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Bjerklöv, Kristian, DocentORCID iD iconorcid.org/0000-0003-4368-2833
Publications (10 of 12) Show all publications
Bjerklöv, K. (2022). Circle maps driven by a class of uniformly distributed sequences on T. Bulletin of the London Mathematical Society, 54(3), 910-928
Open this publication in new window or tab >>Circle maps driven by a class of uniformly distributed sequences on T
2022 (English)In: Bulletin of the London Mathematical Society, ISSN 0024-6093, E-ISSN 1469-2120, Vol. 54, no 3, p. 910-928Article in journal (Refereed) Published
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.). 

Place, publisher, year, edition, pages
Wiley, 2022
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:kth:diva-322572 (URN)10.1112/blms.12603 (DOI)000777021800001 ()2-s2.0-85127433590 (Scopus ID)
Note

QC 20221222

Available from: 2022-12-22 Created: 2022-12-22 Last updated: 2022-12-22Bibliographically approved
Bjerklöv, K. & Krikorian, R. (2021). Coexistence of absolutely continuous and pure point spectrum for kicked quasiperiodic potentials. Journal of Spectral Theory, 11(3), 1215-1254
Open this publication in new window or tab >>Coexistence of absolutely continuous and pure point spectrum for kicked quasiperiodic potentials
2021 (English)In: Journal of Spectral Theory, ISSN 1664-039X, E-ISSN 1664-0403, Vol. 11, no 3, p. 1215-1254Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
European Mathematical Society - EMS - Publishing House GmbH, 2021
Keywords
Spectral theory, smooth dynamics, quasi-periodic cocycles, reducibility of cocycles, Lyapunov exponents
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:kth:diva-303888 (URN)10.4171/JST/370 (DOI)000704993600011 ()2-s2.0-85116839193 (Scopus ID)
Note

QC 20211021

Available from: 2021-10-21 Created: 2021-10-21 Last updated: 2022-06-25Bibliographically approved
Bjerklöv, K. (2020). On the Lyapunov Exponents for a Class of Circle Diffeomorphisms Driven by Expanding Circle Endomorphisms. Journal of Dynamics and Differential Equations
Open this publication in new window or tab >>On the Lyapunov Exponents for a Class of Circle Diffeomorphisms Driven by Expanding Circle Endomorphisms
2020 (English)In: Journal of Dynamics and Differential Equations, ISSN 1040-7294, E-ISSN 1572-9222Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
Springer, 2020
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:kth:diva-285372 (URN)10.1007/s10884-020-09876-x (DOI)000552267700001 ()2-s2.0-85088447734 (Scopus ID)
Note

QC 20201130

Available from: 2020-11-30 Created: 2020-11-30 Last updated: 2024-01-10Bibliographically approved
Bjerklöv, K. & Eliasson, H. (2020). Positive fibered lyapunov exponents for some quasi-periodically driven circle endomorphisms with critical points. Astérisque, 415, 181-193
Open this publication in new window or tab >>Positive fibered lyapunov exponents for some quasi-periodically driven circle endomorphisms with critical points
2020 (English)In: Astérisque, ISSN 0303-1179, Vol. 415, p. 181-193Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
Societe Mathematique de France, 2020
Keywords
Dynamical systems, Lyapunov exponents, quasi-periodicity
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:kth:diva-278929 (URN)10.24033/ast.1104 (DOI)000551832400009 ()2-s2.0-85096987229 (Scopus ID)
Note

QC 20201118

Available from: 2020-11-18 Created: 2020-11-18 Last updated: 2022-06-25Bibliographically approved
Bjerklöv, K. (2020). Positive Lyapunov Exponent for Some Schrödinger Cocycles Over Strongly Expanding Circle Endomorphisms. Communications in Mathematical Physics, 379(1), 353-360
Open this publication in new window or tab >>Positive Lyapunov Exponent for Some Schrödinger Cocycles Over Strongly Expanding Circle Endomorphisms
2020 (English)In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 379, no 1, p. 353-360Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
Springer Science and Business Media Deutschland GmbH, 2020
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:kth:diva-288069 (URN)10.1007/s00220-020-03810-4 (DOI)000549310600007 ()2-s2.0-85088049023 (Scopus ID)
Note

