Cantor Spectrum for Schrödinger Operators with Potentials Arising from Generalized Skew-shifts

We consider continuous SL(2,R)-cocycles over a strictly ergodic homeomorphism which fibers over an almost periodic dynamical system (generalized skew-shifts). We prove that any cocycle which is not uniformly hyperbolic can be approximated by one which is conjugate to an SO(2,R)-cocycle. Using this, we show that if a cocycle’s homotopy class does not display a certain obstruction to uniform hyperbolicity, then it can be C0-perturbed to become uniformly hyperbolic. For cocycles arising from Schrödinger operators, the obstruction vanishes and we conclude that uniform hyperbolicity is dense, which implies that for a generic continuous potential, the spectrum of the corresponding Schrödinger operator is a Cantor set. 1. Statement of the Results Throughout this paper we letX be a compact metric space. Furthermore, unless specified otherwise f : X → X will be a strictly ergodic homeomorphism (i.e., f is minimal and uniquely ergodic) that fibers over an almost periodic dynamical system. This means that there exists an infinite compact abelian group G and an onto continuous map h : X → G such that h( f (x)) = h(x) + α for some α ∈ G. Examples of particular interest include: • minimal translations of the d-torus T, for any d ≥ 1; • the skew-shift (x, y) 7→ (x + α, y + x) on T, where α is irrational. 1.1. Results for SL(2,R)-Cocycles. Given a continuous map A : X → SL(2,R), we consider the skew-product X × SL(2,R) → X × SL(2,R) given by (x, 1) 7→ ( f (x),A(x) · 1). This map is called the cocycle ( f ,A). For n ∈ Z, A is defined by ( f ,A) = ( f ,A). We say a cocycle ( f ,A) is uniformly hyperbolic if there exist constants c > 0, λ > 1 such that ‖A(x)‖ > cλ for every x ∈ X and n > 0. This is equivalent to the usual hyperbolic splitting condition: see [Y]. Recall that uniform hyperbolicity is an open condition in C(X, SL(2,R)). We say that two cocycles ( f ,A) and ( f , Ã) are conjugate if there exists a conjugacy B ∈ C(X, SL(2,R)) such that Ã(x) = B( f (x))A(x)B(x). Our first result is: Date: April 22, 2008. This research was partially conducted during the period A. A. served as a Clay Research Fellow. J. B. was partially supported by a grant from CNPq–Brazil. D. D. was supported in part by NSF grant DMS–0653720. We benefited from CNPq and Procad/CAPES support for traveling. 1Some authors say that the cocycle has an exponential dichotomy. 2If A is a real 2×2 matrix, then ‖A‖ = sup‖v‖,0 ‖A(v)‖/‖v‖, where ‖v‖ is the Euclidean norm of v ∈ R 2. 1 2 AVILA, BOCHI, AND DAMANIK Theorem 1. Let f be as above. If A : X → SL(2,R) is a continuous map such that the cocycle ( f ,A) is not uniformly hyperbolic, then there exists a continuous à : X → SL(2,R), arbitrarily C-close to A, such that the cocycle ( f , Ã) is conjugate to an SO(2,R)-valued cocycle. Remark 1. A cocycle ( f ,A) is conjugate to a cocycle of rotations if and only if there exists C > 1 such that ‖A(x)‖ ≤ C for every x ∈ X and n ∈ Z (here it is enough to assume that f is minimal); see [Cam, EJ, Y]. Remark 2. In Theorem 1, one can drop the hypothesis of unique ergodicity of f (still asking f to be minimal and to fiber over an almost periodic dynamics), as long as X is finite dimensional. See Remark 8. Next we focus on the opposite problem of approximating a cocycle by one that is uniformly hyperbolic. As we will see, this problem is related to the important concept of reducibility. To define reducibility, we will need a slight variation of the notion of conjugacy. Let us say that twococycles ( f ,A) and ( f , Ã) are PSL(2,R)-conjugate if there existsB ∈ C(X,PSL(2,R)) such that Ã(x) = B( f (x))A(x)B(x) (the equality being considered in PSL(2,R)). We say that ( f ,A) is reducible if it is PSL(2,R)-conjugate to a constant cocycle. Remark 3. Reducibility does not imply, in general, that ( f ,A) is conjugate to a constant cocycle, which would correspond to taking B ∈ C(X, SL(2,R)). For example, letX = T, f (x) = x+α. LetH = diag(2, 1/2), and define A(x) = R−π(x+α)HRπx. 3 Notice A is continuous, ( f ,A) is PSL(2,R)-, but not SL(2,R)-, conjugate to a constant. (For an example where ( f ,A) is not uniformly hyperbolic, see Remark 9.) Let us say that a SL(2,R)-cocycle ( f ,A) is reducible up to homotopy if there exists a reducible cocycle ( f , Ã) such that the maps A and à : X → SL(2,R) are homotopic. Let Ruth be the set of all A such that ( f ,A) is reducible up to homotopy. Remark 4. In the case that f is homotopic to the identity map, it is easy to see that Ruth coincides with the set of maps A : X → SL(2,R) that are homotopic to a constant. It is well known that there exists an obstruction to approximating a cocycle by a uniformly hyperbolic one: a uniformly hyperbolic cocycle is always reducible up to homotopy (see Lemma 4). Our next result shows that, up to this obstruction, uniform hyperbolicity is dense. Theorem 2. Uniform hyperbolicity is dense in Ruth. This result is obtained as a consequence of a detailed investigationof theproblem of denseness of reducibility: Theorem 3. Reducibility is dense in Ruth. More precisely: a) If ( f ,A) is uniformly hyperbolic, then it can be approximated by a reducible cocycle (which is uniformly hyperbolic). b) If A ∈ Ruth, but ( f ,A) is not uniformly hyperbolic and A∗ ∈ SL(2,R) is nonhyperbolic (i.e., |trA∗| ≤ 2), then ( f ,A) lies in the closure of thePSL(2,R) conjugacy class of ( f ,A∗). Rθ indicates the rotation of angle θ. CANTOR SPECTRUM FOR GENERALIZED SKEW-SHIFTS 3 Proof of Theorem 2. The closure of the set of uniformly hyperbolic cocycles is obviously invariant under PSL(2,R) conjugacies, and clearly contains all constant cocycles ( f ,A∗) with trA∗ = 2. The result follows from the second part of Theorem 3. Remark 5. It would be interesting to investigate also the closure of an arbitrary PSL(2,R) conjugacy class. Even the case of the PSL(2,R) conjugacy class of a constant hyperbolic cocycle already escapes our methods. Let us say a few words about the proofs and relation with the literature. In the diffeomorphism and flow settings, Smale conjectured in the 1960’s that hyperbolic dynamical systems are dense. This turned out to be false in general. However, there are situations where denseness of hyperbolicity holds; see, for example, the recent work [KSS] in the context of one-dimensional dynamics. Cong [C] proved that uniform hyperbolicity is (open and) dense in the space of L(X, SL(2,R))-cocycles, for any base dynamics f . So our Theorem 2 can be seen as a continuous version of his result. Cong’s proof involves a tower argument to perturb the cocycle and produce an invariant section for its action on the circle P. We develop a somewhat similar technique, replacingP with other spaces. Special care is needed in order to ensure that perturbations and sections be continuous. Another related result was obtained by Fabbri and Johnson who considered continuous-time systems over translation flows on T and proved for a generic translation vector that uniform hyperbolicity occurs for an open and dense set of cocycles; see [FJ]. 1.2. Results for Schrödinger Cocycles. We say ( f ,A) is a Schrödinger cocyclewhen A takes its values in the set