (σ) =
K + σ + G0ε (σ). By using that P ◦ G0ε = 0 and W ◦ G0ε = K, we see that U ◦ Fε is the
identity map on b. Thus, by the implicit function theorem, the equation (U ◦ F )(σ) = 0
has a unique solution σ = σF , for any family F sufficiently close to Fε in F(b), and this
solution depends analytically on F . The assertion now follows, with (zf , vf , sf ) = φ−1 (σF )
and cf = (1 + zf ).
QED
5. A normal form theorem
Here we state and prove a normal form theorem that can be applied e.g. to the problem
(2.8) of eliminating nonresonant modes in the renormalization of vector fields on Td × R` .
18
HANS KOCH and SAŠA KOCIĆ
The changes of variables are chosen in such a way as to preserve certain symmetries. This
aspect will be discussed at the end of this section.
Let D0 be a non-empty set in some complex Banach space X . For every r in some
fixed interval [ρ0 , ρ] of positive real numbers, consider the set Dr of all points x ∈ X whose
distance from (any point in) D0 is less than r, and let Ar be a Banach space of vector
fields Y : Dr → X , that contains the inclusion map I from Dr into X .
Assumption 5.1. The following is assumed to hold, whenever ρ0 ≤ r < s ≤ ρ. Let B be
the open ball in Ar of radius s − r, centered at the origin. If X is any vector field in As ,
then Z 7→ X ◦(I+Z) defines an analytic function from B to Ar , and kX ◦(I+Z)kr ≤ kXks .
This implies in particular that As ⊂ A0r whenever r < s, where A0r denotes space
of vector fields X ∈ Ar whose derivative DX defines a continuous linear operator Z 7→
(DX)Z on Ar . The norm of this linear operator will be denoted by kDXkr .
Let now P be a fixed but arbitrary projection operator on Aρ0 , whose restriction to
each of the spaces Ar is a partial isometry on that space. Our aim is to find conditions
under which vector fields satisfying P X = 0 can be considered normal forms. To be more
precise, let K be a fixed vector field in A0ρ satisfying P K = 0. Given a vector field X near
K in A0ρ , we are looking for a change of variables UX : Dρ0 → Dρ , such that the pullback
UX∗ X of X under UX belongs to Aρ0 and satisfies
P UX∗ X = 0 .
(5.1)
In order to see what conditions may be needed, consider writing UX to first order in X − K
as the time-one flow Φ1Z for some vector field Z = P Z. In this approximation, equation
(5.1) becomes
P (X + [Z, X]) = 0 ,
(5.2)
where [Z, X] = (DX)Z−(DZ)X. This motivates the following condition on K. In addition
to P K = 0, we assume that the image of P A0r under Z 7→ P [Z, K] is dense in P Ar , and
that there exists a positive real number κ < 1, such that
P [Z, K] ≥ κkZk0r ,
r
Z ∈ P A0r ,
(5.3)
whenever ρ0 ≤ r ≤ ρ. Here, kZk0r = kDZkr + kZkr , which we consider from now on to be
the norm on A0r . The main result of this section is the following.
Theorem 5.2. (normal form) Let r = ρ − ρ0 . Under the abovementioned conditions on P
and K, if X is a vector field in A0ρ such that
kX − Kk0ρ ≤ 2−3 κ ,
kP Xkρ ≤ ε ,
(5.4)
with ε > 0 satisfying the condition (5.20) given below, then there exists an analytic change
of coordinates UX : Dρ0 → Dρ , such that UX∗ X belongs to Aρ0 and satisfies equation (5.1).
The map X 7→ UX − I takes values in Aρ0 , is continuous in the region defined by equation
19
Renormalization and Diophantine Invariant Tori
(5.4), analytic in the interior of this region, and satisfies the bounds
3
kP Xkρ ,
κ
er
≤ 32R kP Xkρ ,
κr
r
er
11 e
≤ 2
+ 1 28R
kP Xk2ρ .
2
κr
(κr)
kUX − Ikρ0 ≤
kUX∗ X − Xkρ0
∗
UX X − X − [Z, X] 0
ρ
(5.5)
Here, Z ∈ P A0ρ is defined by (5.2) and satisfies the bound kZkρ ≤ κ2 kP Xkρ .
