RESULTS ON INVARIANT WHISKERED TORI FOR FIBERED HOLOMORPHIC MAPS AND ON COMPENSATED DOMAINS. A Dissertation Presented to The Academic Faculty

Transcription

1 RESULTS ON INVARIANT WHISKERED TORI FOR FIBERED HOLOMORPHIC MAPS AND ON COMPENSATED DOMAINS. A Dissertation Presented to The Academic Faculty By Mikel J. de Viana In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the School of Mathematics Georgia Institute of Technology December 217 Copyright c Mikel J. de Viana 217

2 RESULTS ON INVARIANT WHISKERED TORI FOR FIBERED HOLOMORPHIC MAPS AND ON COMPENSATED DOMAINS. Approved by: Rafael de la Llave, Advisor School of Mathematics Georgia Institute of Technology Chongchun Zeng School of Mathematics Georgia Institute of Technology Professor Federico Bonetto School of Mathematics Georgia Institute of Technology Professor Michael Loss School of Mathematics Georgia Institute of Technology Professor Àngel Jorba Departament de Matemàtiques i Informàtica Universitat de Barcelona Date Approved: 15 May 217

3 A la memoria de Milagros, Came, Casilda, mi Parina, Santos y Juan. iii

4 ACKNOWLEDGEMENTS I want to thank my advisor, Rafael de la Llave, without whom this thesis would not have been possible. Many thanks to the members of the dissertation committee, for their valuable input and for making themselves available on such short notice. I have been inspired by the many friends I made here at Georgia Tech. Their names are far too many to write here without forgetting some. You know who you are: Thank you! Finally, I would like to thank specially my family, both those present and those gone, for their support and their love. This work is dedicated to the loving memory of my grandparents. Every day I come to think of the very trying circumstances under which they formed our family. I live in awe of their accomplishments. iv

5 TABLE OF CONTENTS DEDICATION iii ACKNOWLEDGEMENTS iv LIST OF TABLES ix LIST OF FIGURES x SUMMARY xi I INTRODUCTION II SET UP OF THE PROBLEM AND MOTIVATION OF THE AS- SUMPTIONS The case of maps The invariance equation The iterative procedure for Euclidean M Geometric assumptions on A Solution of the Newton equation 7) in the hyperbolic directions Elliptic directions: Approximate Reducibility and first Melnikov condition Linear equations for the invariance of the splitting Linear equation for the reducibility: The role of parameters, second Melnikov condition and a non-degeneracy condition Extension to general manifolds III FUNCTION SPACES, NUMBER-THEORETICAL CONDITIONS, AND OTHER PRELIMINARIES Function Spaces that we will use Domains Function spaces Some elementary estimates on analytic functions The Grassmannian G k n Analytic vector bundles over T d ρ v

6 3.1.6 Analytic Vector bundle homomorphisms over a translation T α on T d ρ Analytic splittings over T d ρ Charts for V k ρ Projections associated to E S Coordinates of a Whitney sum of bundles Matrices for the projections Cocycles and Adapted Cocycles over T ω Whiskered splittings Number Theoretic conditions IV STATEMENT OF THE MAIN RESULTS AND OUTLINE OF THEIR PROOFS Invariance and Reducibility equations Invariance equation for K Invariance equation for the splitting Reducibility equation Motivation for the definition of the functionals I, S σ, R Statement of Theorem Outline of the proof of Theorem A Nash-Moser iterative procedure with floating parameters The Newton equations for all unknowns The algorithm Relations with previous results in the literature Fibered dynamics and strange non-chaotic attractors Lower dimensional elliptic tori The parameterization method V SOLUTIONS AND APPROXIMATE SOLUTIONS FOR THE LIN- EARIZED EQUATIONS Solution of the linearized invariance equation in the stable and unstable directions vi

7 5.2 Solution of a cohomology equation associated to the linear invariance equation in the central directions Approximate solution of the linear invariance equation in the central directions Solution of the linearized equations for S Solution of a cohomology equation associated to the linear reducibility equation Approximate solution of the linear equation corresponding to the Reducibility VI ZEHNDER S APPROXIMATE RIGHT INVERSES FOR DI, DS, DR Approximate right inverse for DI Right inverse for DS Approximate right inverse for DR Linearization of R Non-linear estimates VII CONVERGENCE OF THE QUASI-NEWTON ITERATION Proof of Theorem Condition number for geometric condition 1) Condition number for geometric condition 2) Condition number for geometric condition 3) Condition number for geometric condition 4) Condition number for geometric condition 5) VIIIINTRODUCTION IX CONDITIONS IMPLYING COMPENSATION IN FINITE DIMEN- SIONS Notation and preliminaries Uniformly planar domains in R n are compensated X COUNTEREXAMPLES FOR TENTATIVE GENERALIZATIONS OF THEOREM vii

8 1.1 Counterexamples for more general domains in finite dimension Counterexample for uniformly planar manifolds in infinite dimension. 136 REFERENCES viii

9 LIST OF TABLES 1 Rule for choosing 66) or 68) ix

10 LIST OF FIGURES 1 The sum of all terms inside the green and purple boxes is e 1. The Newton equation 7) guarantees that the purple box vanishes. Thus, 8) holds The splitting E Ξ and its projections Two non- opposing slices inside an ɛ -uniform ball The Splitting point at γt ) An illustration of T with the tubular neighbourhood of γ highlighted The first steps in the construction of T T E 2k is homeomorphic to this planar region x

11 SUMMARY This thesis is composed of two independent papers: Construction of whiskered invariant tori for fibered holomorphic maps In this paper we present a very general theory that includes results on the persistence of quasi-periodic orbits of systems subject to quasi-periodic perturbations. Such quasi-periodic systems appear naturally in many applications where systems are subject to external perturbations of quasi-periodic nature. This is joint work with Rafael de la Llave. The paper takes up Chapters I - VII of the thesis. We give a brief motivation in Chapter I. In Chapter II we introduce informally the spaces of analytic functions and some number-theoretic and non-degeneracy conditions which are rigorously defined in Chapter III. In Chapter IV we present the Main result of the paper, and present the pseudocode for an Algorithm Algorithm 1) which efficiently computes the invariant objects that have been introduced in Chapter II. A brief review of the most important references in the literature which are related with our work is in Section 4.5. The proof of the main result, theorem 1, takes up Chapters V-VII. We present an outline for the proof of theorem 1 in Section 4.3. xi

