CLASSIFICATION OF LINEAR SKEW-PRODUCTS OF THE COMPLEX PLANE AND AN AFFINE ROUTE TO FRACTALIZATION

. Linear skew products of the complex plane, where θ ∈ T , z ∈ C , ω 2 π is irrational, and θ (cid:55)→ a ( θ ) ∈ C \ { 0 } is a smooth map, appear naturally when linearizing dynamics around an invariant curve of a quasi-periodically forced complex map. In this paper we study linear and topological equivalence classes of such maps through conjugacies which preserve the skewed structure, relating them to the Lyapunov exponent and the winding number of θ (cid:55)→ a ( θ ). We analyze the transition between these classes by considering one parameter families of linear skew products. Finally, we show that, under suitable conditions, an aﬃne variation of the maps above has a non-reducible invariant curve that undergoes a fractalization process when the parameter goes to a critical value. This phenomenon of fractalization of invariant curves is known to happen in nonlinear skew products, but it is remarkable that it also occurs in simple systems as the ones we present.

where n = wind(a, 0), m is an arbitrary integer (that depends on the choice of change of variables), b ∈ C and log |b| is the Lyapunov exponent (see Proposition 2). On the other hand, topological conjugacy classes are also determined by the winding number but only by the sign of the Lyapunov exponent, see Theorems 3.5 and 3. 6.
In order to analyze the transitions between different conjugacy classes, we shall also consider one-parameter families of linear skew products of the form µ being a real parameter. Since we want the winding number to change when moving the parameter, we must allow a to have zeros, which means that in these cases the skew product will not be invertible. By means of suitable normal forms, we study how the Lyapunov exponent depends on the parameters of the system. In particular, we show that the dependence of the Lyapunov exponent w.r.t. parameters is only continuous (and never differentiable) when the winding number changes (see Section 4 for details). The dynamics of general (nonlinear) skew products is a well known topic in dynamical systems that has been considered by several authors (see, for instance, [25,15,8]), and very often specifically to study the existence of invariant curves, the fractalization phenomenon and the existence of Strange Non-chaotic Attractors (SNAs) [22,23,12,11,13,14,16,2,7,6]. In linear systems, there always is an invariant curve given by z = 0. For this reason, let us consider a small modification of a linear skew product, given by the so called affine skew products, θ → θ + ω, z → a(θ, µ)z + c(θ, µ). ( If (2) has an invariant curve, then it can be reduced to the form (1) by translating the curve to the origin and, hence, the classification we have obtained for linear systems can be extended to affine systems (see Section 5). An interesting situation happens when (2), for some value of the parameter µ, has no invariant curve. A natural question is then the following: if the parameter moves from a value for which there is an invariant curve to another value for which there is no invariant curve, how does this curve dissapear? We will see that, in this case, the curve may exhibit a fractalization process (see Section 5 for a precise definition) as the parameter varies. Even further, we show that this process appears in a family of affine skew products as simple as where ω is the golden mean. The parameter µ is used to control the Lyapunov exponent and the value c = 0 controls the existence of a nontrivial invariant curve: as we will see, this system has an attracting invariant curve z µ for |µ| < 1 that we display in Figure 1 for several values of µ, and no invariant curves when µ = 1.
As indicators of the behaviour of the invariant curve when µ approaches 1 from below, we numerically compute its norm, the norm of its derivative, its length and its winding number w.r.t. the origin. All of them go to infinity and it is remarkable that their respective asymptotic behaviour seems very well fitted by quite simple functions, as shown in Figure 2. Note that the linearization of the dynamics at the invariant curve (which in this case is just the linear part of the system) is non-reducible. We also study this fractalization phenomenon in a rigorous way, giving a proof of the asymptotic behaviour displayed in Figure 2. As a side result, to illustrate the "wild" behaviour of this curve, we prove that µ0<µ<1 {z µ (ψ) | ψ ∈ T} = C for any µ 0 ∈ [0, 1). In this direction note that if instead of the value c = 1 in (3) we use c = √ 1 − µ, then the invariant curve is scaled so that it stays bounded and then the union of curves for µ 0 < µ < 1 fills up a bounded domain for any value µ 0 < 1. It is also remarkable that the curve disappears when µ reaches the value 1. Of course, the same phenomenon happens when µ approaches 1 from above, but then the invariant curve is repelling.