We start with some basic estimates on flows. The flow t 7→ ΦtX associated with a
d t
vector field X ∈ Aρ is obtained by solving dt
ΦX = X ◦ ΦtX with initial condition Φ0X = I.
t
Writing ΦX = I + Y (t), this amounts to solving the integral equation
Z
t
X ◦ [I + Y (s)] ds .
Y (t) =
(5.6)
0
In what follows, any reference to a space A% implicitly assumes that ρ0 ≤ % ≤ ρ.
Proposition 5.3. Let %0 , % and τ be positive real numbers, such that %0 + τ kXk% < %.
Then the equation (5.6) has a unique continuous solution t 7→ Y (t) ∈ A%0 on the interval
|t| ≤ τ , and
kΦtX − Ik%0 ≤ ktXk% .
(5.7)
We will omit the proof of this proposition since it is standard: First, equation (5.6)
is solved locally, using the contraction mapping principle. Due to the uniqueness of these
local solutions, they combine into a continuous solution on all of [−τ, τ ]. What makes
things straightforward is that, by Assumption 5.1, the derivative of X 7→ X ◦ (I + Z)
admits a uniform bound on the entire domain needed. As a result, the intervals for the
local proof can be taken of uniform size.
Proposition 5.4. Let 0 < r < % and t ∈ R. Let Z and X be two vector fields in A% ,
satisfying ktZk% ≤ rε and ktDZk% ≤ sε, with ε ≤ 1/6. Then (ΦtZ )∗ X belongs to A%−r ,
and
t ∗
(ΦZ ) X − X ≤ 3es kXk% ε ,
%−r
t ∗
(5.8)
s
2
(ΦZ ) X − X − t[Z, X]
≤
7e
kXk
ε
.
%
%−r
Proof. It suffices to consider t = 1, since we can always rescale time. Let n be a fixed
positive integer. By using Proposition 2.1, and Cauchy’s formula with contour |z| = 1, to
estimate
z d
X ◦ I+
Z
,
(5.9)
(DX)Z = nε
dz
nε
z=0
we obtain the bound
b %0 −r/n ≤ (nε + sε)kXk% ,
kZXk
(5.10)
20
HANS KOCH and SAŠA KOCIĆ
b denotes the map Y 7→ [Z, Y ]. This bound can
where %0 = %. Here, and in what follows, Z
0
be iterated n times, with % decreasing by r/n after each step, and we find
1
1
b nX Z
≤ (n + s)n εn kXk%
%−r
n!
n!
nn s n
1
≤
e ε kXk% ≤ (eε)n es kXk% .
n!
2
In the last inequality, we have used Stirling’s formula. Now
∞
X
1 ∗
n ε es+1
1
b X
(ΦZ ) X − X Z
≤
·
kXk% ,
=
%−r
n!
2 1 − eε
n=1
(5.11)
(5.12)
%−r
and the first bound in (5.8) follows. The second bound is obtained analogously, with the
sum in (5.12) starting at n = 2.
QED
Returning to the main problem (5.1), our first step is to solve this equation to first
order in the size of X − K.
Proposition 5.5. Let 0 < r < %. Let X be a vector field in A0% , satisfying
kX − Kk0% ≤ 14 κ ,
kP Xk% ≤
1
12 κr .
(5.13)
Then the equation (5.2) has a unique solution Z ∈ P A0% . The vector field Z satisfies
kZk% ≤ κ2 kP Xk% . Furthermore, (Φ1Z )∗ X belongs to A%−r and satisfies
1 ∗
6er
(ΦZ ) X − X ≤
kP Xk% kXk% ,
%−r
κr
1 ∗
28er
(ΦZ ) X − X − [Z, X]
≤
kP Xk2% kXk% .
%−r
(κr)2
(5.14)
Proof. The first condition in (5.13) implies that
k[Z, X − K]k% ≤ 2kZk0% kX − Kk0% ≤
κ
kZk0% ,
2
(5.15)
for every Z ∈ A0% . As a consequence, we have
P [Z, X] ≥ kP [Z, K] − kP [Z, X − K] ≥ κ kZk0% ,
%
%
%
2
(5.16)
whenever Z belongs to P A0% . These inequalities, together with our assumption that the
b is dense in P A% , implies that the linear operator P X
b : P A0 → P A% has a
range of P XP
%
bounded inverse, and in particular, that the equation (5.2) has a unique solution Z ∈ P A0% .
