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Bidimensional linear systems
with singular dynamics


Authors: Sylvia Novo and Rafael Obaya
Journal: Proc. Amer. Math. Soc. 124 (1996), 3163-3172
MSC (1991): Primary 28D05, 58F11; Secondary 34C11
DOI: https://doi.org/10.1090/S0002-9939-96-03411-9
MathSciNet review: 1328366
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Abstract: We analyze a class of bidimensional linear systems for which the following characteristics are generic: the system is recurrent and there exists a unique ergodic measure which is concentrated in one ergodic sheet. The trajectories exhibit an oscillatory behaviour from one to the other side of the ergodic sheet which assures the proximal character of the flow.


References [Enhancements On Off] (What's this?)

  • 1. A. I. Alonso and R. Obaya, Ergodic structure of bidimensional linear systems with a linear invariant measure, Univ. of Valladolid preprint, 1993.
  • 2. R. Ellis and R. Johnson, Topological dynamics and linear differential systems, J. Differential Equations 44 (1982), no. 1, 21--39.MR 83c:54058
  • 3. A. M. Fink, Almost periodic differential equations, Lecture Notes in Mathematics 377, Springer-Verlag, Heidelberg, 1974. MR 57:792
  • 4. H. Furstenberg, Strict ergodicity and transformations of the torus, Amer. J. Math. 85 (1961), 573--601. MR 24:A3263
  • 5. S. Glasner and B. Weiss, On the construction of minimal skew products, Israel J. Math 34 (1979), no. 4, 321--336. MR 82f:54068
  • 6. R. Johnson, On a Floquet theory for almost-periodic, two-dimensional linear systems, J. Differential Equations 37 (1980), 184--204. MR 81j:58069
  • 7. ------, Almost-periodic functions with unbounded integral, Pacific J. Math. 87 (1980), 347--362. MR 82e:42013
  • 8. ------, Two-dimensional, almost periodic linear systems with proximal and recurrent behavior, Proc. Amer. Math. Soc. 82 (1981), no. 3, 417--422. MR 83c:34048
  • 9. ------, Exponential dichotomy, rotation number, and linear differential operators with bounded coefficients, J. Differential Equations 61 (1986), 54--78. MR 87e:47065
  • 10. M. G. Nerurkar, Recurrent-proximal linear differential systems with almost periodic coefficients, Proc. Amer. Math. Soc. 100 (1987), no. 4, 729--743. MR 88i:58150
  • 11. ------, On the construction of smooth ergodic skew-products, Ergod. Th. $ \& $ Dynam. Sys. 8 (1988), 311--326. MR 89m:58123
  • 12. S. Novo and R. Obaya, An ergodic classification of bidimensional linear systems, to be published in J. Dynamics Differential Equations.
  • 13. R. J. Sacker and G. R. Sell, Lifting properties in skew-product flows with applications to differential equations, Mem. Amer. Math. Soc. 190, Amer. Math. Soc., Providence, 1977. MR 56:6632
  • 14. S. Schwartzman, Asymptotic cycles, Ann. of Math 66 (1957), 270--284. MR 19:568

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Additional Information

Sylvia Novo
Affiliation: Departamento de Matematica Aplicada a la Ingenieria, E.T.S. de Ingenieros Industriales, Universidad de Valladolid, 47011, Valladolid, Spain
Email: sylnov@wmatem.eis.uva.es

Rafael Obaya
Affiliation: Departamento de Matematica Aplicada a la Ingenieria, E.T.S. de Ingenieros Industriales, Universidad de Valladolid, 47011, Valladolid, Spain
Email: rafoba@wmatem.eis.uva.es

DOI: https://doi.org/10.1090/S0002-9939-96-03411-9
Keywords: Ergodic sheet, recurrent-proximal system, uniquely ergodic flow, singular dynamics
Received by editor(s): November 4, 1994
Received by editor(s) in revised form: April 6, 1995
Additional Notes: Partially supported by Junta de Castilla y León under project VA57/94
Communicated by: Mary Rees
Article copyright: © Copyright 1996 American Mathematical Society

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