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Limit theorems in averaging for dynamical systems

Published online by Cambridge University Press:  14 October 2010

Yuri Kifer
Yuri Kifer
Affiliation:
Institute of Mathematics, Hebrew University, Jerusalem, Givat Ram 91904, Israel
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Abstract

This paper yields diffusion and moderate deviation type asymptotics for solutions of differential equations of the form dZε(t)/dt = εB(Zε(t), fty) where ft is a suspension flow (in particular, a hyperbolic flow) over a sufficiently fast mixing transformation. Such problems emerge in the study of perturbed Hamiltonian systems. These exhibit a new class of limit theorems for dynamical systems and extend a number of previously known results.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 15 , Issue 6 , December 1995 , pp. 1143 - 1172
Copyright
Copyright © Cambridge University Press 1995

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References

REFERENCES

[A]Arnold, V. I.. Geometric Methods in the Theory of Ordinary Differential Equations. Springer: Berlin, 1983.CrossRefGoogle Scholar
[B]Bowen, R.. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. (Lecture Notes in Mathematics, 470) Springer: Berlin, 1975.CrossRefGoogle Scholar
[Bi]Billingsley, P.. Convergence of Probability Measures. Wiley: New York, 1968.Google Scholar
[Br]Bryc, W.. A remark on the connection between the large deviation principle and the central limit theorem. Statist. Probab. Lett. 18 (1993), 253–256.CrossRefGoogle Scholar
[BF]Borodin, A. N. and Freidlin, M. I.. Fast oscillating random perturbations of dynamical systems with conservation laws. Ann. de L'I. H. P. (probab. et stat.) 31 (1995), 485–525.Google Scholar
[BR]Bowen, R. and Ruelle, D.. The ergodic theory of Axiom A flows. Invent. Math. 29 (1975), 181–202.CrossRefGoogle Scholar
[CP]Coelho, Z. and Parry, W.. Central limit asymptotics for shifts of finite type. Isr. J. Math. 69 (1990), 235–249.CrossRefGoogle Scholar
[CE]Cogburn, R. and Ellison, J. A.. A stochastic theory of adiabatic invariance. Comm. Math. Phys. 149 (1992) 97–126.CrossRefGoogle Scholar
[D]Denker, M.. The central limit theorem for dynamical systems. Dynamical Systems and Ergodic Theory, Banach Center Publications 23 PWN-Polish Sci. Publ.: Warszawa, 1989.Google Scholar
[DP]Denker, M. and Philipp, W.. Approximation by Brownian motion for Gibbs measures and flows under a function. Ergod. Th. & Dynam. Sys. 4 (1984), 541–552.CrossRefGoogle Scholar
[Fr]Franko, E.. Flows with unique equilibrium state. Am. J. Math. 99 (1977), 486–514.CrossRefGoogle Scholar
[Fre]Freidlin, M. I.. The averaging principle and theorems on large deviations. Russ. Math. Surv. 33 (5) (1978), 107–160.CrossRefGoogle Scholar
[FW]Freidlin, M. I. and Wentzel, A. D.. Random Perturbations of Dynamical Systems. Springer: Berlin, 1984.CrossRefGoogle Scholar
[G]Gartner, J.. On large deviations from the invariant measure. Theory Probab. Appl. 22 (1977), 24–39.CrossRefGoogle Scholar
[I]Ibragimov, I. A.. Some limit theorems for stochastic processes. Theory Probab. Appl. 7 (1962), 349–382.CrossRefGoogle Scholar
[Kh]Khasminskii, R. Z.. On stochastic processes defined by differential equations with a small parameter. Theory Probab. Appl. 11 (1966), 211–228.CrossRefGoogle Scholar
[K]Kato, T.. Perturbation Theory for Linear Operators. Springer: Berlin, 1966.Google Scholar
[Kl]Kifer, Y.. Large deviations in dynamical systems and stochastic processes. Trans. Am. Math. Soc. 321 (1990), 505–524.CrossRefGoogle Scholar
[K2]Kifer, Y.. Averaging in dynamical systems and large deviations. Invent. Math. 110 (1992), 337–370.CrossRefGoogle Scholar
[K3]Kifer, Y.. Large deviations, averaging, and periodic orbits of dynamical systems. Comm. Math. Phys. 162 (1994), 33–46.CrossRefGoogle Scholar
[L]Lalley, S. P.. Distribution of periodic orbits of symbolic and Axiom A flows. Adv. Appl. Math. 8 (1987), 154–193.CrossRefGoogle Scholar
[P]Pollicott, M.. Large deviations, Gibbs measures, and closed orbits for hyperbolic flows. Preprint 1993.Google Scholar
[PP]Parry, W. and Pollicott, M.. Zeta functions and the periodic orbit structure of hyperbolic dynamics. Asterisque 187–188 (1990).Google Scholar
[R]Ratner, M.. The central limit theorem for geodesic flows on n-dimensional manifolds of negative curvature. Isr. J. Math. 16 (1973), 181–197.CrossRefGoogle Scholar
[Ru]Ruelle, D.. Thermodynamic Formalism. Addison-Wesley: Reading, MA, 1978.Google Scholar
[RV]Rozanov, Yu. A. and Volkonskii, A.. Some limit theorems for random functions I. Theory Probab. Appl. 4 (1959), 178–197.Google Scholar
[SV]Sanders, J. A. and Verhulst, F.. Averaging Methods in Nonlinear Dynamical Systems. Springer: Berlin, 1985.CrossRefGoogle Scholar
[W]Waddington, S.. Large deviation asymptotics for Anosov flows. Preprint 1993.Google Scholar