Lyapunov Exponents For Some Quasi-Periodic Cocycles

We consider SL(2,R)-valued cocycles over rotations of the circle and prove that they are likely to have Lyapunov exponents ≈ ± logλ if the norms of all of the matrices are ≈ λ. This is proved for λ sufficiently large. The ubiquity of elliptic behavior is also observed. Consider an area preserving diffeomorphism f of a compact surface. Assume that f is not uniformly hyperbolic, but that it has obvious hyperbolicity properties on a large part of phase space. We are interested in whether or not f has positive Lyapunov exponents on a positive measure set. A widely shared belief among workers in the subject is that positive exponents are quite prevalent, and numerical evidence seems to substantiate that view. Yet so far little has been proved beyond systems with continuous families of invariant cones. One reason why it is hard to obtain a lower bound for ‖Dfn x ‖ is that even when both Df x and Df m fnx are strongly hyperbolic matrices, it can happen that ‖Dfn+m x ‖ ≪ ‖Dfm fnx‖ · ‖Dfn x ‖. This phenomenon has been used to prove the C genericity of zero exponents (see [M]). It is also related to the fact that small perturbations near homoclinic tangencies can produce elliptic periodic orbits (see [N]). Elliptic behavior, when present, further complicates the task of proving positive exponents. In this paper we consider a model problem. Let T : (X,m) be a measure preserving transformation, and let A : X → GL(n,R) be an arbitrary mapping. We are interested in the Lyapunov exponents of · · ·A(T x) ·A(Tx) ·A(x) for m− a.e. x. Abusing language slightly we call (T ;A) a cocycle. Sometimes we will also refer to A as a cocycle over T . Clearly, the cocycle setting is more general; it includes among other things random matrices and diffeomorphisms. It is easier, however, to work with the space of cocycles than the space of diffeomorphisms, because for cocycles one can vary the dynamics and matrix maps independently. Our theorems imply the following picture. Let λ ∈ R be a large number. We consider a 2-parameter family of cocycles (Tα;At) where Tα : S 1 is rotation by 2πα and {At} is a generic C family of maps from S to SL(2,R) satisfying ||At(x)|| ≈ λ for all x, t. For simplicity let us rule out the possibility that for some open set in t-space (Tα;At) is uniformly hyperbolic for every α. Then (1) the set of parameters (α, t) for which the Lyapunov exponents of (Tα;At) are ≈ ± log λ has nearly full measure; *This research is partially supported by the National Science Foundation. The author wishes also to acknowledge the hospitality and support of the Mittag Leffler Institute, where part of this work was done. Typeset by AMS-TEX 1 2 (2) the closure of the set of (α, t) where α ∈ Q and (Tα;At) has some elliptic behavior also has nearly full measure. Both sets tend to full measure as λ → ∞. Precise formulations of these results are given in Section 1. In broad outline there is much in common between our proofs of (1) and those in [J] and [BC], particularly the latter. We consider a 1-parameter family of cocycles, inductively identify certain regions of criticality, study orbit segments that begin and end near these regions, and try to concatenate long blocks of matrices that have been shown to be hyperbolic. Parameters are deleted to ensure the hyperbolicity of the concatenated blocks, and the induction moves forward. The idea of inductively constructing a “critical set” was first used in [BC]. Among the many known results on the positivity of Lyapunov exponents, we mention in particular those for random matrices (see e.g. [F]), the Schrödinger operator in 1-dimension (see e.g. [FSW], [Ko] and [S]), results using the analytic techniques of Herman (see [H] and e.g. [Kn], [SS]), and those for dynamical systems for which invariant cones have been identified (see e.g. [W]). See also [You] for some examples of nonuniformly hyperbolic cocycles. §1 Precise statement of results This paper is about cocycles in which the norms of the matrices are uniformly large. More precisely, for C, λ ≥ 1, let AC,λ := { A : S → SL(2,R) s.t. A is a C map and (i) C−1λ ≤ ‖A(x)‖ ≤ Cλ ∀x ∈ S , (ii) ‖ dx ‖ , ‖ −1 dx ‖ ≤ Cλ } . We consider A ∈ AC,λ where C is thought of as O(1) and λ is as large as need be. It will be shown in Section (2.1) that if T : S is a rotation and A ∈ AC,λ, then (T ;A) is equivalent to another cocycle (T ;A′) where A′ = ( λ 0 0 λ ) ◦ B and ‖B±‖, ‖dB± dx ‖ ≤ O(1). Our theorems deal exclusively with cocycles in this canonical form. For v 6= 0 ∈ R, let v̄ ∈ P denote the projectivization of v. Similarly, if A : R → R is a linear map with detA 6= 0, let Ā : P → P be the projectivization of A. We coordinatize P as P = [0, π]/{0, π} by positively orienting S and letting θ = 0 correspond to (