DENSITY OF POSITIVE LYAPUNOV EXPONENTS FOR QUASIPERIODIC SL(2,R) COCYCLES IN ARBITRARY DIMENSION

We show that given a fixed irrational rotation of the d-dimensional torus, any analytic SL(2, R) cocycle can be perturbed so that the Lyapunov exponent becomes positive. This result strengthens and generalizes previous results of Krikorian [K] and Fayad-Krikorian [FK]. The key technique is the analiticity of m-functions (under the hypothesis of stability of zero Lyapunov exponents), first observed and used in the solution of the Ten Martini Problem [AJ]. In the appendix, we discuss the smoothness of m-functions for a larger class of systems including the skew-shift.

The Lyapunov exponent is defined by where A n ∈ C 0 (R d /Z d , SL(2, R)) is given by so that (1.4) (α, A) n = (nα, A n ).
Our main result is the following.Let us say that α ∈ R d is irrational if x → x + α is transitive on R d /Z d .
Theorem 1.1.Let α ∈ R d be irrational.Then there is a dense set of A ∈ C ω (R d /Z d , SL(2, R)) (in the usual inductive limit topology) such that L(α, A) > 0.
We will carry the proof assuming irrationality of α to simplify the exposition.
In [FK], this result was proved for d = 1 and in the C ∞ -topology (improving on [K] where an additional arithmetic condition on α, stronger than Diophantine, was imposed).Our result implies theirs, since the inclusion from C ω to C ∞ is continuous with dense image.But the methods are quite different: [FK] uses renormalization techniques (as in [K]), which seem to be bound to the one-dimensional case, while we use a complex analytic approach (as in [AJ]).We notice that the arithmetic properties of α are almost irrelevant in our approach.* CNRS UMR 7599.I would like to thank Bassam Fayad and Raphaël Krikorian for telling me about their result, which prompted me to write this note.
It is reasonable to expect that positive Lyapunov exponent is dense in the much broader context of cocycles over smooth dynamical systems. 1 This is known to be the case for "fast systems" (expanding/hyperbolic), and more generally for systems which present periodic orbits of arbitrarily large period.The proof in those cases can be carried out using a result of Kotani (originally stated for Schrödinger cocycles, but which generalizes to the context of SL(2, R) cocycles): stability of zero Lyapunov exponent implies a continuous conjugation to a cocycle of rotations.
The case of quasiperiodic systems will be treated here via an improvement of Kotani result: in this setting Kotani's continuous conjugacy to rotations turns out to be analytic (this was first remarked in [AJ] in the particular case of "almost Mathieu" cocycles as a step in the solution of the "Ten Martini Problem").This fact easily leads to the density of positive Lyapunov exponent result, since analytic cocycles over rotations are easy to perturb.
Though we are unable to treat at this point the problem of density of positive Lyapunov exponent for general dynamical systems, we would like to point out that one can still improve Kotani's Theorem for a large class of "slow systems" (including the skew-shift (x, y) → (x + α, y + x)).We will give this argument in the appendix.
If α is irrational and L(α, A) > 0 then Oseledets theorem implies that there exists a measurable decomposition of ) and E u (x) are actually continuous functions of x then (α, A) is said to be uniformly hyperbolic.Otherwise it is said to be non-uniformly hyperbolic.Thus our result states that hyperbolicity is dense in this context, though it does not specify whether one gets uniform or non-uniform hyperbolicity.Notice that uniform hyperbolicity is not dense in general, since there are topological obstructions (since uniform hyperbolicity implies that the cocycle is homotopic to the identity).It seems likely that uniform hyperbolicity is not dense in d ≥ 2 frequencies even in the case of cocycles homotopic to the identity (this is indicated by the work of Chulaevsky-Sinai [CS]).It is an interesting open problem to show that uniform hyperbolicity is dense in the one-frequency, homotopic to the identity context.1.1.Outline.We consider perturbations of a fixed cocycle obtained by postcomposition with a rotation.If the Lyapunov exponent is zero for all such (small) perturbation, Kotani Theorem implies that the cocycle is continuously conjugate to a cocycle of rotations.(This is also the starting point of [K] and [FK].) An analytic extension argument (first used in this context in [AJ]) improves this result: the cocycle is actually analytically conjugate to a cocycle of rotations.After a small perturbation (needed to treat non-Diophantine frequencies), such a cocycle can be put into a normal form.It is easy to perturb directly the normal form to get a positive Lyapunov exponent.
Our aim is to prove the following result.
1 For our purposes, a smooth dynamical systems is assumed to be supplied with a probability measure with full support.
Consider the action of SL(2, C) on C by Moebius transformations: Then there exists a unique analytic map m : . We may assume that (θ, x) ∈ Ω implies that (θ, x + y) ∈ Ω, y ∈ R d .The Schwarz Lemma then implies that . Moreover, m is the limit of the sequence of holomorphic functions (2.7) and hence is holomorphic.
The following lemma gives a well-known generalization of a theorem of Kotani for Schrödinger cocycles (see [Sim]), adapted to the general context of analytic cocycles.We point the reader to [AK] for the full proof (of an interesting generalization).
Lemma 2.3.Let α, A, m be as in the previous lemma.If there exists an open interval 0 ∈ I ⊂ R such that L(α, R θ A) = 0 for θ ∈ I then there exists a continuous extension m : (2.9) Obviously a(θ) ≤ 0 for all θ, and a(θ) < 0 exactly when x → m(θ, x) is analytic in x (in particular if θ ∈ H).We must show that a(0) < 0. Let r > 0 be small so that D r is compactly Remark A.1.By the previous theorem, the question of (C ∞ ) density of positive Lyapunov exponents (for subexponential dynamics) is thus reduced to whether one can approach a C ∞ cocycle of rotations by a SL(2, R) cocycle with positive Lyapunov exponent.Though this is very easy to do in the case of rotations, the case of the skew-shift is already (as far as we know) open.
contained in C \ (R \ I)).Then |m(θ, x)| < C for θ ∈ D r , x ∈ R d /Z d .Thus a|D r is the lim sup of a sequence of non