The bound (5.16) also shows that this solution satisfies
kZk% ≤
2
kP Xk% ,
κ
kDZk% ≤
2
kP Xk% .
κ
(5.17)
21
Renormalization and Diophantine Invariant Tori
The remaining claims now follow from Proposition 5.4, setting ε =
2
κr kP Xk%
and s = r.
QED
Now we iterate the map X 7→ (Φ1Z )∗ X described in Proposition 5.5, by starting with
a vector field X = X0 and setting
Xn+1 = (Φ1Zn )∗ Xn ,
P (Xn + [Zn , Xn ]) = 0 ,
(5.18)
for n = 0, 1, . . .. The expectation is that the sequence of maps
Un = Φ1Z0 ◦ Φ1Z1 ◦ . . . ◦ Φ1Zn−1
(5.19)
converges to a solution UX of equation (5.1), as n tends to infinity.
Proof of Theorem 5.2. Setting r = ρ − ρ0 and R = kKkρ + κ, choose ε > 0 such that
ε ≤ 2−6 κr ,
ε ≤ 2−9 κ2 e−r (1 + r)−1 R−1 .
(5.20)
Let ρ0 = ρ, and for m = 0, 1, . . . define ρm+1 = ρm − 2rm , where rm = 2−m−2 r. Our first
goal is to prove that (5.18) defines a sequence of vector fields Xm ∈ A0ρm , satisfying
kXm − Xm−1 k0ρm ≤ 2−m−3 κ ,
kP Xm kρm ≤ 8−m ε .
(5.21)
If we define X−1 = K and X0 = X, then these bounds hold for m = 0 by (5.4). Assume
now that (5.21) holds for m ≤ n. Then, by summing up the bounds on Xm − Xm−1 for
m ≤ n, we obtain the first inequality in
kXn − Kk0ρn ≤
1
κ,
4
kP Xn kρn ≤ 4−n−2 κrn .
(5.22)
The second inequality follows from (5.21), by substituting the first bound in (5.20) on ε.
Thus, Proposition 5.5 guarantees a unique solution to (5.18), and it yields the bounds
kXn+1 −Xn kρn −rn ≤ 6
er −n+1
4
Rε ,
κr
kP Xn+1 kρn −rn ≤ 7
er −2n+3 2
4
Rε .
(κr)2
(5.23)
Here, we have used also that kXn kρn ≤ R, which follows from the first inequality in
(5.22). By using the second condition in (5.20), together with the fact that kF k0ρn −2rn ≤
rn−1 kF kρn −rn , we now obtain (5.21) for m = n + 1 from the bounds (5.23).
Next, consider the functions φk = Φ1Zk − I. By Proposition 4.1 and Proposition 5.5,
kφk kρk+1 ≤ kZk kρk ≤
2
kP Xk kρk < rk .
κ
(5.24)
This shows that Um,n = Φ1Zm ◦ Φ1Zm+1 ◦ . . . ◦ Φ1Zn−1 defines a function in I + Aρn that takes
values in Dρm . Here, and in what follows, it is assumed that 0 ≤ m < n. Setting Uk,k = I,
we have the bound
n−1
X
kUn − Um kρ0 = φk ◦ Uk+1,n k=m
ρ0
(5.25)
n−1
n−1
X
X 2
8−k ε .
≤
kφk kρk+1 ≤
κ
k=m
k=m
22
HANS KOCH and SAŠA KOCIĆ
This shows that n 7→ Un converges in I + Aρ0 to a limit UX that takes values in Dρ , and
that satisfies the first inequality in (5.5) if we set ε = kP Xkρ . Clearly, Xn → UX X in
Aρ0 . The second inequality in (5.5) is now obtained by using the first bound in (5.23).
∗
∗
Since UX
X = UX
X1 with X1 = (Φ1Z )∗ X, we can write
1
∗
∗
UX
X − X − [Z, X] = UX
X1 − X1 + (Φ1Z )∗ X − X − [Z, X] .