QC 20201228

Available from: 2020-12-28 Created: 2020-12-28 Last updated: 2022-06-25Bibliographically approved
Bjerklöv, K. (2020). Some remarks on the dynamics of the almost Mathieu equation at critical coupling*. Nonlinearity, 33(6), 2707-2722
Open this publication in new window or tab >>Some remarks on the dynamics of the almost Mathieu equation at critical coupling*
2020 (English)In: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 33, no 6, p. 2707-2722Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
IOP PUBLISHING LTD, 2020
Keywords
quasi-periodic cocycle, almost Mathieu operator, discrete Schrodinger operator
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-273488 (URN)10.1088/1361-6544/ab7636 (DOI)000528626500001 ()2-s2.0-85085650489 (Scopus ID)
Note

QC 20200525

Available from: 2020-05-25 Created: 2020-05-25 Last updated: 2022-06-26Bibliographically approved
Bjerklöv, K. (2019). On some generalizations of skew-shifts on T-2. Ergodic Theory and Dynamical Systems, 39, 19-61
Open this publication in new window or tab >>On some generalizations of skew-shifts on T-2
2019 (English)In: Ergodic Theory and Dynamical Systems, ISSN 0143-3857, E-ISSN 1469-4417, Vol. 39, p. 19-61Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
CAMBRIDGE UNIV PRESS, 2019
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-239960 (URN)10.1017/etds.2017.19 (DOI)000451398100002 ()2-s2.0-85018981635 (Scopus ID)
Funder
Swedish Research Council, 2012-3090
Note

QC 20181211

Available from: 2018-12-11 Created: 2018-12-11 Last updated: 2022-06-26Bibliographically approved
Bjerklöv, K. (2019). Quasi-periodic kicking of circle diffeomorphisms having unique fixed points. Moscow Mathematical Journal, 19(2), 189-216
Open this publication in new window or tab >>Quasi-periodic kicking of circle diffeomorphisms having unique fixed points
2019 (English)In: Moscow Mathematical Journal, ISSN 1609-3321, E-ISSN 1609-4514, Vol. 19, no 2, p. 189-216Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
Independent University of Moscow, 2019
Keywords
Lyapunov exponents, quasi-periodic forcing, non-uniform hyperbolicity, cocycles
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-255587 (URN)10.17323/1609-4514-2019-19-2-189-216 (DOI)000475756300002 ()2-s2.0-85067058396 (Scopus ID)
Note

QC 20190805

Available from: 2019-08-05 Created: 2019-08-05 Last updated: 2022-06-26Bibliographically approved
Bjerklov, K. (2018). A note on circle maps driven by strongly expanding endomorphisms on T. Dynamical systems, 33(2), 361-368
Open this publication in new window or tab >>A note on circle maps driven by strongly expanding endomorphisms on T
2018 (English)In: Dynamical systems, ISSN 1468-9367, E-ISSN 1468-9375, Vol. 33, no 2, p. 361-368Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
TAYLOR & FRANCIS LTD, 2018
Keywords
Lyapunov exponents, synchronization, forced circle diffeomorphisms
National Category
Algebra and Logic
Identifiers
urn:nbn:se:kth:diva-227244 (URN)10.1080/14689367.2017.1386161 (DOI)000430481600011 ()2-s2.0-85031396752 (Scopus ID)
Note

QC 20180504

Available from: 2018-05-04 Created: 2018-05-04 Last updated: 2024-03-15Bibliographically approved
Bjerklöv, K. (2015). The Dynamics of a Class of Quasi-Periodic Schrödinger Cocycles. Annales de l'Institute Henri Poincare. Physique theorique, 16(4), 961-1031
Open this publication in new window or tab >>The Dynamics of a Class of Quasi-Periodic Schrödinger Cocycles
2015 (English)In: 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) Published
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.

National Category
Materials Engineering
Identifiers
urn:nbn:se:kth:diva-163944 (URN)10.1007/s00023-014-0330-8 (DOI)000350669300002 ()2-s2.0-85027949378 (Scopus ID)
Funder
Swedish Research Council
Note

QC 20150507

Available from: 2015-05-07 Created: 2015-04-13 Last updated: 2024-03-15Bibliographically approved
Organisations
Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0003-4368-2833

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