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

TL;DR

This paper demonstrates that certain analytic quasi-periodic cocycles on higher dimensional tori with nontrivial Lyapunov spectra cannot be homotoped to cocycles with dominated splitting, revealing topological obstructions in higher dimensions.

Contribution

It provides explicit examples showing the failure of dominated splitting in higher dimensional tori, challenging previous results valid in the one-dimensional case.

Findings

Existence of cocycles with nontrivial Lyapunov spectrum

Homotopy classes lack cocycles with dominated splitting

Counterexamples to previous theorems in higher dimensions

Abstract

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.

Equations

$$F(x,v)=(Tx,A(x)v).$$

$$\frac{\Vert A^{(n)}(x)v_i\Vert }{\Vert A^{(n)}(x)v_j\Vert }\ge {\lambda }^n\text{ for all }n\in \mathbb{N}.$$

$$\begin{array}{ccc}E& \stackrel{f}{\to }& E\\ \pi \downarrow & & \downarrow \pi \\ F& \underset{h}{\to }& F\end{array}$$

$$\begin{array}{ccc}E& \stackrel{f}{\to }& E\\ \pi \downarrow & & \downarrow \pi \\ E/K& \underset{\bar{f}}{\to }& E/K\end{array}$$

$$F:M\times X\to M\times X,F(x,p):=(Tx,A(x)p).$$

$$\begin{array}{ccc}M\times X& \stackrel{F}{\to }& M\times X\\ \pi \downarrow & & \downarrow \pi \\ M& \stackrel{T}{\to }& M\end{array}$$

$$\begin{array}{ccc}{H}_i(M\times X,\mathbb{F})& \stackrel{F_{\ast }}{\to }& H_i(M\times X,\mathbb{F})\\ {\pi }_{\ast }\downarrow & & \downarrow {\pi }_{\ast }\\ H_i(M,\mathbb{F})& \stackrel{T_{\ast }}{\to }& H_i(M,\mathbb{F})\end{array}$$

$$H_k(M\times X,\mathbb{F})$$

$$\simeq H_k(M,\mathbb{F})\oplus H_k(X,\mathbb{F}).$$

$$\ker ({\pi }_{\ast })\simeq H_0(M,\mathbb{F})\otimes H_k(X,\mathbb{F})\simeq H_k(X,\mathbb{F}).$$

$$\ker ({\pi }_{\ast }^{\prime })\simeq H_k(M,\mathbb{F})\otimes H_0(X,\mathbb{F})\simeq H_k(M,\mathbb{F}).$$

$$H_k(M\times X,\mathbb{F})=\ker ({\pi }_{\ast })\oplus G.$$

$$0\ne (A_p{)}_{\ast }$$

$$={\pi }_{\ast }^{\prime }\circ (i_p{)}_{\ast }=({\pi }^{\prime }\circ i_p{)}_{\ast }=0.$$

$${\mathbb{R}}^3\setminus \{0\}\ni (x,y,z)\mapsto \frac{(x,y,z)}{\Vert (x,y,z)\Vert }\in {\mathbb{S}}^2$$

$$A(x)=\left(\begin{array}{cc}a(x)& -\overline{b(x)}\\ b(x)& \overline{a(x)}\end{array}\right)$$

$$\tilde{A}(x):=\left(\begin{array}{ccc}\lambda I_{k-1}& 0& 0\\ 0& A(\pi (x))& 0\\ 0& 0& \mu I_{m-k-1}\end{array}\right)\in {\mathrm{SL}}_m(\mathbb{C})$$

$${\tilde{A}}_{V_k}(x)$$

$$=H(A_{e_1}(\pi (x)))=(\iota \circ H\circ A_{e_1}\circ \pi )(x)$$

$$\{V_k\}={\Sigma }_0\subset \Sigma ={\Sigma }_1\subset {\Sigma }_2\subset \cdots \subset {\Sigma }_N={\mathrm{Gr}}_k({\mathbb{C}}^m).$$

$$0=H_1({\Sigma }_i,{\Sigma }_{i-1},\mathbb{K})\to H_2({\Sigma }_{i-1},\mathbb{K})\to H_2({\Sigma }_i,\mathbb{K})\to \cdots$$