12 Regularity properties of the boundary of a domain that imply it is compensated A Domain D in a Banach space B, B ) is called compensated if the distance d D x, y) on D defined by the infimum of the lengths of arcs joining points x, y D is comparable to x y B. This condition is very important for the function theory in D. This is joint work with Rafael de la Llave. This paper consists of Chapters VIII- X of the thesis. In Chapter VIII, we introduce and motivate the problem briefly. In Chapter IX, we prove Theorem 2, that shows that certain regularity conditions on the boundary of D guarantee that D is compensated when B is finite dimensional. In Chapter X, we present examples showing that these conditions cannot be relaxed and that they do not necessarily imply compensation for infinite dimensional B. xii

13 Part I: Construction of whiskered invariant tori for fibered holomorphic maps 1

14 CHAPTER I INTRODUCTION The goal of this paper is to present a very general theory of persistence for quasiperiodic orbits of systems subject to quasi-periodic perturbations. We will present results for discrete time systems maps). Such quasi-periodic systems appear naturally in many applications where systems are subject to external perturbations of quasi-periodic nature. Notably, astronomical systems are subject to perturbations with several frequencies, corresponding to different planets. Other examples arise in biological systems which are subject to seasonal perturbations, machinery subject to perturbation of several parts vibrating with different frequencies, etc. This problem, of course, has a very long history in the literature, with the main breakthroughs in the theoretical literature happening in the 5s and 6s the KAM theory). The systematic numerical treatment is more recent. In Subsection 4.5 we have tried to give a brief summary of the main precedents of this work. Certainly, it is quite incomplete as a survey of the field which would require at least a book. Our approach is based on the so-called parameterization method. We derive a functional equation that expresses the fact that an embedding corresponds to an invariant curve. This functional equation is then studied by methods of functional analysis. We note that it will become apparent that the invariance equation will involve adjustment of parameters. These parameters are part of the unknowns of the 2

15 equation. The role of parameters has been emphasized in the theory of quasi-periodic solutions in [46, 47, 45, 44, 43, 16]. The main technical result is Theorem 1. This is an a-posteriori" theorem, that is, we assume that there is a sufficiently approximate solution satisfying some nondegeneracy conditions and we conclude that there is a true solution. To make precise the meaning of approximate" will entail defining norms. There are several definitions analytic, finitely differentiable, Sobolev) that will apply: We will only consider the analytic case, see Section 3. The approximate solutions assumed by Theorem 1 can be produced in a variety of ways. For example, if the system we consider is close enough to integrable", we can take as approximate solutions the exact solutions in the integrable system. In the quasi- integrable case, we can also obtain improved approximate solutions using formal expansions. Finally, we note that we can take as approximate solutions the results of a numerical computation. To validate the numerical solutions, we do not need to study the algorithm, we only need to verify that they indeed satisfy very approximately the invariance equation and estimate the condition numbers. The proof of Theorem 1 will consist in showing that an iterative procedure of Newton type the error of the improved solution is quadratic in the original error) converges. This algorithm has mathematical consequences. It has been known since [36] that the quadratic convergence can overcome small divisors. The iterative procedure we present also has numerical applications it lends itself to efficient numerical implementations) since it is not based on transformation theory 3

16 but rather in applying corrections which ameliorate notably the curse of dimensionality: The algorithm presented here see Algorithm 1) is based on manipulating functions of as many variables as the dimension of the invariant object, whereas the transformation theory requires to deal with functions of as many variables as the dimension of phase space. The book [26] is devoted to the parameterization method and its applications to a new generation of algorithms for computing invariant objects. In an ideal computer with arbitrary precision and unlimited memory, these algorithms could compute arbitrarily close to breakdown. In a real computer they can come extremely close. The a-posteriori format of the theorem allows us to compute with confidence even very close to breakdown. We hope this implementation will be done in a future work. 4

17 CHAPTER II SET UP OF THE PROBLEM AND MOTIVATION OF THE ASSUMPTIONS. In this Chapter we present informally the formulation of the problem and we motivate the assumptions. For the moment, we will just formulate the equations and describe the geometric properties. We, however, postpone the discussion of analysis issues until after we have discussed spaces and norms for several objects. Of course, the analytic definitions are driven by the geometric formulation which we now develop. The set up we present is very similar to that in [23, 24, 25], but these papers considered only the hyperbolic case. In this paper, see Subsection 2.3.3, we allow also elliptic directions. This will require also adding more parameters, see Section The case of maps. We now formulate the problem for quasi-periodic maps. The results for maps imply the results for differential equations, by taking sections taking sections is numerically efficient too). The same geometric ideas apply to differential equations, but they are easier to interpret in the case of diffeomorphisms since they just involve comparing points, not points and vector fields. Let T d be the d-dimensional torus R d /Z d. We will consider families of skew-product diffeomorphisms over a translation on T d, that is, we study maps of the form F : T d M C c T d M θ, x, β) θ + ω, fθ, x, β) ). 1) 5

18 Where: < d, c n are integers 1, M is an n-dimensional complex analytic manifold, ω T d is a fixed frequency, which satisfies some number-theoretic conditions stated in Subsection f : T d M C c M is analytic, and we denote fθ, z, β) := f β θ, z) Skew-product maps 1) have the physical interpretation that there is a large system described by motion with several frequencies in the variable θ). This large system affects but is not affected by) the evolution of the variable x. One concrete example is a small satellite in the field of several planets, each moving with its own period. Since we will be working with analytic regularity, it will be useful to assume that M is a complex manifold of complex) dimension n. Reference [52] also considers this complex manifold case. If the manifold M in 1) is real, then we need to consider a complex analytic extension F of F with the property that F maps real values into real values. When working with analytic extensions, we will consider that the angle variable θ takes values on a complex strip T d ρ containing T d : T d ρ := { θ C d /Z d, θ = θ 1,..., θ d ) t : Im θ i ρ, 1 i d } 1 Here, c is the dimension of the parameter space C c, and it coincides with the dimension of the central bundle of a Whiskered splitting, to be defined in Subsection We remark that d is possibly strictly smaller than n. 6

