ON HYPERBOLICITY IN THE RENORMALIZATION OF NEAR-CRITICAL AREA-PRESERVING MAPS

. We consider MacKay’s renormalization operator for pairs of area-preserving maps, near the (cid:12)xed point obtained in [1]. Of particular interest is the restriction R 0 of this operator to pairs that commute and have a zero Calabi invariant. We prove that a suitable extension of R 30 is hyperbolic at the (cid:12)xed point, with a single expanding direction. The pairs in this direction are presumably commuting, but we currently have no proof for this. Our analysis yields rigorous bounds on various \universal" quantities, including the expanding eigenvalue.


Introduction.
We consider the operator ⟨F, G⟩ → ⟨G, F G⟩ acting on equivalence classes of pairs of area-preserving maps of the plane. A pair (F ′ , G ′ ) belongs to the class ⟨F, G⟩ if F ′ = Λ −1 F Λ and G ′ = Λ −1 GΛ, for some map Λ : R 2 → R 2 whose derivative has a constant determinant. For a concrete analysis, it is convenient to represent each equivalence class by a suitable member P = (F, G). Restricting the similarity transformations Λ to a specific subgroup, we end up with the following operator R, . (1.1) Numerically, the corresponding invariant circle is observed to persist as σ is increased, up to some value σ ∞ where the circle breaks up. As σ is increased further, one immediately passes through an infinite number of bifurcations involving periodic orbits whose rotation numbers are the continued fraction approximants r 1 = 1 2 , r 2 = 2 3 , r 3 = 3 5 , r 4 = 5 8 , . . . of the number γ. Namely, at values σ n that approach σ ∞ from above as n → ∞, the r n -periodic orbits for P σ = (F σ , G σ ) change stability, from elliptic to hyperbolic. By a p q -periodic point for a commuting pair (F, G) we mean a fixed point for F p G q . Interestingly, the ratio (σ n −σ n−1 )/(σ n+1 −σ n ) is observed to have a limit as n → ∞, namely the number δ. Even more interestingly, the value of δ is "universal", in the sense that a large class of cylinder-map families exhibit the same phenomenon, with the exact same value of δ. Two other universal numbers λ * and µ * describe the accumulation of r n -periodic orbits at the critical golden torus, for the pair P σ with σ = σ ∞ .
The standard framework (coming from statistical mechanics) for studying this type of universal behavior involves renormalization. The renormalization operator R defined in (1.1) was proposed and studied first in [14,15,11]. What makes it relevant to the problem at hand is the fact that if ω is a r n -periodic point for P , then Λ −1 ω is a r n−1 -periodic point for R(P ). The standard explanation of the above-mentioned observations involves the existence of a fixed point P * for R 0 , with the property that the linearization of R 0 at P * has an eigenvalue δ and no other spectrum in the open unit disk. Furthermore, the scaling Λ * associated with P * should be conjugate to diag(λ * , µ * ).
W s P * P * P * σ → Pσ σ → Pσ σ → Pσ  1 depicts the expected action of R 0 , assuming that (some extension of) this operator acts differentiably on a manifold of map-pairs: A family σ → P σ intersects the "critical surface" W s at the parameter value σ = σ ∞ where the golden torus breaks up. W s is the codimension-1 stable manifold of R 0 at the critical fixed point P * . The "bifurcation surfaces" Σ(r n ) accumulate at W s with an asymptotic rate ∝ δ −n .
We start our analysis by discussing the fixed point problem for R. A proof for the existence of a fixed point P * whose scaling Λ * is consistent with numerical observations was given already in [1]. But our analysis of the operator M ′ requires domains that are different from the ones considered in [1], so we first need prove a similar result: , in the sense that SF * S = F −1 * and SG * S = G −1 * . Our proof of this theorem involves estimates that have been carried out by a computer. As a by-product, we obtain highly accurate bounds on the various quantities involved [23]. In particular, we determine enclosures λ ⋄ ∋ λ * and µ ⋄ ∋ µ * that are open intervals of diameter less than 2 −253 . Denote by Λ ⋄ the set of all 2 × 2 matrices Λ = diag(λ, µ) with λ ∈ λ ⋄ and µ ∈ µ ⋄ . Then the domains D F and D G mentioned in Theorem 1.1 satisfy the relations for G = G * . The conditions (1.5) ensure that, if two continuous maps F and G are well-defined on D F and D G , respectively, then so are the renormalized maps Λ −1 GΛ and Λ −1 F GΛ, respectively, if Λ is sufficiently close to Λ * . Additional conditions with stronger implications are discussed in [16] but not considered or used here. Since our proof of Theorem 1.1 is very indirect, let us describe some of the direct information that we have about the fixed point P * . As in [1], the map G * is obtained as a fixed point of the operator N defined by the equation N(G) = Λ −1 F G 0 Λ, where G 0 = Λ −1 F GΛ and F = Λ −1 GΛ. The map G is normalized by fixing the value G(0), and the scaling Λ = diag(λ, µ) is then determined by the condition G 0 (0) = G(0). To be more precise, we solve the equation N(G) = G for maps by solving the corresponding equation N (g) = g for generating functions.
A map G is obtained from its generating function g via the equation , (1.6) where g j = ∂ j g. Assuming that g satisfies a twist condition g 1,2 > 0, the corresponding map G is uniquely defined. Since the one-form wdy−zdx is the differential of g and thus closed, this map G is area-preserving. Domains will be specified later.
Notice that the operator N preserves reversibility. It is not hard to show that, if G is a reversible fixed point of N, then G −1 0 G is the identity map near the origin [1]. So modulo domain questions, a reversible fixed point of N yields a fixed point for R. Notice also that, if (F * , G * ) is a reversible fixed point of R, then F * and G * commute.
Theorem 1.1 is proved by finding a fixed point g * for N , defining the associated map G * via (1.6), and setting F * = Λ −1 * G * Λ * . The resulting domains D G and D F for G * and F * will be described in Section 4. The domain of F * G * is defined as usual as . Similarly for the domain D GF of G * F * . When saying that F * and G * commute, we mean that F * G * and G * F * agree on the intersection D FG ∩ D GF . This intersection is shown to be non-empty.
Whenever two injective maps F and G commute, we can define an extension G of G in such a way that GF is defined on the domain of F G and satisfies GF = F G on this domain. The map F can be extended similarly. This procedure is used to extend F * and G * to the larger domains These domains are then shown to satisfy the relations (1.5).
A related fixed point problem M(P ) = P was considered by Stirnemann in [17], where M is a "palindromic" modification of the third iterate of R, A useful feature of M is that it preserves reversibility with respect to S. It was shown in [17] that the corresponding fixed point equation for a reduced generating function has a solution.
What was left open is the question of whether the corresponding maps F and G commute (which, modulo domain questions, would yield a fixed point for R 3 ), and whether they are area-preserving.
We have tried to work with the operator R directly, but we were unable to find appropriate function spaces for such an analysis. The problem is that R is defined only for map-pairs whose generating functions have relatively large and/or unwieldy domains. This lead us to consider R 3 . By itself, this transformation is still difficult to analyze. What helps is that we already have a commuting fixed point of R, namely the pair P * obtained by solving the fixed point problem for N. Fortunately, we were able to find domains that work simultaneously for both N and M. As a result, P * is also a fixed point of M. Given that are interested only in commuting pairs, we can now work with M instead of R 3 .
Concerning domains, we only discuss here what is needed to define a proper renormalization transformation. Additional considerations arise when studying dynamical properties of commuting pairs, but this goes beyond the scope of this paper. There is however one dynamic quantity associated with area-preserving cylindermaps that we need to consider: the Calabi invariant [8]. If this invariant is nonzero then the map has a "net motion" in the non-periodic direction of the cylinder. This is a trivial obstruction to the existence of homotopically nontrivial invariant circles. An explicit expression for the Calabi invariant C(P ) of a commuting pair P will be given in Subsection 3.1. We are mostly interested in commuting pairs with zero Calabi invariant, which includes e.g. all reversible pairs. For such pairs, Theorem 1.2 below implies that R 3 has at most one unstable direction at P * , and that all other directions are exponentially attracting.
If B is a set of pairs of area-preserving maps, and if Ω a subset of R 2 , we say that Ω is a commutator domain for B if the composed maps F G and GF are defined on Ω, for every pair P = (F, G) belonging to B. For such a domain Ω we define B c (Ω) to be the set of all pairs P ∈ B with the property that F G and GF agree when restricted to Ω, and that C(P ) = 0. Concerning the last statement of Theorem 1.2, it is clear numerically that P * does not attract all nearby pairs in B c (Ω). We believe that this alternative can be eliminated within the framework of this paper, requiring only technical changes. A possible approach is described at the end of Section 8.
A shortcoming of the map-pair approach is that sets of commuting pairs like B c (Ω) are most likely not manifolds. For the renormalization of critical circle maps, an approach that avoids this problem has been introduced and used successfully in [18]. For the problem considered here, commuting pairs can be avoided by studying Hamiltonian flows on T 2 × R 2 instead of area-preserving maps.
An analogue of Theorem 1.1 for a renormalization operator acting on Hamiltonians was proved in [9], and the scaling of the fixed point Hamiltonian H * is compatible with (1.4). Unfortunately, the renormalization operator for Hamiltonians (or vector fields) is substantially more involved than the operator R. So a Hamiltonian analogue of Theorem 1.2 seems currently out of reach. On the other hand, it was shown in [10] that every Hamiltonian on the strong local stable manifold at H * has a golden invariant torus that is "critical", in the sense that the flow on this torus is not differentiably conjugate to a linear flow. A similar argument should work for the operator R 3 , but we have not considered this problem here.
In order to connect these results to the breakup of golden invariant tori, one would also have to prove that, in some open neighborhood of the fixed point P * , all commuting zero-Calabi pairs on one side of the local stable manifold of M ′ have a smooth golden invariant torus, while the pairs on the other side have no golden invariant torus. Two methods that have been developed and studied in connection with this problem are Greene's criterion [6,4,12] and the obstruction method [13,2]. It should be possible to make some progress on these questions by using the methods developed in this paper.
The remaining part of this paper is organized as follows. In Section 3 we give a formal definition of the operator M ′ that implements the above-mentioned operator M ′ in terms of generating functions. Some facts that have guided our choices are described in Section 2. Sections 4 and 5 are concerned with the proof of Theorem 1.1, while Sections 6,7, and 8 are devoted to the proof of Theorem 1.2. To be more precise, both proofs are reduced to the task of proving several technical lemmas. Our proof of these lemmas is computer-assisted, using some estimates that are given in Section 9. In Section 10 we describe how the remaining estimates are translated into a sequence of trivial computations (with rigorous error estimates) that can be, and have been, carried out by a computer. The complete details of this last part, written in the programming language Ada [19], can be found in [23].

