LOCAL STUDY OF A RENORMALIZATION OPERATOR FOR 1D MAPS UNDER QUASIPERIODIC FORCING

. The authors have recently introduced an extension of the classical one dimensional (doubling) renormalization operator to the case where the one dimensional map is forced quasiperiodically. In the classic case the dynamics around the ﬁxed point of the operator is key for understanding the bifurcations of one parameter families of one dimensional unimodal maps. Here we perform a similar study of the (linearised) dynamics around the ﬁxed point for further application to quasiperiodically forced unimodal maps. ,

Abstract. The authors have recently introduced an extension of the classical one dimensional (doubling) renormalization operator to the case where the one dimensional map is forced quasiperiodically. In the classic case the dynamics around the fixed point of the operator is key for understanding the bifurcations of one parameter families of one dimensional unimodal maps. Here we perform a similar study of the (linearised) dynamics around the fixed point for further application to quasiperiodically forced unimodal maps.

1.
Introduction. Given a one-dimensional dynamical system, one can consider a quasiperiodic forcing of its dynamics, given place to quasiperiodically forced one dimensional map of the form where T = R/Z, f ∈ C r (T × R, R) with r ≥ 1 and ω ∈ T \ Q. In other words, a skew map where the dynamics on one of its components is given by a solid rotation of angle ω. Therefore, a map of this form can be identified with a pair (ω, f ) with ω and f as before.
In [16] we showed the existence of some universality and self-renormalization properties in the parameter space of the Forced Logistic Map, although the phenomena of universality and self-renormalization are a bit different with respect to the one-dimensional case. Concretely, the rotation number is shown to have a crucial role. In [7] we introduced an extension of the classical one dimensional (doubling) renormalization operator to cover the quasiperiodic forcing of the map. In the classic operator the dynamics around the fixed point of the operator plays a crucial role to understand the bifurcations of one-parametric families of one dimensional unimodal maps [2]. Here we perform an equivalent study of the (linearised) dynamics around the fixed point for further application to quasiperiodically forced unimodal maps.
In the remainder of this section we review the extension of the operator introduced in [16]. In Section 2 we study, both analytically and numerically, the properties of the linearized operator. Section 3 is devoted to a numerical study of the dynamics of the linearised operator. If the dynamics is projected on the unit sphere (of the corresponding function space) it seems that there exists an attractor that looks like a dyadic solenoid. A detailed study is actually work in progress.

