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 Trudy Mat. Inst. Steklova, 2004, Volume 244, Pages 65–86 (Mi tm443)

Cramér Asymptotics in the Averaging Method for Systems with Fast Hyperbolic Motions

V. I. Bakhtin

Belarusian State University, Faculty of Physics

Abstract: A dynamical system $w'=S(w,z,\varepsilon )$, $z'=z+\varepsilon v(w,z,\varepsilon )$ is considered. It is assumed that slow motions are determined by the vector field $v(w,z,\varepsilon )$ in the Euclidean space and fast motions occur in a neighborhood of a topologically mixing hyperbolic attractor. For the difference between the real and averaged slow motions, the central limit theorem is proved and sharp asymptotics for the probabilities of large deviations (that do not exceed $\varepsilon ^\delta$) are calculated; the exponent $\delta$ depends on the smoothness of the system and approaches zero as the smoothness increases.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2004, 244, 58–79

Bibliographic databases:
UDC: 517.987+519.21

Citation: V. I. Bakhtin, “Cramér Asymptotics in the Averaging Method for Systems with Fast Hyperbolic Motions”, Dynamical systems and related problems of geometry, Collected papers. Dedicated to the memory of academician Andrei Andreevich Bolibrukh, Trudy Mat. Inst. Steklova, 244, Nauka, MAIK «Nauka/Inteperiodika», M., 2004, 65–86; Proc. Steklov Inst. Math., 244 (2004), 58–79

Citation in format AMSBIB
\Bibitem{Bak04} \by V.~I.~Bakhtin \paper Cram\'er Asymptotics in the Averaging Method for Systems with Fast Hyperbolic Motions \inbook Dynamical systems and related problems of geometry \bookinfo Collected papers. Dedicated to the memory of academician Andrei Andreevich Bolibrukh \serial Trudy Mat. Inst. Steklova \yr 2004 \vol 244 \pages 65--86 \publ Nauka, MAIK «Nauka/Inteperiodika» \publaddr M. \mathnet{http://mi.mathnet.ru/tm443} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2075113} \zmath{https://zbmath.org/?q=an:1078.37019} \transl \jour Proc. Steklov Inst. Math. \yr 2004 \vol 244 \pages 58--79 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. D. I. Dolgopyat, “Averaging and invariant measures”, Mosc. Math. J., 5:3 (2005), 537–576
2. Kifer Yu., “Another proof of the averaging principle for fully coupled dynamical systems with hyperbolic fast motions”, Discrete Contin. Dyn. Syst., 13:5 (2005), 1187–1201
3. Bakhtin V., Kifer Yu., “Nonconvergence examples in averaging”, Geometric and Probabilistic Structures in Dynamics, Contemporary Mathematics Series, 469, 2008, 1–17
4. Kifer Yu., Large deviations and adiabatic transitions for dynamical systems and Markov processes in fully coupled averaging, Mem. Amer. Math. Soc., 201, no. 944, 2009, viii+129 pp.
5. De Simoi J., Liverani C., “Limit Theorems For Fast-Slow Partially Hyperbolic Systems”, Invent. Math., 213:3 (2018), 811–1016
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