Spectral properties of renormalization for area-preserving maps

Area-preserving maps have been observed to undergo a universal period-doubling cascade, analogous to the famous Feigenbaum-Coullet-Tresser period doubling cascade in one-dimensional dynamics. A renormalization approach has been used by Eckmann, Koch and Wittwer in a computer-assisted proof of existence of a conservative renormalization fixed point. Furthermore, it has been shown by Gaidashev, Johnson and Martens that infinitely renormalizable maps in a neighborhood of this fixed point admit invariant Cantor sets with vanishing Lyapunov exponents on which dynamics for any two maps is smoothly conjugate. This rigidity is a consequence of an interplay between the decay of geometry and the convergence rate of renormalization towards the fixed point. In this paper we prove a result which is crucial for a demonstration of rigidity: that an upper bound on this convergence rate of renormalizations of infinitely renormalizable maps is sufficiently small.


Introduction
Following the pioneering discovery of the Feigenbaum-Coullet-Tresser period doubling universality in unimodal maps (Feigenbaum 1978), (Feigenbaum 1979), (Tresser and Coullet 1978), universality -independence of the quantifiers of the geometry of orbits and bifurcation cascades in families of maps of the choice of a particular family -has been demonstrated to be a rather generic phenomenon in dynamics.
Universality problems are typically approached via renormalization. In a renormalization setting one introduces a renormalization operator on a functional space, Date: 2014-12-01. and demonstrates that this operator has a hyperbolic fixed point. This approach has been very successful in one-dimensional dynamics, and has led to explanation of universality in unimodal maps (Epstein 1989), (Lyubich 1999), (Martens 1999), critical circle maps (de Faria 1992, de Faria 1999, Yampolsky 2002, Yampolsky 2003 and holomorphic maps with a Siegel disk (McMullen 1998, Yampolsky 2007, Gaidashev and Yampolsky 2007. There is, however, at present no complete understanding of universality in conservative systems, other than in the case of the universality for systems "near integrability" (Abad et al 2000, Abad et al 1998, Koch 2002, Koch 2004, Koch 2008, Gaidashev 2005, Kocić 2005, Khanin et al 2007. Period-doubling renormalization for two-dimensional maps has been extensively studied in (Collet et al 1980, de Carvalho et al 2005. Specifically, the authors of ( de Carvalho et al 2005) have considered strongly dissipative Hénon-like maps of the form where f (x) is a unimodal map (subject to some regularity conditions), and is small. Whenever the one-dimensional map f is renormalizable, one can define a renormalization of F , following (de Carvalho et al 2005), as where U is an appropriate neighborhood of the critical value v = (f (0), 0), and H is an explicit non-linear change of coordinates. (de Carvalho et al 2005) demonstrates that the degenerate map F * (x, y) = (f * (x), x), where f * is the Feigenbaum-Collet-Tresser fixed point of one-dimensional renormalization, is a hyperbolic fixed point of R dCLM . Furthermore, according to (de Carvalho et al 2005), for any infinitelyrenormalizable map of the form (1), there exists a hierarchical family of "pieces" {B n σ }, organized by inclusion in a dyadic tree, such that the set C F = n σ B n σ is an attracting Cantor set on which F acts as an adding machine. Compared to the Feigenbaum-Collet-Tresser one-dimensional renormalization, the new striking feature of the two dimensional renormalization for highly dissipative maps (1), is that the restriction of the dynamics to this Cantor set is not rigid. Indeed, if the average Jacobians of F and G are different, for example, b F < b G , then the conjugacy F | C F ≈ h G| C G is not smooth, rather it is at best a Hölder continuous function with a definite upper bound on the Hölder exponent: The theory has been also generalized to other combinatorial types in (Hazard 2011), and also to three dimensional dissipative Hénon-like maps in (Nam 2011).