1.1.
A setup for the 1D renormalization operator. Let us review the definition of the 1D renormalization operator in a set up that is suitable for introducing a quasiperiodic perturbation to the 1D map.
Given a small value δ ≥ 0, let M δ denote the space of C r (r ≥ 1) even maps ψ of the interval Set a = ψ(1), a = (1 + δ)a and b = ψ(a ). We define D(R δ ) as the set of ψ ∈ M δ such that D1. a < 0, Conditions M1 and M2 require the map to be unimodal with 0 as the critical point. Conditions D1, D2 and D3 ensure that the intervals [−a , a ] and [b , 1] do not overlap and each one is mapped into the other. In Figure 1 we include a schematic plot of a map in M δ where the geometric meaning of the values a, a and b is shown.
We define the (1D doubling) renormalization operator R δ : with a = ψ(1). Let D n (R δ ) denote the set of functions which are n times renormalizable: The setup for the renormalization operator given above is a small modification of the one introduced in [12], which is recovered for δ = 0. The modification done here is to ensure that one dimensional maps can be quasiperiodically perturbed further on and remain well defined. In [7] it is shown that the operator R δ is well defined, and that any fixed point of R 0 extends to a fixed point of R δ for δ small enough.
1.2. The quasiperiodically forced case. Consider a quasiperiodically forced map, with its domain restricted to the compact cylinder T × I δ : with ω an irrational number, f a C r function and I δ = [−(1 + δ), 1 + δ]. Given F = (ω, f ) a quasiperiodically forced map, its renormalization R(F ) is defined as an affine transformation applied to F 2 = F • F (the self-composition of F ). The map F 2 is of the form (2ω, f 2 ), with f 2 (θ, x) = f (θ + ω, f (θ, x)). The extension of the renormalization operator proposed in [7] is of the form R(F ) = (2ω, T ω (f )), with T ω (f ) an affine transformation of f 2 . In other words, the affine transformation applied to F 2 is just a multiplication by two in the ω-component. With this definition we obtain an operator that preserves the skew structure of the maps. We introduce now some notation for the definition of T ω (f ). Let us identify C r (I δ , I δ ) with its natural inclusion in C r (T×I δ , I δ ) defined as [i(f )](θ, x) = f (x) for any f ∈ C r (I δ , I δ ). With this identification C r (I δ , I δ ) is a subspace of C r (T×I δ , I δ ) and the operator defines a projection. Then, we can consider M δ and D(R δ ) as sets in both C r (I δ , I δ ) and C r (T × I δ , I δ ) depending on the context. Consider also X δ , the set defined as Given a function g ∈ X δ , we define the (quasiperiodic doubling) renormalization of g as whereâ = 1 0 g(θ, 1)dθ.
The operator T ω restricted to the set D(R δ ) coincides with the operator R δ . Therefore, any fixed point of R δ extends to a fixed point of T ω . Consider the set 1174ÀNGEL JORBA, PAU RABASSA AND JOAN CARLES TATJER D(T ω ) = {g ∈ X δ | T ω (g) ∈ X δ }, then the renormalization operator T ω is defined from D(T ω ) to X δ . Actually, D(T ω ) contains an open neighbourhood (in X δ ) of D(R δ ), where the operator is well defined [7].
At this point we can go back to the definition of R, the renormalization operator for a quasiperiodically forced map F = (ω, f ) like (1). Consider We define the quasiperiodically forced renormalization operator as Let D n (R) denote the domain of maps (ω, f ) which are n-times renormalizable, in other words Consider Φ the Feigenbaum fixed point of R δ given by [12]. Then Φ is also fixed by T ω and the set T × {Φ} is invariant by R and the dynamics in ω is determined by the expansive map ω n+1 = 2ω n .
1.3. Differentiability of the operator and its tangent map. Given φ ∈ D(T ω ), φ ∈ C r+s , there exists an open neighbourhood U of φ in D(T ω ) in the C r+s topology, such that T ω : U → X δ with the C r topology is a C s operator [7]. This gives place to an obvious "loss of differentiability". An alternative to fix this problem is to consider a setup of the operator on the analytic functions.
Let B ρ be a complex band of width ρ around the real numbers (B ρ = {z = x + iy ∈ C such that |y| < ρ}) and W be an open, bounded and simply connected set in C containing the real interval I δ . Consider B(B ρ , W ) the space of functions f : B ρ × W → C such that: 1. f is holomorphic in B ρ × W and continuous in the closure of B ρ × W . 2. f is real analytic (it maps real numbers to real numbers).
The space B(B ρ , W ) endowed with the supremum norm is a Banach space. Let us define B(W ) as the space of real analytic functions that are holomorphic in W and continuous in its closure, equipped with the supremum norm. This is the one dimensional counterpart of B(B ρ , W ), and we can understand B(W ) as a subset of B(B ρ , W ).
We want to consider the operator T ω in the new topology of B(B ρ , W ). Given In [12] it is shown that there exist a set W and a function Φ such that Φ ∈ D(R 0 ) ∩ B(W ) and Φ is a fixed point of R 0 (hence a fixed point of T ω as well). In [7] it is shown that, for a sufficiently small ρ, there exists U ⊂ B(B ρ , W ), an open neighbourhood of Φ, such that T ω (Ψ) is well defined for any Ψ ∈ U . It is also shown that T ω is Fréchet differentiable for any Ψ ∈ U and its derivative is equal to One of the reasons to introduce the (quasiperiodically forced) renormalization operator is to study the image by R of functions of the form ε is a small parameter and h 0 ∈ T ω0,f0 X. In other words, we are interested only in (infinitesimal) perturbations δF ∈ T F X that preserve the skew product structure of maps like (1). For this reason we will only consider tangent vectors of the form δF = (0, δf ). We can incorporate this in the tangent bundle of X and (re)define it as: We can define now S, the tangent map associated to R, in the C r topology as 2. Properties of DT ω .

2.1.
Fourier expansion of DT ω . Let Ψ be a function in a neighbourhood of Φ (the fixed point of T ω ) where the operator T ω is differentiable. Additionally, assume that Ψ ∈ D(T ω ).
Given a function f ∈ B(B ρ , W ) we can consider its complex Fourier expansion in the periodic variable Then we have that DT ω "diagonalizes" with respect to the complex Fourier expansion, in the sense that we have where and with ψ = p 0 (Ψ) and a = ψ(1). An immediate consequence of this diagonalization is the following.
then we have that the spaces B k are invariant by DT ω (Ψ) for any k > 0. Moreover Proof Using formula (4) on the function u(z) = c(z) +c(z) and doing some algebra it is easy to see that and, doing a similar calculation for Finally, due to the natural isomorphism between B k and B(W ) ⊕ B(W ), it is easy to see the conjugacy between DT ω (Ψ) and L kω .