Full text

1. 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 ${\rm SL}_{2}(\mathbb{R})$-valued cocycle over a circle map, M. Herman remarked that the topological degrees of the base map $T\colon\mathbb{T}\to\mathbb{T}$ and of the matrix valued function $A\colon\mathbb{T}\to{\rm SL}_{2}(\mathbb{R})$ 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 $p\in\mathbb{P}(\mathbb{R}^{2})$, $A_{p}\colon\mathbb{T}\to\mathbb{P}(\mathbb{R}^{2})$ 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 $A\colon\mathbb{T}\to{\rm GL}_{m}(\mathbb{C})$ over irrational translations on the one dimensional torus $\mathbb{T}$ 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 $\mathbb{T}^{d}$, $d\geq 2$. 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 $C_{r}^{\omega}({\mathbb{T}},{\rm Mat}(m,\mathbb{C}))$ of analytic cocycles111We regard functions on the torus $\mathbb{T}=\mathbb{R}/\mathbb{Z}$ as $1$-periodic functions on the real line. Then $C_{r}^{\omega}({\mathbb{T}},{\rm Mat}(m,\mathbb{C}))$ is the space of functions $A\colon\mathbb{T}\to{\rm Mat}(m,\mathbb{C})$ admitting a holomorphic extension to the complex strip \bigl{|}\Im z\bigr{|}<r, continuous up to the boundary and endowed with the uniform norm on the strip. 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 $\epsilon>0$, the complexified cocycle $A(x+iy)$ has dominated splitting for a.e. $y$ with \bigl{|}y\bigr{|}<\epsilon (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 $\mathbb{K}=\mathbb{R}$ or $\mathbb{K}=\mathbb{C}$ refer to either the real or the complex field. Let $\mathbb{T}^{d}=(\mathbb{R}/\mathbb{Z})^{d}$ with $d\geq 2$ be the higher dimensional torus.

A continuous function $A\colon\mathbb{T}^{d}\to{\rm GL}_{m}(\mathbb{K})$ and an ergodic translation $T:\mathbb{T}^{d}\to\mathbb{T}^{d}$ determine the skew-product map $F\colon\mathbb{T}^{d}\times\mathbb{K}^{m}\to\mathbb{T}^{d}\times\mathbb{K}^{m}$,

[TABLE]

We call the new dynamical system $F$ a linear cocycle over the base transformation $T$. Its iterates are $F^{(n)}(x,v)=(T^{n}x,A^{(n)}(x)v)$, where $A^{(n)}(x):=A(T^{n-1}x)\ldots A(Tx)\,A(x)$.

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 $\mathbb{K}$ (or that its Oseledets decomposition is dominated) if there exists a continuous $F$-invariant decomposition $\mathbb{K}^{m}=E_{1}(x)\oplus\ldots\oplus E_{l}(x)$, where $2\leq l\leq m$ and each $E_{i}$ is an $F$-invariant continuous $\mathbb{K}$ sub-bundle of the trivial bundle $\mathbb{T}^{d}\times\mathbb{K}^{m}$ such that for some $\lambda>1$, for any $1\leq i<j\leq l$ and for any unit vectors $v_{i}\in E_{i}(x)$, $v_{j}\in E_{j}(x)$,

[TABLE]

In particular, as $l\geq 2$, the Oseledets decomposition of $A$ is nontrivial (its components are proper subspaces of $\mathbb{K}^{m}$) so the Lyapunov exponents of $A$ are not all equal.

For ${\rm SL}_{2}(\mathbb{R})$-valued cocycles, the dominated splitting property is equivalent to uniform hyperbolicity.