Finally, the authors of (de Carvalho et al 2005) show that the geometry of these Cantor sets is rather particular: the Cantor sets have universal bounded geometry in "most" places, however there are places in the Cantor set were the geometry is unbounded. Rigidity and universality as we know from one-dimensional dynamics has a probabilistic nature for strongly dissipative Hénon like maps. See  for a discussion of probabilistic universality and probabilistic rigidity.
It turns out that the period-doubling renormalization for area-preserving maps is very different from the dissipative case.
A universal period-doubling cascade in families of area-preserving maps was observed by several authors in the early 80's (Derrida and Pomeau 1980, Helleman 1980, Benettin et al 1980, Bountis 1981, Collet et al 1981, Eckmann et al 1982. The existence of a hyperbolic fixed point for the period-doubling renormalization operator is an F -dependent linear change of coordinates, has been proved with computer-assistance in (Eckmann et al 1984).
We have proved in (Gaidashev and Johnson 2009b) that infinitely renormalizable maps in a neighborhood of the fixed point of (Eckmann et al 1984) admit a "stable" Cantor set, that is the set on which the Lyapunov exponents are zero. We have also shown in the same publication that the conjugacy of stable dynamics is at least bi-Lipschitz on a submanifold of locally infinitely renormalizable maps of a finite codimension. Furthermore, (Gaidashev et al 2013) improves this conclusion in the following way.
Rigidity for Area-preserving Maps. The period doubling Cantor sets of areapreserving maps in the universality class of the Eckmann-Koch-Wittwer renormalization fixed point are smoothly conjugate.
A crucial ingredient of the proof in (Gaidashev et al 2013) is a new tight bound on the spectral radius of the renormalization operator. The goal of the present paper is to prove this new bound.
We demonstrate that the spectral radius of the action of DR EKW , evaluated at the Eckmann-Koch-Wittwer fixed point F EKW , restricted to the tangent space T F EKW W of the stable manifold W of the infinitely renormalizable maps, is equal exactly to the absolute value of the " horizontal" scaling parameter Furthermore, we show that the single eigenvalue λ F EKW in the spectrum of DR EKW [F EKW ] corresponds to an eigenvector, generated by a very specific coordinate change. To eliminate this irrelevant eigenvalue from the renormalization spectrum, we introduce an F -dependent nonlinear coordinate change S F into the period-doubling renormalization scheme the spectral radius of the restriction of the spectrum of DR c [F * ] to its stable subspace T F * W at the fixed point F * of R c , and obtain the following spectral bound, which is of crucial importance to our proof of rigidity.

Acknowledgment
This work was started during a visit by the authors to the Institut Mittag-Lefler (Djursholm, Sweden) as part of the research program on "Dynamics and PDEs". The hospitality of the institute is gratefully acknowledged. The second author was funded by a postdoctoral fellowship from the Institut Mittag-Lefler, he is currently funded by a postdoctoral fellowship from Vetenskapsrådet (the Swedish Research Council).

Renormalization for area-preserving reversible twist maps
An "area-preserving map" will mean an exact symplectic diffeomorphism of a subset of R 2 onto its image.
