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Renormalization of Vector Fields
1
Hans Koch 2
Abstract. These notes cover some of the recent developments in the renormalization of quasiperi- odic flows. This includes skew flows over tori, Hamiltonian flows, and other flows on Td^ ×R`. After stating some of the problems and describing alternative approaches, we focus on the definition and basic properties of a single renormalization step. A second part deals with the construction of conjugacies and invariant tori, including shearless tori, and non-differentiable tori for critical Hamiltonians. Then we discuss properties related to the spectrum of the linearized renormaliza- tion transformation, such as the accumulation rates for sequences of closed orbits. The last part describes extensions from “self-similar” to Diophantine rotation vectors. This involves sequences of renormalization transformations that are related to continued fractions expansions in one and more dimensions. Whenever appropriate, the discussion of details is restricted to special cases where inessential technical complications can be avoided.
1 Expanded notes from a mini-course given at the Fields Institute in Toronto, Canada, November 2005 (^2) Department of Mathematics, The University of Texas at Austin, 1 University Station C1200, Austin, TX
78712-
1
2 HANS KOCH
- Background Content
- 1.1 Invariant tori
- 1.2 Two direct approaches
- 1.3 Hamiltonians
- 1.4 KAM theory
- 1.5 Scales
- 1.6 Breakup of invariant tori
- Renormalization of flows
- 2.1 Hamiltonian systems
- 2.2 Resonant and nonresonant Hamiltonians
- 2.3 The change of variables UH
- 2.4 Other vector fields
- 2.5 Skew systems
- A single renormalization group step
- 3.1 Skew systems: definitions
- 3.2 Skew systems: estimates
- 3.3 More general vector fields
- 3.4 A general elimination procedure
- A nontrivial RG fixed point
- 4.1 Observations and result
- 4.2 Strategy of proof
- 4.3 Non-twist flows
- Invariant tori
- 5.1 Some ideas and results
- 5.2 Renormalization of invariant tori
- 5.3 Existence
- 5.4 Critical invariant tori
- 5.5 Shearless tori
- Scaling
- 6.1 Spectrum of the linearized RG transformation
- 6.2 Accumulation of periodic orbits
- 6.3 Choice of the manifold Σ
- 6.4 The manifolds Σn and orbits γn
- Sequences of RG transformations
- 7.1 Diophantine and Brjuno numbers
- 7.2 Multidimensional continued fractions
- 7.3 Composing different RG transformations
- 7.4 An invariant manifold theorem
- Reduction of skew flows
- 8.1 A general result
- 8.2 The stable manifold
- 8.3 Conjugacy to a linear flow
- 8.4 The special case G=SL(2,R)
- 8.5 Excluding hyperbolicity
- References
4 HANS KOCH
where D = ω · ∇ − DX ◦ Γ 0. Iterating the map γ 7 → γ 0 + W (γ) yields a formal power series for the torus, also referred to as Lindstedt series. This series is in general highly divergent, but there are nontrivial situations where resummation techniques can be used to obtain γ from its Lindstedt series [39]. Interestingly, the problems that one encounters are similarly to those found for Feyn- man graph expansions in quantum field theory. It is these expansions that lead to the development of renormalization methods [118]. The divergencies can be associated with different “scales” in the problem, and by re-normalizing the expansion parameters appro- priately, the divergencies at any given scale cancel. Applications of renormalization ideas from quantum field theory to the resummation problem for Lindstedt series can be found e.g. in [51, 57,58,59]. In this context, renormalization can be viewed as a method for deal- ing with combinatorial problems and cancellations in certain highly nontrivial perturbation expansions. Equation (1.3) also has a vague resemblance to field equations in quantum field theory. In some special cases, it is possible to make this connection more precise and write (1.3) as the Euler-Lagrange equation for some functional γ 7 → L 1 (γ). The modern way of ana- lyzing such fields is via functional integrals. Expanding these integrals in powers of small coupling constants yields the above-mentioned Feynman graphs. (Integrals of the same type also appear in statistical mechanics, where there are usually no small parameters.) The approach taken in non-perturbative renormalization is to perform the integration one scale at a time, transforming a Lagrangian Lk at scale k to a new Lagrangian Lk+1 at scale k + 1, making the map Lk 7 → Lk+1 a dynamical system, if possible. Inspired by this approach, Bricmont et. al. have devised a renormalization scheme that applies to the problem of constructing invariant tori [8]. The formalism itself is non-perturbative, but in practice, the analysis can be carried out only for X close to constant, where γ is small. Similar ideas have also been applied successfully to the study of PDEs [9]. In the context described here, renormalization can be viewed as a procedure for solving certain difficult “scale free” problems iteratively, one scale at a time.