Following the terminology in [2], given $1\leq k<m$, we say that a linear cocycle $A\colon\mathbb{T}^{d}\to{\rm GL}_{m}(\mathbb{K})$ is $k$-dominated if it admits a dominated decomposition $\mathbb{K}^{m}=E^{+}\oplus E^{-}$ with $\dim E^{+}=k$ and where the Lyapunov exponents of $A|_{E^{+}}$ are strictly larger than all Lyapunov exponents of $A|_{E^{-}}$.

It is clear that if the linear cocycle $A$ has the dominated splitting $\mathbb{K}^{m}=E_{1}(x)\oplus\ldots\oplus E_{l}(x)$, then $A$ is $k$-dominated for every dimension $k=\dim(E_{1})+\ldots+\dim(E_{i})$ with $1\leq i\leq l-1$.

We are now ready to formulate the main result of this paper.

Theorem 1**.**

Given integers $d\geq 2$ and $1\leq k<m$ there exist analytic quasi-periodic cocycles $A\colon\mathbb{T}^{d}\to{\rm GL}_{m}(\mathbb{C})$ with an invariant measurable decomposition $\mathbb{C}^{m}=E^{+}\oplus E^{-}$ such that

1. (1)

$\dim E^{+}=k$, 2. (2)

all Lyapunov exponents of $A|_{E^{+}}$ are positive, 3. (3)

all Lyapunov exponents of $A|_{E^{-}}$ are negative, 4. (4)

no continuous cocycle $B\colon\mathbb{T}^{d}\to{\rm GL}_{m}(\mathbb{C})$ in the homotopy class of $A$ is $k$-dominated.

Remark 1*.*

This theorem shows that the dichotomy in [2, Theorem 1.1] does not hold for analytic quasi-periodic cocycles over a torus $\mathbb{T}^{d}$ of dimension $d\geq 2$. In fact any sufficiently small neighborhood $\mathscr{V}$ of $A$ is contained in the homotopy class of $A$. In this neighborhood $\mathscr{V}$, by our continuity result [6, Theorem 6.1], assuming that the translation vector satisfies a generic Diophantine condition, the Oseledets decomposition $\mathbb{C}^{m}=E^{+}\oplus E^{-}$ persists with $\dim E^{+}=k$. However, in view of Theorem 1, this decomposition is never $k$-dominated.

Consider now the projective space $\mathbb{P}(\mathbb{K}^{m})$ where the group ${\rm GL}_{m}(\mathbb{K})$ acts transitively. More generally let ${\rm Gr}_{k}(\mathbb{K}^{m})$ be the Grassmannian space of all $k$-dimensional $\mathbb{K}$-linear subspaces of $\mathbb{K}^{m}$, which reduces to the projective space when $k=1$.

The cocycle $F$ determines the skew-product map $\hat{F}\colon\mathbb{T}^{d}\times{\rm Gr}_{k}(\mathbb{K}^{m})\to\mathbb{T}^{d}\times{\rm Gr}_{k}(\mathbb{K}^{m})$ defined by $\hat{F}(x,V):=(Tx,A(x)V)$. Clearly the $k$-domination property implies the existence of a continuous invariant section $E^{+}\colon\mathbb{T}^{d}\to{\rm Gr}_{k}(\mathbb{K}^{m})$ for the bundle map $\hat{F}$. The strategy to prove Theorem 1 is to derive topological obstructions to the existence of continuous invariant sections $\sigma\colon\mathbb{T}^{d}\to{\rm Gr}_{k}(\mathbb{C}^{m})$ of the cocycle $A$.

Remark 2*.*

The statement of Theorem 1 hods also for ${\rm GL}_{m}(\mathbb{R})$-valued cocycles over a torus $\mathbb{T}^{d}$ with dimension $d\geq 1$. 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 ${\rm Gr}_{k}(\mathbb{R}^{m})$ are not simply connected, something which is not true about the complex Grassmannians ${\rm Gr}_{k}(\mathbb{C}^{m})$.

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.

