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  • 1.
    Bjerklov, Kristian
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    A note on circle maps driven by strongly expanding endomorphisms on T2018In: Dynamical systems, ISSN 1468-9367, E-ISSN 1468-9375, Vol. 33, no 2, p. 361-368Article in journal (Refereed)
    Abstract [en]

    We investigate the dynamics of a class of smooth maps of the two-torus T2 of the form T(x, y) = (Nx, f(x)(y)), where f(x) : T -> T is a monotone family (in x) of orientation preserving circle diffeomorphisms and N is an element of Z(+) is large. For our class of maps, we show that the dynamics essentially is the same as that of the projective action of non-uniformly hyperbolic SL(2, R)-cocycles. This generalizes a result by L.S. Young [6] to maps T outside the (projective) matrix cocycle case.

  • 2.
    Bjerklöv, Kristian
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Attractors in the quasi-periodically perturbed quadratic family2012In: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 25, no 5, p. 1537-1545Article in journal (Refereed)
    Abstract [en]

    We give a geometric description of an attractor arising in quasi-periodically perturbed maps T x [0, 1] (sic) (theta, x) bar right arrow (theta + omega, c(theta)x(1 - x)) is an element of T x [0, 1] for certain choices of smooth c : T -> [1.5, 4] and Diophantine omega. The existence of the 'strange' attractor was established in Bjerklov 2009 Commun. Math. Phys. 286 137.

  • 3.
    Bjerklöv, Kristian
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Circle maps driven by a class of uniformly distributed sequences on T2022In: Bulletin of the London Mathematical Society, ISSN 0024-6093, E-ISSN 1469-2120, Vol. 54, no 3, p. 910-928Article in journal (Refereed)
    Abstract [en]

    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.). 

  • 4.
    Bjerklöv, Kristian
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    On some generalizations of skew-shifts on T-22019In: Ergodic Theory and Dynamical Systems, ISSN 0143-3857, E-ISSN 1469-4417, Vol. 39, p. 19-61Article in journal (Refereed)
    Abstract [en]

    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.

  • 5.
    Bjerklöv, Kristian
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    On the Lyapunov Exponents for a Class of Circle Diffeomorphisms Driven by Expanding Circle Endomorphisms2020In: Journal of Dynamics and Differential Equations, ISSN 1040-7294, E-ISSN 1572-9222Article in journal (Refereed)
    Abstract [en]

    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.

  • 6.
    Bjerklöv, Kristian
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Positive Lyapunov Exponent for Some Schrödinger Cocycles Over Strongly Expanding Circle Endomorphisms2020In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 379, no 1, p. 353-360Article in journal (Refereed)
    Abstract [en]

    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.

  • 7.
    Bjerklöv, Kristian
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Quasi-periodic kicking of circle diffeomorphisms having unique fixed points2019In: Moscow Mathematical Journal, ISSN 1609-3321, E-ISSN 1609-4514, Vol. 19, no 2, p. 189-216Article in journal (Refereed)
    Abstract [en]

    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.

  • 8.
    Bjerklöv, Kristian
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Quasi-periodic perturbation of unimodal maps exhibiting an attracting 3-cycle2012In: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 25, no 3, p. 683-741Article in journal (Refereed)
    Abstract [en]

    We study a class of smooth maps Phi : T x [0, 1]. T x [0, 1] of the form theta bar right arrow theta + omega x bar right arrow c(theta)h(x) where h : [0, 1] --> [0, 1] is a unimodal map exhibiting an attracting periodic point of prime period 3, and omega is irrational (T = R/Z). We show that the following phenomenon can occur for certain h and c : T --> R: There exists a single measurable function psi : T --> [0, 1] whose graph attracts (exponentially fast) a.e. (theta, x) is an element of T x [0, 1] under forward iterations of the map Phi. Moreover, the graph of psi is dense in a cylinder M subset of T x [0, 1]. Furthermore, for every integer n >= 1 there exists n distinct repelling continuous curves Gamma(k) : (theta, phi(k)(theta))(theta is an element of T), all lying in M, such that Phi(Gamma(k)) = Gamma(k+1) (k < n) and Phi(Gamma(n)) = Gamma(1). We give concrete examples where both c(theta) and h(x) are real-analytic, but in the analysis we only need that they are C-1. In our setting the function c(theta) will be very close to 1 for all theta outside a tiny interval; on the interval c(theta) > 1 makes a small bump. Thus we cause the perturbation of h by rare quasi-periodic kicking.

