Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Entropy via random perturbations


Author: Yuri Kifer
Journal: Trans. Amer. Math. Soc. 282 (1984), 589-601
MSC: Primary 58F15; Secondary 58F30, 58G32
MathSciNet review: 732108
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The entropy of a dynamical system $ {S^t}$ on a hyperbolic attractor with respect to the Bowen-Ruelle-Sinai measure is obtained as a limit of entropy characteristics of small random perturbations $ x_t^\varepsilon $ of $ {S^t}$. Both the case of perturbations only in some neighborhood of an attractor and global perturbations of a flow with hyperbolic attracting sets are considered.


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

  • [1] D. G. Aronson, Non-negative solutions of linear parabolic equations, Ann. Scuola Norm. Sup. Pisa (3) 22 (1968), 607–694. MR 0435594
  • [2] Patrick Billingsley, Ergodic theory and information, John Wiley & Sons, Inc., New York-London-Sydney, 1965. MR 0192027
  • [3] Rufus Bowen and David Ruelle, The ergodic theory of Axiom A flows, Invent. Math. 29 (1975), no. 3, 181–202. MR 0380889
  • [4] J. L. Doob, Stochastic processes, John Wiley & Sons, Inc., New York; Chapman & Hall, Limited, London, 1953. MR 0058896
  • [5] R. Z. Has′minskiĭ, The averaging principle for parabolic and elliptic differential equations and Markov processes with small diffusion, Teor. Verojatnost. i Primenen. 8 (1963), 3–25 (Russian, with English summary). MR 0161044
  • [6] N. Ikeda and A. Watanabe, Stochastic differential equations and diffusion processes, North-Holland, Amsterdam, 1981.
  • [7] A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math. 51 (1980), 137–173. MR 573822
  • [8] Yu. Kifer, On small random perturbations of some smooth dynamical systems, Math. USSR-Izv. 8 (1974), 1083-1107.
  • [9] Yuri Kifer, Stochastic stability of the topological pressure, J. Analyse Math. 38 (1980), 255–286. MR 600787, 10.1007/BF02807217
  • [10] Ricardo Mañé, A proof of Pesin’s formula, Ergodic Theory Dynamical Systems 1 (1981), no. 1, 95–102. MR 627789
  • [11] Ja. B. Pesin, Characteristic Ljapunov exponents, and smooth ergodic theory, Uspehi Mat. Nauk 32 (1977), no. 4 (196), 55–112, 287 (Russian). MR 0466791
  • [12] A. D. Ventcel′ and M. I. Freĭdlin, Small random perturbations of dynamical systems, Uspehi Mat. Nauk 25 (1970), no. 1 (151), 3–55 (Russian). MR 0267221

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 58F15, 58F30, 58G32

Retrieve articles in all journals with MSC: 58F15, 58F30, 58G32


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1984-0732108-0
Keywords: Hyperbolic attractor, diffusion process, entropy
Article copyright: © Copyright 1984 American Mathematical Society