2. Existence of invariant sections

We call factor of linear maps any commutative diagram

[TABLE]

where $E$, $F$ are vector spaces, $f\colon E\to E$, $h\colon F\to F$ are linear endomorphisms and $\pi\colon E\to F$ is a linear epimorphism. We call splitting of a factor (1) any linear map $\sigma\colon F\to E$ such that $\pi\circ\sigma={\rm id}_{F}$ and $f\circ\sigma=\sigma\circ h$. In other words $\sigma$ is an $f$-invariant section of the vector bundle $\pi\colon E\to F$.

Letting $K=\ker(\pi)$, by the fundamental theorem on homomorphisms, the linear epimorphism $\pi\colon E\to F$ induces an isomorphism $\bar{\pi}\colon E/K\simeq F$ through which the factor (1) can be expressed as

[TABLE]

where $K$ stands for an $f$-invariant vector subspace of $E$. From these considerations it follows easily that

Proposition 1**.**

The factor (1) has a splitting   if and only if  the vector space $E$ admits an $f$-invariant decomposition $E=G\oplus\ker(\pi)$.

Let $M$ be a compact connected manifold. Consider a continuous map $T:M\to M$ and a transitive action $G\times X\to X$ of a connected Lie group $G$ on some compact connected homogeneous space $X$. A continuous function $A:M\to G$ determines the skew-product map

[TABLE]

By definition, letting $\pi\colon M\times X\to M$ stand for the canonical projection $\pi(x,p)=x$, the following diagram commutes

[TABLE]

We call $F$-invariant section any continuous map $\sigma\colon M\to X$ such that $F(x,\sigma(x))=(Tx,\sigma(Tx))$ for all $x\in M$.

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).

Proposition 2**.**

Given a number field $\mathbb{F}$, if the map (3) admits an invariant section then for each $0\leq i\leq\dim M$ the homological factor

[TABLE]

admits a splitting.

Proof.

By the Künneth theorem the map $\pi_{\ast}\colon H_{i}(M\times X,\mathbb{F})\to H_{i}(M,\mathbb{F})$ is surjective. If $\sigma\colon M\to X$ is an $F$-invariant section then its homology $\sigma_{\ast}\colon H_{i}(M,\mathbb{F})\to H_{i}(M\times X,\mathbb{F})$ is a splitting of the homological factor (5). ∎

The next proposition specializes the previous criterion to the case where the map $T\colon M\to M$ is homotopic to the identity.

Proposition 3**.**

Let $T\colon M\to M$ be a continuous map homotopic to the identity. Let $\mathbb{F}$ be a number field, $A\colon M\to G$ a continuous function and $1\leq k\leq\dim M$ a dimension such that:

1. (1)

$H_{k}(M,\mathbb{F})\neq\{0\}$, 2. (2)

$H_{i}(M,\mathbb{F})=\{0\}$*   or   $H_{k-i}(X,\mathbb{F})=\{0\}$, for all $0<i<k$,* 3. (3)

For some $p\in X$ the map $A_{p}\colon M\to X$, $A_{p}(x):=A(x)p$, induces a non-zero homology map in dimension $k$, i.e., the linear map $(A_{p})_{\ast}\colon H_{k}(M,\mathbb{F})\to H_{k}(X,\mathbb{F})$ is non zero.

Then $F$ admits no $F$-invariant section.

Proof.

By the Künneth theorem and assumptions (1)-(2),

[TABLE]

We are also using here that $M$ and $X$ are connected so that $H_{0}(M,\mathbb{F})\simeq H_{0}(X,\mathbb{F})\simeq\mathbb{F}$. Hence the epimorphism $\pi_{\ast}\colon H_{k}(M\times X,\mathbb{F})\to H_{k}(M,\mathbb{F})$ has kernel

[TABLE]

Similarly, the projection $\pi^{\prime}\colon M\times X\to X$, $\pi^{\prime}(x,p)=p$, induces a homology map $\pi^{\prime}_{\ast}\colon H_{k}(M\times X,\mathbb{F})\to H_{k}(X,\mathbb{F})$ with kernel

[TABLE]

Because $G$ is connected, each element $A(x)\in G$ induces an action $A(x)\colon X\to X$ which is isotopic to the identity. Therefore the homology map $F_{\ast}\colon H_{k}(M\times X,\mathbb{F})\to H_{k}(M\times X,\mathbb{F})$ acts as the identity on $\ker(\pi_{\ast})$.