2.2.
Spectrum of L ω . In the previous section we have shown that the operator DT ω (Ψ) "diagonalizes" into the infinite sum of operators L α with α = ω, 2ω, 3ω, ... plus the differential of the 1D renormalization operator. Therefore the understanding of L α with respect to α is crucial for the understanding of the derivative of the (quasiperiodically extended) renormalization operator. This section is devoted to the study of the spectral properties of L ω .
Given a value γ ∈ T, consider the rotation R γ defined as Then we have the following result.
Proposition 2.2. For any ω, γ ∈ T we have that L ω and R γ commute.
Proof. It follows from L 1 and L 2 being linear and the fact that any pair of rotations commute.
This proposition has the following consequences on the spectrum of L ω .
is also an eigenvector of the same eigenvalue for any γ ∈ T.
Proof. Suppose that (u, v) is an eigenvector of eigenvalue λ, we have λ(u, v) = L ω (u, v). Composing in both parts by R γ and using the last proposition the result follows.
Corollary 2.4. Let λ be a real eigenvalue of L ω different from zero and with finite geometric multiplicity. Then, the geometric multiplicity of λ is even.
Proof. Assume that λ has geometric multiplicity odd. Then its eigenspace is generated by n vectors y 1 , y 2 , . . . , y n , with n odd. We can consider R γ y i for any i, which will also be in the eigenspace of the eigenvalue. Since the vector R γ y i is linearly independent with y i but it is in the eigenspace, we have that it is generated by the other eigenvectors. Then one of the original vectors can be replaced by R γ y i . Rearranging the vectors if necessary we can suppose that y 2 = R γ y 1 . Doing this process repeatedly we will end up with an even number of vectors.
On the other hand we have the following result on the dependence of the operator with respect to ω Proposition 2.5. The operator L ω depends analytically on ω.
Proof. It follows from the fact that L ω is the sum of two bounded linear operators (which do not depend on ω) times an entire function on ω.
This result allows us to apply theorems III-6.17 and VII-1.7 of [10]. These results imply that, as long as the eigenvalues of L ω do not cross each other, the eigenvalues and their associated eigenspaces depend analytically on the parameter ω.
We want to show that the spectrum of L ω is a countable set of eigenvalues with no accumulation points different from zero. In other words, this is the spectrum we would have if L ω were a compact operator (Theorem III-6.26 of [10]). In this direction, we show that if we consider L ω acting on a space of functions defined on a smaller domain, then it is compact. Proposition 2.6. Assume that W 1 ⊂ W is an open and simply connected set that verifies W 1 ⊂ W , aW 1 ⊂ W 1 and ψ(aW 1 ) ⊂ W 1 , where ψ = p 0 (Ψ). Then the operator Proof. Since the operators L 1 and L 2 are well defined, it is enough to prove that they are compact.
Let us define the set K = W 1 , which is compact since W is bounded, and the set As B contains all the polynomials with real coefficients, we can use the Mergelyan theorem (see, for instance, [17]) to show that B = B(W 1 ). Then, it is enough to see that L 1 and L 2 are compact as operators on B (see (11.2.9) in [3]).
Consider U the unit ball of B. To prove that L i is compact it is enough to prove that L i (U ) is relatively compact (for i = 1, 2). To use (9.13.1) in [3] we note that, for each compact set L ⊂ W , there exists a constant m L such that |[L i (f )](z)| ≤ m L for all z ∈ L. Taking L = W 1 , (9.13.1) in [3] shows that L i (U ) is relatively compact in C 0 (W 1 , C). To finish the proof, we note that: i) each sequence of elements of L i (U ) has a convergent subsequence to an element of C 0 (W 1 , C); ii) a uniformly convergent sequence of analytic functions on any compact subset of W 1 converges to an analytic function on W 1 . This shows that L i (U ) is relatively compact in B.   with respect to the order of the discretization. See the text for more details.

