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Communications in Mathematical Physics

, Volume 300, Issue 2, pp 411–433 | Cite as

(Non)Invariance of Dynamical Quantities for Orbit Equivalent Flows

  • Katrin Gelfert
  • Adilson E. Motter
Article

Abstract

We study how dynamical quantities such as Lyapunov exponents, metric entropy, topological pressure, recurrence rates, and dimension-like characteristics change under a time reparameterization of a dynamical system. These quantities are shown to either remain invariant, transform according to a multiplicative factor or transform through a convoluted dependence that may take the form of an integral over the initial local values. We discuss the significance of these results for the apparent non-invariance of chaos in general relativity and explore applications to the synchronization of equilibrium states and the elimination of expansions.

Keywords

Lyapunov Exponent Hausdorff Dimension Topological Entropy Borel Probability Measure Time Transformation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Abramov L.: On the entropy of a flow. Amer. Math. Soc. Transl. Ser. 2 49, 167–170 (1966)MATHGoogle Scholar
  2. 2.
    Anosov D.V., Sinai Ya.G.: Certain smooth ergodic systems. Russ. Math. Surv. 22, 103–167 (1967)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Barbaroux J.-M., Germinet F., Tcheremchnatsev S.: Generalized fractal dimensions: Equivalences and basic properties. J. Math. Pures Appl. 80, 977–1012 (2001)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Barreira, L., Pesin, Y.: Smooth ergodic theory and nonuniformly hyperbolic dynamics, with an appendix by O. Sarig. In: Handbook of Dynamical Systems 1B, B. Hasselblatt, A. Katok eds., Amsterdam: Elsevier, 2006Google Scholar
  5. 5.
    Barreira L., Radu L., Wolf C.: Dimension of measures for suspension flows. Dyn. Syst. 19, 89–107 (2004)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Benini R., Montani G.: Frame independence of the inhomogeneous mixmaster chaos via Misner-Chitré-like variables. Phys. Rev. D 70, 103527 (2004)CrossRefMathSciNetADSGoogle Scholar
  7. 7.
    Berger B.K.: Comments on the computation of Liapunov exponents for the Mixmaster universe. Gen. Relativ. Gravit. 23, 1385–1402 (1991)CrossRefADSGoogle Scholar
  8. 8.
    Burd A.B., Buric N., Ellis G.F.R.: A numerical analysis of chaotic behaviour in Bianchi IX models. Gen. Relat. Gravit. 22, 349–363 (1990)MATHCrossRefMathSciNetADSGoogle Scholar
  9. 9.
    Chitré, D.M.: Investigations of Vanishing of a Horizon for Bianchy Type X (the Mixmaster) Universe. Ph.D. Thesis, University of Maryland, 1972Google Scholar
  10. 10.
    Contopoulos G., Grammaticos B., Ramani A.: The mixmaster universe model, revisited. J. Phys. A 27, 5357–5361 (1994)MATHCrossRefMathSciNetADSGoogle Scholar
  11. 11.
    Cornfeld I., Fomin S., Sinai Y.: Ergodic Theory. Springer, Berlin-Heidelberg-NewYork (1982)MATHGoogle Scholar
  12. 12.
    Cornish N.J., Levin J.J.: Mixmaster universe: A chaotic Farey tale. Phys. Rev. D 55, 7489–7510 (1997)CrossRefADSGoogle Scholar
  13. 13.
    Cornish N.J., Levin J.J.: The mixmaster universe is chaotic. Phys. Rev. Lett. 78, 998–1001 (1997)CrossRefADSGoogle Scholar
  14. 14.
    Fayad B.: Analytic mixing reparametrizations of irrational flows. Erg. Th. Dynam. Syst. 22, 437–468 (2002)MATHMathSciNetGoogle Scholar
  15. 15.
    Francisco G., Matsas G.E.A.: Qualitative and numerical study of Bianchi IX models. Gen. Relat. Grav. 20, 1047–1054 (1988)CrossRefMathSciNetADSGoogle Scholar
  16. 16.
    Furstenberg H.: Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton University Press, Princeton, NJ (1981)MATHGoogle Scholar
  17. 17.
    Galatolo S.: Dimension via waiting time and recurrence. Math. Res. Lett. 12, 377–386 (2005)MATHMathSciNetGoogle Scholar
  18. 18.
    Grassberger P., Procaccia I.: Characterization of Strange Attractors. Phys. Rev. Lett. 50, 346–349 (1993)CrossRefMathSciNetADSGoogle Scholar
  19. 19.
    Haydn N., Luevano J., Mantica G., Vaienti S.: Multifractal properties of return time statistics. Phys. Rev. Lett. 88, 224502 (2002)CrossRefADSGoogle Scholar
