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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    KTH, Superseded Departments, Mathematics.
    Dynamical Properties of Quasi-periodic Schrödinger Equations2003Doctoral thesis, comprehensive summary (Other academic)
  • 4. Bjerklöv, Kristian
    Dynamics of the quasi-periodic Schrodinger cocycle at the lowest energy in the spectrum2007In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 272, no 2, p. 397-442Article in journal (Refereed)
    Abstract [en]

    In this paper we consider the quasi-periodic Schrodinger cocycle over T-d (d >= 1) and, in particular, its projectivization. In the regime of large coupling constants and Diophantine frequencies, we give an affirmative answer to a question posed by M. Herman [21, p.482] concerning the geometric structure of certain Strange Nonchaotic Attractors which appear in the projective dynamical system. We also show that for some phase, the lowest energy in the spectrum of the associated Schrodinger operator is an eigenvalue with an exponentially decaying eigenfunction. This generalizes [39] to the multi-frequency case (d > 1).

  • 5. Bjerklöv, Kristian
    Explicit examples of arbitrarily large analytic ergodic potentials with zero Lyapunov exponent2006In: Geometric and Functional Analysis, ISSN 1016-443X, E-ISSN 1420-8970, Vol. 16, no 6, p. 1183-1200Article in journal (Refereed)
    Abstract [en]

    We give explicit examples of arbitrarily large analytic ergodic potentials for which the Schrodinger equation has zero Lyapunov exponent for certain energies. For one of these energies there is an explicit solution. In the quasi-periodic case we prove that one can have positive Lyapunov exponent on certain regions of the spectrum and zero on other regions. We also show the existence of 1-dependent random potentials with zero Lyapunov exponent.

  • 6.
    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.

  • 7.
    Bjerklöv, Kristian
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Positive Lyapunov exponent and minimality for a class of one-dimensional quasi-periodic Schrodinger equations2005In: Ergodic Theory and Dynamical Systems, ISSN 0143-3857, E-ISSN 1469-4417, Vol. 25, p. 1015-1045Article in journal (Refereed)
    Abstract [en]

    We study the discrete quasi-periodic Schrodinger equation -(u(n+1) + u(n-1)) + lambda V(theta + n omega)u(n) = Eu-n with a non-constant C-1 potential function V : T -> R. We prove that for sufficiently large k there is a set Omega subset of T of frequencies omega, whose measure tends to 1 as lambda -> infinity, with the following property. For each w e Q there is a 'large' (in measure) set of energies E, all lying in the spectrum of the associated Schrodinger operator (and hence giving a lower estimate on the measure of the spectrum), such that the Lyapunov exponent is positive and, moreover, the projective dynamical system induced by the Schrodinger cocycle is minimal but not ergodic.

  • 8. Bjerklöv, Kristian
    Positive lyapunov exponent and minimality for the continuous 1-d quasi-periodic Schrodinger equation with two basic frequencies2007In: Annales de l'Institute Henri Poincare. Physique theorique, ISSN 1424-0637, E-ISSN 1424-0661, Vol. 8, no 4, p. 687-730Article in journal (Refereed)
    Abstract [en]

    We consider the time-independent quasi-periodic Schrodinger equation [GRAPHICS] with a potential function V : T-2 -> R of class C-2 with a unique non-degenerate global minimum, large coupling constants K-2 and energies E in the bottom of the spectrum of the associated Schrodinger operator. We obtain estimates on the Lyapunov exponents and the Lebesgue measure of the spectrum, as well as localization results. Moreover, we show that the projective flow on T-2 x P-1 induced by the Schrodinger equation often is minimal.

  • 9.
    Bjerklöv, Kristian
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Positive Lyapunov exponents for continuous quasiperiodic Schrodinger equations2006In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 47, no 2Article in journal (Refereed)
    Abstract [en]

    We prove that the continuous one-dimensional Schrodinger equation with an analytic quasi-periodic potential has positive Lyapunov exponents in the bottom of the spectrum for large couplings.

  • 10.
    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.

  • 11.
    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.

  • 12.
    Bjerklöv, Kristian
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    SNA's in the Quasi-Periodic Quadratic Family2009In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 286, no 1, p. 137-161Article in journal (Refereed)
    Abstract [en]

    We rigorously show that there can exist Strange Nonchaotic Attractors (SNA) in the quasi-periodically forced quadratic ( or logistic) map (theta, x) -> (theta + omega, c(theta)x(1 - x)) for certain choices of c : T bar right arrow [3/2, 4] and Diophantine omega.

  • 13.
    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.

  • 14. Bjerklöv, Kristian
    et al.
    Damanik, David
    Johnson, Russell
    Lyapunov exponents of continuous Schrodinger cocycles over irrational rotations2008In: Annali di Matematica Pura ed Applicata, ISSN 0373-3114, E-ISSN 1618-1891, Vol. 187, no 1, p. 1-6Article in journal (Refereed)
    Abstract [en]

    We consider the Lyapunov exponent of those continuous SL( 2, R)-valued cocycles over irrational rotations that appear in the study of Schrodinger operators and prove generic results related to large coupling asymptotics and uniform convergence.

  • 15. Bjerklöv, Kristian
    et al.
    Johnson, Russell
    Minimal subsets of projective flows2008In: Discrete and continuous dynamical systems. Series B, ISSN 1531-3492, E-ISSN 1553-524X, Vol. 9, no 3-4, p. 493-516Article in journal (Refereed)
    Abstract [en]

    We study the minimal subsets of the projective flow defined by a two-dimensional linear differential system with almost periodic coefficients. We show that such a minimal set may exhibit Li-Yorke chaos and discuss specific examples in which this phenomenon is present. We then give a classification of these minimal sets, and use it to discuss the bounded mean motion property relative to the projective flow.

  • 16.
    Bjerklöv, Kristian
    et al.
    University of Toronto, Canada.
    Jäger, Tobias
    Rotation numbers for quasiperiodically forced circle maps mode-locking vs. strict monotonicity2009In: Journal of The American Mathematical Society, ISSN 0894-0347, E-ISSN 1088-6834, Vol. 22, no 2, p. 353-362Article in journal (Refereed)
  • 17. Bjerklöv, Kristian
    et al.
    Saprykina, Maria
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Universal asymptotics in hyperbolicity breakdown2008In: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 21, no 3, p. 557-586Article in journal (Refereed)
    Abstract [en]

    We study a scenario for the disappearance of hyperbolicity of invariant tori in a class of quasi-periodic systems. In this scenario, the system loses hyperbolicity because two invariant directions come close to each other, losing their regularity. In a recent paper, based on numerical results, Haro and de la Llave (2006 Chaos 16 013120) discovered a quantitative universality in this scenario, namely, that the minimal angle between the two invariant directions has a power law dependence on the parameters and the exponents of the power law are universal. We present an analytic proof of this result.

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