2.3.
Numerical computation of the spectrum of L ω . In this section we introduce a discretization of the tangent map S for its numerical study. The discretization described here is the same used in [7], which is a slight modification of the one introduced by Lanford in [12] (see also [13]).
As before, let W be an open set in C and consider B(W ) the Banach space of real analytic functions, holomorphic on W and continuous on its closure, equipped with the supremum norm.
Let D(z 0 , ρ) be the complex disc centred on z 0 with radius ρ. Given a function ξ ∈ B(D(z 0 , ρ)), we can consider the following Taylor expansion of ξ around z 0 , The truncation of this Taylor series at order N induces a projection defined as p (N ) : B(D(z 0 , ρ)) → R N +1 ξ → (ξ 0 , ξ 1 , . . . , ξ N ).   Table 1 we have the first twenty-four eigenvalues of L Given a general linear bounded operator T , we can compute the eigenvalues of a discretization T (N ) of the operator in order to study the spectrum of T , but in general the eigenvalues of T (N ) might have nothing to do with the spectrum of T . For example an infinite-dimensional operator does not need to have eigenvalues, but a finite-dimensional one will always have the same number of eigenvalues (counted with multiplicity) as the dimension of the space. For this reason we do a couple of numerical test on the results obtained in the discretization of L (N ) ω . Consider that we have a real eigenvalue of multiplicity two, or a pair of complex eigenvalues which are persistent for different values of N (the order of the discretization). The first test done to the eigenvalues is to check if the distance between the associated eigenvectors decreases when N is increased. In the left panel of Figure 2 we have the distance between the eigenvectors associated to the same eigenvalue of the operators L (N ) ω and L (110) ω as a graph of N , with N varying from 40 to 100. We have plotted this distance for the first twenty-four eigenvalues. To compute the distance between eigenvectors we have estimated the supremum norm of the difference between the real function represented by each of the vectors, in other words we have computed i (N ) (v (N ) ) − i (110) (v (110) ) ∞ in the interval (z 0 − ρ, z 0 + ρ) = W ∩ R. Note that the distance seems to go to zero and this suggests that the eigenvectors, namely v (N ) , converge to a limit v * . One should expect these eigenvalues to be in the spectrum of L ω , but nothing ensures that v * belongs to the domain of L ω .
Let us remark that with the numerical computations done so far, we have only checked that the eigenvectors as elements of B(D(z 0 , ρ)) ⊕ B (D(z 0 , ρ)) converge on the segment (z 0 − ρ, z 0 + ρ) ⊂ R but not on the whole set D(z 0 , ρ). We have done a second test to check that the approximate eigenvectors have a domain of analyticity containing D(z 0 , ρ).
Consider that we have a function ξ holomorphic in a domain of the complex plane containing D(z 0 , ρ). If the we consider the expansion of ξ given by equation (8), we have that r the radius of convergence of the series around z 0 is given as With the discretization considered here we have an approximation of the terms ξ n , hence these can be used to compute a numerical estimation of r.
Consider v an eigenvector of the operator L ω . We have that 2 ) a numerical approximation of the eigenvector, we can use the procedure described above to estimate the radius of convergence of each v 1 and v 2 . We have done this for the eigenvectors associated to each of the first twenty-four eigenvalues of L ω with ω = √

5−1 2
(keeping only the smaller of the two radius obtained). The results are displayed on the right panel of Figure 2, where the estimated radius has been plotted with respect to N , the order of the discretization. Note that the estimations give a radius bigger than ρ = 3 2 , which indicates that the eigenvectors are analytic in D(z 0 , ρ), and continuous on its closure, for z 0 = 1 5 and ρ = 3 2 . Up to this point, we have considered ω fixed to √ 5−1 2 , but the same computations can be done to study the spectrum of L ω with respect to the parameter ω. In Figure  3 we display first twenty-four eigenvalues of the map with respect to ω. The set T has been discretized in a equispaced grid of 1280 points. Recall that the operator L ω depends analytically on ω (Proposition 2.5), therefore the spectrum also does (as long as the eigenvalues do not collide, see Theorems III-6.17 and VII-1.7 in [10]).
For this computation we have also made the same test as before to the eigenvalues. The results of these tests are shown in Figure 4. To estimate the convergence of the eigenvectors we have compared the eigenspaces of the eigenvalues of L (90) ω with the eigenspaces associated to L (100) ω for each value of ω in the cited grid of points on T. The estimation of the radius of convergence has been also done with respect to ω for N = 90. We have plotted the estimated error and convergence radius for the first twenty-four eigenvalues in the same figure. Both result indicate that the eigenvalues obtained are reliable.