2.
Preliminaries. This section is a compendium of results and definitions that will be used during the work. This section is added in order to facilitate the reading of the paper.
The first notions we want to introduce concern the arithmetic properties of the frequency ω (see e.g. [19,18]). The set of Diophantine numbers is defined as follows.
for all p q ∈ Q. We denote by D γ,τ the set of numbers that satisfy (4) for fixed γ > 0 and τ ≥ 2.
We define, as well, the set of numbers of constant type. The numbers of constant type are Diophantine of type (γ, 2). The Diophantine condition shall be used widely during Section 3. We shall require ω to be of constant type in Theorem 5.4 and related results.
We define, as well, the quantity If Λ happens to be finite, then the Birkhoff Ergodic Theorem tells us that, for Lebesgue-a.e. θ ∈ T, the lim sup (5) is in fact a limit and λ(θ) = Λ. If a(θ) never vanishes, lim sup is again a lim and coincides with Λ for all θ ∈ T. In this last case (5) converges uniformly. This follows from the fact that irrational rotations on T are uniquely ergodic.
The Argument principle is a standard result in complex analysis, see, for instance, [1]. This result will be used in Sections 4 and 5.   Another result we want to mention is the generalization to complex-valued functions of the Malgrange collocation Theorem given in [20]. This will be used to construct a normal form at some specific critical values. Theorem 2.5 (Malgrange-Nirenberg). Let U ⊂ R × R n be an open set containing the origin. Consider f : (t, x) ∈ U → f (t, x) ∈ C of C ∞ class. Let p > 0 be the first integer such that ∂ p ∂t p f (0, 0) = 0. Then, in an neighbourhood of the origin, one has the factorization and Q and λ j are C ∞ complex-valued functions with Q(0, 0) = 0. If f is real, Q and P can be chosen to be real.
When classifying linear skew products, we will use the following concept.
such that 1. G t is a homeomorphism of C for all t ∈ [0, 1], 2. G 0 = g and G 1 is the identity map.
3. Linear invertible skew-products. In this section we focus on linear skewproducts on the complex plane, where a(θ) ∈ C \ {0} for all θ ∈ T. This means that (7) is an invertible map. Moreover, we assume that the map θ → a(θ) is of class C r (r ≥ 1) and that ω ∈ D γ,τ . We are interested in classifying these linear skew-products and, to this end, we consider two different types of conjugacies. where ν ∈ T and, for each θ, H(θ, ·) is a homeomorphism of the plane verifying H(θ, 0) = 0 and such that When H(θ, z) can be chosen to be linear w.r.t. z, i.e. H(θ, z) = c(θ)z, with c(θ) continuous and different from zero for all θ, then F a and F b are said to be linearly conjugate as skew products up to an angle translation. If ν = 0 we simply say that F a and F b are linearly conjugate as skew products. Note that this is equivalent to In what follows, we will refer to "conjugacies as skew products" simply as "conjugacies". Generally speaking, we will also refer to conjugacies of any kind as changes of variables. is said to be reducible iff there exists a linear change of variables, (θ, z) = (θ, e(θ)u) such that the transformed system becomes uncoupled. That is, the transformed system takes the form 3.1. Linear conjugacy classes. Next, we give necessary and sufficient conditions for two systems to be linearly conjugate. Let wind(a(θ), 0) denote the winding number of the closed curve a(θ) with respect to the point z = 0.
Proposition 1 (Linear conjugacy classes). Let ω ∈ D γ,τ . Then there exists r = r(τ ) > 0 such that if a(θ) and b(θ) are of class C r then F a and F b are linearly conjugate if and only if the following two conditions are satisfied: There exists m ∈ Z and a branch of the logarithm such that Moreover, if such m exists, it is unique and the linear change of coordinates H(θ, z) = c(θ)z satisfies that wind(c(θ), 0) = m.
It is easy to check that (a) and (b) are independent conditions.