(5.26)
1
The first term on the right hand side of this equation can be estimated in the same way
as UX∗ X − X, which yields the bound
r
∗
UX X1 − X1 0 ≤ 32R e kP X1 kρ1 .
1
ρ
κr1
(5.27)
Since kP X1 kρ1 ≤ k(Φ1Z )∗ X − X − [Z, X]kρ1 , the third inequality in (5.5) now follows from
the second inequality in (5.14).
The analyticity of the map X 7→ UX follows from the uniform convergence of Un → UX .
QED
Some special cases
We conclude this section with a discussion of invariance properties of UX that result from
choosing projections of the type described in Definition 2.2, acting on the spaces Ar defined
in Section 2.
Consider a projection P (J), with J chosen in such a way that the condition (5.3)
−
holds for P = P (J). A possible choice for J is the set I introduced in Definition 2.2,
as Proposition 2.4 shows. It is easy to check that P maps Hamiltonian vector fields to
Hamiltonian vector fields. (Similarly for divergence free vector fields.) This includes vector
fields that are Hamiltonian with respect to the pullback of the standard symplectic form
under linear transformations C(x, y) = (x, Cy), where C can be any real nonsingular ` × `
matrix. This follows from the fact that C ∗ commutes with P . Thus, since [Z, X] is
Hamiltonian whenever Z and X are, the entire analysis in this section could be restricted
to Hamiltonian vector fields. Since the solution of (5.2) is unique, we find that the change
of coordinates UX is symplectic whenever K and X are Hamiltonian. The same type of
argument shows that UX is volume preserving if the vector fields K and X are divergence
free.
Next, consider a linear map G on Cd × C` that leaves the domains Dr invariant.
Assume that G∗ is an isometry on each of the spaces Ar , and that it commutes with P .
An example of such a map is G(x, y) = (x, −y). If Z and X are symmetric with respect
to G, then so is [Z, X]. Thus, the analysis of this section can be restricted to symmetric
vector fields. This shows that UX commutes with G∗ whenever K and X are symmetric.
Assume in addition that G ◦ G = I, and that K, X are reversible with respect to G.
Then [Z, X] is reversible whenever Z is symmetric (both with respect to G). Thus, the
b maps the symmetric subspace of P A0ρ to the reversible subspace of P Aρ .
operator P X
As the proof of Proposition 5.5 shows, this operator has (under the given assumptions) a
bounded inverse, so the solution Z of equation (5.2) is symmetric. Consequently, the flows
Φ1Zk defining UX commute with G, and the same is true for UX .
23
Renormalization and Diophantine Invariant Tori
In summary, if X is of the type described above (Hamiltonian, divergence free, symmetric, or reversible), then the same is true for UX∗ X.
6. A stable manifold theorem
Here we state and prove a local stable manifold theorem for sequences of maps of the type
encountered e.g. in renormalization. It allows for a description of two different contraction
rates.
For every integer n ≥ 0 let Xn be a complex Banach space, and let En , Pn be continuous linear projections on Xn , satisfying Pn En = En Pn = Pn and kEn k = kI − En k = 1.
For each n > 0, let Rn be a bounded analytic map, from an open neighborhood Dn−1
of the origin in Xn−1 , to Xn , with the following properties: Rn Pn−1 is linear, and the
restriction Ln of this linear operator to Pn−1 Xn−1 is invertible. Furthermore, there exist
real numbers ϑn ≤ ϑ < 1 and εn ≤ ε = (1 − ϑ)/4, such that for all x ∈ Dn−1 ,
k(I − En )Rn (x)k ≤ εn k(I − En−1 )xk ,
k(I − Pn )Rn (x)k ≤ ϑn k(I − Pn−1 )xk ,
kPn Rn (x) − Ln Pn−1 xk ≤ εk(I − En−1 )xk ,
(6.1)
kL−1
n k ≤ ϑ.
en = Rn ◦ Rn−1 ◦ . . . ◦ R1 . The domain of R
e1 is
Consider now the composed maps R
e 0 = D0 , and for n = 1, 2, . . . , the domain D
e n of R
en+1 is defined inductively
taken to be D
e n−1 that is mapped into Dn by R
en .
as the subset of D
We will assume that the domain Dn−1 of Rn is given by conditions
kPn−1 xk < 1 ,
k(I − Pn−1 )xk < 1 ,
k(I − En−1 )xk < δn−1 ,
(6.2)
where {δk } is a sequence of positive real numbers, such that δk ≥ εk δk−1 for all k > 0.