Recall, that an area-preserving map that satisfies the twist condition everywhere in its domain of definition can be uniquely specified by a generating function S: Furthermore, we will assume that F is reversible, that is For such maps it follows from (2) that It is this "little" s that will be referred to below as "the generating function". If the equation −s(y, x) = u has a unique differentiable solution y = y(x, u), then the derivative of such a map F is given by the following formula: The period-doubling phenomenon can be illustrated with the area-preserving Hénon family (cf. (Bountis 1981)) : Maps H a have a fixed point ((−1 + √ 1 + a)/a, (−1 + √ 1 + a)/a) which is stable (elliptic) for −1 < a < 3. When a 1 = 3 this fixed point becomes hyperbolic: the eigenvalues of the linearization of the map at the fixed point bifurcate through −1 and become real. At the same time a stable orbit of period two is "born" with H a (x ± , x ∓ ) = (x ∓ , x ± ), x ± = (1 ± √ a − 3)/a. This orbit, in turn, becomes hyperbolic at a 2 = 4, giving birth to a period 4 stable orbit. Generally, there exists a sequence of parameter values a k , at which the orbit of period 2 k−1 turns unstable, while at the same time a stable orbit of period 2 k is born. The parameter values a k accumulate on some a ∞ . The crucial observation is that the accumulation rate is universal for a large class of families, not necessarily Hénon. Furthermore, the 2 k periodic orbits scale asymptotically with two scaling parameters To explain how orbits scale with λ and µ we will follow (Bountis 1981). Consider an interval (a k , a k+1 ) of parameter values in a "typical" family F a . For any value α ∈ (a k , a k+1 ) the map F α possesses a stable periodic orbit of period 2 k . We fix some α k within the interval (a k , a k+1 ) in some consistent way; for instance, by requiring that DF 2 k α k at a point in the stable 2 k -periodic orbit is conjugate, via a diffeomorphism H k , to a rotation with some fixed rotation number r. Let p k be some unstable periodic point in the 2 k−1 -periodic orbit, and let p k be the further of the two stable 2 k -periodic points that bifurcated from p k . Denote with d k = |p k − p k |, the distance between p k and p k . The new elliptic point p k is surrounded by (infinitesimal) invariant ellipses; let c k be the distance between p k and p k in t he direction of the minor semi-axis of an invariant ellipse surrounding p k , see Figure 1. Then, where ρ k is the ratio of the smaller and larger eigenvalues of DH k (p k ). This universality can be explained rigorously if one shows that the renormalization operator has a fixed point, and the derivative of this operator is hyperbolic at this fixed point.
It has been argued in (Collet et al 1981) that Λ F is a diagonal linear transformation. Furthermore, such Λ F has been used in (Eckmann et al 1982) and (Eckmann et al 1984) in a computer assisted proof of existence of a reversible renormalization fixed point F EKW and hyperbolicity of the operator R EKW .
We will now derive an equation for the generating function of the renormalized .
If the solution of (10) is unique, then z(x, y) = z(y, x), and it follows from (9) that the generating function of the renormalized F is given by (11)s(x, y) = µ −1 s(z(x, y), λy).
As we have already mentioned, the following has been proved with the help of a computer in (Eckmann et al 1982) and (Eckmann et al 1984): Theorem 1. There exist a polynomial s 0.5 ∈ A 0.5 s (ρ) and a ball B (s 0.5 ) ⊂ A 0.5 s (ρ), = 6.0 × 10 −7 , ρ = 1.6, such that the operator R EKW is well-defined and analytic on B (s 0.5 ).
Furthermore, its derivative DR EKW | B (s0.5) is a compact linear operator, and has exactly two eigenvalues δ 1 = 8.721..., and δ 2 = 1 λ * of modulus larger than 1, while Finally, there is an s EKW ∈ B (s 0.5 ) such that The scalings λ * and µ * corresponding to the fixed point s EKW satisfy Remark 1.3. The bound (16) is not sharp. In fact, a bound on the largest eigenvalue of DR EKW (s EKW ), restricted to the tangent space of the stable manifold, is expected to be quite smaller.
The size of the neighborhood in A β s (ρ) where the operator R EKW is well-defined, analytic and compact has been improved in (Gaidashev 2010). Here, we will cite a somewhat different version of the result of (Gaidashev 2010) which suits the present discussion (in particular, in the Theorem below some parameter, like ρ in A β s (ρ), are different from those used in (Gaidashev 2010)). We would like to emphasize that all parameters and bounds used and reported in the Theorem below, and, indeed, throughout the paper, are numbers representable on the computer.

Theorem 2.