Assume now, by contradiction, that $F$ admits an invariant section. By Proposition 2 there exists an $F_{\ast}$-invariant subspace $G$ such that

[TABLE]

Since $T\colon M\to M$ is homotopic to ${\rm id}_{M}$ we have $T_{\ast}={\rm id}$ on $H_{k}(M,\mathbb{F})$. This implies that $F_{\ast}$ is the identity map on $G$. Hence, because (6) is $F_{\ast}$-invariant, it follows that $F_{\ast}$ is the identity on $H_{k}(M\times X,\mathbb{F})$.

Finally, defining the inclusion map $i_{p}\colon M\to M\times X$, $i_{p}(x):=(x,p)$, since $A_{p}=\pi^{\prime}\circ F\circ i_{p}$ we have at the homology level

[TABLE]

We have used assumption (3) and the fact that the composition $\pi^{\prime}\circ i_{p}$ is a constant map. This contradiction proves that $F$ admits no invariant section. ∎

3. Consequences for quasi-periodic cocycles

Finally we show that for certain homotopy types a continuous quasi-periodic cocycle $A\colon\mathbb{T}^{d}\to{\rm GL}_{m}(\mathbb{C})$ cannot have dominated splitting. The base dynamics is assumed to be an ergodic translation of a torus $\mathbb{T}^{d}$ of dimension $d\geq 2$.

Let ${\rm Gr}_{k}(\mathbb{C}^{m})$ denote the complex Grassmannian of $k$-dimensional complex subspaces of $\mathbb{C}^{m}$.

Proposition 4**.**

Let $A\colon\mathbb{T}^{d}\to{\rm GL}_{m}(\mathbb{C})$ be a continuous function with $d\geq 2$. If the map $A_{V}:\mathbb{T}^{d}\to{\rm Gr}_{k}(\mathbb{C}^{m})$, $A_{V}(x)=A(x)V$, for some $V\in{\rm Gr}_{k}(\mathbb{C}^{m})$, induces a non-zero homology map in dimension two, i.e., the linear map $(A_{V})_{\ast}\colon H_{2}(\mathbb{T}^{d},\mathbb{F})\to H_{2}({\rm Gr}_{k}(\mathbb{C}^{m}),\mathbb{F})$ is non zero for some field $\mathbb{F}$ and some $1\leq k<m$, then the quasi-periodic cocycle $A$ has no continuous invariant section $\sigma\colon\mathbb{T}^{d}\to{\rm Gr}_{k}(\mathbb{C}^{m})$. In particular $A$ is not $k$-dominated.

Proof.

Let us apply Proposition 3 with $M=\mathbb{T}^{d}$, $X={\rm Gr}_{k}(\mathbb{C}^{m})$ and dimension $k=2$. For any field $\mathbb{F}$ we have $\dim H_{0}({\rm Gr}_{k}(\mathbb{C}^{m}),\mathbb{F})=1$ because ${\rm Gr}_{k}(\mathbb{C}^{m})$ is a connected manifold. We have $\dim H_{1}({\rm Gr}_{k}(\mathbb{C}^{m}),\mathbb{F})=0$ and $\dim H_{2}({\rm Gr}_{k}(\mathbb{C}^{m}),\mathbb{F})\geq 1$ (see [9, Section 3.2] or [7, Section 5 of Chapter 1]). We also have $H_{2}(\mathbb{T}^{d},\mathbb{F})\simeq\mathbb{F}^{\binom{d}{2}}\neq\{0\}$ because $d\geq 2$. 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 map $\hat{F}\colon\mathbb{T}^{d}\times{\rm Gr}_{k}(\mathbb{C}^{m})\to\mathbb{T}^{d}\times{\rm Gr}_{k}(\mathbb{C}^{m})$ does not admit any $\hat{F}$-invariant section.