Theorem 6.1. (local stable manifold) Let R1 , R2 , . . . be a sequence of maps with the
T∞ e
properties described above. Then W0 = n=0 D
n is the graph of an analytic function
W0 : (I − P0 )D0 → P0 D0 , satisfying W0 (0) = 0. For every x ∈ W0 ,
em (x)k ≤ ϑ(m) + ε(m) k(I − P0 )xk ,
kR
em (x)k ≤ ε(m) k(I − E0 )xk ,
k(I − Em )R
(6.3)
where ϑ(m) = ϑ1 ϑ2 · · · ϑm and ε(m) = ε1 ε2 · · · εm . Furthermore, if the third condition in
(6.1) is strengthened to
kPn Rn (x) − Ln Pn−1 xk ≤ ϕn k(I − En−1 )xk2 ,
(6.4)
with ϕn δn−1 ≤ ε, then DW0 (0) = 0, and
em (x)k ≤ ϑ(m) 2 k(I − P0 )xk2 .
kPm R
(6.5)
24
HANS KOCH and SAŠA KOCIĆ
Notice that, by our assumptions (6.1), if x belongs to the domain of Rn , and if
Pn Rn (x) has norm less than one, then Rn (x) belongs to the domain of Rn+1 . This shows
that
e n = {x ∈ D
e n−1 : kPn R
en (x)k < 1} ,
D
n = 1, 2, . . . .
(6.6)
Let Sn = Pn Xn , and denote by bn the open unit ball in Sn , centered at the origin.
Define Fn to be the space of analytic functions f : bn → Xn , equipped with the sup-norm
kf k = sups∈bn kf (s)k . Denote by In the inclusion map of bn into Xn . Notice that, if
f ∈ Fn−1 satisfies
Pn−1 f = In−1 ,
kf − In−1 k < 1 ,
k(I − En−1 ) ◦ f k < δn−1 ,
(6.7)
then f (s) belongs to the domain of Rn , for all s ∈ bn−1 . For such functions f , define
Yn,f = Pn (Rn ◦ f ) .
(6.8)
Proposition 6.2. Assume that f ∈ Fn−1 satisfies (6.7). Then Yn,f : bn−1 → Sn has a
−1
unique right inverse Yn,f
: bn → bn−1 . Both Yn,f and its inverse depend analytically on
f , on the domain defined by (6.7). Furthermore,
kYn,f − Ln k ≤ εk(I − En−1 ) ◦ f k ,
−1
kYn,f
− L−1
n k ≤ ϑεk(I − En−1 ) ◦ f k .
(6.9)
Proof. Let U = Yn,f − Ln . By the third condition in (6.1) we have
kU (s)k = kPn Rn (f (s)) − Ln Pn−1 f (s)k ≤ εk(I − En−1 )f (s)k ,
(6.10)
for all s ∈ bn−1 This implies the first bound in (6.9).
By our assumption on f and ε, we have kU k ≤ ε ≤ r/2, where r = (1 − ϑ)/2. If
s ∈ Sn−1 is of norm ≤ ϑ and h ∈ S of norm one, then by Cauchy’s formula
kDU (s)hk ≤ r−1 sup kU (s + zh)k ≤ r−1 kU k ≤ 1/2 .
(6.11)
|z|=r
The equation for a right inverse L−1
n + V of Ln + U can be written as ψ(V ) = V , with
−1
ψ defined by ψ(V ) = −Ln U ◦ (L−1
n + V ). Consider the space of analytic functions
V : bn → Sn−1 , equipped with the sup-norm. Denote by B the closed ball of radius r in
this space, centered at the origin. Then ψ is analytic on B, with derivative given by
−1
Dψ(V )h = −L−1
n (DU ) ◦ (Ln + V ) h .