  20. 20.
    Hobill D., Bernstein D., Welge M., Simkins D.: The Mixmaster cosmology as a dynamical system. Class. Quant. Grav. 8, 1155–1171 (1991)MATHCrossRefMathSciNetADSGoogle Scholar
  21. 21.
    Hobill, D., Burd, A.B., Coley, A.A. (eds.): Deterministic chaos in general relativity. NATO ASI Series B, Vol. 332, London: Plenum Press, 1994Google Scholar
  22. 22.
    Katok, A., Hasselblatt, B.: Introduction to the Modern Theory of Dynamical Systems. In: Encyclopedia of Mathematics and Its Applications 54, Cambridge: Cambridge University Press, 1995Google Scholar
  23. 23.
    Katok A., Knieper G., Weiss H.: Formulas for the derivative and critical points of topological entropy for Anosov and geodesic flows. Commun. Math. Phys. 138, 19–31 (1991)MATHCrossRefMathSciNetADSGoogle Scholar
  24. 24.
    Katok, A., Thouvenot, J.-P.: Spectral properties and combinatorial constructions in ergodic theory. In: Handbook of Dynamical Systems 1B, B. Hasselblatt, A. Katok, eds., Amsterdam: Elsevier, 2006Google Scholar
  25. 25.
    Misner C.W.: Mixmaster universe. Phys. Rev. Lett. 22, 1071–1074 (1969)MATHCrossRefADSGoogle Scholar
  26. 26.
    Motter A.E.: Relativistic chaos is coordinate invariant. Phys. Rev. Lett. 91, 231101 (2003)CrossRefADSGoogle Scholar
  27. 27.
    Motter A.E., Gelfert K.: Time-metric equivalence and dimension change under time reparatererizations. Phys. Rev. E 79, 065202(R) (2009)CrossRefADSGoogle Scholar
  28. 28.
    Motter A.E., Letelier P.S.: Mixmaster chaos. Phys. Lett. A 285, 127–131 (2001)MATHCrossRefMathSciNetADSGoogle Scholar
  29. 29.
    Motter A.E., Letelier P.S.: FRW cosmologies between chaos and integrability. Phys. Rev. D 65, 068502 (2002)CrossRefMathSciNetADSGoogle Scholar
  30. 30.
    Motter A.E., Saa A.: Relativistic invariance of Lyapunov exponents in bounded and unbounded systems. Phys. Rev. Lett. 102, 184101 (2009)CrossRefADSGoogle Scholar
  31. 31.
    Ohno T.: A weak equivalence and topological entropy. Publ. Res. Inst. Math. Sci. 16, 289–298 (1980)MATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Parry W.: Topics in Ergodic Theory. Cambridge University Press, Cambridge (1981)MATHGoogle Scholar
  33. 33.
    Parry W.: Synchronisation of canonical measures for hyperbolic attractors. Commun. Math. Phys. 106, 267–275 (1986)MATHCrossRefMathSciNetADSGoogle Scholar
  34. 34.
    Pesin Y.: On rigorous mathematical definitions of correlation dimension and generalized spectrum for dimensions. J. Stat. Phys. 71, 529–547 (1993)MATHCrossRefMathSciNetADSGoogle Scholar
  35. 35.
    Pesin, Y.: Dimension Theory in Dynamical Systems: Contemporary Views and Applications, Chicago Lectures in Mathematics, Chicago: Chicago University Press, 1998Google Scholar
  36. 36.
    Pollicott M., Sharp R., Tuncel S., Walters P.: The mathematical research of William Parry. Erg. Th. Dynam. Syst. 28, 321–337 (2008)MATHMathSciNetGoogle Scholar
  37. 37.
    Rugh, S.E.: In Ref. [21], p. 359Google Scholar
  38. 38.
    Sun W., Young T., Zhou Y.: Topological entropies of equivalent smooth flows. Trans. Amer. Math. Soc. 361, 3071–3082 (2009)MATHCrossRefMathSciNetGoogle Scholar
  39. 39.
    Szydlowski M.: Chaos hidden behind time parametrization in the Mixmaster cosmology. Gen. Relativ. Gravit. 29, 185–203 (1997)MATHCrossRefMathSciNetADSGoogle Scholar
  40. 40.
    Szydlowski M., Krawiec A.: Description of chaos in simple relativistic systems. Phys. Rev. D 53, 6893–6901 (1996)CrossRefADSGoogle Scholar
  41. 41.
    Totoki H.: Time changes of flows. Mem. Fac. Sci. Kyushu Univ. Ser. A 20, 27–55 (1966)MATHMathSciNetGoogle Scholar
  42. 42.
    Walters, P.: An Introduction to Ergodic Theory. Graduate Texts in Mathematics 79, Berlin-Heidelberg-New York: Springer, 1981Google Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Instituto de MatemáticaUFRJ, Cidade Universitária - Ilha do FundãoRio de JaneiroBrazil
  2. 2.Department of Physics and Astronomy & Northwestern Institute on Complex SystemsNorthwestern UniversityEvanstonUSA

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