Finally, if the quasi-periodic cocycle $A$ is $k$-dominated then the $F$-invariant sub-bundle $E^{+}$ determines an $\hat{F}$-invariant section $E^{+}:\mathbb{T}^{d}\to{\rm Gr}_{k}(\mathbb{C}^{m})$. This contradiction proves that $A$ is not $k$-dominated. ∎

Corollary 1**.**

Consider a quasi-periodic cocycle $A\colon\mathbb{T}^{2}\to{\rm GL}_{2}(\mathbb{C})$. If the map $A_{\hat{v}}:\mathbb{T}^{2}\to\mathbb{P}(\mathbb{C}^{2})$, $A_{\hat{v}}(x)=A(x)\hat{v}$, for some $\hat{v}\in\mathbb{P}(\mathbb{C}^{2})$, is not homotopic to a constant then $A$ does not have dominated splitting.

Proof.

The projective space $\mathbb{P}(\mathbb{C}^{2})$ can be identified with the Riemann sphere $\mathbb{S}^{2}\equiv\mathbb{C}\cup\{\infty\}$. Since $A_{\hat{v}}$ is not homotopic to a constant, by Hopf theorem $\deg(A_{\hat{v}})\neq 0$. Then, making the canonical identifications $H_{2}(\mathbb{T}^{2},\mathbb{F})\simeq\mathbb{F}$ and $H_{2}(\mathbb{P}(\mathbb{C}^{2}),\mathbb{F})\simeq\mathbb{F}$, the homology map $(A_{\hat{v}})_{\ast}\colon H_{2}(\mathbb{T}^{2},\mathbb{F})\to H_{2}(\mathbb{P}(\mathbb{C}^{2}),\mathbb{F})$ is the multiplication by $\deg(A_{\hat{v}})$, and hence it is non zero. ∎

Corollary 2**.**

There are continuous functions $A\colon\mathbb{T}^{2}\to{\rm GL}_{2}(\mathbb{C})$ whose homotopy classes contain no quasi-periodic cocycle with dominated splitting.

Proof.

Consider any analytic map $f\colon\mathbb{T}^{2}\to\mathbb{R}^{3}\setminus\{0\}$. Let $p\colon\mathbb{R}^{3}\setminus\{0\}\to\mathbb{P}(\mathbb{C}^{2})$ be the composition of the projection

[TABLE]

onto the unit sphere $\mathbb{S}^{2}$ with the stereographic projection, which maps $\mathbb{S}^{2}$ diffeomorphically onto the projective space $\mathbb{P}(\mathbb{C}^{2})=\mathbb{C}\cup\{\infty\}$.

Assume that the origin belongs to a bounded connected component of $\mathbb{R}^{3}\setminus f(\mathbb{T}^{2})$. Then the parametric hypersurface $f$ has non zero winding number around [math], which implies that the composition $\phi=p\circ f\colon\mathbb{T}^{2}\to\mathbb{P}(\mathbb{C}^{2})$ has non zero degree.

Write $\phi=a/b$ as the ratio of two real analytic functions $a,b\colon\mathbb{T}^{2}\to\mathbb{C}$, where $b$ vanishes exactly at the points $x\in\mathbb{T}^{2}$ where $\phi(x)=\infty$ and the pair $(a(x),b(x))\neq(0,0)$ for all $x\in\mathbb{T}^{2}$. Then the analytic function $A\colon\mathbb{T}^{2}\to{\rm GL}_{2}(\mathbb{C})$

[TABLE]

satisfies the assumption of Corollary 1 with $\hat{v}=\widehat{(1,0)}$. Hence it cannot have dominated splitting.

