We consider the family of one-frequency quasi-periodic Schrödinger coycles Gω,E​, parametrized by the energy E. For potential functions v(x)=λv0​(x), where v0​∈C2(T,R) is a Morse function with finitely many critical points and λ>0 is large, we show that, for any value of E∈R and for any phase x∗∈T such that ∣v0​(x∗)−E/λ∣ is not too small, there exists a set of frequencies Ω=Ω(E,x∗) of positive measure such that the following hold: (1) for every ω∈Ω, the upper Lyapunov exponent of the cocycle Gω,E​ is ≳logλ and x∗ is (essentially) a typical point in Oseledets’ theorem; (2) either Gω,E​ is uniformly hyperbolic, or there exists a phase x0​∈T such that E is an eigenvalue of the corresponding discrete Schrödinger operator Hx0​,ω​.

We consider monotone families of circle diffeomorphisms forced by the strongly chaotic circle endomorphisms x bar right arrow bx mod 1, where the integer b is large. We obtain estimates of the fibered Lyapunov exponents and show that in the limit as b tends to infinity, they approach the values of the Lyapunov exponents for the corresponding random case. The estimates are based on a control of the distribution of the iterates of almost every point, up to a fixed (small) scale, depending on b.

We use certain uniformly distributed sequences on (Formula presented.), including the sequence (Formula presented.), to drive families of circle maps. We show that: (1) under mild assumptions on the function (Formula presented.), the discrete Schrödinger equation on the half line, with a potential of the form (Formula presented.) where (Formula presented.) is large, has for all energies (Formula presented.) exponentially growing solutions for almost every (a.e.) (Formula presented.); (2) the derivative of compositions (Formula presented.), where (Formula presented.) ((Formula presented.)) grow exponentially fast with (Formula presented.) for a.e. (Formula presented.). 

We consider skew-product maps on T2 of the form F(x, y) = (bx, x+ g(y) ) where g: T→ T is an orientation-preserving C2-diffeomorphism and b≥ 2 is an integer. We show that the fibred (upper and lower) Lyapunov exponent of almost every point (x, y) is as close to ∫ Tlog (g′(η) ) dη as we like, provided that b is large enough.

We introduce a class of real analytic "peaky" potentials for which the corresponding quasiperiodic 1D-Schrodinger operators exhibit, for quasiperiodic frequencies in a set of positive Lebesgue measure, both absolutely continuous and pure point spectrum.

In this paper we give examples of skew-product maps T : T-2 -> T-2 of the form T (x, y) = (x + omega, x + f(y)), where f : T -> T is an explicit C-1-endomorphism of degree two with a unique critical point and omega belongs to a set of positive measure, for which the fibered Lyapunov exponent is positive for a.e. (x, y) is an element of T-2. The critical point is of type f '(+/- e(-epsilon)) approximate to e(-beta s/(ln s)2) for all large s, where beta > 0 is a small numerical constant.

We show that for a large class of potential functions and big coupling constant λ the Schrödinger cocycle over the expanding map x↦bx(mod1) on T has a Lyapunov exponent > (log λ) / 4 for all energies, provided that the integer b≥ λ3.

We show that the quasi-periodic Schrodinger cocycle with a continuous potential is of parabolic type, with a unique invariant section, at all gap edges where the Lyapunov exponent vanishes. This applies, in particular, to the almost Mathieu equation with critical coupling. It also provides examples of real-analytic cocycles having a unique invariant section which is not smooth.

In this paper we investigate maps of the two-torus T-2 of the form T (x, y) = (x + omega, g(x) + f (y)) for Diophantine omega is an element of T and for a class of maps f, g : T -> T, where each g is strictly monotone and of degree 2 and each f is an orientation-preserving circle homeomorphism. For our class of f and g, we show that T is minimal and has exactly two invariant and ergodic Borel probability measures. Moreover, these measures are supported on two T-invariant graphs. One of the graphs is a strange non-chaotic attractor whose basin of attraction consists of (Lebesgue) almost all points in T-2. Only a low-regularity assumption (Lipschitz) is needed on the maps f and g, and the results are robust with respect to Lipschitz-small perturbations of f and g.

We investigate the dynamics of certain homeomorphisms F: T-2 -> T-2 of the form F(x, y) = (x + omega , h(x)+ f (y)), where omega is an element of R\Q, f: T -> T is a circle diffeomorphism with a unique (and thus neutral) fixed point and h: T -> T is a function which is zero outside a small interval. We show that such a map can display a non-uniformly hyperbolic behavior: (small) negative fibred Lyapunov exponents for a.e. (x, y) and an attracting non-continuous invariant graph. We apply this result to (projective) SL(2, R)-cocycles G: (x, u) bar right arrow (x + omega, A(x)u) with A(x) = R phi(x)B, where R-theta is a rotation matrix and B is a parabolic matrix, to get exam ples of non-uniformly hyperbolic cocycles (homotopic to the identity) with perturbatively small Lyapunov exponents.