There exists a polynomial s 0 ∈ A(ρ), ρ = 1.75, such that the following holds. i) The operator R EKW is well-defined and analytic in B R (s 0 ) ⊂ A(ρ) with ii) For all s ∈ B R (s 0 ) with real Taylor coefficients, the scalings λ = λ[s] and µ = µ[s] satisfy Definition 1.4. The set of reversible twist maps F of the form (4) with s ∈ B (s) ⊂ A β s (ρ) will be referred to as F β,ρ (s): . We will also use the notation We will finish our introduction into period-doubling for area-preserving maps with a summary of properties of the fixed point map. In (Gaidashev and Johnson 2009a) we have described the domain of analyticity of maps in some neighborhood of the fixed point. Additional properties of the domain are studied in (Johnson 2011). Before we state the results of (Gaidashev and Johnson 2009a), we will fix a notation for spaces of functions analytic on a subset of C 2 .
Definition 1.5. Denote O 2 (D) the Banach space of maps F : D → C 2 , analytic on an open simply connected set D ⊂ C 2 , continuous on ∂D, equipped with a finite max supremum norm · D : The Banach space of functions y : A → C, analytic on an open simply connected set A ⊂ C 2 , continuous on ∂A, equipped with a finite supremum norm · A will be denoted O 1 (A): If D is a bidisk D ρ ⊂ C 2 for some ρ, then we use the notation The next Theorem describes the analyticity domains for maps in a neighborhood of the Eckmann-Koch-Wittwer fixed point map, and those for functions in a neighborhood of the Eckmann-Koch-Wittwer fixed point generating function. The Theorem has been proved in two different versions: one for the space A 0.5 s (1.6) (the functional space in the original paper (Eckmann et al 1984)), the other for the space A s (1.75) -the space in which we will obtain a bound on the renormalization spectral radius in the stable manifold in this paper. To state the Theorem in a compact form, we introduce the following notation: ρ 0.5 = 1.6, ρ 0 = 1.75, 0.5 = 6.0 × 10 −7 , 0 = 5.79833984375 × 10 −4 , while s 0.5 (as in Theorem 1) and s 0 will denoted the approximate renormalization fixed points in spaces A 0.5 s (1.6) and A s (1.75), respectively. Theorem 3. There exists a polynomial s β such that the following holds for all ii) There exist simply connected open setsD =D(β, β , ρ β ) ⊂ D, such thatD ∩ R 2 is a non-empty simply connected open set, and such that for every (x, u) ∈D and Remark 1.6. It is not too hard to see that the subsets F β,ρ β β (s β ), β = 0 or 0.5, are analytic Banach submanifolds of the spaces O 2 (D(β, β , ρ β ). Indeed, the map where y[s](x, u) is the solution of the equation (20), and h[s](x, u) = (x, y[s](x, u)), is analytic as a map from B β (s β ) to O 2 (D(β, β , ρ β ) according to Theorem 3, and has an analytic inverse where g[F ](x, y) = (x, U (x, y)), and U is as in Theorem 3.
We are now ready to give a definition of the Eckmann-Koch-Wittwer renormalization operator for maps of the subset of a plane. Notice, that the condition P EKW [s](λ, 0) = 0 from Definition 1.1 is equivalent to F (F (λ, −s(z(λ, 0), λ))) = (0, 0), or, using the reversibility λ = π x F (F (0, 0)). On the other hand, Definition 1.7. We will refer to the composition F • F as the prerenormalization of F , whenever this composition is defined: Remark 1.8. Suppose that for some choice of β, β and ρ β , the operator R EKW and the map I, described in Remark 1.6, are well-defined on some B β (s β ) ⊂ A β s (ρ β ). Also, suppose that the inverse of I exists on I(B β (s β )). Then,

Statement of main results
Consider the coordinate transformation for t ∈ C, |t| < 4/(ρ + |β|) (recall Definition 1.2). We will now introduce two renormalization operators, one -on the generating functions, and one -on the maps, which incorporates the coordinate change S t as an additional coordinate transformation.
with G is as in (14), and where λ and µ solve the following equations:

SPECTRAL PROPERTIES OF RENORMALIZATION FOR AREA-PRESERVING MAPS 11
Definition 2.2. Given c ∈ R, set, formally, We are now ready to state our main theorem. Below, and through the paper, s (i,j) stands for the (i, j)-th component of a Taylor series expansion of an analytic function of two variables.
