Topological obstructions to dominated splitting for ergodic translations on the higher dimensional torus

Consider the space of analytic, quasi-periodic cocycles on the higher dimensional torus. We provide examples of cocycles with nontrivial Lyapunov spectrum, whose homotopy classes do not contain any cocycles satisfying the dominated splitting property. This shows that the main result in the recent work"Complex one-frequency cocycles"by A. \'Avila, S. Jitomirskaya and C. Sadel does not hold in the higher dimensional torus setting.


Introduction and statements
It is well known that the homotopy type may prevent a continuous linear cocycle over a base dynamical system from being uniformly hyperbolic. In fact, for an SL 2 (R)-valued cocycle over a circle map, M. Herman remarked that the topological degrees of the base map T : T → T and of the matrix valued function A : T → SL 2 (R) provide topological obstructions to the uniform hyperbolicity of the cocycle. More precisely, this obstruction happens when deg(T ) − 1 does not divide deg(A p ), where for any p ∈ P(R 2 ), A p : T → P(R 2 ) denotes the projective space induced map A p (x) = A(x) p (see [10] or [4]).
In sharp contrast with this, A.Ávila, S. Jitomirskaya and C. Sadel [2] recently proved that analytic cocycles A : T → GL m (C) over irrational translations on the one dimensional torus T are always approximated by cocycles with dominated splitting (a type of uniform projective hyperbolicity), provided the Oseledets filtration is nontrivial. In particular, every homotopy class of such cocycles contains analytic cocycles with dominated splitting.
In dynamical systems, the Bochi-Mañé dichotomy refers to a generic (low regularity) dichotomy between zero Lyapunov exponents and uniform hyperbolicity, or dominated splitting in higher dimensions. This dichotomy, proved by J. Bochi [3], was first announced by R. Mañé in the context of C 1 -area preserving diffeomorphisms of a surface. Later J. Bochi and M. Viana generalized it to C 1 -volume preserving diffeomorphisms of any compact manifold [4]. These works [3,4] also include versions of the dichotomy for classes of C 0 -cocycles. Because the low regularity is essential here, it is quite surprising that the same type of dichotomy can hold in [2] for a class of analytic cocyles.
The purpose of this note is to show that the main result in the aforementioned paper [2] does not hold for cocycles over ergodic translations on the higher dimensional torus T d , d ≥ 2. We obtain this by developing a simple homological obstruction to the existence of continuous invariant sections of the skew product map induced by the cocycle at the level of the Grassmannian space of a certain dimension.
A somewhat related topic is that of the regularity of the Lyapunov exponents under small perturbations of the cocycle in certain topological spaces of cocycles. In [2] the authors prove continuity of the Lyapunov exponents on the space C ω r (T, Mat(m, C)) of analytic cocycles 1 over irrational translations on the one dimensional torus. Dominated splitting plays a crucial role in their proof, more precisely, the fact that if the Oseledets filtration of the cocycle A(x) is nontrivial, then for small enough > 0, the complexified cocycle A(x + iy) has dominated splitting for a.e. y with y < (see [2,Lemma 4.1]). As a consequence of our main result, the analogue of this statement for ergodic translations on the higher dimensional torus does not hold (see Remark 3). However, in [5] we established by other means the continuity of the Lyapunov exponents for analytic cocycles over such translations.
We now introduce the main concepts more formally. Let K = R or K = C refer to either the real or the complex field. Let T d = (R/Z) d with d ≥ 2 be the higher dimensional torus.
A continuous function A : T d → GL m (K) and an ergodic translation We call the new dynamical system F a linear cocycle over the base transformation T . Its iterates are Since T is usually fixed, we identify the linear cocycle F with the matrix-valued function A, and its iterates F (n) with A (n) .
The Lyapunov exponents of a linear cocycle A measure the average exponential rate of growth of the iterates A (n) (x) along the invariant subspaces given by the Oseledets theorem.
We say that a linear cocycle A has dominated splitting with respect to K (or that its Oseledets decomposition is dominated) if there exists a continuous F -invariant decomposition K m = E 1 (x) ⊕ . . . ⊕ E l (x), where 2 ≤ l ≤ m and each E i is an F -invariant continuous K subbundle of the trivial bundle T d × K m such that for some λ > 1, for any 1 ≤ i < j ≤ l and for any unit vectors In particular, as l ≥ 2, the Oseledets decomposition of A is nontrivial (its components are proper subspaces of K m ) so the Lyapunov exponents of A are not all equal.
For SL 2 (R)-valued cocycles, the dominated splitting property is equivalent to uniform hyperbolicity.
Following the terminology in [2], given 1 ≤ k < m, we say that a linear cocycle A : It is clear that if the linear cocycle A has the dominated splitting We are now ready to formulate the main result of this paper. Theorem 1. Given integers d ≥ 2 and 1 ≤ k < m there exist analytic quasi-periodic cocycles A : Remark 1. This theorem shows that the dichotomy in [2, Theorem 1.1] does not hold for analytic quasi-periodic cocycles over a torus T d of dimension d ≥ 2. In fact any sufficiently small neighborhood V of A is contained in the homotopy class of A. In this neighborhood V, by our continuity result [6, Theorem 6.1], assuming that the translation vector satisfies a generic Diophantine condition, the Oseledets decomposition C m = E + ⊕ E − persists with dim E + = k. However, in view of Theorem 1, this decomposition is never k-dominated.
Consider now the projective space P(K m ) where the group GL m (K) acts transitively. More generally let Gr k (K m ) be the Grassmannian space of all k-dimensional K-linear subspaces of K m , which reduces to the projective space when k = 1.
The cocycle F determines the skew-product mapF : . Clearly the kdomination property implies the existence of a continuous invariant section E + : T d → Gr k (K m ) for the bundle mapF . The strategy to prove Theorem 1 is to derive topological obstructions to the existence of continuous invariant sections σ : Remark 2. The statement of Theorem 1 hods also for GL m (R)-valued cocycles over a torus T d with dimension d ≥ 1. This can be proven analogously or more simply using M. Herman's method described in [10]. The topological obstructions there use first homotopy groups and are applicable because the real Grassmannians Gr k (R m ) are not simply connected, something which is not true about the complex Grassmannians Gr k (C m ).
The paper is organized as follows. In Section 2 we provide a necessary condition for the existence of a continuous invariant section of a skew product map. In Section 3 we use the previous abstract result to provide topological obstructions to the existence of continuous invariant sections for quasi-periodic cocycles on the higher dimensional torus. This in particular implies our main theorem.
We are grateful to Christian Sadel for posing the question regarding dominated splitting for quasi-periodic cocycles on the torus of several variables, to Gustavo Granja for a valuable suggestion on using the nonexistence of homological splitting as a topological obstruction to dominated splitting and to Marcelo Viana for providing us with several references on this subject.

