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Triple collisions of invariant bundles
1. | Department of Mathematics, Uppsala University, Box 480, 75106 Uppsala, Sweden |
2. | Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via de les Corts Catalanes 585, 08007 Barcelona |
References:
[1] |
A. Avila and S. Jitomirskaya, The ten martini problem, Annals of Mathematics (2), 170 (2009), 303-342.
doi: 10.4007/annals.2009.170.303. |
[2] |
J. Bourgain, "Green's Function Estimates for Lattice Schrödinger Operators and Applications," Annals of Mathematics Studies, 158, Princeton University Press, Princeton, NJ, 2005. |
[3] |
R. Calleja and J.-Ll. Figueras, Collision of invariant bundles of quasi-periodic attractors in the dissipative standard map, Chaos, 22 (2012), 033114, 10 pp.
doi: 10.1063/1.4737205. |
[4] |
M. Canadell and A. Haro, Parameterization method for computing quasi-periodic normally hyperbolic invariant tori, preprint, (2013). |
[5] |
C. Chicone and R. C. Swanson, Spectral theory for linearizations of dynamical systems, J. Differential Equations, 40 (1981), 155-167.
doi: 10.1016/0022-0396(81)90015-2. |
[6] |
M. D. Choi, G. A. Eliott and N. Yui, Gauss polynomials and the rotation algebra, Invent. Math., 99 (1990), 225-246.
doi: 10.1007/BF01234419. |
[7] |
U. Feudel, S. Kuznetsov and A. Pikovsky, "Strange Nonchaotic Attractors. Dynamics Between Order and Chaos in Quasiperiodically Forced Systems," World Scientific Series on Nonlinear Science, Series A: Monographs and Treatises, 56, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006. |
[8] |
J.-Ll. Figueras, "Fiberwise Hyperbolic Invariant Tori in Quasi-Periodically Forced Skew Product Systems," Ph.D. thesis, Universitat de Barcelona, 2011. |
[9] |
Á. Haro and R. de la Llave, Manifolds on the verge of a hyperbolicity breakdown, Chaos, 16 (2006), 013120, 8 pp.
doi: 10.1063/1.2150947. |
[10] |
Á. Haro and R. de la Llave, A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: Numerical algorithms, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 1261-1300.
doi: 10.3934/dcdsb.2006.6.1261. |
[11] |
Á. Haro and J. Puig, Strange nonchaotic attractors in Harper maps, Chaos, 16 (2006), 033127, 7 pp.
doi: 10.1063/1.2259821. |
[12] |
P. G. Harper, Single band motion of conduction electrons in a uniform magnetic field, Proceedings of the Physical Society, Section A, 68 (1955), 874.
doi: 10.1088/0370-1298/68/10/304. |
[13] |
M.-R. Herman, Une méthode pour minorer les exposants de Lyapounov et quelques exemples montrant le caractère local d'un théorème d'Arnold et de Moser sur le tore de dimension $2$, Comment. Math. Helv., 58 (1983), 453-502.
doi: 10.1007/BF02564647. |
[14] |
M. W. Hirsch, C. C. Pugh and M. Shub, "Invariant Manifolds," Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977. |
[15] |
T. H. Jäger, On the structure of strange non-chaotic attractors in pinched skew products, Ergodic Theory Dynam. Systems, 27 (2007), 493-510.
doi: 10.1017/S0143385706000745. |
[16] |
A. Yu Jalnine and A. H. Osbaldestin, Smooth and nonsmooth dependence of Lyapunov vectors upon the angle variable on a torus in the context of torus-doubling transitions in the quasiperiodically forced Hénon map, Phys. Rev. E (3), 71 (2005), 016206, 14 pp.
doi: 10.1103/PhysRevE.71.016206. |
[17] |
Russell A. Johnson, The Oseledec and Sacker-Sell spectra for almost periodic linear systems: An example, Proc. Amer. Math. Soc., 99 (1987), 261-267.
doi: 10.1090/S0002-9939-1987-0870782-7. |
[18] |
G. Keller, A note on strange nonchaotic attractors, Fund. Math., 151 (1996), 139-148. |
[19] |
J. A. Ketoja and I. I. Satija, Self-similarity and localization, Phys. Rev. Lett., 75 (1995), 2762-2765.
doi: 10.1103/PhysRevLett.75.2762. |
[20] |
John N. Mather, Characterization of Anosov diffeomorphisms, Nederl. Akad. Wetensch. Proc. Ser. A 71 = Indag. Math., 30 (1968), 479-483. |
[21] |
V. Millionshchikov, Proof of the existence of non-irreducible systems of linear differential equations with almost periodic coefficients, J. Differential Equations, 6 (1968), 149-158. |
[22] |
V. I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat. Obšč., 19 (1968), 179-210. |
[23] |
J. Puig, Cantor spectrum for the almost Mathieu operator, Comm. Math. Phys., 244 (2006), 297-309.
doi: 10.1007/s00220-003-0977-3. |
[24] |
R. J. Sacker and G. R. Sell, Existence of dichotomies and invariant splittings for linear differential systems. I, J. Differential Equations, 15 (1974), 429-458.
doi: 10.1016/0022-0396(74)90067-9. |
[25] |
Robert J. Sacker and George R. Sell, Existence of dichotomies and invariant splittings for linear differential systems. I, J. Differential Equations, 15 (1974), 429-458.
doi: 10.1016/0022-0396(74)90067-9. |
[26] |
J. B. Sokoloff, Unusual band structure, wave functions and electrical conductance in crystals with incommensurate periodic potentials, Physics Reports, 126 (1985), 189-244.