Main Theorem. (Existence and Spectral properties) There exists a polynomial iii) The linear operator DR c0 [s * ] has two eigenvalues outside of the unit circle: iv) The complement of these two eigenvalues in the spectrum is compactly contained in the unit disk: The Main Theorem implies that there exist codimension 2 local stable manifolds W Rc 0 (s * ) ⊂ A s (1.75), such that the contraction rate in W Rc 0 (s * ) is bounded from above by ν: i) The set of reversible twist maps of the form (4) such that s ∈ W Rc 0 (s * ) ⊂ A s (1.75) will be denoted W , and referred to as infinitely renormalizable maps.
Naturally, these sets are invariant under renormalization if is sufficiently small.
Notice, that, among other things, this Theorem restates the result about existence of the Eckmann-Koch-Wittwer fixed point and renormalization hyperbolicity of Theorem 1 in a setting of a different functional space. We do not prove that the fixed point s * , after an small adjustment corresponding to the coordinate change S t , coincides with s EKW from Theorem 1, although the computer bounds on these two fixed points differ by a tiny amount on any bi-disk contained in the intersection of their domains.
The fact that the operator R c0 as in (26) contains an additional coordinate change does not cause a problem: conceptually, period-doubling renormalization of a map is its second iterate conjugated by a coordinate change, which does not have to be necessarily linear.

Coordinate changes and renormalization eigenvalues
Let D andD be as in the Theorem 3. Consider the action of the operator , with λ * and µ * being the fixed scaling parameters corresponding to the Collet-Eckmann-Koch as in Theorem 1.
According to Theorem 1 this operator is analytic and compact on the subset F 0.5,1.6 (s 0.5 ), = 6.0 × 10 −7 , of O 2 (D), and has a fixed point F EKW . In this paper, we will prove the existence of a fixed point s * of the operator R EKW in a Banach space different from that in Theorem 1. Therefore, we will state most of our results concerning the spectra of renormalization operators for general spaces A β s (ρ) and sets F β,ρ β β (s * ), under the hypotheses of existence of a fixed point s * , and analyticity and compactness of the operators in some neighborhood of the fixed point. Later, a specific choice of parameters β, ρ and will be made, and the hypotheses -verified.
Let S = id + σ be a coordinate transformation of the domain D of maps F , satisfying DS • F = DS. In particular, these transformations preserve the subset of area-preserving maps. Notice, that Suppose that the operator R * has a fixed point F * in some neighborhood B ⊂ O 2 (D), on which R * is analytic and compact. Consider the action DR * [F ]h F,σ of the derivative of this operator.
and clearly, h F * ,σ is an eigenvector, if τ = κσ, of eigenvalue κ. In particular, is an eigenvalue of multiplicity (at least) 2 with eigenvectors h F * ,σ generated by respectively. Next, suppose S σ t , S σ 0 = Id, is a transformation of coordinates generated by a function σ as in (29)-(30), associated with an eigenvalue κ of DR * [F * ]. In addition to the operator (27), consider where the parameter t σ [F ] is chosen as E(κ) being the Riesz spectral projection associated with κ: (γ -a Jordan contour that enclose only κ in the spectrum of DR * [F * ]). We will now compare the spectra of the operators R * and R σ . The result below should be interpreted as follows: if h F * ,σ is an eigenvector of DR * [F * ] generated by a coordinate change id + σ, and associated with some eigenvalue κ, then this eigenvalue is eliminated from the spectrum of DR σ [F * ], if its multiplicity is 1. Moreover, if the multiplicity of κ is 1, then Proof. Since DR σ [F * ] and DR * [F * ] are both compact operators acting on an infinite-dimensional space, their spectra contain {0}.