(6.12)
By equation (6.11), we see that kDψ(V )k < 1/2, for all V ∈ B. Since kψ(0)k ≤ r/2, the
map ψ is a contraction on B, and thus has a (unique) fixed point in B. This fixed point
V satisfies kV k = kψ(V )k ≤ kL−1
n U k, which implies the second inequality in (6.9). The
Renormalization and Diophantine Invariant Tori
25
analyticity of U 7→ V follows form the uniform convergence of ψ n (0) → V for kU k ≤ r/2.
QED
This proposition allows us to define the maps
−1
Rn (f ) = Rn ◦ f ◦ Yn,f
,
e n = Rn ◦ Rn−1 ◦ . . . ◦ R1 .
R
(6.13)
Notice that Pn Rn (f ) = In . In particular, since Rn ◦ Pn−1 = Pn ◦ Rn ◦ Pn−1 by the second
condition in (6.1), we have Rn (In−1 ) = In . The domain of Rn is the set of all f ∈ Fn−1
satisfying (6.7).
e n (f0 ) is well defined for all n ≥ 1,
Lemma 6.3. If f0 belongs to the domain of R1 , then R
and
e n (f0 ) − In k ≤ ϑ(n) kf0 − I0 k .
kR
(6.14)
Proof. Let n ≥ 1. Let f be an arbitrary function in the domain of Rn , and define f 0 =
−1
Rn (f ). Consider a fixed but arbitrary s ∈ bn and define s0 = Yn,f
(s). By Proposition 6.2,
0
s belongs to bn−1 . Thus, the second condition in (6.1) implies that
kf 0 (s) − sk = k(I − Pn )Rn (f (s0 ))k
≤ ϑn k(I − Pn−1 )f (s0 )k = ϑn kf (s0 ) − s0 k .
This shows that kf 0 −In k ≤ ϑn kf −In−1 k. In addition, we have Pn f 0 = In by the definition
of Rn , and k(I − En ) ◦ f 0 k ≤ εn δn−1 by the first inequality in (6.1). Thus, since ϑn < 1 and
εn δn−1 ≤ δn , the function f 0 belongs to the domain of Rn+1 . This proves Lemma 6.3.
QED
e n can be taken to be the domain of R1 . If
This lemma shows that the domain of R
f0 is any function in this domain, define
e n (f0 ) ,
fn = R
Yn = Yn,fn−1 ,
−1
−1
Zm,n = Ym+1
◦ . . . ◦ Yn−1
◦ Yn−1 ,
(6.15)
whenever 0 ≤ m < n.
Proposition 6.4. For every f in the domain of R1 , there exists a unique sequence m 7→
zm ∈ bm satisfying
zm−1 = Ym−1 (zm ) ,
m = 1, 2, . . . ,
(6.16)
and this sequence is given by the limits zm = limn→∞ Zm,n (0). The maps f 7→ zm are
analytic on the domain of R1 .
Proof. First, we note that it suffices to prove the claims for m ≥ N , where N is fixed but
arbitrary positive integer.
n
Let f0 = f . Since kYn−1 − L−1
n k ≤ ϑ by Proposition 6.2 and Lemma 6.3, there exist
an integer N > 0, and two positive real numbers r, r0 < 1, both independent of f0 , such
that Yn−1 maps bn into to r0 bn−1 and contracts distances by a factor ≤ r, whenever n ≥ N .
In what follows, we assume that N ≤ m < n.
26
HANS KOCH and SAŠA KOCIĆ
Consider now an arbitrary sequence n 7→ sn ∈ bn , with the property that sn belongs
to the closure of r0 bn . Notice that if a sequence n 7→ zn ∈ bn satisfies (6.16), then it
automatically has this property. Define sm,n = Zm,n (sn ). Then ksm,k − sm,n k < 2rn−m
whenever k > n. This shows that n 7→ sm,n converges as n → ∞, and that the limit ŝm is
independent of the sequence {sn }. In particular, we see that ŝm = zm by choosing sn = 0
for all n. The identities (6.16) are obtained by choosing sn = zn for all n.
By Proposition 6.2, the maps f 7→ sm,n = Zm,n (0) are analytic on the domain of R1 .
The analyticity of f 7→ zm now follows from the uniform convergence of sm,n → zm . QED
Corollary 6.5. Let f be a family in the domain of R1 , and let s ∈ b0 . Then f (s) belongs
to W0 if and only if s = z0 (f ).