19 We will denote by T ω the translation on the strip: T ω : T d ρ T d ρ θ θ + ω. Note that when the quasi-periodic effect is small we have that f β θ, x) is almost independent of θ, that is f β θ, x) f β x), θ T d ρ Hence, we will refer to situations where D 1 f β θ, x) 1 2) as the perturbative case. The perturbative case has been considered in [52] when n = d = 1. Most of our results will be independent of the perturbative assumption, but if it holds we will obtain sharper results. We say that ω T d is non-resonant if ω k / Z, k Z d \ {} When the frequency ω is non-resonant, the map 1) does not admit a periodic orbit. The invariant objects which organize the dynamics generated by 1) are invariant tori, which are the graph of a map K : T d ρ M, i.e. tori of the form T K = { θ, Kθ)) θ T d ρ } 3) 7

20 2.2 The invariance equation. Since a point θ, Kθ)) maps under 1) to θ + ω, f β θ, Kθ) ), we see that the torus T K is invariant if and only if the map K satisfies f β θ, Kθ)) = Kθ + ω), θ T d ρ 4) The equation 4) will be the centerpiece of our analysis. We can think of it as a functional analysis problem. As we will see later, obtaining solutions of equation 4) requires that we consider, rather than a single map, a family of maps indexed by the parameter β, and that we adjust the parameter. The role of the parameter β in obtaining solutions of the invariance equation 4) was emphasized already in [46], and we explain it informally in Subsection We remark that the unknowns of the invariance equation 4) are both K and β. We will use the shorthand notation f β Kθ) := f β θ, Kθ)) 2.3 The iterative procedure for Euclidean M This is a case that appears often in applications and it leads to valuable intuition. If M is an Euclidean manifold, we can write 4) as f β K K T ω = 5) Of course, in a general manifold one cannot compare points by substracting them, but one can use exponential mappings, so that adapting the results of this paper from Euclidean manifolds to general Riemannian manifolds is essentially straightforward and mainly typographical effort see Section 2.4 for details). 8

21 We will try to solve the parameterized equation 5) by a Newton method. We start from an approximate solution K : T d ρ C n, β C c of 5) f β K K T ω = e, 6) where e : T d ρ C n, the error of the approximate solution K, β ), will be thought of as small 2. We try to find : T d ρ C n, a correction to K, and δ C c, a correction to β, in such a way that, δ ) eliminate e in first order, that is, D 2 f β K + β f β K δ T ω = e 7) If, β ) satisfy 7), then the corrected torus K 1 := K + and parameter β 1 := β + δ are much improved approximate solutions of 5): f β1 K 1 K 1 T ω = E 2 f) [K, β ) ;, δ ], 8) where we denoted by E 2 f) [K, β ;, δ ] := f β1 K 1 f β K + D 2 f β K + β f β K δ ) the error of the linearization of f at K, β ), in the direction, δ ) see diagram 2.3). If the correction, δ ), obtained as a solution of equation 7), is of the same order of magnitude as the error e and if, furthermore, the image of T d ρ under the improved torus K 1 still lies in the domain of analyticity of f then, by Taylor s estimate, the error e 1 := f β1 K 1 K 1 T ω will be quadratic with repect to e. We remark that equation 7) has two unknowns: We will use an additional equation in order to determine δ, see Section 2.4. Given an analytic map K : T d ρ C n, we will denote 2 In order to make rigorous sense of the meaning of "small" here, we need to introduce function spaces and norms. We delay this until Chapter 3. 9

22 Figure 1: The sum of all terms inside the green and purple boxes is e 1. The Newton equation 7) guarantees that the purple box vanishes. Thus, 8) holds. A θ) := D 2 f β Kθ) The Newton equation 7) relates the corrections T ω and by an affine transformation whose linear part is A K,β. Applying 7) recursively, we obtain n 1 T nω = A n) K,β + A j) K,β T n j)ω e + β f β K δ ) T n j 1)ω, 9) j= where we have introduced the products A n) := A T n 1)ω A T n 2)ω... A A n) := A 1 T n 1)ω... A 1 T ω A 1 1) Note that, for k, l Z, A k+l) = A k) T lω A l). It is customary to refer to An) as cocycles over a translation: A discrete GLn, C)-valued cocycle over T ω is a map B : Z T d ρ GLn, C) Bk, θ) := B k) θ) such that B k+l) = B k) T lω B l) holds for all k, l Z. 1

23 2.3.1 Geometric assumptions on A It is now clear that we will need to make assumptions on the asymptotic behavior of the products A ±n) : We now introduce informally three assumptions H1, H2, H3 below) which we will require for the cocycle A K,β H1 There exist analytic vector bundles E s, E u, E c over T d ρ denoted, respectively, as the stable, unstable and central bundles) such that, for all θ T d ρ, E s θ) E u θ) E c θ) C n where, for σ {s, u, c}, we have denoted by E σ θ) the fiber of E σ over θ. We will say that the bundles E s, E c, E u ) form a splitting E over T d ρ. We will furthermore assume that: Each of the bundles E σ, σ {s, u, c} is sufficiently approximately invariant with respect to the natural action of A K,β. The splitting E is whiskered with respect to A K,β : For the full definition of whiskered splitting, see Subsection We now present an informal definition which suffices for the purposes of this introduction. Informally, we will think of a whiskered splitting as an analytic splitting E s, E c, E u ) such that the standard exponential trichotomy holds 3 : There exist positive real numbers C h, µ s, µ u, µ c,+, µ c, satisfying: µ s < 1, µ u < 1, 1 < µ c,+, 1 < µ c,, µ s µ c,+ < 1, µ u µ c, < 1 3 the existence of an exponential trichotomy has been used very often see e.g. [58, 18]). 11

24 such that v E s A n) v Ch v µ n s, n v E u A n) v Ch v µ n u, n v E c A n) A n) C h v µ n c,+, n C h v µ n c,, n We will need to modify this preliminary definition of Whiskered splitting which is too strong for approximately invariant splittings) by considering projected cocycles see Subsection ) Solution of the Newton equation 7) in the hyperbolic directions The Newton equation 7) is equivalent to a system of linear equations A K,β σ σ T ω = e σ β f β K δ σ, σ {s, u, c} 11) where The unknown σ is a section of the bundle E σ. e σ = e σ,σ + ẽ σ, where e σ,σ is the projection of e onto E σ, and where the nondiagonal error ẽ σ is zero if E σ is invariant under A K,β. Note that we have set the unknown δ σ related to the correction of the parameter) in the right-hand side of 11): We explain briefly why we choose to present the system 11) in this way in Subsection Assuming a solution of the system 11) is found, we then set := s + u + c, δ := δ s + δ u + δ c 12