Irrelevant eigenvalues.
In order to determine a proper choice of scaling Λ = Λ P and a proper modification of the operator M, it is useful to consider what happens if the scaling is fixed. Much of the discussion in this section is purely formal; but part of it will be made rigorous later.
Let P * = (F * , G * ) be the fixed point of R described in Theorem 1.1. Denote by R * the operator R with a fixed scaling Λ = Λ * . That is, where we have used the notation V (F, G)U = (V F U, V GU ) for maps V, F, G, U of the plane. As with other renormalization operators in dynamics, fixing the scaling introduces "irrelevant" eigenvalues at the fixed point. These eigenvalues, and the corresponding eigenvectors, can be computed explicitly: Consider changes of coordinates U ε = I + εT + O(ε 2 ) with constant determinant det(DU ε ). Then where P T = (DP * )DT − T P * . Notice that the map T → P T is linear. Applying R * to the left hand side of (2.2) we get a similar conjugacy, but with U ε replaced by Λ −1 * U ε Λ * . This shows that In particular, if Λ −1 * T Λ = κT then P T is an eigenvector of DR * (P * ) with eigenvalue κ. Restricting to coordinate changes that are analytic near the origin, these eigenvalues are all of the form κ = λ m−1 * µ n * or κ = λ m * µ n−1 , with m and n nonnegative integers.
The non-contracting eigenvalues κ and the corresponding functions T are listed in Table 1. In this table, the eigenvalue with label 0N is the N-th largest eigenvalue in modulus in the even subspace (defined later for generating functions). Similarly, 1N labels the N-th largest |eigenvalue| in the odd subspace. The same labels are also used in our programs [23].

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The goal is to choose the scaling Λ = Λ P in the definition (1.1) of R in such a way that DR(P * )P T = 0 for the maps T listed in Table 1. Then we can expect DR(P * ) to have a simple eigenvalue δ > 1 and no other spectrum outside the open unit disk.
The above-mentioned expectation assumes that R is restricted to commuting pairs with zero Calabi invariant. If we allow "general" pairs, then DR(P * ) has other expanding eigenvalues, as was already described in [11]. They can again be computed explicitly: If Θ = F G(GF ) −1 is the commutator for a pair P = (F, G), then the corresponding commutatorΘ for the renormalized pairP = R(P ) is given byΘ The derivative of the map Θ →Θ at the trivial commutator Θ * = I is given by θ → −Λ −1 * θΛ * . To see how this relates to the spectrum of DR(P * ), consider a one-parameter family of pairs P * + εP ′ + O(ε 2 ) and the corresponding family of commutators I + εθ + O(ε 2 ). If P ′ is an eigenvector of DR(P * ) with eigenvalue κ, then we must have Λ −1 * θΛ * = κθ. So either θ = 0, or else κ has to be an eigenvalue of the linear operator θ → −Λ −1 * θΛ * . When restricted to functions θ that are analytic near the origin, the eigenvalues of this operator are of the form , with m and n nonnegative integers. Numerically, these values do indeed appear as eigenvalues of DR(P * ).
Consider now the operator M defined by (1.7), but with a fixed scaling Λ = Λ * . This operator will be denoted by M * . When restricted to commuting directions, DM * (P * ) has the same eigenvalues and eigenvectors as DR 3 (P * ). But other eigenvalues and eigenvectors need not agree. In particular, consider the commutatorΘ for the pairP = M(P ). A computation similar to (2.4) yields (2.5) To first order in P − P * , the map Θ →Θ is given by θ → −A −1 θA, where A = Λ 3 * F −1 * . As we will show, the scaling map A has a fixed pointω, which belongs to the domain of Θ for the maps considered here. Numerically, the derivative DA(ω) has two eigenvalues α 3 and β 3 , where Thus, the operator θ → −A −1 θA has eigenvalues κ 3 with κ = −α m−1 β n or κ = α m β n−1 . Notice that αβ = λ * µ * since F * is area-preserving. For reasons analogous to those described after (2.4), any eigenvalue of M * (P * ) corresponding to a noncommuting direction has to take one of these values κ 3 . The largest (in modulus) five eigenvalues 1/3 of this type are listed in Table 2. In addition, we list as first entry the eigenvalue that corresponds to a commuting direction with nonzero Calabi invariant. The coordinates (a, b) used in the the second column of this table are those that diagonalize A to diag(α, β). We note that, for reversible pairs, a relation analogous to (2.5) holds if the commutator Θ is replaced by the map Ξ = F GS. This should not be surprising, since Θ = Ξ 2 in this case. Restricting to commuting pairs, this implies that the fixed point of Ξ * A plays a role similar to the fixed point ω of A. Thus, one would expect that Ξ * (ω) = ω. (Notice that Ξ * is an involution and thus has a curve of fixed points.) This is indeed observed numerically.