Finally, if $B\colon\mathbb{T}^{2}\to{\rm GL}_{2}(\mathbb{C})$ is another continuous function homotopic to $A$ then the functions $A_{\hat{v}}\colon\mathbb{T}^{2}\to\mathbb{P}(\mathbb{C}^{2})$, $A_{\hat{v}}(x)=A(x)\hat{v}$, and $B_{\hat{v}}\colon\mathbb{T}^{2}\to\mathbb{P}(\mathbb{C}^{2})$, $B_{\hat{v}}(x)=B(x)\hat{v}$, are also homotopic. Hence $B_{\hat{v}}$ 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.

Proposition 5**.**

Given integers $d\geq 2$ and $1\leq k<m$ there exist analytic quasi-periodic cocycles $A\colon\mathbb{T}^{d}\to{\rm GL}_{m}(\mathbb{C})$ with an invariant measurable decomposition $\mathbb{C}^{m}=E^{+}\oplus E^{-}$ such that

1. (1)

$\dim E^{+}=k$, 2. (2)

all Lyapunov exponents of $A|_{E^{+}}$ are positive, 3. (3)

all Lyapunov exponents of $A|_{E^{-}}$ are negative, 4. (4)

no continuous cocycle $B\colon\mathbb{T}^{d}\to{\rm GL}_{m}(\mathbb{C})$ in the homotopy class of $A$ admits a continuous invariant section $\sigma\colon\mathbb{T}^{d}\to{\rm Gr}_{k}(\mathbb{C}^{m})$.

Proof.

Consider an analytic quasi-periodic cocycle $A\colon\mathbb{T}^{2}\to{\rm SL}_{2}(\mathbb{C})$ in the homotopy class of a cocycle given by Corollary 2. By [1, Theorem 1], possibly perturbing it, we can assume that $A$ admits an invariant measurable decomposition $\mathbb{C}^{m}=E^{+}\oplus E^{-}$ with $\dim E^{-}=\dim E^{+}=1$ and having non-zero Lyapunov exponents, w.r.t. an ergodic translation with frequency vector $\omega\in\mathbb{T}^{2}$.

Take positive numbers $\lambda>1>\mu$ such that $\lambda^{k-1}\,\mu^{m-k-1}=1$ and let $\tilde{A}\colon\mathbb{T}^{d}\to{\rm SL}_{m}(\mathbb{C})$ be the cocycle

[TABLE]

where $I_{k-1}$ and $I_{m-k-1}$ stand for identity matrices of the specified dimensions, and $\pi\colon\mathbb{T}^{d}\to\mathbb{T}^{2}$ denotes the projection $\pi(x_{1},\ldots,x_{d}):=(x_{1},x_{2})$. By construction the cocycle $\tilde{A}\colon\mathbb{T}^{d}\to{\rm SL}_{m}(\mathbb{C})$ satisfies properties (1)-(3), w.r.t. any ergodic translation with frequency vector $\tilde{\omega}\in\mathbb{T}^{d}$ such that $\pi(\tilde{\omega})=\omega$.

We are going to use Proposition 3 to prove item (4). For each $1\leq i\leq m$, let $V_{i}\in{\rm Gr}_{k}(\mathbb{C}^{m})$ be the complex $i$-plane generated by the first $i$ vectors $e_{1},\ldots,e_{i}$ of the canonical basis of $\mathbb{C}^{m}$. We claim that the map $\tilde{A}_{V_{k}}\colon\mathbb{T}^{d}\to{\rm Gr}_{k}(\mathbb{C}^{m})$, $\tilde{A}_{V_{k}}(x):=\tilde{A}(x)\,V_{k}$, induces a non zero linear map $(\tilde{A}_{V_{k}})_{\ast}\colon H_{2}(\mathbb{T}^{d},\mathbb{K})\to H_{2}({\rm Gr}_{k}(\mathbb{C}^{m}),\mathbb{K})$. By construction this is true about the map $A_{e_{1}}\colon\mathbb{T}^{2}\to{\rm Gr}_{k}(\mathbb{C}^{2})$, $A_{e_{1}}(x):=A(x)\,\hat{e}_{1}$, which induces a non zero linear map at the second homology level. To relate the homologies of $\tilde{A}_{V_{k}}$ and $A_{e_{1}}$ we factor the first, $\tilde{A}_{V_{k}}$, as a composition of several maps which include the second, $A_{e_{1}}$.

