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October  2013, 18(8): 2069-2082. doi: 10.3934/dcdsb.2013.18.2069

Triple collisions of invariant bundles

Jordi-Lluís Figueras 1, and  Àlex Haro 2,

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

Received  July 2012 Revised  May 2013 Published  July 2013

We provide several explicit examples of 3D quasiperiodic linear skew-products with simple Lyapunov spectrum, that is with $3$ different Lyapunov multipliers, for which the corresponding Oseledets bundles are measurable but not continuous, colliding in a measure zero dense set.
Keywords: Oseledets theory., Nonuniformly hyperbolic linear skew-products.
Mathematics Subject Classification: Primary: 37D20; Secondary: 37D4.
Citation: Jordi-Lluís Figueras, Àlex Haro. Triple collisions of invariant bundles. Discrete and Continuous Dynamical Systems - B, 2013, 18 (8) : 2069-2082. doi: 10.3934/dcdsb.2013.18.2069
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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