A Variational Approach to the Random Diffeomorphisms Type Perturbations of a Hyperbolic Diffeomorphism

  • Yuri Kifer
Conference paper


Let f be a diffeomorphism of a compact Riemannian manifold M. The standard model of random perturbations of f is generated by a particle which jumps from x to fx and smears with some distribution close to the δ — function at fx. This model has the continuous time version where a flow is perturbed by a small diffusion. This approach leads to a Markov chain X n ε (or diffusion X t ε , in the continuous time case) with a small parameter ε > 0 and one is interested whether invariant measures of X n ε converge as ε → 0 to a particular invariant measure of the diffeomorphism f. To describe limiting measures I employed in [5] the Donsker-Varadhan variational formula
$$ {\lambda ^\varepsilon }(V) = \mathop {\sup }\limits_{\mu \in p(M)} (\int {Vd\mu } - {I^\varepsilon }(\mu )) $$
for the principal eigenvalues λ ε (V) of the operators P V ε g = P ε (e V g) where P ε is the transition operator of the Markov chain X n ε , V is a continuous function, and I(μ) is certain lower semicontinuous convex functional on the space P(M) of probability measures on M. It turns out that if f is a hyperbolic diffeomorphism then λε(V) converges as ε → 0 to the topological pressure Q(V + φ u ) of f corresponding to the function V + φ u where φu = -log J u (x) and J u (x) is the Jacobian of the differential Df restricted to the unstable subbundle.


Invariant Measure Random Perturbation Compact Riemannian Manifold Principal Eigenvalue Random Dynamical System 
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  1. 1.
    Bowen, R.: Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Lecture Notes in Math., 470, Springer-Verlag (1975)MATHGoogle Scholar
  2. 2.
    Bogenschütz, T.: Entropy, pressure, and a variational principle for random dynamical systems. PreprintGoogle Scholar
  3. 3.
    Bowen, R., Ruelle, D.: The ergodic theory of Axiom A flows. Invent.Math. 29, 181–202 (1975)MathSciNetCrossRefMATHADSGoogle Scholar
  4. 4.
    Kifer, Y.: Ergodic Theory of Random Transformations. Birkhäuser, Boston, 1986CrossRefMATHGoogle Scholar
  5. 5.
    Kifer, Y.: Principal eigenvalues, topological pressure, and stochastic stability of equilibrium states. Israel J.Math. 70, 1–47 (1990)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Kifer, Y.: Large deviations in dynamical systems and stochastic processes. Trans.Amer.Math.Soc. 321, 505–524 (1990)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Kifer, Y., Newhouse, S.: A global volume lemma and applications. Israel J.Math. 74, 209–223 (1991)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Ledrappier, F., Young, L.-S.: Entropy formula for random transformations. Probab.Theory Related Fields 80, 217–240 (1988)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Young, L.-S.: Stochastic stability of hyperbolic attractors. Ergodic Theory Dynamical Systems 6, 311–319 (1986)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • Yuri Kifer
    • 1
  1. 1.Institute of MathematicsThe Hebrew UniversityJerusalemIsrael

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