A formal definition of the operator M ′ .
The main goal in this section is to describe the operator M ′ that represents M ′ in terms of generating functions. We also compute the Calabi invariant C(P ). The descriptions are formal in the sense that we do not specify any domains at this point.

Generating functions for composed maps.
Consider an area-preserving map G defined by a generating function g via (1.6). For reference below, we note that the identity wdy − zdx = dg mentioned after (1.6) can be written as where θ is the one-form θ = z dx on the domain of G, and where G * θ denotes the pushforward of θ under the map G.
The composition of G with a map F generated by f can be represented as . (3.1) So the generating function f ⊙g for the composed map F G is where V = V(x, y) is determined by the second equation in (3.2), assuming that this equation has a unique solution. V will be referred to as the midpoint function for f ⊙g. Denote by V and W the midpoint functions for f ⊙g and g⊙f , respectively. Let x(x, y) = x and y(x, y) = y. We define the commutator for the pair p = (f, g) as Assume now that F and G are well defined on some planar domain D, and that D/∼ defines a cylinder, where "∼" denotes the equivalence relation on D whose equivalence classes are the orbits of F . Assume furthermore that G commutes with F , so that G defines a map on the cylinder D/∼. Our goal is to compute the Calabi invariant for this map. To this end, we need to choose a differentiable curve γ in D, from a point ω to its image under F . When regarded as a curve on the cylinder, γ is closed. By definition, the Calabi invariant is the signed area of the region between the closed curves G • γ and γ. When lifted to D, this region is bounded by two other curves: a curve σ connecting ω to G(ω), and its image under the map F .
To obtain an explicit formula, denote by (x, y) points in the domain of both f ⊙g and g⊙f .
The signed area defining the Calabi invariant for P can now be written as This is just the commutator −C(p), evaluated at (x, y). Since we have not been precise here about domains, we take C(P ) = −C(p) as a definition of the Calabi invariant of P for the purpose of our renormalization analysis. Notice that its value is independent of the point (x, y): having assumed that F G and GF agree, the generating functions f ⊙g and g⊙f have the same derivatives, so C(p) is a constant in this case.

Definition of the operator M ′ .
Consider the operator R with scaling maps Λ of the form Λ(x, z) = (λx, µz). Formally, the corresponding operator for pairs of generating functions p = (f, g) is given by where ℓ(x, y) = (λx, λy), with λ and µ to be determined. Similarly, the operator M * is represented in terms of generating functions by where ℓ * (x, y) = (λ * x, λ * y). We will prove later that this operator M * is welldefined in an open neighborhood of p * in a space B p = B f × B g of pairs p = (f, g).
As we have seen in Section 2, the coordinate-related eigenvalues of modulus ≥ 1 arise from coordinate transformations of the form , Here τ ′ denotes the derivative of τ , and t is a vector in R 7 close to the origin. If F is an area-preserving map with generating function f , then Conditioned on the choice of a continuous linear map T : B p → R 7 , define now for all p close to p * . Here p * is the fixed point of M * obtained from the fixed point g * of N . The derivative of M at p * is given by the equation . , E 7 be the seven coordinate-related eigenvectors of DM * (p * ) for the eigenvalues listed in Table 1. Then the equation (3.10) may be written in the form where Φ 1,k : B f → R and Φ 2,k : B g → R are continuous linear functionals. In fact, it is these functionals that we choose (as described below); the map T is implicitly defined by this choice.
Recall that DM(p * ) also has six undesired eigenvalues in non-commuting directions, namely the cubes of the values listed in Table 2. In order to "eliminate" these eigenvalues, we compute polynomial approximations E 8 , E 9 , . . . , E 13 for the corresponding eigenvectors, and define after choosing suitable linear functionals Φ 3,k for k = 8, 9, . . . , 13. Clearly, if p is a commuting pair in the sense that All of our functionals Φ j,k are of course chosen in such a way that DM ′ (p * )E k ≈ 0 for k = 1, 2, . . . , 13. For simplicity, they are taken to be of the form with Φ j,k,n ∈ R, and with ξ j,k,n ∈ R 2 well inside the domain of the functions h being considered. The precise values of these quantities can be found in [23]. By definition, the choices made in defining M ′ are "suitable" if DM ′ (p * ) has a single expanding eigenvalue and no other eigenvalues outside the open unit disk. We note that the last eigenvalue in Table 2 is not expanding; but we choose to eliminate it as well, since it is uncomfortably close to 1.