  • 9.
    Bjerklöv, Kristian
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Some remarks on the dynamics of the almost Mathieu equation at critical coupling*2020In: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 33, no 6, p. 2707-2722Article in journal (Refereed)
    Abstract [en]

    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.

  • 10.
    Bjerklöv, Kristian
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    The Dynamics of a Class of Quasi-Periodic Schrödinger Cocycles2015In: Annales de l'Institute Henri Poincare. Physique theorique, ISSN 1424-0637, E-ISSN 1424-0661, Vol. 16, no 4, p. 961-1031Article in journal (Refereed)
    Abstract [en]

    Let f : T -> R be a Morse function of class C-2 with exactly two critical points, let omega is an element of T be Diopharitine, and let lambda > 0 be sufficiently large (depending on f and omega). For any value of the parameter E is an element of R, we make a careful analysis of the dynamics of the skew-product map Phi(E)(theta, r) = (theta + omega, lambda f(theta) - E - 1/r), acting on the "torus" T x (R) over cap. Here, (R) over cap denotes the projective space R boolean OR {infinity}. The map Phi(E) is intimately related to the quasi-periodic Schrodinger cocycle (omega, A(E)) : T x R-2 -> T x R-2, (theta, x) -> (theta + omega, A(E)(theta) . x), where A(E) : T -> SL(2, R) is given by A(E)(theta) = ((0)(-1) 1(lambda f(theta) - E)), E is an element of R. More precisely, (omega, A(E)) naturally acts on the space T x (R) over cap, and Phi(E) is the map thus obtained. The cocycle (omega, A(E)) arises when investigating the eigenvalue equation H(theta)u = Eu, where H-theta is the quasi-periodic Schrodinger operator (H(theta)u)(n) = -(u(n+1) + u(n-1)) + lambda f (theta + (n - 1)omega)u(n), (1) The (maximal) Lyapunov exponent of the Schrodinger cocycle (omega, A(E)) is greater than or similar to log lambda, uniformly in E is an element of R. This implies that the map PE has exactly two ergodic probability measures for all E is an element of R; (2) If E is on the edge of an open gap in the spectrum sigma(H), then there exist a phase 0 is an element of T and a vector u is an element of l(2)(Z), exponentially decaying at +/-infinity, such that H(theta)u = Eu;acting on the space l(2) (Z). It is well known that the spectrum of H-theta, sigma(H), is independent of the phase theta is an element of T. Under our assumptions on f, omega and lambda, Sinai (in J Stat Phys 46(5-6):861-909, 1987) has shown that sigma(H) is a Cantor set, and the operator H-theta has a pure-point spectrum, with exponentially decaying eig,enfunctions, for a.e. theta is an element of T The analysis of Phi(E) allows us to derive three main results: (3) The map Phi(E) is minimal iff E E is an element of sigma(H)\ {edges of open gaps}. In particular, Phi(E) is minimal for all E is an element of R for which the fibered rotation number alpha(E) associated with (omega, A(E)) is irrational with respect to omega.

  • 11.
    Bjerklöv, Kristian
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Eliasson, Håkan
    UPMC Univ Paris 06, Univ Paris Diderot, Sorbonne Univ,CNRS,UMR 7586, Sorbonne Paris Cite,Inst Math Jussieu Paris Rive, F-75013 Paris, France..
    Positive fibered lyapunov exponents for some quasi-periodically driven circle endomorphisms with critical points2020In: Astérisque, ISSN 0303-1179, Vol. 415, p. 181-193Article in journal (Refereed)
    Abstract [en]

    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.

  • 12.
    Bjerklöv, Kristian
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Krikorian, Raphael
    Univ Cergy Pontoise, Dept Math, CNRS, UMR 8088, 2 Av Adolphe Chauvin, F-95302 Cergy Pontoise, France..
    Coexistence of absolutely continuous and pure point spectrum for kicked quasiperiodic potentials2021In: Journal of Spectral Theory, ISSN 1664-039X, E-ISSN 1664-0403, Vol. 11, no 3, p. 1215-1254Article in journal (Refereed)
    Abstract [en]

    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.

1 - 12 of 12
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