Proof. (⇐=) Because of (a), the curve a(θ) b(θ) has winding number 0 and so does e −imω a(θ) b(θ) . Hence the curve defined by is also a closed C r curve. Let us consider its Fourier series where α 0 = 0 because of (b). As l(θ) ∈ C r we know that |α k | = O(1/|k| r ). We now define for any k = 0 and set for example c 0 = 0. Note that the Diophantine condition on ω implies that | c k | = O(|k| τ −r ). So, if r > τ + 1, the series k c k e ikθ is absolutely and uniformly convergent and defines a 2π−periodic function L, It is easy to check by comparing the coefficients in the Fourier series that, by construction, Let c(θ) = exp(L(θ)) which has winding number zero with respect to z = 0 because L(θ) is a closed curve. It follows that and therefore Hence, c(θ) provides the linear change and wind(c(θ), 0) = m.
From now on, we will denote by r(τ ) ∈ N a value of r for which the previous proposition holds. We shall use the previous proposition to find canonical forms for these linear skew products. We need to differentiate cases depending on the winding number of the curve θ → a(θ).
Proposition 2 (Linear normal form). Assume ω ∈ D γ,τ , a(θ) is C r(τ ) and wind(a(θ), 0) = n. Then, for any m ∈ Z, there exists a linear change, of winding number −m, which conjugates F a to for any determination of the logarithm. Moreover, two such systems (θ + ω, b 1 e inθ z) and (θ + ω, b 2 e inθ z), with b 1 , b 2 ∈ C are linearly conjugate if and only if b 1 = b 2 e imω for some m ∈ Z.
Note that log |b| is the Lyapunov exponent of the skew product.
Proof. Consider m ∈ Z fixed and choose a branch l(θ) of log(a(θ)e −inθ ) which exists since this expression has winding number 0. We want to find b = |b|e iρ such that condition (b) in Proposition 1 is satisfied (observe that condition (a) is already fulfilled). That is, we want that T log e imω e i(−mω−ρ) a(θ) |b|e inθ dθ = 0, for some branch of the logarithm. Given that l(θ) − log |b| − iρ is such a branch, we obtain T l(θ)dθ = 2π log |b| + 2πiρ.
Separating real and imaginary part, for some branch of the logarithm. But this is equivalent to requiring e −ipω b1 b2 = 1 for some p ∈ Z.
From the proposition above we obtain a trivial corollary about the reducibility of these systems, in the case of winding number 0.
Corollary 1 (Zero index and reducibility). Assume ω ∈ D γ,τ , a(θ) is C r(τ ) . If wind(a(θ), 0) = 0, then the system is reducible. Moreover, the system is reducible to a system of the form (θ + ω, bz) with b ∈ R, if and only if there exists m ∈ Z and a branch of the argument such that T arg(a(θ))dθ − mω = 0.
In such case, the change has winding number equal to −m.
In the nonzero winding number case, it turns out that we can always reduce to the case of b ∈ R by changing the phase. More precisely we have the following statement.

3.2.
Topological conjugacy classes. We now proceed to classify linear invertible skew-products of class C r(τ ) from a topological point of view. We recall from Definition 3.1 that if two linear skew products F a and F b are topologically conjugate then there exist a constant ν ∈ T and a continuous map H : In the case that H(θ, ·) is isotopic to the identity we say that F a and F b are topologically conjugate via a homeomorphism isotopic to the identity. for all x, y : T → C \ {0} continuous.
Proposition 4 (Winding number and topological conjugacy). If two linear skew products F a and F b are topologically conjugate via a homeomorphism isotopic to the identity then wind(a(θ), 0) = wind(b(θ), 0).
Next, in Theorems 3.5 and 3.6 we give a topological classification of linear skew products, depending on the winding number and the Lyapunov exponent.
To show (d) observe that Proposition 3 gives that F a is linearly conjugate (up to an angle translation) to F be inθ (θ, z) = (θ + ω, be inθ z).
It is then easy to see that, if the Lyapunov exponent is negative, the following change of variables produces the desired result. Case (e) is analogous. Items (c) and (f) follow from Propositions 2 and 3.
Theorem 3.6. Assume that ω ∈ D γ,τ . Then, the skew-products (10), (11), (12), (13), (14) and (15) belong to different topological conjugacy classes. Moreover, a) Consider a linear skew product of type (12) such that ρ, ω and 2π are linearly independent over Z, and let us consider a second linear skew product of type (12). Then, they are topologically conjugate via a homeomorphism isotopic to the identity iff they are linearly conjugate. b) Two linear skew products of the types (13), (14) or (15) with two different values of n are not topologically conjugate via a homeomorphism isotopic to the identity.