1.3. Hamiltonians
The renormalization group approach that we will focus on later is much closer to KAM theory than to the approaches sketched above. In order to simplify the discussion, we will restrict our attention to Hamiltonian flows. Consider M = Td^ × B, where B is some open neighborhood of the origin in Rd. A Hamiltonian vector field in action-angle variables is of the form X = J∇H, with J =
[ 0 I
−I 0
]
, where H is the corresponding Hamiltonian, a differentiable function on M. In other words, the equation ˙u = X(u), with u = (q, p), can be written as q˙ = ∇pH , p˙ = −∇q H. (1.5)
The corresponding flow will be denoted by ΦH. Some basic facts and notation: H is invariant under the flow. The maps Φ Ht are symplectic, in the sense that they preserve the symplectic form
dqj ∧ dpj. If U is any symplectic diffeomorphism of M, then the pushforward of X under U is again a Hamiltonian vector field, with Hamiltonian H ◦ U. Furthermore, U preserves the Poisson bracket {f, g} = ∇ 1 f · ∇ 2 g − ∇ 2 f · ∇ 1 g.
Renormalization of Vector Fields 5
A change of coordinates (q′, p′) = U (q, p) is canonical if and only if the one-form p′^ · dq′^ − p · dq is closed. Locally, this one-form can be written as the differential of some function, which we will write as p′^ · q′^ − φ. We will only be interested in cases where φ is defined globally, as a function of q and p′. In this case, φ will be referred to as the generating function of U. It satisfies q′^ = ∇ 2 φ(q, p′) and p = ∇ 1 φ(q, p′). In particular, if
U = I + u , u(q, p) =
Q(q, p), P (q, p)
then we have
Q(q, p) = (∇ 2 φ)
q, p + P (q, p)
, P (q, p) = −(∇ 1 φ)
q, p + P (q, p)
Conversely, given φ not too large, these two equations determine a canonical transformation of the form (1.6). If φ is small, say of “size ε”, then so are P, Q, and we have
H ◦ U = H(. + ∇ 2 φ ,. − ∇ 1 φ) + O(ε^2 ) = H + {H, φ} + O(ε^2 ). (1.8)
1.4. KAM theory
KAM theory [82,5,110,121,26] is concerned with with small perturbations of integrable Hamiltonians, such as K(q, p) = ω · p + 12 (M p) · p , (1.9)
with ω ∈ Rd^ and M a symmetric d×d matrix. The dynamics for K is given by ˙q = ω +M p and ˙p = 0. Notice that surfaces of constant p are invariant tori for the flow generated by K , with frequency vector w(p) = ω + M p. The goal is to construct an invariant torus for H = K + h with rotation vector ω, by iterating the following procedure. Assuming that h is small, say of “size ε”, we have
(K + h) ◦ U = K + {K, φ} + h + O(ε^2 ) = K −
[
w · ∇ 1 φ − h
]
Now we try to solve [.. .] = 0 near p = 0, up to an error of order ε^2. The equation for the ν-th Fourier mode of φ is iw · νφν = hν + O(ε^2 ). (1.11)
So among other things, the average h 0 has to be small near p = 0. Assuming that the matrix M is nonsingular, this can be achieved by a p-translation, and a restriction to p near zero. Assuming in addition that ω satisfies a Diophantine condition, equation (1.11) can be solved for frequencies ν that are not too large. Finally, if we also assume that h is analytic, so that hν → 0 rapidly as |ν| → ∞, then the equation (1.11) can be solved for all ν. Thus, the new Hamiltonian (K + h) ◦ U is of the form K + g, with g of order ε^2. Now the procedure is iterated. The KAM theorem for this situation states that, under the assumptions made above, the invariant torus with frequency vector ω persists under small perturbations of K.