Construction of the fixed point.
As mentioned in the introduction, the first component G * of the fixed point P * = (F * , G * ) described in Theorem 1.1 is constructed as a fixed point of the operator N defined by . The corresponding condition for generating functions is that g 01 (0, −1) = 0 and g 02 (0, −1) = −1.
Applying the identity (3.2) twice, we see that the generating function for GF G is given by These equations simplify if we assume that f and g generate maps F and G that are reversible.
To be more precise, notice that the generating function for G −1 is (x, y) → −g(y, x). This can be seen directly from (1.6). Thus, the generating function Sg for the reversed map SG = SG −1 S is given by Using that Sg 1 = −(Sg) 2 and Sf 2 = −(Sf ) 1 , the equations determining the midpoint functions V 0 and W 0 can be written as Assume now that f and g are reversible, in the sense that Sf = f and Sg = g. Then (4.3) reduces to the single equation This relation between V 0 and W 0 also implies that g⊙f ⊙g is reversible. The representation of N in terms of generating functions is now given by If we express our generating functions in the variables t = x + y and s = x − y, then f and g are reversible if and only if they are even functions of t. As was found in [1], the generating functions arising in the analysis of N are well approximated by polynomials in the variables u and v, if t 0 , s 0 , b ∈ R are chosen appropriately. This suggested the following choice of functions spaces. Given a pair ρ = (ρ u , ρ v ) of positive real numbers, denote by D ρ the set of all points (u, v) ∈ C 2 such that |u| < ρ u and |v| < ρ v . Define A ρ to be the space of all all analytic functions ϕ : D ρ → C, that extend continuously to the boundary of D ρ and have a finite norm where ϕ and ψ are functions on D ρ . Definition 4.1. We define B ϱ to be Banach space of all functions (4.7) with ϕ and ψ belonging to A ρ , equipped with the norm The Taylor coefficient ϕ 0,0 of ϕ will be referred to as the constant term of f . We will call f even if ψ = 0, or odd if ϕ = 0. The subspace of even and odd functions in B ϱ will be denoted by B 0 ϱ and B 1 ϱ , respectively. Clearly, B 0 ϱ is isometrically isomorphic to A ρ . It is not hard to see that both B 0 ϱ and B ϱ are Banach algebras.
Remark. As defined above, B ϱ is a Banach space over C. When discussing maps associated with generating functions in B ϱ , we only consider the corresponding Banach space over R, where a function is assumed to take real values for real arguments. Since it should be clear from the context which number field is being used, we will denote both spaces by B ϱ . Similarly for the spaces A ρ . If we wish to stress that a function takes real values for real arguments, we will refer to this function as being real.
For the domain of the function g in (4.4) we use parameter values with r to be specified. Here, the subscript g in ϱ g is just a symbol and does not refer to a specific function g. This is part of a convention that is used throughout this paper: any subscript in non-slant font is an abstract symbol and not an argument. The domain D ϱ and space B ϱ with ϱ = ϱ g (r) will be denoted by D g (r) and B g (r), respectively. For the function f = (λµ) −1 g • ℓ we use the domain D f (r) and space B f (r), defined by the parameter values In addition, define real domains D g (r) = R 2 ∩D g (r) and D f (r) = R 2 ∩D f (r). For the values of r considered here, D g (r) is a simply connected region η − r (x) < y < η + r (x) between the graphs of two continuous functions y = η ± r (x), defined over an interval ξ − r < x < ξ + r . The values ξ ± r and the functions η ± r can be computed explicitly. Similarly for the domain D f (r).

Theorem 4.2. The operator N is well-defined and analytic on an open set in
). This open set contains a unique real fixed point g * of N . The associated scaling constants λ * and µ * satisfy the bounds (1.4). The corresponding function f * belongs to B 0 f ( 9 8 ). On D g ( 31 32 ) and D f ( 31 32 ), the functions f = f * and g = g * satisfy the twist condition g 1,2 , f 1,2 > 1 and the bounds g 1,1 , g 2,2 , f 1,1 , f 2,2 < −1.
Concerning our proof of this theorem: The existence of the fixed point g , and the bounds on λ * and µ * , are proved the same way as the corresponding parts of Theorem 4.1 in [1]. In particular, the fixed point problem for N is reduced to a fixed point problem for a quasi-Newton map associated with N . The second derivative inequalities are then easy to verify, since our construction of f * and g * yields highly accurate polynomial approximations with rigorous error bounds. For details we refer to [1,23]. The task of controlling g⊙f ⊙g also appears in connection with the operator M * and will be discussed in Sections 6,9, and 10.

Map domains.
When considering map-pairs (F, G) associated with pairs (f, g) of generating functions, the domains of f and g are always taken to be D g = D g (r) and D f = D f (r), with r = 15 16 .
In the remaining part of this section, f, g, F, G stand for f * , g * , F * , G * . The inequalities in Theorem 4.2 imply that f ⊙g and g⊙f are real analytic and satisfies a twist condition on their domains D f⊙g and D g⊙f , respectively. By definition, D f⊙g is the set of all points (x, y) ∈ R 2 for which there exists a unique V ∈ R such that (x, V) ∈ D g and (V, y) ∈ D f and g 2 (x, V) + f 1 (V, y) = 0. Similarly for D g⊙f . The domain of the map G is defined by the equation ] .
Notice that, by the twist property of g, D G is a simply connected region ζ − (x) < z < ζ + (x) between the graphs of two continuous functions z = ζ ± (x), defined over This lemma is proved easily by including two additional domain-checks in the proof of Theorem 4.2. Namely, using the estimates on the midpoint function V 0 and W 0 obtained in the proof of Theorem 4.2, we verify that the composed maps on the right hand side of (4.1) are well defined if f and g are restricted D f (r) and D g (r), respectively, for some r < 15 16 . No additional work is needed to allow arbitrary scalings Λ ∈ Λ ⋄ in Lemma 5.1, since by definition, Λ ⋄ is the enclosure used for Λ * in the proof of Theorem 4.2. For the complete details we refer to [23].
Notice that where G 0 = Λ −1 * F GΛ * and F 0 = Λ −1 * G 0 Λ * . Given that Λ 2 ⋄ D G ⊂ D G , the map G 0 is well-defined on Λ ⋄ D G , and F 0 is well-defined on D G . This has motivated our choice For monodromy arguments it would be useful to know that (the union of) the domains D f⊙g and D g⊙f are simply connected. We saw no easy way to prove this, but the sets W and SW given below achieve the same purpose. We will now give a sketch of our proof and refer to [1,23] for details. The claim that g 0 = g near (0, −1) follows from a simple computation [1], which shows that J = G −1 0 G satisfies J = Λ −1 * J −1 Λ * near the origin. Using that J is analytic, and that |λ| 3 < |µ| < |λ| 4 by Theorem 4.2, it follows that J = ±I near the origin. Using our computer-generated bound on J, the case J = −I is easily excluded.
The remaining part of the proof is an exercise in interval arithmetic. We outline it here since it is qualitatively different from our other proofs: As mentioned earlier, our construction of f * and g * yields highly accurate approximations (polynomials in u and v) with rigorous error bounds. When analyzing the domain of f ⊙g, this allows us to decide e.g. whether the function V → g 2 (x, V) + f 1 (V, y) has a zero V = V(x, y) with (x, V) ∈ D g and (V, y) ∈ D f . If (x, y) is an arbitrary (unknown) point in a given square, the answer can be either True, False, or Uncertain. The squares being considered are from a finite collection of closed diadic squares that cover a "sufficiently large" region. Denote by A and B the union of squares yielding answer=True and answer=Uncertain, respectively. Then B covers the boundary of D f⊙g . By inspection (after determining the boundary of B) we find that the complement of B is the disjoint union of two connected sets, with one of them being the interior of the bounded set A.
Concerning the extension of g 0 , notice that the inequalities in Theorem 4. The above-mentioned domains, scaled by a factor λ * , are depicted in Figure  2. The larger contour is the boundary of D f⊙g or (indistinguishably at the given resolution) the set B described above. The smaller contour is the boundary of λ * D g . The straight line shown in Figure 1 is the line of fixed points for the reflection S, where x + y = 0. Seven of the eight isolated points that can be seen near (or on) this line are the points ξ 3,k,n . The eighth point (x,ȳ) will be described below. What remains to be proved is (1.5). We will use Lemma 5.1 repeatedly without mentioning it each time.
This implies the third inclusion relation in (1.5) and completes the proof of Theorem 1.1.
The following will be needed in our discussion of the operator M ′ . Proving this lemma is a straightforward rigorous "computation". The existence of a set C with the asserted properties can be seen by inspection (using the set B described after Lemma 5.2). For the complete details of the proof we refer to [23].
We note thatx +ȳ = 0, since the maps F and G commute. The point (x,ȳ) is shown in Figure 2: it is the point on the line x + y = 0 near the lower boundary of λ * D g .
6. The midpoint equations for even pairs. The definition (3.6) of the operator M * involves the composed functions g⊙f ⊙g and g⊙f ⊙g⊙f ⊙g. If we denote these two functions byf andĝ, respectively, then where V j = V j (x, y) and W j = W j (x, y) are determined by the condition that the right hand side of (6.1) and (6.2) be stationary with respect to variations of each V j and each W j . The resulting equations for the functions V j and W j will be referred to as the midpoint equations. For j = 0 they are given by (4.3), and for j = 1, 2 they are

HANS KOCH
For the functions V 0 and W 0 we use spaces B Vf (r) = B ϱ ′ (r) with domain param- . The exact definition of the scaling σ can be found in [23]. For the functions V j and W j with j = 1, 2 we use spaces B Vg (r) = B ϱ ′ (r) with domain parameter ϱ ′ = σ(λ 3 , ϱ g ).
The midpoint equations need to be solved in two different situations. First, we need to compute and estimate the midpoint functions for the fixed point p * , since the same functions appear in the derivative DM * (p * ). In this case, the functions f and g involved are even. Then we also need to estimate M * in an open neighborhood of p * . In this case, we will use perturbation theory, since the odd parts of f and g can be chosen arbitrarily small. This will be discussed in the next section. As it turns out, controlling odd perturbations involves the same estimates as controlling even perturbations. Thus, we assume in this section that f and g are even. In addition, we make here the ansatz W j = −SV j . This is justified a-posteriori by the fact that it yields a solution for (4.3) and (6.3), and that these solutions are unique due to the twist properties of f * and g * .
The equation for the midpoint function V 0 , as described after (4.3), can be written as Formally, the derivative of K ′ is given by Similarly, the midpoint equations (6.3) can be written as . (6.6) Here, we have used that the second and fourth equations in (6.3) become redundant in the case considered here, where f and g are even and W j = −SV j . Formally, the derivative of K ′′ is given by with The operators L ′ 1 and L ′′ 1 will be needed in the next section. In what follows, if X and Y are Banach spaces, the product space X × Y is assumed to be equipped with the norm ∥(x, y)∥ = ∥x∥ X + ∥y∥ Y .

Lemma 6.1.
There exist open sets V 0 ⊂ B Vf (1) and V 1 , V 2 ⊂ B Vg (1) such that the following holds. For j = 0, 1, 2 define W j = −SV j . Then for every f ∈ B f (1) and every g ∈ B g (1), each term on the right hand side of (6.1) defines a function in B Vf (1), and each term on the right hand side of (6.2) defines a function in B Vg (1).  (1), respectively; and the equations K ′ (V 0 ) = 0 and K ′′ (V 1 , V 2 ) = 0 have unique solutions V j ∈ V j . These solutions depend analytically on f ∈ U f and g ∈ U g .
Our proof of this lemma is computer-assisted and will be described in Sections 9,10.
Concerning the invertibility of the operators L ′ σ and L ′′ σ for σ = ±1, our programs only check explicitly the case σ = −1. This suffices for the following reason. The basic identity being used is 3) for functions f and g that are not necessarily even. However, the non-even part can be taken arbitrarily small; so we can use simple perturbation theory, as we will now describe.
The system of equations (4.3) can be written in the form K ′ (p, q) = 0, where p = (f, g) and q = (V 0 , W 0 ). Here p is restricted to some open neighborhood U p of p * in B p (1) = B f (1) × B g (1), and q is restricted to U q = V 0 × W 0 , with V 0 and W 0 as specified in Lemma 6.1. Then K is analytic on U p × U q and vanishes at (p * , q * ), where q * is the pair of midpoint functions (V 0 , W 0 ) for p * . Next we consider pairs p * + p ′ of generating functions with p ′ close to zero. To simplify notation, let us write (p, q) in place of (p * , q * ). To first order in the perturbation p ′ , the equation K ′ (p + p ′ , q + q ′ ) = 0 for the midpoint functions q + q ′ becomes D 1 K ′ (p, q)p ′ + D 2 K ′ (p, q)q ′ = 0. Assume first that p ′ has a fixed parity: Sf ′ = σf ′ and Sg ′ = σg ′ , with σ = ±1. Then a straightforward computation shows that the equation where L ′ ±1 are the operators defined in (6.5). By Lemma 6.1, these operators are invertible, and by linearity, this implies that D 2 K ′ (p, q) is invertible. Thus, by the implicit function theorem, the equation K ′ (p, q) = 0 defines an analytic map p → q in some open neighborhood of p * , taking values in V 0 × W 0 . Similarly, the system (6.3) can be written in the form K ′′ (p, q) = 0, where p = (f, g) and q = (V 1 , W 1 , V 2 , W 2 ). For fixed parity σ, the equations where L ′′ ±1 are the operators defined in (6.7). By the same argument as above, we conclude that the equation K ′′ (p, q) = 0 defines an analytic map p → q in some open neighborhood of p * , taking values in In the following theorem we use the fact [23] that the points ξ 1,k,n and ξ 2,k,n in (3.13) have been chosen to lie in D f and D g , respectively. The choice of the constants Φ j,k,n and polynomials E 8 , E 9 , . . . , E 13 is not relevant at this point. 8. Hyperbolicity. In order to simplify notation, we write B p instead of B p (r) whenever r = 1. Similarly for all other spaces and domains that depend on a choice of r in (4.9) or (4.10).
Denote by V j and W j = −SV j the midpoint functions for the fixed point p * = (f * , g * ) of the operator M * . For every pair p = (f, g) in B p we have wheref andĝ are the functions defined by (6.1) and (6.2), respectively. The derivative of M at p * is given by (3.10), and for the derivative of M ′ we have Here, DC is its derivative of the commutator (3.3). With V and W as defined above, y) .
A straightforward computation shows that DM * (p * ) preserves parity, in the sense that the even subspace B 0 p = B 0 f ×B 0 g and the odd subspace B 1 p = B 1 f ×B 1 g of B p are both invariant under DM * (p * ). By our choice of polynomials E 8 , E 9 , . . . , E 13 and constants in (3.13), the derivatives of M and M ′ at p * are parity-preserving as well. And of course, these quantities have been chosen in such a way that, numerically, the derivative of M ′ at p * has the desired spectrum. For the precise values we refer to [23]. By Theorem 7.2 the linear operator DM ′ (p * ) is compact on B p . Since we had mentioned it only in passing, let us recall that the norm on B p is given by Denote by A the restriction of DM ′ (p * ) to the (invariant) even subspace B 0 p . In order to prove that A has no eigenvalues outside the open unit disk, except for a simple eigenvalue δ 3 > 1, we consider a map F that can be expected to be a contraction (for some norm) and whose fixed point is the normalized eigenvectorp of A for the eigenvalue δ 3 . A simple candidate for such a map is p → ∥Ap∥ − . Our proof of this lemma is computer-assisted and will be described in Sections 9,10. With the same tools we also prove Lemma 8.2. The fifth power of DM ′ (p * ), when restricted to the (invariant) odd subspace B 1 p , is a contraction. As an immediate consequence of these two lemmas we have the Proof. By Lemma 8.2 is suffices to prove the claim for the operator A.
From (8.5) we see that any fixed point p of F m satisfies φ(p) = 1 and is an eigenvector of A m with eigenvalue φ(A m p). By uniqueness, every eigenvector of A m nearp is a constant multiple ofp. This applies in particular top + εAp for ε ̸ = 0 close to zero, sop is an eigenvector of A; and the corresponding eigenvalue is ϑ.
Consider the operator B defined by Bp = Ap − φ(Ap)p. Using (8.5) we have And ∥B m ∥ < 1 by Lemma 8.1. Assume for contradiction that the eigenvalue ϑ of A is not simple. Then there exists p ∈ B p such that (A − ϑ)p =p. A straightforward computation shows that this p satisfies B n p = ϑ n [p − φ(p)p] for all n. But this is incompatible with B m being a contraction. Similarly, it is clear from the second identity in (8.6) that A cannot have an eigenvalue of modulus ≥ 1, besides ϑ. Our next goal is to relate the operator M for pairs of generating functions to the operator M for pairs of area-preserving maps. At this point it becomes relevant that two generating functions that differ only by a constant yield the same map. We have ignored this fact up to now in order to simplify the description. Denote by B ′ f the subspace of B f consisting of all function f ∈ B f whose constant term (see We may assume that p * ∈ B ′ p . Since the the constant-term issue is trivial, let us simply redefine M to take values in B ′ p , by setting the constant terms in M(p) equal to zero. Similarly for the operator M ′ .
Denote by Υ g the map g → G defined by the equation (1.6).
To be more precise, we assume that g belongs to B ′ g and is close enough to g * to satisfy a twist condition. Then Υ g (g) is well-defined as a map from D G to R 2 . And it is clear from (1.6) that Υ g is one-to-one near g * . Similarly, denote by Υ f the map that associates to a generating function f ∈ B ′ f close to f * the corresponding map F : D F → R 2 . Now we need to check that the maps F = Υ f (f ) and G = Υ g (g) can be composed as required by (1.7). 15 16 such that the following holds. Denote by V j and W j the midpoint functions for p * . Let f ∈ B f (r) and g ∈ B f (r). Then the function f given by (6.1) is well-defined on the domain of V 0 , and the functionĝ given by

Lemma 8.4. There exists r <
This lemma is proved easily by using the estimates on the midpoint function V j and W j obtained in the proof of Lemma 6.
From the discussion preceding Theorem 7.2 we know that these midpoint functions depend analytically on p ′ . Thus, Lemma 8.4 remains true if we replace the functions V j and W j by V ′ j and W ′ j , respectively, assuming (which we shall) that the radius of B ′ is sufficiently small.
As described earlier, to every pair of generating functions p = (f, g) in some open ball B ⊂ B ′ p centered at p * , we can associate two area-preserving maps F = Υ f (f ) and G = Υ g (g). Using these two maps, define Υ p (p) = (F, G).
Choose another ball B 0 ⊂ B ′ ∩ B centered at p * , such that M(B 0 ) and M ′ (B 0 ) are included in B. Assuming (which we shall) that the radius of B 0 has been chosen sufficiently small, the above-mentioned generalization of Lemma 8.4 implies that if p ∈ B 0 and P = Υ p (p), then M(P ) is well-defined and agrees with Υ p (M(p)). We note that the change of coordinates Λ in the expression (1.7) for M(P ) depends on p and can be expressed explicitly in terms of the change of coordinates U (t) given in (3.7), with t = T (p − p * ).
Clearly Υ p is one-to-one on B. Equip B = Υ p (B) with the topology that makes Υ p a homeomorphism. Then B is a Banach manifold with a single chart Υ −1 On the open neighborhood B 0 = Υ p (B 0 ) of P * = Υ p (p * ) we can define M ′ = Υ p M ′ Υ −1 p . By Theorem 7.2 the operators M and M ′ are differentiable on B 0 , and in fact real analytic. The assertion in Theorem 1.2 concerning the spectrum of DM ′ (P * ) follows from Corollary 8.3.
Next we consider map-pairs that commute near the fixed pointω of A = Λ 3 * F −1 * . Consider the associated point (x,ȳ) and the set C described in Lemma 5.3. Pick ϵ > 0 such that the closure of the square R = {(x, y) ∈ R 2 : |x −x| < ϵ , |y −ȳ| < ϵ} is included in the domain of both f ⊙g and g⊙f , for every pair p ∈ B. This is possible if the radius of B has been chosen sufficiently small. By decreasing ϵ > 0, if necessary, the commutators C(p) are uniformly bounded on R. To prove the last statement in Theorem 1.2, denote by W s and W u the local stable and unstable manifolds, respectively, for the operator M ′ at p * . Consider first the case where M is not expanding on B c . That is, there exists an open ball Given that M agrees with M ′ on B c ∩ B 0 , and that M ′ is hyperbolic, this implies that Next, consider the case where M is expanding on B c . Then there exists N > 0 such that the following holds. For every n ≥ N there exists a sequence of pairs p 1 , p 2 , . . . ∈ B c converging to p * , such that the orbit of p k contains a point p ′ k at a distance between 2 −n−1 and 2 −n from p * . So some subsequence of k → p ′ k converges to a point p ′′ n ∈ W u , and 2 −n−1 ≤ ∥p ′′ n − p * ∥ ≤ 2 −n . Since B c is closed in B, we have p ′′ n ∈ B c . Now, given any ξ ∈ R, consider the function h : W u → R defined by h(p) = C(p)(ξ). We have h(p ′′ n ) = 0 for all n ≥ N . Since W s is real analytic and p ′′ n → p * , it follows that h = 0. Given that ξ ∈ R was arbitrary, we conclude that W u ⊂ B c .
It seems surprising that we cannot exclude the possibility that p * is an attracting fixed point for the restriction of M to B c . The underlying problem is that we know very little about how the set B c embeds into the space B p . The most useful pieces of information are probably the scaling relation (2.5) for the commutator Θ, and the corresponding scaling of the Calabi invariant (see Subsection 3.1). One way to take advantage of these scaling properties would be the following.
Assume that the eigenvectorp of DM ′ (p * ) for the eigenvalue ϑ = δ 3 belongs to the subspace B c . Thenp is also an eigenvector of DM(p * ), with eigenvalue ϑ. We claim that this implies that the local unstable manifold of M is included in B c . Here is just a sketch of a proof: Since no power ϑ n with n ≥ 2 is an eigenvalue of DM(p * ), there exists a real analytic M-invariant curve t → p(t) such that p(0) = p * and p ′ (0) =p, and M(p(ϑ −1 t)) = p(t) for t near zero. Now consider the Taylor series of (t, ω) → P (t)(ω) about the point (0,ω). Using the above-mentioned scaling properties, together with our bounds on α, β, and δ, one finds that all pairs P (t) for t near zero are necessarily commuting and have a zero Calabi invariant.
Within the framework developed in this paper, the "cleanest" way to prove the above assumption would be to replace the approximate eigenfunctions E 8 , E 9 , . . . , E 13 in our definition (3.12) of M ′ by the true eigenfunctions. If Corollary 8.3 remains true for this operator M ′ , then we havep ∈ B c , as is easy to see.
In practice, we cannot determine these eigenfunctions exactly. But it suffices to have good bounds (enclosures), as is the case for the function g * and related quantities. Obtaining such bounds is a purely technical problem. The remaining part of our analysis should not change. Our only reason for working with approximate eigenfunctions was to avoid adding yet another level of complexity to the problem.

Reduction to simpler estimates.
Here we give some estimates that are used in our programs. They concern mostly simple operations like the differentiation and composition of functions, and the norm and inverses of certain operators. We also describe why the remaining problem reduces to such estimates.

Derivatives.
Derivatives of functions in B ϱ reduce via (4.7) to derivatives of functions in A ρ . For these derivatives we use the following estimate from [1]. Given positive real numbers σ < τ , and a non-negative integer k, define In the special case B = C we get the estimate |f (x, y)| ≤ ∥f ∥ ϱ , for all (x, y) ∈ D ϱ . The proof of this proposition is a simple exercise in power series. In particular, from the representation (4.6) one gets (9.2) and the last sum is bounded by ∥ϕ∥ ρ with ρ = (ρ u , ρ v ), if ∥U ∥ ≤ ρ u and ∥V ∥ ≤ ρ v . The second inequality in (9.2) can be far from optimal if B is a function space and the norm of U (or V ) is significantly larger than its sup-norm. In our program we deal with this problem as follows. For k > 1 define if U is nonzero. Notice that lim k ∥U ∥ k is the spectral radius of the operator "multiplication by U ", which in turn is the sup-norm of U . Let us now fix k > 1. Writing a nonnegative integer m as m = qk + r with 0 ≤ r < k, we have Similarly we have ∥V n ∥ ≤ C V ∥V ∥ n k . These are the bounds that we use (for large m or n) to estimate the first sum in (9.2). Notice also that ϕ(U, V ) ∈ B whenever ϕ ∈ A ρ and ∥U ∥ k ≤ ρ u and ∥V ∥ k ≤ ρ v .
for all x ∈ B. Then there exists a unique x ∈ B such that F (x) = 0.
The proof is a straightforward: apply the contraction mapping theorem to the map x → x − M 0 F (x). where c m,n is defined in such a way that e m,n has norm one. Linear equations Lx = y are considered in the context of Proposition 9.3. The linear operator L : X → Y is assumed to be bounded, and M 0 : Y → X is an approximate inverse of L. Applying Proposition 9.3 to the function F defined by F (x) = Lx − y, we obtain the Colollary 9.4. Assume that M 0 is one-to-one, and that Then there exists a unique x ∈ X such that ∥x − x 0 ∥ ≤ r and Lx = y.
This corollary is also used (with y = I and x 0 = M 0 ) to estimate the inverse of an operator L. Similarly, we estimate the reciprocal of a function f by inverting the multiplication operator L : h → hf . In this case, or whenever M 0 commutes with L, the one-to-one condition on M 0 can be dropped.

Contracting iterates.
The following proposition, with a suitable choice of parameters, is used to prove the claim in Lemma 8.1 concerning the contraction property of the map F 6 defined in (8.5).
Let B be any Banach space, and let m ≥ 1.
Assume that for n = 1, 2, . . . , m, for all p ∈ B in the domain of F n . Then F n maps B into B ′ , for all n ≤ m.
The proof of this proposition is straightforward and thus will be omitted. We note that r ′ ≫ r in our application of this proposition, since our map F is far from being a contraction (for the given norm).
9.6. Analyticity. The analyticity properties mentioned in Lemma 6.1 follow "by construction". In order to explain why, let us recall some facts about analytic maps.
Let X and Y be Banach spaces over C, and let B ⊂ X be open. A map F : B → Y is said to be analytic if it is Fréchet differentiable. Thus, sums, products, and compositions of analytic maps are analytic. Equivalently, F is analytic if it is locally bounded, and if ψ • F • ϕ is analytic for arbitrary continuous linear maps ϕ : C → X and ψ : Y → C. This shows in particular that uniform limits of analytic functions are analytic. These and other useful facts can be found e.g. in [7].
So the reason why e.g. the solution V 0 of the midpoint equation K ′ (V 0 ) = 0 depends analytically on the functions f and g is that K ′ (V 0 ) depends analytically (in fact linearly) on f and g, and that the solution is obtained via an iteration V → V − M 0 K ′ (V) that converges uniformly. The same applies to all of our implicit equations: We always use the contraction mapping theorem, which yields a solution via uniform limits. 9.7. Remaining tasks. In Sections 4-8 we have reduced the proofs of Theorems 1.1 and 1.2 to the task of proving Theorem 4.2 and several technical lemmas. Among these lemmas, only Lemmas 6.1, 8.1, and 8.2 are sufficiently involved to require arguments that go beyond straightforward "computations". What we have done in the preceding subsections was to reduce the proof of these three lemmas to computations as well.
The objects in these computations are elements in one of our Banach spaces, such as R, A ρ or B ϱ . By "computing" an element x in some space X we mean finding a set X ⊂ X that contains x. The types of enclosures X that we use, and the procedures for finding them, will be described in the next section.
One task that is potentially infinite is the computation of an operator norm. However, the operators L whose norms we have to estimate are compact. In fact, they owe their compactness to the scaling f → f • λ 3 . So in the notation of (9.5), the norm ∥Le m,n ∥ ρ tends to zero as m + n increases. In this case, it suffices to compute Le m,n for m + n ≤ N , plus the image LE of a set E that encloses all functions e m,n with m + n > N . To get an accurate bound on ∥L∥, if desired, it suffices to choose N in such a way that ∥LE∥ ρ ≤ ∥Le m,n ∥ ρ for some pair (m, n) with m + n ≤ N .
Concerning Theorem 4.2, we recall that a similar result has already been proved in reference [1], so we refer to [1,23] for details.

Organization of the programs.
In this section we describe how the remaining proofs are organized. The precised definitions, and all other details of the proof, can be found in the source code of our programs [23], written in the programming language Ada [19]. Similar techniques have been used in other computer-assisted proofs, including [3,17,1,5], to list just a few on area-preserving maps.
10.1. Basic strategy. The remaining task is compute the quantities that appear in any of the yet-unproved lemmas, and to verify that they have the desired properties. It is important to notice that the original claims all have been reduced to inequalities. At this point, the main objects besides numbers are functions in the spaces A ρ or B ϱ , and no operation involved is more complex than the composition of such functions.
The first step is to implement procedures that allows us to work at the level of these main objects. In particular, if F and G represent enclosures for two functions F ∈ B ϱ and G ∈ B ϱ , respectively, then we want F*G to yield a rigorous enclosure for the product F G. Once such basic operations are implemented, we can use the propositions from Section 9 to do the same for more complex operations, such as quotients F/G or composed maps Comp(Phi,U,V) or partial derivatives DerX(F) etc. Then it becomes possible to verify a claim like ∥M 0 K ′ (V)∥ < ε by executing a command like Norm(M0*KPrime(V))<Eps and checking that the result is True.
Our choice of enclosures will be described in detail below. This sets the framework for implementing bounds like Comp or Norm. The actual implementation is best (most efficiently and accurately) described by the source code of our programs [23]. As a rough guide to this code we will mention the program units (Ada packages) where the entities being discussed are defined.