25 The Whitney sum E s E u is informally called the hyperbolic directions of the splitting. It has been noted in [2] that the system 11), for σ {s, u}, can be solved via a geometric iteration, which we implement in Section 5.1. Since the method of 5.1 does not require to work in coordinates, we will not need to assume that the bundles E s, E u are trivilizable Elliptic directions: Approximate Reducibility and first Melnikov condition The Newton equation 7) along the center directions will be dealt with by nongeometric methods such as Fourier series, which require taking coordinates. The second geometric condition guarantees that this can be done: H2 The central bundle E c is trivializable over T d ρ via a vector bundle isomorphism ψ : T d ρ C c E c Let Π c E be the projection onto E c, and let A c,c K,β := Π c E T ω A K,β Π c E Using ψ, we can find a matrix representation of A c,c as follows: A c,c K,β := ψ 1 T ω A c,c K,β ψ Assumption H3 appears already in [46], but [46] assumes that it holds in all the linearization and not only along the central directions. H3 For fixed λ 1,... λ c C which satisfy some number-theoretic conditions stated in Subsection ), there exists parameter β C c and an analytic bundle of frames U : T d ρ GLc, C) 13

26 in such a way that U 1 T ω A c,c K,β U Λ = e R 12) where Λ = diag[λ 1,..., λ c ], and e R GLc, C) is sufficiently small. If 12) holds with e R, it is said that A c,c K,β we will say that A c,c K,β is approximately reducible. is reducible to Λ. If 12) holds, Remark 1. The geometric meaning of 12) is that we can find a linear change of variables in such a way that the coordinate expression of the linearized equations A c,c K,β is expressed as multiplication by a constant matrix Λ, and that Λ remains fixed throughout the iterative procedure. This is a strong assumption, and one cannot expect it to hold for a fixed map. However, we will show see Subsection 2.3.5) that it can hold throughout the iterative procedure if f is embedded in a family {f β } β C c depending on a c-dimensional parameter β and if, furthermore, a non-degeneracy condition see hypothesis [H4] below) is satisfied. Using the trivialization ψ and the approximate reducibility identity 12) we see that the Newton equation 11) for σ = c is equivalent to: Λ c c T ω = U 1 T ω [ e c β f β K δ c ] e R U 1 c 13) where c := U 1 ψ 1 c. It is standard in KAM theory to argue that the term e R U 1 c is of quadratic order and hence deleting it will not affect the quadratic convergence of the Newton iteration we will indeed show that this is the case). The resulting equation: Λ c c T ω = U 1 T ω [ e c β f β K δ c ] 14) is a constant coefficients cohomology equation. Equating Fourier coefficients of ω, we readily obtain a formal solution see equation 47) in Section 5.5). A small divisor 14

27 problem, involving both the spectrum of Λ and the frequency ω, is present in the formal solution. To overcome the small divisors, we will require as is standard in the theory of lower dimensional elliptic tori: See Section 4.5) that Λ and ω jointly satisfy a number theoretic condition called the first Melnikov condition see Subsection ) Linear equations for the invariance of the splitting The system of linear equations 11) involves non-diagonal errors because the splitting E, s E, c E u ) is only approximately invariant. Hence, at each step of the iterative procedure, we must also construct a more approximately invariant splitting from E, s E, c E u ), and reformulate system 11) with respect to the improved splitting. We thus need to complement system 11) with a Newton equation for the improvement of the splitting. To this end, we introduce the well known coordinates of the Grassmannian modelled after a space of linear transformations) in Subsection Using the coordinates in Subsection 3.1.4, a neighborhood of the splitting E, s E, c E u ) is parameterized by an open set of a Banach space see 3.1.5). This, in turn, allows to introduce functional equations for the invariance of the splitting in Subsection Geometrically speaking, a more approximate solution of the functional equations in correspond to a more approximately invariant splitting see Proposition 1). The additive corrections for the splitting E, s E, c E u ) will be obtained as the solutions of the Newton equations associated to the functionals presented in Subsection The Newton equations are found in Section 6.2. We solve the Newton equations for the invariance of the splitting using a geometric iteration see Section 5.4) similar to that used for the Newton equation 11) along the hyperbolic directions. 15

28 2.3.5 Linear equation for the reducibility: The role of parameters, second Melnikov condition and a non-degeneracy condition We now observe that the trivialized Newton equation for the invariance of the torus along the central directions 13) involves the error e R of the approximate reducibility identity 12). Hence, at each step of the iterative procedure, we must also compute a frame U under which A c,c reformulate equation 13) with the improved frame U. is more approximately reducible. At each step, we will We implement a quasi-newton method for U: Let W be the additive correction i.e. we will set U = U + W ). Proceeding as we did to obtain equation 14), we obtain the following cohomology equation for W Λ W W T ω Λ = U 1 T ω e R η R [δ] 15) where η R is an affine function 4 which appears because of the linear terms in the linearization of A c,c. See Proposition 2 for the expression of η R. Using the Fourier method, we obtain a formal solution see equation 75) in Section 5.5). A new small divisor problem appears, now involving also the differences of points in the spectrum of Λ. To overcome these small divisors, we will require that the pair Λ, ω) satisfy the second Melnikov condition see Subsection ). Remark 2. It is not strictly necessary to impose the second Melnikov condition on the pair Λ, ω) in order to just compute an invariant torus near K : An almostreducibility scheme can be implemented see [14], [1]). However, the tori produced by an almost-reducibility scheme may fail to be reducible see [15]). An additional obstruction arises for the diagonal terms of equation 15): We must 4 For an affine function η, we will denote by η[δ] the evaluation of η at δ. 16

29 ensure that [ U 1 T ω e R η R [δ] ] ) =, 1 j c 16) and that 16) is satisfied throughout the Newton iteration. If the non - degeneracy condition jj H4 The linear part of the affine function η R is an isomorphism. is satisfied, then there exists a unique δ such that 16) holds. Thus, the parameter correction δ is determined by the linear equation 16). Note that [H1], [H2], [H3], [H4] are open conditions and, assuming that certain condition numbers are satisfied for the initial data of the iterative procedure, we can ensure that [H1], [H2], [H3], [H4] are satisfied at each step. 2.4 Extension to general manifolds. The above considerations can be formulated for M a Riemannian manifold using standard tools: Exponential mappings and connectors [28]. The error in the invariance equations can be measured by e = exp 1 K T ω f K We can introduce the correction as a section of T M restricted to the image of K. K = exp K Tω The analogue of the Newton equation 7) is 17