Renormalization of Vector Fields 7
The best known examples in dynamical systems may be the composition operators R : F 7 → F k^ (modulo rescaling). Here, and in what follows, F k^ denotes the k-th iterate of a map F. These operators have been studied in great detail [128], after the observation of universality and scaling in one-parameter families of interval maps undergoing period doubling bifurcations 1 → 2 →... → 2 n−^1 → 2 n^ →.. .. The k = 2 version of R lifts the inverse cascade to the space of maps, in the sense that F ◦ F has a period 2n−^1 whenever F has a period 2n. In problems dealing with irrational rotations, the scales come from the arithmetic properties of the rotation numbers. Consider e.g. the number α = 1/(k + 1/(k +.. .))), with k some fixed positive integer. (For k = 1, α is the inverse golden mean.) Its continued fraction approximants un/vn may be obtained as follows: [ un vn
]
= T n
[
]
, T =
[
1 k
]
Consider a circle C, defined by a strictly monotone function C on R, by identifying C(x) with x for any real x. A simple example would be C 0 (x) = x − 1. A map on C is a function M : R → R that commutes with C, and a point x has rotation number u/v for this map if Cu^ ◦ M v^ (x) = x. In this formulation, a renormalization operator that takes a pair F =
[ C
M
]
with rotation number un− 1 /vn− 1 to a pair with rotation number un/vn, is given by
R :
[
C
M
]
[
C^0 ◦ M 1
C^1 ◦ M k
]
(modulo rescaling). (1.15)
The pair F 0 =
[ C 0
M 0
]
, with M 0 (x) = x + α, is a fixed point of R , and it clearly has rotation number α. Notice that the “exponents” in (1.15) are precisely the matrix elements of T. This suggests of course a generalization to problems with more than one frequency. This renormalization operator R (for any k) has been studied in great detail [129]. To give a very simple application, it can be shown e.g. that if F is a small perturbation of F 0 with rotation number α, then Rn(F ) → F 0 as n → ∞. This in turn can be used to establish a conjugacy between F and F 0. The analogous operator can be defined also for other types of maps. Such an operator was studies in connection with the breakup of invariant circles in area-preserving maps of the plane [68, 100, 52, 101, 117].
1.6. Breakup of invariant tori
Let now d = 2 and ω =
[
ϑ−^1 1
]
, with ϑ the golden mean (^12)
5 + 12. By the KAM theorem, a
Hamiltonian H close to an integrable Hamiltonian like (1.9) has a smooth invariant torus Γ with winding numbers ωj /ωd. The proof also shows that near this torus, H is essentially integrable, and so the motion is highly ordered and stable. (The same is true in higher dimensions.) In the case d = 2, an invariant 2-torus has an even stronger stabilizing effect, as it divides the (3 dimensional) energy surface containing it into disjoint invariant regions. Consider a one-parameter family β 7 → Hβ^ of Hamiltonians on T^2 × R^2 , such as
Hβ^ (q, p) = ω · p +
p^21 + β
[
cos(q 1 ) + cos(q 1 − q 2 )
]
8 HANS KOCH
which is essentially the Hamiltonian used in [47]. For values of β close to 0, this Hamiltonian has a golden invariant torus, that is, a smooth invariant torus with winding number ϑ−^1. This torus is observed to persist as β is increased, up to some value β = β∞, where it breaks up. The breakup is also seen to promote chaotic motion in the form of hyperbolic orbits with golden mean rotation number. Before the critical point β∞ , the system has “prominent” periodic orbits (symmetric Birkhoff orbits) for each of the rotation numbers 12 , 23 , 35 , 58 ,... associated with the continued fraction expansion of θ−^1. Past the critical point, these orbit with rotation number un/vn turn unstable, at parameter values βn that converge to β∞ in the limit n → ∞. The convergence is observed to be asymptotically geometric, with
lim n→∞
βn+1 − βn βn − βn− 1
= δ−^1 , δ = 1. 6279.... (1.17)
This ration appears to be universal, in the sense that the same values is observed within a large class of one-parameter families. The “critical” Hamiltonian Hβ∞^ appears to have an invariant torus Γ that is non- smooth. Near this torus, the motion for Hβn+1^ looks like that of Hβn^ , modulo a scaling of time (by ϑ) and a scaling of space. The observed eigenvalues of the spatial scaling are
λτ = ϑ, λx = μ∗/λz λy = μ∞/λτ , λz = − 0. 32606... , (1.18)
with μ∗ = 0. 23046.. .. Again, these values appear to be universal.