Proof. The first claim is obvious. To prove item a), we assume that are topologically conjugate and that, for instance, ρ 1 , ω and 2π are linearly independent over Z. As they are topologically conjugate, by definition, there exists a continuous map H such that where z = re iϕ . Let us fix the value of r (for instance, r = 1) and let us look at H as a continuous function of two angles. Expanding both sides of the last equality in Fourier series w.r.t. ϕ, we obtain As H cannot be the zero function, there exists k such that h k does not vanish. Hence, h k (θ + ω) = e i(ρ2−kρ1) h k (θ). Expanding h k in a Fourier series w.r.t. θ, and selecting a non-zero Fourier coefficient h kj , we have that h kj e ijω = e i(ρ2−kρ1) h kj , which implies that jω = ρ 2 − kρ 1 mod 2π. Applying the same calculation for the inverse conjugation, we have that there exist integer valuesĵ andk such that jω = ρ 1 −kρ 2 mod 2π. Using this last two equations we obtain that for some m ∈ Z. As ρ 1 , ω and 2π are linearly independent over Z we have that 1 − kk = 0,ĵ + jk = 0 and m = 0. The solutions of these equations are: k =k = 1,ĵ = −j and k =k = −1,ĵ = j, which implies that H has to be of the form H(θ, e iϕ ) = h −1 (θ)e −iϕ + h 1 (θ)e iϕ . Notice that these conditions cannot hold at the same time. Indeed, if such is the case, there exist j 1 and j 2 verifying the following: Adding these equations, it follows (j 1 + j 2 )ω = 2ρ 1 + 2π(m 1 + m 2 ). This case is out of the study as it implies that ρ 1 , ω and 2π are linearly dependent over Z which leads to a contradiction with the hypothesis assumed in (a). Then, if k =k = −1, as H(θ, ·) restricted to z = e iϕ is h −1 (θ)e −iϕ , it cannot be isotopic to the identity as it is reversing the orientation of the unit circle. Therefore, the only remaining possibility is k =k = 1 and then, it is immediate to check that (θ, z) → (θ, h 1 (θ)z) is a linear conjugacy between the two skew products.
Item b) follows from Proposition 4 and from the fact that the attracting or repelling character of the origin is preserved by a topological conjugacy.
Remark 3. The dynamics of the maps (10), (11), (13) and (14) are locally robust in a neighbourhood of the origin under generic perturbations, because the origin is attracting or repelling. This is not the case for (12) and (15).
r is an invariant torus for the maps (12) and (15). These invariant foliations could be destroyed by a generic perturbation of the map. Moreover, if we consider coordinates (θ, ϕ) in the torus, the map (12) restricted to T 2 r satisfies (θ, ϕ) → (θ + ω, ϕ + ρ) which is a translation in the torus and the map (15) restricted to T 2 r satisfies (θ, ϕ) → (θ + ω, nθ + ϕ), which is sometimes called a skew shift. The second map is uniquely ergodic if ω 2π is irrational, with the Lebesgue measure as the unique invariant measure and the first map is uniquely ergodic if ω, ρ and 2π are rationally independent.
4. Normal forms and Lyapunov exponents. Let us consider a linear quasiperiodic skew product as defined in (7), given by a ∈ C r (T, C), r ≥ 0. We have shown that the winding number of a is preserved by linear changes of variables (see Proposition 1) so that it can be seen as an invariant of the cocycle. In this section we shall study how the Lyapunov exponent varies when introducing a new real parameter µ. In particular we are interested in the regularity of the Lyapunov exponent as µ crosses a critical value for which the skew-product is not invertible. Notice that, up to now, all the skew-products have been invertible. To carry out this study, we use Λ µ as in Definition 2.3. Recall that, if Λ µ is finite, it coincides with the Lyapunov exponent. Roughly speaking, the next result shows that Λ µ depends smoothly on µ except when a changes its winding number.
Then, the Lyapunov exponent Λ(µ) is a continuous function of µ such that 1. Λ is C ∞ at any µ = µ 0 . 2. Λ is C 0 at µ = µ 0 and there exist constants A + and A − , such that, when µ → µ 0 , the following expression holds: where A + is used when µ > µ 0 and A − when µ < µ 0 . The values A + and A − never coincide.