Existence of invariant sections
We call factor of linear maps any commutative diagram where E, F are vector spaces, f : E → E, h : F → F are linear endomorphisms and π : E → F is a linear epimorphism. We call splitting of a factor (1) any linear map σ : In other words σ is an f -invariant section of the vector bundle π : E → F . Letting K = ker(π), by the fundamental theorem on homomorphisms, the linear epimorphism π : E → F induces an isomorphism π : E/K F through which the factor (1) can be expressed as where K stands for an f -invariant vector subspace of E. From these considerations it follows easily that By definition, letting π : M ×X → M stand for the canonical projection π(x, p) = x, the following diagram commutes We call F -invariant section any continuous map σ : M → X such that F (x, σ(x)) = (T x, σ(T x)) for all x ∈ M .
An obvious necessary condition to the existence of an F -invariant section is the splitting property of the factor (4) at the level of homology (the reader may consult [8] for a general reference on singular homology).
admits a splitting.
Proof. By the Künneth theorem the map π * : We are also using here that M and X are connected so that H 0 (M, F) H 0 (X, F) F. Hence the epimorphism π * : Similarly, the projection π : M × X → X, π (x, p) = p, induces a homology map π * : Because G is connected, each element A(x) ∈ G induces an action A(x) : X → X which is isotopic to the identity. Therefore the homology map F * : H k (M ×X, F) → H k (M ×X, F) acts as the identity on ker(π * ). Assume now, by contradiction, that F admits an invariant section. By Proposition 2 there exists an F * -invariant subspace G such that Since T : M → M is homotopic to id M we have T * = id on H k (M, F). This implies that F * is the identity map on G. Hence, because (6) is We have used assumption (3) and the fact that the composition π •i p is a constant map. This contradiction proves that F admits no invariant section.

Consequences for quasi-periodic cocycles
Finally we show that for certain homotopy types a continuous quasiperiodic cocycle A : T d → GL m (C) cannot have dominated splitting. The base dynamics is assumed to be an ergodic translation of a torus T d of dimension d ≥ 2.
Let Gr k (C m ) denote the complex Grassmannian of k-dimensional complex subspaces of C m .

Proposition 4. Let
, induces a non-zero homology map in dimension two, i.e., the linear map (A V ) * : H 2 (T d , F) → H 2 (Gr k (C m ), F) is non zero for some field F and some 1 ≤ k < m, then the quasi-periodic cocycle A has no continuous invariant section σ : T d → Gr k (C m ). In particular A is not k-dominated.
Proof. Let us apply Proposition 3 with M = T d , X = Gr k (C m ) and dimension k = 2. For any field F we have dim H 0 (Gr k (C m ), F) = 1 because Gr k (C m ) is a connected manifold. We have dim H 1 (Gr k (C m ), F) = 0 and dim H 2 (Gr k (C m ), F) ≥ 1 (see [9,Section 3.2] or [7, Section 5 of Chapter 1]). We also have Therefore assumption (1) and (2) of Proposition 3 hold. On the other hand, our hypothesis implies assumption (3) of that proposition. Therefore, by Proposition 3, the mapF : T d ×Gr k (C m ) → T d ×Gr k (C m ) does not admit anyF -invariant section.
Finally, if the quasi-periodic cocycle A is k-dominated then the Finvariant sub-bundle E + determines anF -invariant section E + : T d → Gr k (C m ). This contradiction proves that A is not k-dominated. Corollary 1. Consider a quasi-periodic cocycle A : T 2 → GL 2 (C). If the map Av : T 2 → P(C 2 ), Av(x) = A(x)v, for somev ∈ P(C 2 ), is not homotopic to a constant then A does not have dominated splitting.
Proof. The projective space P(C 2 ) can be identified with the Riemann sphere S 2 ≡ C ∪ {∞}. Since Av is not homotopic to a constant, by Hopf theorem deg(Av) = 0. Then, making the canonical identifications H 2 (T 2 , F) F and H 2 (P(C 2 ), F) F, the homology map (Av) * : H 2 (T 2 , F) → H 2 (P(C 2 ), F) is the multiplication by deg(Av), and hence it is non zero.