-1.6 1.
v*q
z
Fig. 2. Orbits for a critical Hamiltonian [2]
10 HANS KOCH
choice for the change of variables H 7 → H ◦ U. In other problems where renormaliza- tion has been applied, the analogue of U can be guessed from the observed scaling. This scaling is typically a contraction on phase space, so it may be reduced to a simple (finite codimension) normal form. But the torus cannot be contracted. As it turns out [74], it is still possible to find a RG transformation of the form
R(H) = H ◦ T (mod G), (2.2)
where G is some “group” of similarity transformations. But it appears that this group needs to be infinite dimensional. Among the possible similarity transformations are
- Scaling of time or energy, H 7 → η−^1 H − E.
- Scaling of momenta H 7 → μ−^1 H(., μ.)
- Change of variables H 7 → H ◦ U , with U canonical and homotopic to the identity.
H∗ ◦ T
H ◦ T
H∗ = R(H∗) R(H)^
H
orbit of H under G
Hamiltonians in normal form
I+ A
Fig. 3. General form of the RG transformation R
The goal is to find a suitable “normal form” for Hamiltonians, and a map H 7 → GH ∈ G, such that GH (H ◦ T ) is in normal form whenever H is. Using for GH a composition of the three similarity transformations listed above, we have
R(H) = H′^ ◦ UH′ , H′^ =
ημ
H ◦ Tμ − E , (2.3)
where UH′ is a canonical change of variables,
Tμ(q, p) =
T q, μ(T ∗)−^1 p
and E, η, μ are normalization constants that may depend on H. We will set E = 0 from now on, and mostly ignore constant terms in Hamiltonians, since such terms do not change
Renormalization of Vector Fields 11
the vector field. The constants η and μ can be determined e.g. by prescribing the value of two coefficients in the Taylor expansion for the torus average of R(H). Choosing a suitable normal form that also determines UH′^ is more delicate. This problem will be discussed in Subsections 2.2 and 2.3. We expect R to have at least one integrable fixed point, describing the flow near smooth invariant tori. Such fixed points are in fact easy to find. To be more specific, let T be a matrix in SL(d, Z) that has two eigenvectors T ω = ϑ 1 ω and T Ω = ϑ 2 Ω, for two eigenvalues satisfying ϑ 1 > 1 > |ϑ 2 |. Consider the integrable Hamiltonians
K(q, p) = f (p) = (ω · p) +
m 2
(Ω · p)^2 , (2.5)
with m > 0, unless specified otherwise. Starting with this Hamiltonian K, computing K ◦ T , and then applying a momentum and energy scaling (including an angle-dependent change of variables would be counter-productive here), we obtain
R(K)(q, p) = η−^1 μ−^1 f
μ(T ∗)−^1 p
− E
= η−^1 ϑ− 1 1 (ω · p) + η−^1 μϑ− 22
m 2
(Ω · p)^2 − E.
For K to be a fixed point for R, we need an energy scaling η = ϑ− 1 1 , and E = 0. Ignoring for the time being the case m = 0, where the momentum scaling μ is undetermined, we have μ = ϑ− 1 1 ϑ^22. Notice that |μ| < 1, due to our condition ϑ 1 > 1 > |ϑ 2 |, meaning that the momenta p are contracted by the scaling H 7 → μ−^1 H(., μ.). So far so good. The problem arises when we try to extend R to Hamiltonians H that depend on the angle variables q as well. Consider e.g. a space Aρ of Hamiltonians that are analytic in the domain Dρ , defined by |Imqj | < ρ and |pj | < ρ, with ρ some fixed positive real number. If H is analytic on Dρ then H′^ = H ◦ Tμ is analytic on T (^) μ− 1 Dρ.
But this new the domain is narrower than Dρ in the angular direction ω, by a factor ϑ− 1 1. The question is whether this loss of analyticity can be restored by a canonical change of variables H′^7 → H′^ ◦ UH′^. At first, this seems unlikely, since UH′^ should be close to the identity for H close to K, if we want R to be a smooth map on Aρ. And a fixed domain loss cannot be restored by changes of variables arbitrarily close to the identity. What will save the situation are cancellations.
