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Topological Fiber Entropy for Linear Flows on Vector Bundles

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Abstract

For linear flows on vector bundles, it is shown that the topological entropy of lower dimensional subspaces in the fibers is determined by the Morse spectrum over chain recurrent components of the induced flows on Grassmann bundles.

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References

  1. Alongi J, Nelson G. Recurrence and topology. Am Math Soc. 2007.

  2. Arnold L. Random dynamical systems. Springer-Verlag; 1998.

  3. Bogenschütz T. Entropy, pressure and a variational principle for random dynamical systems. Random and Computational Dynamics Unknown Month 1992;1(1):99–116.

    Google Scholar 

  4. Bowen R. Entropy for group endomorphisms and homogeneous spaces. Trans Am Math Soc. 1971;153. Erratum, 1973;181:509–510.

    Google Scholar 

  5. Braga Barros C, San Martin LAB. Chain transitive sets for flows on flag bundles. Forum Math. 2007;19:19–60.

    MATH  MathSciNet  Google Scholar 

  6. Colonius F, Fabbri R, Johnson R. Chain recurrence, growth rates and ergodic limits. Ergodic Theory and Dynamical Systems 2004;27:1509–1524.

    MathSciNet  Google Scholar 

  7. Colonius F, Kliemann W. The Morse spectrum of linear flows on vector bundles. Trans Am Math Soc. 1996;348:4355–4388.

    Article  MATH  MathSciNet  Google Scholar 

  8. Colonius F, Kliemann W. The dynamics of control. Boston: Birkhäuser; 2000.

    Book  Google Scholar 

  9. Colonius F, Kliemann W. Morse decompositions and spectra on flag bundles. J Dyn Diff Equat. 2002;14:719–741.

    Article  MATH  MathSciNet  Google Scholar 

  10. Dunford N, Schwartz JT. Linear operators, Part I: general theory. Wiley-Interscience; 1977.

  11. Handel M, Kitchens B. Metrics and entropy for non-compact spaces. IsraelJ Math. 1995;91:253–271.

    MATH  MathSciNet  Google Scholar 

  12. Johnson RA, Palmer KJ, Sell GR. Ergodic properties of linear dynamical systems. SIAM J Math Anal. 1987;18:1–33.

    Article  MATH  MathSciNet  Google Scholar 

  13. Karoubi M. K-Theory, an introduction. Springer-Verlag; 1978.

  14. Kolyada S, Snoha L. Topological entropy for nonautonomous dynamical systems. Random and Comput Dyn. 1996;4:205–233.

    MATH  MathSciNet  Google Scholar 

  15. Patrão M. Entropy and its variational principle for non-compact metric spaces. Ergodic Theor Dyn Syst. 2010;30:1529–1542.

    Article  MATH  Google Scholar 

  16. Patrão M, San Martin LAB.Chain recurrence of flows and semiflows on fiber bundles. Discrete Contin Dynam Syst A. 2007;17:113–139.

    Google Scholar 

  17. Petersen K. Ergodic theory.Cambridge University Press; 1989.

  18. Robinson C. Dynamical systems. Stability, symbolic dynamics, and chaos.CRC Press; 1995.

  19. Sacker RJ, Sell GR. A spectral theory for linear differential systems. J Diff Equ. 1978;27:320–358.

    Article  MATH  MathSciNet  Google Scholar 

  20. Salamon D, Zehnder E. Flows on vector bundles and hyperbolic sets. Trans Amer Math Soc. 1988;306:623–649.

    Article  MATH  MathSciNet  Google Scholar 

  21. San Martin LAB, Seco L. Morse and Lyapunov spectra and dynamics on flag bundles. Ergodic Theor Dyn Syst. 2009;30:893–922.

    Article  MathSciNet  Google Scholar 

  22. Selgrade J. Isolated invariant sets for flows on vector bundles. Trans Amer Math Soc. 1975;203:259–390.

    Article  MathSciNet  Google Scholar 

  23. Zhang J-L, Chen L-X.Lower bounds of the topological entropy for nonautonomous dynamical systems. Appl Math J Chinese Univ. 2009;24(1):76–82.

    Article  MATH  MathSciNet  Google Scholar 

  24. Walters P. An introduction to Ergodic theory.Springer-Verlag; 1982.

  25. Warga J. Optimal control of differential and functional equations.Academic; 1972.

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Acknowledgments

F. Colonius is supported by DFG grant Co 124/17-2 within DFG Priority Program 1305 Control Theory of Digitally Networked Dynamical Systems. L.A.B. San Martin is supported by CNPq grant n o 303755/2009-1 and FAPESP grant n o 07/06896-5. A.J. da Silva is supported by CAPES grant n o 4229/10-0.

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Authors and Affiliations

  1. Institut für Mathematik, Universität Augsburg, 86135, Augsburg, Germany

    Fritz Colonius

  2. Instituto de Matemática, Universidade Estadual de Campinas, Cx. Postal 6065, 13.081-970, Campinas, SP, Brasil

    Luiz A.B. San Martin

  3. Institut für Mathematik, Universität Augsburg, 86159, Augsburg, Germany

    Adriano J. da Silva

Authors
  1. Fritz Colonius
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  2. Luiz A.B. San Martin
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  3. Adriano J. da Silva
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Correspondence to Fritz Colonius.

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Colonius, F., San Martin, L.A. & da Silva, A.J. Topological Fiber Entropy for Linear Flows on Vector Bundles. J Dyn Control Syst 20, 475–490 (2014). https://doi.org/10.1007/s10883-014-9217-8

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  • DOI: https://doi.org/10.1007/s10883-014-9217-8

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