30 S K Tω f K D 2f K T ω = e 17) where S y x is the connector, i.e. an identification between T x M and T y M when x and y are close). We can take: Sx y = D exp x ) exp 1 x y and we think of S y x as parallel transport from T x M to T y M along the closest geodesic joining x and y. Then, the linearized equations read again: A T ω = e but A : T K M T K Tω M is defined by S K Tω f K D 2f K Of course, this is completely consistent with our treatment of the Euclidean manifolds. In that case all the tangent spaces can be identified and the connectors are the identity. We have now informally introduced all the basic objects involved in the proof of our main result, Theorem 1. An outline of the proof of Theorem 1 is given in Section 4.3, where we emphasize some formal aspects of the iterative procedure. A pseudocode for the iterative procedure is given in Algorithm 1. 18

31 CHAPTER III FUNCTION SPACES, NUMBER-THEORETICAL CONDITIONS, AND OTHER PRELIMINARIES. In this Chapter we introduce the function spaces and the norms that will be used to measure how approximately are the Invariance and Reducibility equations satisfied. We introduce the number theoretic Melnikov) conditions that play a role in the proof of Theorem 1. We also fix several notations. Remark 3. In this work there will appear many constants which depend only on quantities that are fixed in the statement of Theorem 1. Following standard practice in order to avoid an abundance of such constants, we set C to be a generic constant which does not depend on k, the step of the Nash-Moser iteration in Algortithm 1. The optimal value of C may change from line to line. For simple model problems, careful bookkeeping of these constants appear e.g. in [8], [19] and are used to obtain effective estimates. 3.1 Function Spaces that we will use Domains Given ρ >, we define the complex strip: T d ρ := { θ C d /Z d, θ = θ 1,..., θ d ) : Imθ j ) ρ, j } Let η >, ρ >, and β C k. We denote by B k η β ) C k the open ball of radius η centered at β. Given a function K : T d ρ C n, we define the following domains: D ρ,η K, β) := T d ρ D η K) B c ηβ) T d ρ C n C c 19

32 where D η K) is the domain D η K) := Bη n z) z KT d ρ) Function spaces. Let the domain Υ be either T d ρ or D ρ,η K, β). Given a complex manifold X, let A X Υ := { f : Υ X : f is analytic on Int Υ ), and continuous on Υ } The spaces A X T d ρ will appear often in this paper, and we will use the shorthand notation A X T d ρ When X, X ) is a Banach space, A X ρ := A X ρ is a Banach space 1 with the norm g ρ := sup θ T d ρ gθ) Any g A Cc ρ admits a Fourier series: gθ) = k Z d ĝk) exp2πik θ), θ T d ρ where ĝk) is the k-th Fourier coefficient ĝk) := gθ) exp 2πi k θ) T d Some elementary estimates on analytic functions. We now recall two basic estimates for g A Cc ρ see [59]): The Cauchy- Fourier estimate is: ĝ k ) C exp 2πρ k ) g ρ 18) The Cauchy estimate for the derivative is: Dg ρ ζ C ζ 1 g ρ 19) 1 In this paper we will only consider some specific spaces X, which we introduce below. 2

33 3.1.4 The Grassmannian G k n. Given k n, let G k n be the Grassmann manifold of k-dimensional complex subspaces of C n. The manifold G k n is metrizable with the metric d G k n V 1, V 2 ) := sup v V 1, v =1 inf { v w : w V 2 } Let V G k n, and V G n k n, a complementary subspace i.e. V V = C n ). We introduce the subset of G k n defined by: U V G k n V ) := { W G k n W V = { } } The set U V V ) is a neighborhood of V G k G n. It is parameterized by the vector k n space: L V, V ) := { linear maps V V } The parameterization is as follows: Given S LV, V ), it is easy to check that there exists a unique V S U V V ) such that G k n V S := {v + Sv, v V } A chart for U V V ) is G k n Φ V,V G k n : U V G k n V ) LV, V ) V S S. We will not use the well-known fact that the charts Φ V,V, V G k n, endow G k n with a complex analytic manifold structure. See [38] for details. G k n We will denote by LV,V ) the operator norm on LV, V ). The neighborhood U V V ) is also metrizable with the metric G k n d V U V G k V ) S 1, V S2 ) := S 1 S 2 LV,V ) n 21

34 Remark 4. Given r >, let U V, r V ) be the following neighborhood of V in G k G n: k n { } U V, r V ) := W U V G k n G V ) d V, W ) < r k n U V V ) Note that the restrictions of d U V on U V, r V ). G k n Analytic vector bundles over T d ρ. G k n G k n and d V ) G on U V, r V ) are equivalent metrics k n G k n Let V k ρ be the set of analytic vector bundles of rank k over T d ρ: π E σ : E T d ρ { π 1 E θ) } := Eθ) G k n and such that E A Gk n ρ, where we have denoted also by E the function E : T d ρ G k n θ Eθ). We endow V k ρ with the metric d V k ρ E 1, E 2 ) := sup θ T d ρ d G k n E 1 θ), E 2 θ) ) Given α T d, we introduce the vector space of analytic sections over the translation T α and onto the vector bundle E σ : A Eσ ρ,α := { f A Cn ρ fθ) E σ θ + α), θ T d ρ } with the norm f ρ := sup θ T d ρ fθ), A Eσ ρ, α is a Banach space Analytic Vector bundle homomorphisms over a translation T α on T d ρ Given E 1 V k 1 ρ, E 2 V k 2 ρ, α T d we denote by A LE 1, E 2 ) ρ,α the vector space of analytic vector bundle homomorphisms E 1 E 2 over T α : It is the set of analytic functions A : E 1 E 2 22

35 such that, for any θ T d ρ: 1. The image of the fiber E 1 θ) under A is contained in the fiber E 2 θ + α). 2. Given θ T d ρ, let Aθ) := A E1 θ). Then, Aθ) L E 1 θ), E 2 θ + α)). Let E V j ρ. We denote by A GLE) ρ,α the vector space of analytic vector bundle isomorphisms E E over T α : A GLE) ρ,α := { A A LE, E) ρ,α Aθ) GL Eθ), Eθ + α)), θ T d ρ } LE, E) With the norm A ρ := sup θ T d ρ Aθ) LEθ), Eθ+α)), Aρ,α is a Banach space. We will also use the following Banach algebra type inequality: Let α, β T d, E 1 V k 1 ρ, E 2 V k 2 ρ, E 3 V k 3 ρ A A LE 1, E 2 ) ρ,α, B A LE 2, E 3 ) ρ,β. Then, A B A LE 1,E 3 ) ρ,α+β and A B ρ A ρ B ρ Analytic splittings over T d ρ We will assume that E A Gs n Gu n Gc n ρ, an analytic splitting over T d ρ, is given as data for the iterative procedure recall the discussion in Subsection 2.3.4). The splitting E consists of three vector bundles E σ V σ ρ, σ {s, u, c} 2 such that for all θ T d ρ, it holds that E s θ) E u θ) E c θ) = C n 2 We use the letter σ to denote both the vector bundles s for "stable", u for "unstable", c for "central") and their rank. Note that s + u + c = n. 23