2.2. Resonant and nonresonant Hamiltonians
Motivated by the above, we start by trying to identify the “good” and “bad” terms in the Fourier-Taylor series
H(q, p) =
(ν,α)∈I
Hν,αeiν·q^ pα^ , pα^ =
j
p αj j ,^ (2.7)
where I = Zd^ × Zd +. The Hamiltonian H is analytic on Dρ if and only if the series (2.7) converges on Dρ , which is roughly equivalent to
|Hνα| / e−ρ|ν|ρ−|α|^ , (ν, α) ∈ I. (2.8)
Renormalization of Vector Fields 13
sufficiently small. This shows that the term [.. .] in (2.12) is non-positive. These arguments clearly extend to r < ρ sufficiently close to ρ. Thus, if H ∈ I+ Ar then
‖H ◦ Tμ‖ρ ≤
(ν,α)∈I+
|Hν,α| ‖Eν,α ◦ Tμ‖ρ ≤
(ν,α)∈I+
|Hν,α| ‖Eν,α‖r = ‖H‖r.
The assertion now follows by taking r < ρ′^ < ρ and using the fact that the inclusion map from Aρ′ into Ar is compact. QED
Notice that the resonant modes, which are essentially the ones that cause small de- nominator problems in KAM theory, are easy to deal with in this approach. It should be noted also that the smallness conditions in Proposition 2.1 can easily be replaced by concrete inequalities. To give a concrete example: Using a slightly different definition of I
, an analogue of this proposition is proved in [1] for
|ϑ 2 | + σ
ϑ 1 − |ϑ 2 |
ρ′ ρ
μ ϑd
∣ e
ρκ(ϑ 1 −|ϑ 2 |) (^) < ρ ′ ρ
2.3. The change of variables UH
Proposition 2.1 suggests that we take the resonant Hamiltonians as our “normal form”. This requires that the change of variables UH′^ in equation (2.3) can be chosen in such a way that I
H′^ ◦ UH′
which makes R(H) again resonant. In other words, the role of UH′ would be to eliminate nonresonant modes. Now why should this equation be solvable? Roughly speaking, the reason is that the equation deals mainly with nonresonant functions, which should avoid small denominator problems. To be more precise, let K 0 (q, p) = ω · p, and consider a Hamiltonian H = K 0 + h not too far from K 0. Denote by h+^ and h−^ the resonant and nonresonant parts of h, respectively, and assume that ε = ‖h−‖ρ is small. If U is a canonical transformation with nonresonant generating function φ of order ε, then
H ◦ U = H + {H, φ} + O(ε^2 ) = K 0 +
[
h − ω · ∇ 1 φ + {h, φ}
]
Let us try to solve I − (H ◦ U ) = 0 to first order in ε. The resulting equation for φ is I − [.. .] = 0, which can be written as
ω · ∇ 1 φ + I
hφ = h−^ ,
where ̂g denotes the Hamiltonian vector field associated with a Hamiltonian g, that is, ̂ gf = {f, g}. The formal solution of this equation is
(ω · ∇ 1 )φ = (I + L − )−^1 h−^ , (2.16)
14 HANS KOCH
with
L − = I
h(ω · ∇ 1 )−^1 = I −
[
(∇ 2 h) ·
ω · ∇ 1
− (∇ 1 h) ·
ω · ∇ 1
]
Here L − is a linear operator on I − Aρ. Now by the definition of I − , the two operators (written as fractions) in square brackets are bounded in norm by σ−^1 and (ρκ)−^1 , respec- tively. Thus, if ‖∇h‖ρ is sufficiently small, such that ‖L − ‖ < 1, then equation (2.16) can be solved by a Neumann series. The solution ψ = ω · ∇ 1 φ belongs to Aρ and is of order ε. The next step would be to solve equation (1.7) for the functions P and Q defining the canonical transformation U generated by φ. Notice that what enters this equation is not φ directly, but its gradient. This gradient is (ω · ∇ 1 )−^1 ∇ψ, a function for which we have again convenient bounds. By construction, the new Hamiltonian H ◦ U has a nonresonant part of order ε^2. Thus we can solve equation (2.14) by iterating the step H 7 → H ◦ U described above. A generalization of this procedure will be described in detail in Subsection 7.4.
Remarks.
◦ This elimination procedure shrinks domains. However, the domain loss tends to zero with the size of h−. Thus, for near-resonant Hamiltonians, the subsequent step H 7 → H ◦Tμ more than compensates for this domain loss, making R analyticity improving.
◦ It should be stressed that only the nonresonant part of h needs to be small for this procedure to work. The condition ‖L − ‖ < 1 allows for Hamiltonians that are not close to being integrable.