Corollary 2.
There are continuous functions A : T 2 → GL 2 (C) whose homotopy classes contain no quasi-periodic cocycle with dominated splitting.
Proof. Consider any analytic map f : onto the unit sphere S 2 with the stereographic projection, which maps S 2 diffeomorphically onto the projective space P(C 2 ) = C ∪ {∞}. Assume that the origin belongs to a bounded connected component of R 3 \f (T 2 ). Then the parametric hypersurface f has non zero winding number around 0, which implies that the composition φ = p • f : T 2 → P(C 2 ) has non zero degree.
Write φ = a/b as the ratio of two real analytic functions a, b : T 2 → C, where b vanishes exactly at the points x ∈ T 2 where φ(x) = ∞ and the pair (a(x), b(x)) = (0, 0) for all x ∈ T 2 . Then the analytic function A : satisfies the assumption of Corollary 1 withv = (1, 0). Hence it cannot have dominated splitting.
Finally, if B : T 2 → GL 2 (C) is another continuous function homotopic to A then the functions Av : T 2 → P(C 2 ), Av(x) = A(x)v, and Bv : T 2 → P(C 2 ), Bv(x) = B(x)v, are also homotopic. Hence Bv is not homotopic to a constant and by Corollary 1 the cocycle B cannot have dominated splitting either. Theorem 1 follows from the following proposition.
We are going to use Proposition 3 to prove item (4). For each 1 ≤ i ≤ m, let V i ∈ Gr k (C m ) be the complex i-plane generated by the first i vectors e 1 , . . . , e i of the canonical basis of C m . We claim that the map . By construction this is true about the map A e 1 : T 2 → Gr k (C 2 ), A e 1 (x) := A(x)ê 1 , which induces a non zero linear map at the second homology level. To relate the homologies ofÃ V k and A e 1 we factor the first,Ã V k , as a composition of several maps which include the second, A e 1 .
Let Σ := {V ∈ Gr k (C m ) : This is a complex analytic submanifold of the Grassmannian space Gr k (C m ), which is diffeomorphic to the complex projective line P(C 2 ). Let p : C m → C 2 be the linear projection p(z 1 , . . . , z m ) = (z k , z k+1 ), and define H : where ι stands for the inclusion map ι : Σ → Gr k (C m ).
By Künneth theorem, the linear map π * : Because H is a diffeomorphism, the homology map H * is an isomorphism. We are left to prove that ι * : H 2 (Σ, K) → H 2 (Gr k (C m ), K) is injective. This will imply that (Ã V k ) * is non zero and, by Proposition 3, that no cocycle homotopic toÃ admits a continuous invariant section with values in Gr k (C m ).
Let us now turn to prove the injectivity of ι * . The Grassmannian Gr k (C m ) is an analytic manifold of dimension k (m − k). By Schubert Calculus (see [9,Section 3.2] or [7, Section 5 of Chapter 1]), the manifold Gr k (C m ) admits a class of standard cell decompositions, whose cells are referred as Schubert cells. The closures of these cells are analytic subvarieties known as Schubert cycles. The submanifold Σ is itself a Schubert cycle with complex dimension 1 which can be integrated in a cell decomposition {V k } = Σ 0 ⊂ Σ = Σ 1 ⊂ Σ 2 ⊂ · · · ⊂ Σ N = Gr k (C m ).
Each space Σ i is an analytic subvariety obtained from Σ i−1 by joining a cell with (real) even dimension and boundary contained in Σ i−1 . This implies that H 1 (Σ i , Σ i−1 , K) = 0 for all i and all fields K. Hence, by the long exact sequence of the pair (Σ i , Σ i−1 ), is an exact sequence. Therefore, because ι can be factored as the composition of the inclusions Σ i−1 → Σ i with i = 2, . . . , N , the map ι is injective at the second homology level.
Remark 3. Given a cocycle A ∈ C ω r (T d , GL m (R)) in one of the homotopy classes from Proposition 5, the cocycle A y : T d → GL m (C), A y (x) = A(x + iy), cannot have dominated splitting for any y ∈ R d . a Ciência e a Tecnologia, under the project: UID/MAT/04561/2013.
The second author was supported by the Norwegian Research Council project no. 213638, "Discrete Models in Mathematical Analysis" and by the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq, Brazil).