3.
A numerical aided study of the dynamics of the operator. In the one dimensional renormalization theory it is well known that the dynamics around Φ the fixed point of R 0 plays a major roll in the bifurcations of unimodal 1D maps [2]. For any uniparametric family of 1D maps, the accumulation ratio of consecutive period doubling bifurcations is equal to δ, the dominant eigenvalue of DR 0 . By analogy, we believe that a better understanding of the dynamics of R around Φ can help the understanding of the bifurcations in quasiperiodically forced 1D maps [16].
Recall that, in the quasiperiodic version of the renormalization operator that we proposed, instead of a fixed point we have an invariant set {(ω, Φ)}, with ω ∈ T. This set includes fixed points and periodic orbits, but these correspond to rational values of ω, which are not of interest.  To study the linearized behaviour around the set {(ω, Φ)} we could consider S the associated tangent map given by (3) on points of the form (ω, Φ, h), for some h ∈ B. We have shown before DT ω (Φ) leaves the spaces B k defined by (5) invariant. Moreover, the operator restricted to these spaces is conjugated to L kω , which has its dominant eigenvalue outside the unit circle for any ω. Therefore, one can expect the map S to have a expanding behaviour on the third component. Nevertheless, we might investigate numerically if there exists a dominant invariant direction of expansiveness and its shape. With this aim we consider the map A defined as: 1184ÀNGEL JORBA, PAU RABASSA AND JOAN CARLES TATJER We can use the discretization of L ω described in Section 2.3 to study numerically the map A. As B 1 can be identified with the space B(W ) ⊕ B(W ), we consider the coordinates of v = (x, y) given by this splitting. Following the discretization, each function x ∈ B(W ) is approximated by a vector (x 0 , x 1 , x 2 , ..., x N ) ∈ R N +1 where x i is the i-th coefficient of the Taylor expansion for x around 0. This also holds for y, the second component of v. Therefore, each element v in B 1 can be approximated by a vector (x 0 , x 1 , ..., x N , y 0 , ..., y N ) in R 2(N +1) . We can use this discretization to study the dynamics of A. Let A (N ) denote this approximation, and we call N the order of the discretization.
Given an initial point v 0 = (x 0 , y 0 ) = 0,we have iterated this point by the map for a certain transient N 1 and then we have plotted the following N 2 iterates. Figures 5  and 6 show different projections of the resulting attracting set. The values taken to elaborate this particular set of figures are are N = 30, N 1 = 2000 and N 2 = 80000. We display the coordinates corresponding to the first even Taylor coefficients of the functions x and y. The odd Taylor coefficients obtained were all equal to zero, so we have omitted them. This last observation suggest that the attractor is contained in the set of even functions (note that the subspace of B 1 consisting of all the even functions is invariant by L ω ).
The same computations have been done for bigger values of N leading to the same results. This indicates that the set obtained is stable with respect to the order of discretization, therefore it can be expected to be close to the true attracting set of the original system.
Let us remark that we have not made explicit the initial values of w 0 and v 0 taken for the computations. Indeed, the results seem to be independent of these values. We have repeated this computation taking as initial value of v 0 all the elements of the canonical base of the discretized space R 2(N +1) and we have always obtained the same results. We have also repeated the computations for several irrational values of ω 0 obtaining always the same results.
The numerical approximation of the attractor displayed in Figure 5 and Figure 6 reveal the rotational symmetry of the attractor.
3.1. Rotational symmetry elimination and the dyadic solenoid. Given γ ∈ T, consider the following auxiliary function Given a one dimensional map, the effects caused by a quasiperiodic perturbation h and by t γ (h) should be essentially the same. This is what causes the rotational symmetry in the attracting set of A.
Let B 1 be the subspace of B defined by (5) for k = 1. Note that t γ restricted to B 1 corresponds indeed to R γ given by (7), which commutes with L ω .
In Figure 7 and Figure 8 we display different projections of the attracting set obtained iterating the map B.
Note that the different projections of the attracting set displayed in Figure 7 keep a big resemblance with the plots of the dyadic solenoid displayed in Figure 5 of [14]. Indeed we believe that the attractor is the inclusion of a dyadic solenoid in B 1 . For more details on the definition and the dynamics of the solenoid map see [1,11,14,18].
In Figure 9 we display a numerical approximation of the operator L ω defined above. We can observe that for every value of ω there exists a single dominant eigenvalue. We believe that this dyadic solenoid similarity is explained by the  Figure 9. Numerical approximation of the spectrum of L ω with respect to the parameter ω. From left to right we have the real part, the imaginary part and the modulus of the first eight eigenvalues of L ω with respect to ω.
fact that the second component of (10) contracts all point towards the dominant eigenspace (which depends on ω) while the first component is expansive in ω. This is essentially the same mechanism that creates the dyadic solenoid in R 3 .