36 3.1.8 Charts for V k ρ Let {σ, σ, σ } be a permutation of the set {s, u, c}. We denote the Whitney sum of the vector bundles E σ and E σ by E σ σ := E σ E σ V σ +σ ρ Note that the bundles E σ σ and E σ are complementary: E σ σ θ) E σ θ) = C n, θ T d ρ We introduce the set { U V σ ρ E σ ) := E σ, Eσ σ ) } E σ Vρ σ E σ θ) E σ σ θ) = { }, θ T d ρ Given S σ A L ρ,, there exists a unique ES σ U Vρ σ Eσ ) such that E σ Sθ) = { v + S σ θ) v v E σ θ) }, θ T d ρ Since T d ρ is compact, U V σ ρ E σ ) is a neighborhood of E σ V σ ρ. It is parameterized via the chart σ Φ Eσ,Eσ Vρ σ : U V σ ρ E σ ) A L E σ, Eσ σ ) ρ, E σ S S σ. The neighborhood U V σ ρ E σ ) is also metrizable with the metric d UV σ ρ E σ ) Eσ S 1, E σ S 2 ) := S 1 S 2 ρ Remark 5. Given r >, let UV r ρ σ Eσ ) be the following neighborhood of E σ in Vρ σ : { } UV r ρ σ Eσ ) := ES σ U V σ ρ E σ ) d U V V ) Eσ S, E σ ) < r Note that the restrictions of d U V are equivalent metrics on U r V σ ρ. G k n G k n V ) Eσ S, Eσ ) and d V σ ρ on U r V σ ρ 24

37 Remark 6. Given three vector bundle homomorphisms E σ, Eσ σ ) S σ A L ρ,, σ {s, u, c}, we will denote by E S the splitting E s S, Eu S, Ec S ). Remark 7. The initial data E A Gs n Gu n Gc n ρ is a geometric object which is not well suited for the operations that will be performed in Algorithm 1. In particular, A Gs n Gu n Gc n ρ is not a vector space, so it does not make sense to perform additive corrections as will be done at each step of the Nash Moser iteration. σ To overcome this difficulty, we will apply the chart Φ Eσ,Eσ Vρ σ to the splitting E, which is given as data for the first step of the iterative procedure. Note that σ Φ Eσ,Eσ Vρ σ E σ ) =, σ {s, c, u} Then we will work for the rest of the Nash-Moser iteration in the vector spaces E A L σ, Eσ σ ) ρ,, where it makes sense to solve linearized equations and perform additive corrections. Hence, at each step of the iterative procedure, we are given as data three vector bundle linear homomorphisms E σ, Eσ σ ) S σ A L ρ,, σ {s, u, c} we will then compute additive corrections 3 E σ, Eσ σ ) χ σ A L ρ,, σ {s, u, c} such that the bundles ES+χ σ := σ Φ Eσ,Eσ Vρ σ ) 1 S σ + χ σ ), σ {s, u, c} 3 The corrections χ σ are obtained as solutions of linear equations, see Lemma 4, for which the E σ, Eσ σ ) fact that A L ρ, is a vector space is also used. 25

38 are more approximately invariant under the action of the cocycle A recall the discussion in Subsection ). The corrections S σ + χ σ will be improved solutions of a functional equation, introduced in Subsection 4.1.2, which expresses the invariance of the splitting. The invariance equation for the splitting in Subsection admits a geometric interpretation, which we present in Proposition Projections associated to E S. The splitting E S defines a bundle of projections onto each E σ S σ {s, u, c}): Π σ E S := Π σ S A LCn T d ρ,c n T d ρ ) ρ, Π σ S is characterized as follows : Given v Cn θ) let Π σ Sθ) v := if v E σ σ S θ) v if v ES σθ) and extend linearly. We also introduce the bundle of projections Π σ σ S, which is characterized as follows: Given v C n θ) let Π σ σ S θ) v := if v ES σθ) v if v E σ σ S θ) Coordinates of a Whitney sum of bundles In this Subsection, given σ, σ {s, u, c} with σ σ and two vector bundle homomorphisms E σ, Eσ σ ) S σ A L ρ,, S σ A L ) E σ, Eσ σ ρ, we compute Φ Eσ Eσ,Eσ Vρ σ+σ sum of the bundles E σ S, Eσ S ES σ Eσ S ), the coordinate corresponding to the Whitney under the chart ΦEσ Eσ 26,Eσ Vρ σ+σ.

39 We remark that the coordinate Φ Eσ Eσ V σ+σ ρ,eσ E σ S Eσ S ) is not just Sσ S σ the range of S σ S σ has a non-trivial projection on E σ E σ. since To find the correct expression for Φ Eσ Eσ parameterized by E σ E σ as follows: E σ S E σ S V σ+σ ρ,eσ ES σ Eσ S ), note that Eσ S Eσ S ) θ) = v + w + S σ θ)v + S σ θ)w, v E σ θ), w E σ θ) = ) v + S σ,σ θ)w } {{ } call this v 1 + ) S σ,σ σ,σ θ)v + w + S θ)v + S σ,σ θ)w }{{}}{{} call this v call this v 3 2 where we have denoted S σ,σ θ) := Π σ S θ)sσ θ). Note that v 1 E σ θ), v 2 E σ θ) and v 3 E σ θ) and that Φ Eσ Eσ E σ σ,eσ V ) ρ σ+σ ES σ Eσ S ) is the unique vector bundle homomorphism in A L ρ, such that v 1, v 2 ) is mapped to v 3. Hence, we begin by, E σ solving for v, w in terms of v 1, v 2 : Id E σ S σ,σ S σ,σ Id E σ where we denoted by Id E σ, Id E σ we obtain: Φ Eσ Eσ,Eσ Vρ σ+σ E σ S E σ S ) = S σ,σ Π σ Matrices for the projections 1 v 1 = v 2 v w the identity maps in E σ, E σ, respectively. Hence, Id E σ S σ,σ S σ,σ Id E σ 1 + S σ,σ Π σ Id E σ S σ,σ S σ,σ Id E σ We will need to use the matrix of Π σ S with respect to the splitting E. In this subsection, we use the coordinates is 1 C n θ) = E σ θ) E σ σ θ) 27