doi: 10.1016/0370-1573(85)90088-2. |
show all references
References:
[1] |
A. Avila and S. Jitomirskaya, The ten martini problem, Annals of Mathematics (2), 170 (2009), 303-342.
doi: 10.4007/annals.2009.170.303. |
[2] |
J. Bourgain, "Green's Function Estimates for Lattice Schrödinger Operators and Applications," Annals of Mathematics Studies, 158, Princeton University Press, Princeton, NJ, 2005. |
[3] |
R. Calleja and J.-Ll. Figueras, Collision of invariant bundles of quasi-periodic attractors in the dissipative standard map, Chaos, 22 (2012), 033114, 10 pp.
doi: 10.1063/1.4737205. |
[4] |
M. Canadell and A. Haro, Parameterization method for computing quasi-periodic normally hyperbolic invariant tori, preprint, (2013). |
[5] |
C. Chicone and R. C. Swanson, Spectral theory for linearizations of dynamical systems, J. Differential Equations, 40 (1981), 155-167.
doi: 10.1016/0022-0396(81)90015-2. |
[6] |
M. D. Choi, G. A. Eliott and N. Yui, Gauss polynomials and the rotation algebra, Invent. Math., 99 (1990), 225-246.
doi: 10.1007/BF01234419. |
[7] |
U. Feudel, S. Kuznetsov and A. Pikovsky, "Strange Nonchaotic Attractors. Dynamics Between Order and Chaos in Quasiperiodically Forced Systems," World Scientific Series on Nonlinear Science, Series A: Monographs and Treatises, 56, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006. |
[8] |
J.-Ll. Figueras, "Fiberwise Hyperbolic Invariant Tori in Quasi-Periodically Forced Skew Product Systems," Ph.D. thesis, Universitat de Barcelona, 2011. |
[9] |
Á. Haro and R. de la Llave, Manifolds on the verge of a hyperbolicity breakdown, Chaos, 16 (2006), 013120, 8 pp.
doi: 10.1063/1.2150947. |
[10] |
Á. Haro and R. de la Llave, A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: Numerical algorithms, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 1261-1300.
doi: 10.3934/dcdsb.2006.6.1261. |
[11] |
Á. Haro and J. Puig, Strange nonchaotic attractors in Harper maps, Chaos, 16 (2006), 033127, 7 pp.
doi: 10.1063/1.2259821. |
[12] |
P. G. Harper, Single band motion of conduction electrons in a uniform magnetic field, Proceedings of the Physical Society, Section A, 68 (1955), 874.
doi: 10.1088/0370-1298/68/10/304. |
[13] |
M.-R. Herman, Une méthode pour minorer les exposants de Lyapounov et quelques exemples montrant le caractère local d'un théorème d'Arnold et de Moser sur le tore de dimension $2$, Comment. Math. Helv., 58 (1983), 453-502.
doi: 10.1007/BF02564647. |
[14] |
M. W. Hirsch, C. C. Pugh and M. Shub, "Invariant Manifolds," Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977. |
[15] |
T. H. Jäger, On the structure of strange non-chaotic attractors in pinched skew products, Ergodic Theory Dynam. Systems, 27 (2007), 493-510.
doi: 10.1017/S0143385706000745. |
[16] |
A. Yu Jalnine and A. H. Osbaldestin, Smooth and nonsmooth dependence of Lyapunov vectors upon the angle variable on a torus in the context of torus-doubling transitions in the quasiperiodically forced Hénon map, Phys. Rev. E (3), 71 (2005), 016206, 14 pp.
doi: 10.1103/PhysRevE.71.016206. |
[17] |
Russell A. Johnson, The Oseledec and Sacker-Sell spectra for almost periodic linear systems: An example, Proc. Amer. Math. Soc., 99 (1987), 261-267.
doi: 10.1090/S0002-9939-1987-0870782-7. |
[18] |
G. Keller, A note on strange nonchaotic attractors, Fund. Math., 151 (1996), 139-148. |
[19] |
J. A. Ketoja and I. I. Satija, Self-similarity and localization, Phys. Rev. Lett., 75 (1995), 2762-2765.
doi: 10.1103/PhysRevLett.75.2762. |
[20] |
John N. Mather, Characterization of Anosov diffeomorphisms, Nederl. Akad. Wetensch. Proc. Ser. A 71 = Indag. Math., 30 (1968), 479-483. |
[21] |
V. Millionshchikov, Proof of the existence of non-irreducible systems of linear differential equations with almost periodic coefficients, J. Differential Equations, 6 (1968), 149-158. |
[22] |
V. I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat. Obšč., 19 (1968), 179-210. |
[23] |
J. Puig, Cantor spectrum for the almost Mathieu operator, Comm. Math. Phys., 244 (2006), 297-309.
doi: 10.1007/s00220-003-0977-3. |
[24] |
R. J. Sacker and G. R. Sell, Existence of dichotomies and invariant splittings for linear differential systems. I, J. Differential Equations, 15 (1974), 429-458.
doi: 10.1016/0022-0396(74)90067-9. |
[25] |
Robert J. Sacker and George R. Sell, Existence of dichotomies and invariant splittings for linear differential systems. I, J. Differential Equations, 15 (1974), 429-458.
doi: 10.1016/0022-0396(74)90067-9. |
[26] |
J. B. Sokoloff, Unusual band structure, wave functions and electrical conductance in crystals with incommensurate periodic potentials, Physics Reports, 126 (1985), 189-244.
doi: 10.1016/0370-1573(85)90088-2. |
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