For completeness, let us state a concrete result about the transformation R. Let ω = (1, ω 2 ,... , ωd) be a fixed vector in Rd, whose components span an algebraic number field of degree d. We will call such vectors self-similar, for the following reason. It can be shown [74] that there exists a matrix T ∈ SL(d, Z) with simple eigenvalues ϑj satisfying ϑ 1 > 1 > |ϑ 2 | ≥... ≥ |ϑd|, such that T ω = ϑ 1 ω. Consider now a fixed matrix T with these properties.
Theorem 2.2. [1] Let 0 < ρ < σ/κ. If ρ′^ < ρ is sufficiently close to ρ and μ ∈ C satisfies (2.13), then there exists an open neighborhood B ∈ Aρ′^ of K 0 such that R : B → Aρ is well defined, analytic, and compact.
The version of this theorem given in [1] contains additional information about the domain B.
2.4. Other vector fields
Flows ˙q = X(q) on the torus Td^ are a special case of the above, as can be seen by restricting the flow for a Hamiltonian H = p · X(q) to the invariant torus p = 0. For flows that are not described by a generating function, we have to renormalize the vector field directly. Let X be a vector field on a manifold M. Consider a change of coordinates x = U(y) on M. Then ˙x = DU(y) ˙y. So the pullback of X under U is
U∗X = (DU)−^1 (X ◦ U). (2.17)
16 HANS KOCH
one-dimensional Schr¨odinger equation with quasiperiodic potentials, where G = SL(2, R). The discrete analogue are products of matrices Fi depending quasiperiodically on the index i. One of the problem is to find the spectrum of products Fm · · · F 2 F 1 in the limit of large m. This is trivial in the case where i 7 → Fi is periodic. We can rewrite (2.21) as
y˙(t) = f (q 0 + tω)y(t) , y(0) = y 0 , (2.22)
with f a function on Td, taking values in A. This equation, together with ˙q = ω defines the a vector field X on the manifold M = Td^ × G,
X(q, y) =
ω, f (q)y
, f (q) ∈ A , (q, y) ∈ M. (2.23)
The flow for X is given by
Φ Xt (q 0 , y 0 ) =
q 0 + tω, Ψ Xt (q 0 )y 0
, (q 0 , y 0 ) ∈ M , t ∈ R. (2.24)
where t 7 → Ψ Xt (q 0 ) denotes the solution of (2.22) for y 0 ∈ G the identity. Classical Floquet theory shows that if t 7 → q(t) is periodic, and in particular if d = 1, then the system is reducible. To be more precise, the vector field (2.23) is said to be reducible if there exists a function U : Td^ → G, such that
Ψ Xt (q) = U (q + tω)etC^ U (q)−^1 , t ∈ R , q ∈ Td^ , (2.25)
for some constant matrix C ∈ A. If ω ∈ Rd^ is fixed, we will also refer to f as being reducible. For another characterization of reducibility, considering the map U : M → M, defined by U(q, y) =
q, U (q)y
The pullback of X = (ω, f .) under this map is given by the equation
( U∗X
(q, y) =
ω, (U?f )(q)y
, U?f = U −^1 (f − Dω )U , (2.27)
where Dω = ω · ∇. Modulo smoothness assumptions, (2.25) is equivalent to f = U∗C. In the quasiperiodic case, solving V?f ≡ C leads to small divisor problems, as in classical KAM theory. Results based on KAM type methods have been obtained in the case where G = SL(2, R) [34,111,40], and for compact Lie groups [84,85]. Another approach to the reducibility problem involves renormalization methods. For discrete time cocycles over rotations by an irrational angle α, and for G = SU(2), Rychlik introduced in [115] a renormalization scheme based on a rescaling of first return maps, using the continued fractions expansion of α. Improvements of this scheme and global (non-perturbative) results can be found in [86,87,6]. In the context of flows, renormalization techniques were used in [97] to prove a local normal form theorem for analytic skew systems with a Brjuno base flow. Extensions of such RG techniques to skew systems with higher dimensional base maps or flows have become possible with the introduction in [70] of a suitable multidimensional
Renormalization of Vector Fields 17
continued fractions algorithm. A renormalization scheme based on this algorithm was introduced recently in [79]. It applies to Diophantine skew flows on Td^ × G, for arbitrary subgroups of GL(n, C) or GL(n, R), and for arbitrary dimensions d, n. In addition, the smoothness requirements are lowered, from analyticity to a finite degree of differentiability (depending on d and on the Diophantine exponent). The RG transformations themselves are restricted to vector fields X = (ω, f .) with f small. Among the general results are the existence of a stable codimension d manifold (near f = 0) of reducible skew systems. Other near-constant vector fields are mapped to this case by increasing the dimension of the torus. In the case d = 2 and G = SL(2, R), the stable manifold is identified with the set of skew systems having a fixed fibered rotation number (see Subsection 8.4). The precise results, and the techniques used to prove them, will be described (below and) in Section 8.