Proof. Consider first x = f (z0 ). Then x ∈ D0 , since f belongs to the domain of R1 , and
the following holds for n = 1, 2, . . .. Set xn = fn (zn ). By the definition of Rn , and by
en (x). Furthermore, Pn xn = Pn fn (zn ) = zn ∈
Proposition 6.4, we have xn = Rn (xn−1 ) = R
e
bn , and thus x belongs to the set Dn described in (6.6). This shows that x ∈ W0 .
Consider now a fixed s = s0 in b0 , and assume that x0 = f (s0 ) belongs W0 . Then we
e n (x) for all n > 0, and sn = Pn xn belongs to bn . Set f0 = f . Proceeding
can define xn = R
by induction, let n > 0, and assume that xn−1 = fn−1 (sn−1 ). Since sn = Yn (sn−1 ), and
since Yn has a unique right inverse on bn by Proposition 6.2, we have sn−1 = Yn−1 (sn ).
As a result, xn = fn (sn ). This shows that sn−1 = Yn−1 (sn ) holds for all n > 0, and thus
sn = zn by Proposition 6.4.
QED
Proof of Theorem 6.1. Denote by B00 the unit ball in (I − P0 )X0 , centered at the
origin. To a point x ∈ B00 we associate the family f : s 7→ s + x. This family belongs to the
domain of R1 . Now define W0 (x) = z0 (f ). By Corollary 6.5, x + s = f (s) belongs to W0
if and only if s = W0 (x). This shows that W0 is the graph of W0 over B00 . The analyticity
of W0 follows from the analyticity of z0 . Furthermore, we have W0 (0) = z0 (I0 ) = 0.
The second bound in (6.3) follows from the first condition in (6.1). In order to prove
the first bound, consider the family f0 (s) = s+(I−P0 )x, the associated functions fn and Yn
en (x) = fn (zn )
defined in (6.15), and the parameters zn described in Proposition 6.4. Then R
for all n ≥ 0. By Lemma 6.3 we have
kfm (zm ) − zm k ≤ ϑ(m) k(I − P0 )xk ,
and by Proposition 6.2 and the second inequality in (6.3),
n−1
X
−1
−1
−1
−1
−1
−1
kzm − Lm+1 · · · Ln zn k = Lm+1 · · · Lk Yk+1 − Lk+1 (zk+1 )
k=m
≤
n−1
X
ϑk−m ϑε ε(k) k(I − E0 )xk
k=m
≤
ϑε (m)
ε k(I − E0 )xk ,
1 − ϑε
Renormalization and Diophantine Invariant Tori
27
−1
whenever 0 ≤ m < n. These two inequalities, together with the fact that L−1
m+1 · · · Ln zn
tends to zero as n → ∞, imply the first bound in (6.3).
Next, assume that (6.4) holds. Then the equation (6.10) shows that for each n > 0,
the map f 7→ Yn,f has a vanishing derivative at f = In−1 . By the definition of W0 , this
implies that DW0 (0) = 0.
Let now x0 ∈ W0 . Then x0 = u + W0 (u) with u = (I − P0 )x0 . Assume that u 6= 0.
Let ` be a continuous linear functional on X0 of norm one, such that `(W0 (u)) = kW0 (u)k.
Define g(z) = `(W0 (zu/kuk)) for all z in the complex unit disk |z| < 1. Since W0 and
DW0 vanish at the origin, z 7→ z −2 g(z) defines an analytic function on the unit disk, and
by Schwarz’s lemma, this function is bounded in modulus by 1. Here, we have used that
W0 has norm less than one on its domain. This shows that kW0 (u)k = g(kuk) ≤ kuk2 , or
in other words, that kP0 x0 k ≤ k(I − P0 )x0 k2 .
Finally, let m > 0 and consider the stable manifold Wm for the shifted sequence of
em (x0 ) belongs to Wm . The same arguments as
maps Rm , Rm+1 , . . .. Clearly, xm = R
above show that kPm xm k ≤ k(I − Pm )xm k2 . The bound (6.5) now follows from the second
condition in (6.1).
QED
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