40 Note that Id E σ θ) Id E σ θ) Π σ Sθ) = ξ 1,1θ), S σ θ) Id E σ θ) Φ Eσ Eσ,Eσ V Π σ σ S θ) = ξ σ +σ E S σ Eσ S )θ) ρ 1,2θ), Id E σ σ θ) Id E σ θ) Π σ Sθ) = ξ 2,1θ), Id E σ σ θ) S σ θ) Φ Eσ Eσ,Eσ V Π σ σ S θ) = ξ σ +σ E S σ Eσ S )θ) ρ 2,2θ) Id E σ σ θ) Id E σ σ θ) where ξ 1,1, ξ 1,2, ξ 2,1, ξ 2,2 ) is the solution of the linear system Id E σ θ) ξ 1,1 θ) + ξ 1,2θ) S σ θ) Id E σ θ) ξ 2,1 θ) + ξ 2,2θ) S σ θ) Hence, ξ 1,1 ξ 1,2 = ξ 2,1 ξ 2,2 Φ Eσ Eσ V σ +σ ρ Φ Eσ Eσ V σ +σ ρ Id E σ,eσ,eσ E σ S Id E σ σ θ) E σ S Id E σ σ θ) Φ Eσ Eσ V σ +σ ρ Eσ S )θ) Eσ S )θ),eσ E σ S S σ Id E σ σ Id E σ θ) = = Eσ S ) Id E σ σ θ) 1 28

41 Cocycles and Adapted Cocycles over T ω Given A A GLCn T d ρ) ρ, ω and k Z, then A k) A GLCn T d ρ) ρ, kω is defined by equation 1). Let E S be a splitting, and σ, σ {s, u, c}. We will use the adapted to E S cocycles A σ,σ [S] A LCn T d ρ) ρ, ω, defined as follows: A σ,σ [S] := Π σ S T ω A Π σ S We endow A σ,σ [S] with the structure of a cocycle as follows: Given k N, let A[S] σ,σ ) k) := Π σ S T kω A T k 1)ω Π σ S T k 1)ω Π σ S T ω A Π σ S A[S] σ,σ ) k) := Π σ S T kω A 1 T k 1)ω Π σ S T k 1)ω Π σ S T ω A 1 Π σ S Remark 8. Note that A A GLCn T d ρ) ρ, ω acts on A Gσ n ρ by left-multiplication: Given ES σ AGσ n ρ, we define A ES σ AGσ n ρ by A E σ Sθ) := { Av v E σ Sθ) }, θ T d ρ For σ σ it holds that A σ,σ [S] ρ C A ρ d V σ ρ E σ S T ω, A E σ S) Whiskered splittings Given N N, to remain fixed throughout the iterative procedure outlined in Algorithm 1, and positive numbers µ s, µ u, µ c,+, µ c, satisfying µ s < 1, µ u < 1, 1 < µ c,+, 1 < µ c,, µ s µ c,+ < 1, µ u µ c, < 1, 29

42 we will say that the splitting E S is N, µ s, µ u, µ c,+, µ c, - whiskered with respect to the cocycle A if there holds: A s,s [S]) N) ρ µ N s, A u,u [S]) N) ρ µ N u, A c,c [S]) N) ρ µ N c,+, 2) A c,c [S]) N) ρ µ N c, Remark 9. Let C h := C h N, µ s, µ u, µ c,+, µ c,, A, E S ) := max A σ,σ [S]) j) ρ σ {s,u,c}, 21) N j N Then, for kn n < k + 1)N, we have A σ,σ [S]) n) ρ A σ,σ [S]) kn) ρ A σ,σ [S]) n kn) ρ µ kn σ C h C h µ n σ Hence, if E S is N, µ s, µ u, µ c,+, µ c, - whiskered with respect to the cocycle A, it will also hold that, given n N, A s,s [S]) n) ρ C h µ n s, A u,u [S]) n) ρ C h µ n u, A c,c [S]) n) ρ C h µ n c,+, 22) A c,c [S]) n) ρ C h µ n c, 3

43 Conversely, given any ɛ >, if inequalities 22) hold then inequalities 2) hold for any N ɛ 1/ɛ, and the slightly deteriorated exponent µ σ := µ σ + ɛ. We will use the inequalities 22) rather than inequalities 2) in the definition of whiskered splitting. To make reference to the constant C h we will say that the splitting E S is N, C h, µ s, µ u, µ c,+, µ c, - whiskered with respect to the cocycle A, where C h := C h N, µ s, µ u, µ c,+, µ c,, A, E S ) is computed as in 21). Remark 1. The definition of whiskered splitting presented in Subsection which used directly the cocycle A instead of the adapted cocycles A σ,σ [S]) is more common in the literature. It is appropriate for invariant splittings, but it is too strong a condition for approximately invariant splittings. We mention briefly why: Fix θ T d ρ and let, e.g. v E s S θ), with v. Then, if E S is only approximately invariant, it can occur that Π u Sθ + ω) Aθ) v) of course, this cannot occur if E S is invariant under A). Then, since the base T d ρ is compact, it is straightforward to show that A n) v ρ C C h µ n u v, n > where the geometric constant C > depends on the minimum angle between the bundles E s S and Eu S over Td ρ. Hence the condition, stated in Subsection 2.3.3, A n) v ρ C h µ n s v, v E s S in general will hold only if v =. The condition in equation 22 can be thought as the appropriate relaxation for the case of approximately invariant splittings of the definition of whiskered invariant 31

44 splittings in Subsection We mention that no essential changes will be needed in the arguments concerning the partial hyperbolicity of the splitting E S, with respect to the by now standard) techniques found in the literature Number Theoretic conditions. Let ν, τ > to remain fixed throughout the iterative procedure. In solving the cohomology equations that appear at each step of Algorithm 1, we will need to assume that the tangential frequency ω T d satisfies the following non-resonance condition: Diophantine condition: ω DCν, τ) if j k ω ν k τ, k Z d \ { }, j Z We will also need to assume that the normal frequency Λ = λ 1, λ 2,..., λ c ) C c and ω satisfy the following joint non-resonance conditions: First Melnikov condition: Λ DC 1st ω ν, τ) if for 1 j c: λ j exp2πik ω) ν k τ, k Z d \ { } λ j 1 Second Melnikov condition: Λ DC 2nd ω ν, τ) if for 1 l, j c: λ l λ j exp2πik ω) ν k τ, k Z d \ { } λ j λ l if j l 32