3. A single renormalization group step
This section covers the more technical aspects of renormalization. For skew flows, this includes explicit estimates of the type needed (later) to deal with general Diophantine rotation vectors. And for vector fields on Td^ × R`, we give a complete description of the main renormalization step: the elimination of nonresonant modes.
3.1. Skew systems: definitions
Skew flows are ideal for the description of a complete RG step, since the analysis is quite simple: estimating the action of R on resonant frequencies takes a few lines, eliminat- ing nonresonant frequencies involves little more than the implicit function theorem, and combining the two is straightforward. Consider skew flows X = (ω, f ) with f close to zero. Given γ ≥ 0, define Fγ to be the Banach space of integrable functions f : Td^ → GL(n, C), for which the norm
‖f ‖γ = ‖f 0 ‖ +
06 =ν∈Zd
‖fν ‖(2‖ν‖)γ^ (3.1)
is finite. Here, fν denotes the ν-th Fourier coefficient of f. Notice that Fγ is roughly Cγ^. The set of functions in Fγ that take values in G or A will be denoted by Gγ or Aγ , respectively. We will now drop the subscript γ if no confusion can arise. A single RG step for skew flows is associated with a unit vector ω ∈ Rd, and a matrix T in SL(d, Z), which we assume now to be given. Let
T (q, y) =
T (q), y
The pullback of X under T is given by ( T ∗X
(q, y) =
T −^1 ω, (T ?f )(q)y
, T ?f = f ◦ T. (3.3)
Denote by K(r) the set of vectors in Rn^ that are contracted by a factor ≤ r by the matrix T ∗. Choose 0 < σ < τ < 1, if possible, such that
2 σ‖T ‖ < τ , ω⊥^ ⊂ K(τ /2). (3.4)
Renormalization of Vector Fields 19
and if for all continuous linear maps f : C → X and h : Y → C, the function h ◦ G ◦ f is analytic. This shows e.g. that uniform limits of analytic functions are analytic. Assuming that B is a ball of radius r and that F is bounded on B, a third equivalent condition is that G has derivatives of all orders at the center of B, and that the corresponding Taylor series has a radius of convergence at least r and agrees with G on B. See e.g. [62] for more details.
Our next goal is to solve (3.6). Given f = C + h in F with C constant, we seek a solution of the form U = exp(D−^1 u), where u is a function in I− F. We have
I
− U∗f = I
−^ [
e−D
− (^1) u (f − Dω )eD
− (^1) u]
= I
−^ [(
I − D−^1 u
(C + h − Dω )(I + D−^1 u
)]
+ O
‖h‖‖u‖
+ O
‖h‖^2
= I
−^ [
h − Ĉ D−^1 u − Dω D−^1
]
+ O
‖h‖‖u‖
+ O
‖u‖^2
= h − u + O
‖h‖‖u‖
+ O
‖u‖^2
So if ‖h‖ is sufficiently small, then by the implicit function theorem, the equation I− U∗f = 0 has a solution uf = h + O(‖h‖^2 ), and this solution depends analytically on f. Wit a bit more work, one gets an explicit bound on uf for ‖C‖ ≤ σ/6 and ‖h‖ ≤ 2 −^9 σ, and verifies that ‖U f∗ f − I
f ‖ ≤ 24 σ−^1 ‖h‖^2. (3.11)
As a result, we have
Theorem 3.3. [79] Assume that σ and τ satisfy (3.4). Let f = C + h, with C constant and Eh = 0. If ‖C‖ < σ/ 6 and ‖h‖ < 2 −^9 σ, then
N (f ) = η−^1
[
C + ˜h
]
∥˜h
2 τ^
γ (^) ‖h‖ , ∣∣E˜h∣∣ (^) ≤ 24 σ− (^1) τ γ (^) ‖h‖ (^2). (3.12)
N is analytic on the region determined by the given bounds on C and h.