Suppose h is a eigenvector of DR * [F * ] corresponding to some eigenvalue δ, then (we have used the fact that F * satisfies the fixed point equation), where More specifically, Vice verse, suppose h is an eigenvector of DR σ [F * ] corresponding to an eigenvalue δ = κ, then, and by (33) and a similar computation as above, for a ∈ R, Lemma 3.2. Suppose that there are β, , ρ, λ * , µ * and a function s * ∈ A β s (ρ) such that the operator R EKW is analytic and compact as maps from F β,ρ (s * ) to O 2 (D), and where F * is generated by s * .
The 6-th line reduces to after we use the midpoint equation differentiated with respect to x: To summarize, Finally, we use the fact that The Lemma below, whose elementary proof we will omit, shows that λ * is also in the spectrum of DR * [F * ]: Lemma 4.2. Suppose that β, and ρ are such that s * ∈ A β s (ρ) is a fixed point of R * , and the operator R * is analytic and compact as a map from B (s * ) to A β s (ρ). Also, suppose that the map I, described in Remark 1.6, is well-defined and analytic on B (s * ), and that it has an analytic inverse I −1 on I(B (s * )). Then, At the same time, it is straightforward to see that the spectra of and DR EKW [s EKW ] are identical.
Lemma 4.3. Suppose that β, and ρ are such that s * ∈ A β s (ρ), and the operator R EKW is analytic and compact as a map from B (s * ) to A β s (ρ). Also, suppose that the map I, described in Remark 1.6, is well-defined and analytic on B (s * ), and that it has an analytic inverse I −1 on I(B (s * )). Then, The convergence rate in the stable manifold of the renormalization operator plays a crucial role in demonstrating rigidity. It turns out that the eigenvalue λ * is the largest eigenvalues in the stable subspace of DR EKW [F * ], or equivalently DR EKW [s * ]. However, it's value |λ * | ≈ 0.2488 is not small enough to ensure rigidity. At the same time, the eigenspace of the eigenvalue λ * is, in the terminology of the renormalization theory, irrelevant to dynamics (the associated eigenvector is generated by a coordinate transformation). We, therefore, would like to eliminate this eigenvalue via an appropriate coordinate change, as described above.
However, first we would like to identify the eigenvector corresponding to the eigenvalue λ * for the operator R EKW . This vector turns out to be different from ψ s * .
Lemma 4.4. Suppose that β, and ρ are such that the operator R EKW has a fixed point s * ∈ A β s (ρ), and R EKW is analytic and compact as a map from B (s * ) to A β s (ρ). Also, suppose that the map I, described in Remark 1.6, is well-defined and analytic on B (s * ), and that it has an analytic inverse I −1 on I(B (s * )).
Then, the number λ * is an eigenvalue of DR EKW [s * ], and the eigenspace of λ * contains the eigenvector Proof. Notice, thatψ is of the form where ψ x (x, y) = s * 1 (x, y)x + s * 2 (x, y)y is the eigenvector of DR * [s * ] corresponding to the rescaling of the variables x and y, while is the eigenvector corresponding to the rescaling of s. ψ x (x, y) and ψ u (x, y) correspond to the eigenvectors h F * ,σ 1 0,0 and h F * ,σ 2 0,0 , respectively, of DR 0 [F * ]. Recall, that h F * ,σ 1 0,0 and h F * ,σ 2 0,0 are eigenvectors of DR 0 [F * ], with eigenvalue 1, and eigenvectors of DR EKW [F * ] with eigenvalue 0.
By Lemma 4.1 ψ s * is an eigenvector of DR * , the corresponding eigenvector of DR * is h F * ,σ 1 1,0 −2σ 2 1,0 . Thus, ψ s * +ψ corresponds to the vector To finish the proof, it suffices to prove that The result follows if where 0 = s(x, Z(x, y)) + s(y, Z(x, y)), ψ EKW s * is as in (39), G as in (14), and E is the Riesz projection for the operator DR EKW [s * ].