Let $\Sigma:=\{V\in{\rm Gr}_{k}(\mathbb{C}^{m})\,\colon\,V_{k-1}\subset V\subset V_{k+1}\,\}$. This is a complex analytic submanifold of the Grassmannian space ${\rm Gr}_{k}(\mathbb{C}^{m})$, which is diffeomorphic to the complex projective line $\mathbb{P}(\mathbb{C}^{2})$. Let $p\colon\mathbb{C}^{m}\to\mathbb{C}^{2}$ be the linear projection $p(z_{1},\ldots,z_{m})=(z_{k},z_{k+1})$, and define $H\colon\mathbb{P}(\mathbb{C}^{2})\to\Sigma$ by $H(\hat{v}):=V_{k-1}+\mathbb{C}\,p(v)$. Then for all $x\in\mathbb{T}^{d}$,

[TABLE]

where $\iota$ stands for the inclusion map $\iota\colon\Sigma\to{\rm Gr}_{k}(\mathbb{C}^{m})$.

By Künneth theorem, the linear map $\pi_{\ast}\colon H_{2}(\mathbb{T}^{d},\mathbb{F})\to H_{2}(\mathbb{T}^{2},\mathbb{F})$ is surjective. Because $H$ is a diffeomorphism, the homology map $H_{\ast}$ is an isomorphism. We are left to prove that $\iota_{\ast}\colon H_{2}(\Sigma,\mathbb{K})\to H_{2}({\rm Gr}_{k}(\mathbb{C}^{m}),\mathbb{K})$ is injective. This will imply that $(\tilde{A}_{V_{k}})_{\ast}$ is non zero and, by Proposition 3, that no cocycle homotopic to $\tilde{A}$ admits a continuous invariant section with values in ${\rm Gr}_{k}(\mathbb{C}^{m})$.

Let us now turn to prove the injectivity of $\iota_{\ast}$. The Grassmannian ${\rm Gr}_{k}(\mathbb{C}^{m})$ 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 ${\rm Gr}_{k}(\mathbb{C}^{m})$ 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 $\Sigma$ is itself a Schubert cycle with complex dimension $1$ which can be integrated in a cell decomposition

[TABLE]

Each space $\Sigma_{i}$ is an analytic subvariety obtained from $\Sigma_{i-1}$ by joining a cell with (real) even dimension and boundary contained in $\Sigma_{i-1}$. This implies that $H_{1}(\Sigma_{i},\Sigma_{i-1},\mathbb{K})=0$ for all $i$ and all fields $\mathbb{K}$. Hence, by the long exact sequence of the pair $(\Sigma_{i},\Sigma_{i-1})$,

[TABLE]

is an exact sequence. Therefore, because $\iota$ can be factored as the composition of the inclusions $\Sigma_{i-1}\hookrightarrow\Sigma_{i}$ with $i=2,\ldots,N$, the map $\iota$ is injective at the second homology level. ∎

Remark 3*.*

Given a cocycle $A\in C^{\omega}_{r}(\mathbb{T}^{d},{\rm GL}_{m}(\mathbb{R}))$ in one of the homotopy classes from Proposition 5, the cocycle $A_{y}:\mathbb{T}^{d}\to{\rm GL}_{m}(\mathbb{C})$, $A_{y}(x)=A(x+iy)$, cannot have dominated splitting for any $y\in\mathbb{R}^{d}$.

Acknowledgments

The first author was supported by Fundação para 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).