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Deviations of trajectory averages and the defect in Pesin's formula for Anosov diffeomorphisms


Authors: Steven Orey and Stephan Pelikan
Journal: Trans. Amer. Math. Soc. 315 (1989), 741-753
MSC: Primary 60F10; Secondary 28D20, 58F11, 58F15
MathSciNet review: 935534
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Abstract: A large deviation theorem at the Donsker Varadhan level three is obtained for the convergence of trajectory averages of Anosov diffeomorphisms. It is possible to provide an explicit description of the rate function.


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  • [A] D. V. Anosov, Geodesic flows on closed Riemann manifolds with negative curvature., Proceedings of the Steklov Institute of Mathematics, No. 90 (1967). Translated from the Russian by S. Feder, American Mathematical Society, Providence, R.I., 1969. MR 0242194
  • [B] Rufus Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics, Vol. 470, Springer-Verlag, Berlin-New York, 1975. MR 0442989
  • [DV] M. Donsker and S. S. R. Varadhan, Asymptotic evaluation of certain Markov process expectations for large time, part 4, Comm. Pure Appl. Math. (1983), 183-212.
  • [L] F. Ledrappier, Propriétés ergodiques des mesures de Sinaï, Inst. Hautes Études Sci. Publ. Math. 59 (1984), 163–188 (French). MR 743818
  • [ME] Nathaniel F. G. Martin and James W. England, Mathematical theory of entropy, Encyclopedia of Mathematics and its Applications, vol. 12, Addison-Wesley Publishing Co., Reading, Mass., 1981. With a foreword by James K. Brooks. MR 612318
  • [O] Steven Orey, Large deviations in ergodic theory, Seminar on stochastic processes, 1984 (Evanston, Ill., 1984) Progr. Probab. Statist., vol. 9, Birkhäuser Boston, Boston, MA, 1986, pp. 195–249. MR 896730, 10.1007/978-1-4684-6745-1_12
  • [OP] S. Orey and S. Pelikan, Large deviation principles for stationary shifts, IMA preprint no. 227.
  • [P] Ja. B. Pesin, Characteristic Ljapunov exponents, and smooth ergodic theory, Uspehi Mat. Nauk 32 (1977), no. 4 (196), 55–112, 287 (Russian). MR 0466791
  • [Ra] M. Ratner, Markov partitions for Anosov flows on 𝑛-dimensional manifolds, Israel J. Math. 15 (1973), 92–114. MR 0339282
  • [R] David Ruelle, Thermodynamic formalism, Encyclopedia of Mathematics and its Applications, vol. 5, Addison-Wesley Publishing Co., Reading, Mass., 1978. The mathematical structures of classical equilibrium statistical mechanics; With a foreword by Giovanni Gallavotti and Gian-Carlo Rota. MR 511655
  • [S1] Ja. G. Sinaĭ, Construction of Markov partitionings, Funkcional. Anal. i Priložen. 2 (1968), no. 3, 70–80 (Loose errata) (Russian). MR 0250352
  • [S2] Ja. G. Sinaĭ, Construction of Markov partitionings, Funkcional. Anal. i Priložen. 2 (1968), no. 3, 70–80 (Loose errata) (Russian). MR 0250352
  • [T] Yōichirō Takahashi, Entropy functional (free energy) for dynamical systems and their random perturbations, Stochastic analysis (Katata/Kyoto, 1982) North-Holland Math. Library, vol. 32, North-Holland, Amsterdam, 1984, pp. 437–467. MR 780769, 10.1016/S0924-6509(08)70404-5
  • [V] S. R. S. Varadhan, Large deviations and applications, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 46, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1984. MR 758258

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1989-0935534-4
Keywords: Anosov diffeomorphism, trajectory average, large deviations
Article copyright: © Copyright 1989 American Mathematical Society