45 CHAPTER IV STATEMENT OF THE MAIN RESULTS AND OUTLINE OF THEIR PROOFS In this Chapter we state our main result, Theorem 1. Theorem 1 is formulated in an a posteriori format: The objects to be computed are zeroes of suitable functional equations, which we introduce in Subsection 4.1. An outline of the proof of Theorem 1 is presented in Section 4.3. The full proof of Theorem 1 takes up Chapters 5, 6 and Invariance and Reducibility equations. We now introduce five functional equations, involving the functionals I see Subsection 4.1.1), S σ see Subsection 4.1.2), and R see Subsection 4.1.3). The motivation for these functional equations will be discussed in Subsection Invariance equation for K. We start with the functional associated to the invariance of the torus K, which is: I : A Cn ρ C c A Cn ρ I [K, β] = f β K K T ω, If I [K, β], then the graph of K over T d ρ, T K see 3)), is invariant under F β, where: F β : T d ρ C n T d ρ C n θ, z) θ + ω, f β θ, z)). If I [K, β] ρ 1, then we say that T K is approximately invariant under F β. 33

46 Remark 11. For the functional I to be well-defined at K, β), we need to impose that the image of T d ρ under K which we will denote by KT d ρ) ) is contained in the domain of analyticity of f β. We will need to ensure, at each step of the iterative procedure outlined in Algorithm 1, that this condition is verified. Informally, since the corrections produced at each step of Algorithm 1 can be made small with the error of the initial approximate solutions see the discussion on tame estimates in Section 4.3), the condition KT d ρ) Domainf β ) will be automatically satisfied throughout the iterative procedure if the initial errors are sufficiently small Invariance equation for the splitting. Let σ, υ {s, c, u}, {σ, σ, σ } a permutation of {s, u, c}, S σ A and let ES σ := σ Φ Eσ,Eσ Vρ σ ) 1 S σ ) L Es u c, E s u c ) ρ, be the analytic vector bundle that corresponds to S σ := S σ,σ, S σ,σ ) under the parameterization of V σ ρ introduced in Subsection We had introduced the adapted to E S cocycles A σ,υ [S] in Subsection We will use the shorthand notation A σ,υ := Aσ,υ [] = Πυ T ω A Π σ Recall from Subsection that E σ S is modelled after Eσ via the parameterization v v + S σ,σ v + S σ,σ v, v E σ Here, S σ,σ = Π σ S σ. Thus, E σ S is A -invariant if, and only if, given any v A Eσ ρ,, there exists w v A Eσ ρ, ω such that: 34

47 A σ,σ A σ,σ A σ,σ A σ,σ A σ,σ A σ,σ A σ,σ A σ,σ A σ,σ v S σ,σ v S σ,σ v = w v S σ,σ T ω w v S σ,σ T ω w v 23) System 23) holds if, and only if, A σ,σ + Aσ,σ Sσ,σ + A σ,σ S σ,σ = S σ,σ T ω [ A σ,σ + Aσ,σ Sσ,σ + A σ,σ Sσ,σ ] A σ,σ + Aσ,σ S σ,σ + A σ,σ S σ,σ = S σ,σ T ω [ A σ,σ + Aσ,σ Sσ,σ + A σ,σ Sσ,σ ] Thus, the functional associated to the invariance of the bundle E σ S is: S σ : A Cn ρ ) ) ) C c A L E σ,eσ ρ, A L E σ,eσ ρ, A L E σ,eσ ρ, ω A L ρ, ω E σ,eσ ) S σ [K, β, S σ,σ, S σ,σ ] = A σ,σ + Aσ,σ Sσ,σ + A σ,σ S σ,σ S σ,σ T ω [ A σ,σ + Aσ,σ Sσ,σ + A σ,σ Sσ,σ ] A σ,σ + Aσ,σ S σ,σ + A σ,σ S σ,σ S σ,σ T ω [ A σ,σ + Aσ,σ Sσ,σ + A σ,σ Sσ,σ ] If S σ [K, β, S σ,σ, S σ,σ ], then E σ S is invariant under A. If S σ [K, β, S σ,σ, S σ,σ ] 1 ρ we will say that E σ S is approximately invariant under A K, β. Remark 12. For the functionals S σ to be well-defined at K, β, S σ,σ, S σ,σ ), the projected cocycles A σ,σ must be well-defined. This, in turn, requires that E S is a splitting so that the projections onto the bundles E σ S are well-defined). Hence, at each 35

48 step of the iterative procedure outlined in Algorithm 1, it must be verified that E S is a splitting, which is an open condition. As in Remark 11, it will hold automatically throughout the iterative procedure if the initial errors are sufficiently small. We now show that the approximate invariance of the bundle ES σ under A just defined admits a geometric interpretation: Proposition 1. Denote S σ [K, β, S σ,σ, S σ,σ ] = e σ S and let ) ES σ σ 1 := Φ Eσ,Eσ S σ ) V σ ρ be the analytic vector bundle that corresponds to S σ := S σ,σ, S σ,σ ) under the parameterization of V σ ρ introduced in Subsection Then: 1. d V σ ρ E σ S T ω, A E σ S ) < C eσ S ρ we introduced the vector bundle A E σ S in Remark 8). 2. A σ,σ [S] ρ < C A K, β ρ e σ S ρ, for σ σ. Proof. Let e σ S = e σ,σ S, eσ,σ S ) t. Given any θ T d ρ and any v ES σ θ), normalizing if necessary, let v E σ S θ) be such that v A = 1, where v A := v A v) A θ) E σ S θ) is defined as v A := A θ) v + S σ,σ θ) v + S σ,σ θ) v ) t Note that v C A 1 θ). Since A θ) is an isomorphism, given any x A θ) E σ S θ) with x = 1, there exists v Eσ S θ) such that v Av) = x. w v E σ θ + ω) be defined as w v := A σ,σ θ) v + Aσ,σ θ) Sσ,σ θ) v + A σ,σ θ) Sσ,σ θ) v Let Note that S σ [K, β, S σ,σ, S σ,σ ] = e σ S implies v A = w v + v e 36