A particular situation is when the system is reducible for µ < µ 0 , and nonreducible for µ > µ 0 . Similarly, in the real 1D case it is known that the dependence of the Lyapunov exponent w.r.t. the parameter µ is continuous but never differentiable at µ 0 , see [16]. In the real 2D case, there is numerical evidence of the same phenomenon [10,3]. As a matter of fact, the recent preprint [5] contains a rigorous proof of this fact for a class of 2D cocycles arising from the study of the spectrum of some discrete Schrödinger operators.
Proof. First note that, for µ < µ 0 , condition 1 implies that the curves a(·, µ) are homotopic on C \ {0} (the homotopy is given by a(θ, µ) itself) and this shows that wind(a(·, µ), 0) is constant for µ < µ 0 . As the same reasoning applies for µ > µ 0 we can also conclude that wind(a(·, µ), 0) is constant for µ > µ 0 . Note that, if |θ − θ 0 | and |µ − µ 0 | are small enough, where O 2 is a term of order 2 in (θ − θ 0 ) and (µ − µ 0 ) For θ near θ 0 , a(θ, µ 0 ) is close to a straight line passing through the origin at θ = θ 0 , and hence, dividing a conveniently small disk centered at the origin into two almost equal components. Moreover by condition 2, if µ 1 < µ 0 the curve a(·, µ 1 ) lies in one of these components and, if µ 2 > µ 0 , the curve a(·, µ 2 ) lies in the other component. In the situation described, the winding number changes by 1 when µ crosses µ 0 . This can be seen, for instance, by applying the Argument principle to the function f (z) = z.
such that the skew-product takes the form where h is a C ∞ zero-free function and ν(µ) = 1 + µ − µ 0 .

Remark 4.
The assumptions on the winding number are done in order to assure the winding number to increase when µ crosses µ 0 . The fact that changes by 1 follows from Lemma 4.2.
Proof of Theorem 4.1. Using Lemma 4.2 we conclude that the winding number of a(·, µ) around the origin changes by 1 when µ crosses µ 0 . Suppose that wind(a(·, µ), 0) = n if µ < µ 0 and wind(a(·, µ), 0) = n + 1 if µ > µ 0 (the inverse situation can be reduced to this one by reversing the parameter µ w.r.t. µ 0 ). We use Lemma 4.3 to put a(θ, µ) in normal form with a linear change. The system is transformed tõ ϕ = ϕ + ω, ζ = h(µ)e inϕ (e iϕ + ν(µ))ζ, Recall that the Lyapunov exponent is preserved by linear changes. We finally use Lemma 4.4 where the value of the Lyapunov exponent is computed for the normal form.

A fractalization mechanism.
In this section we focus on affine skew products, where a(·, ·) and c(·, ·) are of class C r (r ≥ 1) and ω is Diophantine. They can be seen as a very simple extension of linear skew products. As usual, an invariant curve is defined as the graph of a map θ → z(θ), of class C r (r ≥ 1), which is invariant by the dynamics, i.e., z(θ + ω) = a(θ, µ)z(θ) + c(θ, µ).
We start by observing that the existence of invariant curves is a generic phenomenon.
Proposition 5. With the hypotheses above, the affine system (21) has an invariant curve if the Lyapunov exponent of the linear part (1) is different from zero.
Let E = C 0 (T, C) endowed with the sup norm. If |b| < 1, the operator T : E → E defined as T (ψ)(θ) = |b|e iρ e in(θ−ω) ψ(θ − ω) +c(θ − ω) is a contraction. The Banach fixed point theorem provides the existence of a continuous invariant curve. Finally, Theorem 3.2 in [25] states that any such invariant curve is as smooth as the map itself.
The same argument applies to the case |b| > 1 using the inverse of (21).
When (21) has an invariant curve, translating this curve to the origin we obtain a linear skew product so that, in this case, we can extend the classification of linear systems to affine ones. A natural question is to understand the transition between different conjugacy classes when a parameter varies. In particular, we are interested in studying the fate of an invariant curve when going through a parameter value with zero Lyapunov exponent. In what follows we describe, via a particular example, the phenomenon of fractalization as a possible answer to the question above, when the system is non-reducible.