We will quote a version of a lemma from (Gaidashev 2010) which we will require to demonstrate analyticity and compactness of the operator R. The proof of the Lemma is computer-assisted. Notice, the parameters that enter the Lemma are different from those used in (Gaidashev 2010). As before, the reported numbers are representable on a computer. and s 0 is as in Theorem 2, the prerenormalization P EKW [s] is well-defined and analytic function on the set D r ≡ D r (0) = {(x, y) ∈ C 2 : |x| < r, |y| < r}, r = 0.51853174082497335, with Z r ≤ 1.63160151494042404.
We will now demonstrate analyticity and compactness of the modified renormalization operator in a functional space, different from that used in (Eckmann et al 1984), specifically, in the space A s (1.75). It is in this space that we will eventually compute a bound on the spectral radius of the action of the modified renormalization operator on infinitely renormalizable maps.
Proposition 4.7. There exists a polynomial s 0 ⊂ B R (s 0 ) ⊂ A s (1.75), where R and s 0 are as in Lemma 4.6, such that the operator R is well-defined, analytic and compact as a map from B 0 (s 0 ), 0 = 5.79833984375 × 10 −4 , to A s (1.75), if B 0 (s 0 ) ⊂ B R (s 0 ) contains the fixed point s * .
Proof. The polynomial s 0 has been computed as a high order numerical approximation of a fixed point s * of R.
First, we get a bound on t for all s ∈ B δ (s 0 ): We estimate the right hand side rigorously on the computer and obtain (44) |t| ≤ 2.1095979213715 × 10 −6 .
The condition of the hypothesis that s * ∈ B δ (s 0 ) is specifically required to be able to compute this estimate.
Notice that according to Definition 4.5 and Theorem 2, the maps s → t and, hence, s → ξ t are analytic on a larger neighborhood B R (s 0 ) of analyticity of R EKW . According to Theorem 2 and Lemma 4.6, the prerenormalization P EKW is also analytic as a map from B R (s 0 ) to A s (r), r = 0.516235055482147608. We verify that for all s ∈ B δ (s 0 ) and t as in (44) the following holds: where λ − = −0.27569580078125 is the lower bound from Theorem 2. Furthermore, with t as in (44). Therefore, the map s → P[s] is analytic on B δ (s 0 ). Since the inclusion of sets (45) is compact, R[s] has an analytic extension to a neighborhood of D 1.75 , R[s] ∈ A s (ρ ), ρ > 1.75. Compactness of the map s → R[s] now follows from the fact that the inclusions of spaces A s (ρ ) ⊂ A s (ρ) is compact.
Recall, that according to Lemma 4.2, λ * is an eigenvalue of DR * [F * ] of multiplicity at least 1. According to Lemma 3.2, λ * is in the spectrum of DR EKW [F * ], and according to Lemma 4.3, λ * ∈ DR EKW [s * ]. Proof. First, notice the difference between the definition of λ in (1.1) s(G(λ, 0)) = 0, and in Definition (4.5) s(G(λ + tλ 2 , 0)) = 0 (we will use the notation λ EKW below to emphasize the difference). This implies that if D s λ EKW [s]ψ is an action of the derivative of λ EKW [s] on a vector ψ, then Similarly, where Similarly to Lemma (3.1), we get that if ψ is an eigenvector of DR EKW [s * ] associated with the eigenvalue δ = λ * , then ψ = ψ EKW s * , and Finally, assume that λ * / ∈ spec(DR[s * ]), but that there exists an eigenvector and, by (46), This contradiction finishes the proof. So far we were not able to make any claims about the multiplicity of the eigenvalue λ * in the spectrum of DR EKW [s * ]. However, we will demonstrate in Section 5 that it is indeed equal to 1.