Enclosures.
To every space X considered we associate a finite collection E(X ) of subsets of X that are representable on the computer. For a product space X = X 1 × . . . × X n we choose E(X ) to be the collection of all sets X 1 × . . . × X n with X j ∈ E(X j ) for each j, unless specified otherwise. An "enclosure" for an element x ∈ X is a set X ∈ E(X ) that contains x. A "bound" on a map f : X → Y is a map F : E(X ) → E(Y) ∪ {undefined}, with the property that f (x) ∈ F (X) whenever x ∈ X ∈ E(X ), unless F (X) = undefined. In practice, if F (X) = undefined then the program halts with an error message.
Each collection E(X ) corresponds to a data type in our programs. A simple type of enclosure is associated with a data type Ball2, which consists of all triples S=(S.C,S.R,S.B), where S.C is a representable real number (type Rep), and where S.R and S.B are nonnegative representable real numbers (type Radius). If X is any Banach algebra over F = R or F = C with unit 1, we include in E(X ) all sets S ♭ = (S.C + S.R U F )1 + S.B U X where S can be any Ball2. Here U F and U X denote the closed unit balls in F and X , respectively.
For the type Radius we use a standard Ada type Long Long Float with 64 mantissa bits (with the Gnat compiler [21]). The type Rep is either Long Long Float as well, or an MPFR floating point type [22] with up to 320 mantissa bits, depending on the program. Both types support controlled rounding [20] for the basic arithmetic operations. It is straightforward to implement bounds on the corresponding operations in a Banach algebra X , using enclosures S ♭ with S of type Ball2. For details we refer to the packages Flts.Std.Balls2 and MPFR.Floats.Balls2 in [23]. In the cases X = R and X = C, these are our only sets in E(X ). For the spaces X = A ρ and X = B ϱ we now define additional sets and bounds.
Our collection E(A ρ ) also contains sets T ♭ associated with pairs T=(T.R,T.C) from a data type Taylor2. The component T.R is ρ = (ρ u , ρ v ), which we always take to be a pair of type Radius. To be more specific about the component T.C, let us first describe the representation used in [1]. There, T.C is a two-dimensional array with Next consider a space B ϱ with ϱ = (t 0 , s 0 , b, ρ). For E(B ϱ ) we use a data type Fun, whose members are quadruplets F=(F.A,F.E,F.P,F.Q), where F.A specifies the parameters (t 0 , s 0 , b) and F.E is of type Boolean; while F.P and F.Q are of type Taylor2, with F.P.R and F.Q.R equal to ρ. If F.E=True then F ♭ consists of all functions (4.7) with ϕ ∈ F.P ♭ and ψ ∈ F.Q ♭ . The type Fun with F.E=False exists only for reasons of efficiency. For a discussion of this type we refer to [1]. Fun-based bounds on operations involving functions in the spaces B ϱ are defined in the package Funs.
Bounds on solutions of implicit equations use Proposition 9.3 via one of the generic packages Newton or Zeros. For basic linear algebra we use generic packages Matrices and Vectors etc. The type FunVec is a Vector with components of type Fun. It is used e.g. to store powers (type We should add that the packages Taylors2 and Funs are based on a generic type Scalar, and the above description applies only when these packages are instantiated with Scalar => Ball2.

Main procedures and programs.
The procedures that are directly related to renormalization are defined in the package RG and its children: RG.Fix (used to solve the fixed point problem for N ) and RG.Maps (used to analyze map-domains) and RG.Spec (used for the analysis of M ′ ). Our main programs do little else than invoking a few procedures from these packages.
RG is again based on a generic type Scalar. It uses two instantiations of Funs side by side: the main version Funs(Scalar => Scalar) named SFuns, and a "numeric" version Funs(Scalar => Reps) named NFuns. The latter is used whenever an estimate needs an approximate solution, which is the case for all of our implicit equations. But approximate solutions that are needed repeatedly can be (and have been) saved to file and are read from file if available.
Among the data files in [23] are approximate fixed point for the maps N and F, approximate midpoints functions, approximate inverses for the operators DK ′ (V 0 ) and DK ′′ (V 1 , V 2 ), and some approximate eigenvectors for DM * (p * ). The labeling of eigenvalues and eigenvectors that is used in our programs is the one given in Tables 1, 2. The eigenvalue δ 3 and eigenvectorp are labeled 03.
The proof of Theorem 4.2 uses a quasi-Newton map for the operator N near the approximate fixed point. As in [1], the main work is carried out by the procedures RG.Fix.Renorm and RG.Fix.DContractNorm. The fixed point g * is shown to lie in a ball of radius less that 10 −89 , centered at at a polynomial approximation of degree 200. We note that the enclosure for g * is taken slightly larger than necessary. As a by-product, our enclosure for λ * is wider than necessary, and this ensures e.g. the existence of the ε > 0 in Lemma 7.1.
Lemma 5.2 is proved by running the program CheckMaps, which calls procedures from the package RG.Maps. The same program is also used to verify the inequalities given in Theorem 4.2. The claims in Lemma 5.3 concerning the map A are verified by the program AllEigen3. The proof of Lemma 6.1 uses the procedures MidPointVf and MidPointVg from RG.Spec. They are called from the program WriteRG3Data, which also carries out the domain checks for Lemma 8.4. Lemma 8.1 is proved in three steps. Using an approximate fixed point p 0 for F and fixed constants K n defined by RGParam.XC3, the procedure RG.Spec.Contr3Fix determines suitable values of the remaining constants in Proposition 9.5 and verifies all inequalities in this proposition, except for the bounds ∥DF n (p)∥ ≤ K n . An enclosure for DF(p) for arbitrary p ∈ B ′ is determined in RG.Spec.XContr3Mat, and the necessary bounds on ∥DF n (p)∥ for n = 1, 2, . . . , 6 are computed by the program XMatNorm. The same program is used also to prove Lemma 8.2.
We note that controlling the derivative of M ′ is much more delicate than controlling the map itself, since DM ′ (p * )p is not stationary with respect to variations of the midpoint functions V j . Getting good bounds on these functions was rather challenging. Besides high degrees, large mantissas, and other technicalities [23], it required a careful choice of domains.
What our programs effectively do is to reduce the original problem to a finite sequence of trivial numerical computations (with specified rounding). These computations have been carried out successfully by a standard desktop computer. We used the Gnat compiler [21] to generate the executable code and to link with the MPFR library [22]. On a current desktop computer with four 3.4GHz processors, the total running time of our programs is roughly 240 hours. The files in [23] include the source code of our programs, data files, log files, and a README file with instructions on how to compile and run the programs. Bounds on λ, µ, α, β, and δ that are more accurate than the ones given in this paper can be found in the log file alleigen3.log.