Proof. The function ˜h in equation (3.12) is given by ˜h = T?
[
I+ h + (U f? f − I+ f )
]
. Now we can use Lemma 3.1 and the bound (3.11). In particular, we have
∥ ∥˜h
∥ (^) ≤ τ γ^
‖h‖ + 2^4 σ−^1 ‖h‖^2
≤ 32 τ γ^ ‖h‖ , (3.13)
as claimed. The analyticity of N follows from the analyticity of the map f 7 → uf , the uniform convergence of the exponentials in (3.10), and the chain rule. QED
Notice that, by construction, if f belongs to A then so does N (f ). Similarly, if f is real-valued, then so is N (f ). Since this theorem will be used later with ‖T ‖ very large, requiring σ to be very small by (3.4), it should also be noted that the domain of N is roughly of size σ, which is due to the factor σ−^1 in the estimate (3.9).
20 HANS KOCH
3.3. More general vector fields
We start with some basic estimates for analytic vector fields on M = Td^ × R`, near Z = (ω, 0). Recall that the RG transformation for such vector fields is given by
R(X) = η−^1 U X∗ T (^) μ∗ X , (3.14)
where UX is a change of coordinates that eliminates nonresonant modes. The definition of resonant and nonresonant modes is analogous to the one used for Hamiltonians, and the proof that X 7 → T (^) μ∗ X is analyticity improving (and thus compact) when restricted to the resonant subspace is essentially the same as in the Hamiltonian case. Thus, we will describe here only the elimination procedure X 7 → U X∗ X. But we will do this in full detail, since the elimination of nonresonant modes is probably the most crucial part of R. The version presented here is taken from [78]. We start with some estimates on the flow generated by an analytic vector field Y. On the spaces Cm^ we use the ∞^ norm, and for linear operators we use the operator norm. Given ρ > 0, denote by Dρ the set of all vectors (q, p) in Cd^ × C^ characterized by ‖Imq‖ < ρ and ‖p‖ < ρ. If V is any complex Banach space, an analytic function f : Dρ → V that is 2π-periodic in each of the variables qj can be written as
f (q, p) =
(ν,α)∈I
fν,αeiν·q^ pα^ , ν · q =
j
νj qj pα^ =
j
pα j j, (3.15)
where I = Zd^ × N`. Define Aρ(V ) to be the space of all functions (3.15) for which the norm ‖f ‖ρ =
(ν,α)∈I
‖fν,α‖eρ|ν|ρ|α|^ (3.16)
is finite. Here, |ν| =
j |νj^ |, and^ |α|^ is defined analogously. If no ambiguity can arise, we will simply write Aρ in place of Aρ(V ). The operator norm of a continuous linear map L on Aρ will be denoted by ‖L‖ρ. It is easy to check that if V is a Banach algebra, then so is Aρ(V ). Another basic fact about the spaces Aρ is the following. Let π 1 (q, p) = (q, 0).
Proposition 3.4. Let 0 ≤ ρ′^ ≤ ρ. Let X ∈ Aρ(V ) and Y = (Y ′, Y ′′), with Y ′^ ∈ Aρ′
Cd
and Y ′′^ ∈ Aρ′
C`
. Then (a) (DX)Y ∈ Aρ′^ (V ) and ‖(DX)Y ‖ρ′^ ≤ (ρ − ρ′)−^1 ‖X‖ρ‖Y ‖ρ′^ , if ρ′^ < ρ. (b) X ◦ (π 1 + Y ) ∈ Aρ′ (V ) and ‖X ◦ (π 1 + Y )‖ρ′ ≤ ‖X‖ρ , if ρ′^ + ‖Y ′‖ρ′ , ‖Y ′′‖ρ′ ≤ ρ.
The flow ΦY associated with a vector field Y can be estimated e.g. by comparing it to the flow ΦZ for a constant real vector field Z = (ω, 0). The following bound is obtained by applying a standard contraction mapping argument to the equation
G(t) =
∫ (^) t
0
[(Y − Z) ◦ ΦsZ ] ◦ [I + G(s)] ds , (3.17)
satisfied by the difference G(t) = ΦtY − ΦtZ.