Spectral properties of R. Proof of Main Theorem
We will now describe our computer-assisted proof of Main Theorem. To implement the operator DR[s * ] on the computer, we would have to implement the Riesz projection as well. Unfortunately, this is not easy, therefore, we do it only approximately, using the operator R c introduced in the Definition 2.1. Specifically, the component (0, 3) of the composition s • G will be consistently normalized to be where s 0 is our polynomial approximation for the fixed point.
The operator R c differs from R (cf.4.5) only in the "amount" by which the eigendirection ψ EKW s * is "eliminated". In particular, as the next proposition demonstrates, R c is still analytic and compact in the same neighborhood of s 0 . Furthermore, the operators R c are compact in B R (s 0 ) ⊂ A(ρ), with R c [s] ∈ A(ρ ), ρ = 1.0699996948242188ρ.
Proof. The proof is almost identical to that of Proposition 4.7, with a different (but still sufficiently small) bound on |t c [s]|.
The following Lemma shows that the spectra of the operators R and R c are close to each other. Proof. According to Propositions 4.7 and 5.1, under the hypothesis of the Lemma, R and R c * are analytic and compact as operators from B δ (s 0 ) to A s (1.75).
Recall, that ψ EKW s * is an eigenvector of DR EKW [s * ] corresponding to the eigenvalue λ * .
We consider the action of DR c * [s * ] on a vector ψ. Similarly to (46), and we see that the equation has a unique solution a if For such κ, the vector is an eigenvector of DR c * [s * ] associated with the eigenvalue κ.
The eigenvalues κ as in (48) satisfy |κ| > 0.00124359130859375 We will now describe a rigorous computer upper bound on the spectrum of the operator DR c [s * ].
Proof of part ii) of Main Theorem.
Step 1). Recall the Definition 1.2 of the Banach subspace A s (ρ) of A(ρ). We will now choose a new bases {ψ i,j } in A s (ρ). Given s ∈ A s (ρ) we write its Taylor expansion in the form s(x, y) = where ψ i,j ∈ A s (ρ): and the index set I of these basis vectors is defined as DenoteÃ s (ρ) the set of all sequences Equipped with the l 1 -norm A s (ρ) is a Banach space, which is isomorphic to A s (ρ). Clearly, the isomorphism J : A s (ρ) →Ã s (ρ) is an isometry: We divide the set I in three disjoint parts: with N = 22, M = 60. We will denote the cardinality of the first set as D(N ), the cardinality of I 1 ∪ I 2 as D(M ).
We assign a single index to vectors ψ i,j , (i, j) ∈ I 1 ∪ I 2 , as follows: This correspondence (i, j) → k is one-to-one, we will, therefore, also use the notation (i(k), j(k)).
For any s ∈ A s (ρ), we define the following projections on the subspaces of the where s 0 is some good numerical approximation of the fixed point. Denote for brevity L s c ≡ DR c [s]. We can now write a matrix representation of the finitedimensional linear operator Step 2). We compute the unit eigenvectors e k of the matrix D numerically, and form a D(N ) × D(N ) matrix A whose columns are the approximate eigenvectors e k . We would now like to find a rigorous bound B on the inverse B of A. Step 3 For any s ∈Â s (ρ), we define the following projections on the basis vectors.
We proceed to quantify this claim.
We will use the Contraction Mapping Principle in the following form. Define the following linear operator on A s (ρ) where Kh ≡δ 1 P 1 h +δ 2 P 2 h, andδ 1 andδ 2 are defined via P 1 L s0 c0 e 1 =δ 1 e 1 , P 2 L s0 c0 e 2 =δ 2 e 2 . Consider the operator We can now see that the hypothesis of the Contraction Mapping Principle is indeed verified: Step 5). Notice, that in general, ] is a small number which we have estimated to be (54) |t c0 [s * c0 ]| < 7.89560771750566329 × 10 −12 . Consider the map F * c0 generated by s * c0 . Recall that by Theorem 3, there exists a simply connected open set D such that F * c0 ∈ O 2 (D). The fixed point equation for the map F * c0 is as follows: