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

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Abstract | References | Similar Articles | Additional Information

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

**[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**

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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