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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 & 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.  Google Scholar

[2]

J. Bourgain, "Green's Function Estimates for Lattice Schrödinger Operators and Applications," Annals of Mathematics Studies, 158, Princeton University Press, Princeton, NJ, 2005.  Google Scholar

[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.  Google Scholar

[4]

M. Canadell and A. Haro, Parameterization method for computing quasi-periodic normally hyperbolic invariant tori, preprint, (2013). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

J.-Ll. Figueras, "Fiberwise Hyperbolic Invariant Tori in Quasi-Periodically Forced Skew Product Systems," Ph.D. thesis, Universitat de Barcelona, 2011. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

Á. Haro and J. Puig, Strange nonchaotic attractors in Harper maps, Chaos, 16 (2006), 033127, 7 pp. doi: 10.1063/1.2259821.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

M. W. Hirsch, C. C. Pugh and M. Shub, "Invariant Manifolds," Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

G. Keller, A note on strange nonchaotic attractors, Fund. Math., 151 (1996), 139-148.  Google Scholar

[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.  Google Scholar

[20]

John N. Mather, Characterization of Anosov diffeomorphisms, Nederl. Akad. Wetensch. Proc. Ser. A 71 = Indag. Math., 30 (1968), 479-483.  Google Scholar

[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. Google Scholar

[22]

V. I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat. Obšč., 19 (1968), 179-210.  Google Scholar

[23]

J. Puig, Cantor spectrum for the almost Mathieu operator, Comm. Math. Phys., 244 (2006), 297-309. doi: 10.1007/s00220-003-0977-3.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

J. Bourgain, "Green's Function Estimates for Lattice Schrödinger Operators and Applications," Annals of Mathematics Studies, 158, Princeton University Press, Princeton, NJ, 2005.  Google Scholar

[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.  Google Scholar

[4]

M. Canadell and A. Haro, Parameterization method for computing quasi-periodic normally hyperbolic invariant tori, preprint, (2013). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

J.-Ll. Figueras, "Fiberwise Hyperbolic Invariant Tori in Quasi-Periodically Forced Skew Product Systems," Ph.D. thesis, Universitat de Barcelona, 2011. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

Á. Haro and J. Puig, Strange nonchaotic attractors in Harper maps, Chaos, 16 (2006), 033127, 7 pp. doi: 10.1063/1.2259821.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

M. W. Hirsch, C. C. Pugh and M. Shub, "Invariant Manifolds," Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

G. Keller, A note on strange nonchaotic attractors, Fund. Math., 151 (1996), 139-148.  Google Scholar

[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.  Google Scholar

[20]

John N. Mather, Characterization of Anosov diffeomorphisms, Nederl. Akad. Wetensch. Proc. Ser. A 71 = Indag. Math., 30 (1968), 479-483.  Google Scholar

[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. Google Scholar

[22]

V. I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat. Obšč., 19 (1968), 179-210.  Google Scholar

[23]

J. Puig, Cantor spectrum for the almost Mathieu operator, Comm. Math. Phys., 244 (2006), 297-309. doi: 10.1007/s00220-003-0977-3.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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