Summary
We exhibit random strange attractors with random Sinai-Bowen-Ruelle measures for the composition of independent random diffeomorphisms.
Article PDF
Similar content being viewed by others
References
Breiman, L.: Probability. Reading, Massachusetts: Addison Wesley 1968
Baxendale, P.: Brownian motion in the diffeomorphism group I. Compositio Mathematica 53, 19–50 (1984)
Brin, M., Nitecki, Z.: Absolute continuity of stable foliation in Hilbert space. In preparation
Bowen, R.: Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Lect. Notes Math., vol. 470. Berlin Heidelberg New York: Springer 1975
Carverhill, A.: Flows of stochastic dynamical systems: ergodic theory. Stochastics 14, 273–317 (1985)
Carverhill, A.: Survey: Lyapunov exponents for stochastic flows on manifolds. In: Arnold, L., Wihstutz, V. (eds.) Lyapunov Exponents. Proceedings, Bremen 1984. (Lect. Notes Math., vol. 1186, pp. 292–307) Berlin Heidelberg New York: Springer 1986
Elworthy, K.D.: Stochastic differential equations on manifolds. L.M.S. Lecture Notes. Cambridge: Cambridge University Press 1982
Eckmann, J.P., Ruelle, D.: Ergodic theory of chaos and strange attractors. Rev. Modern Phys. 57, 617–656 (1985)
Ichihara, K., Kunita, H.: A classification of the second order degenerate eliptic operators and its probabilistic characterization. Z. Wahrscheinlichkeitstheor. Verw. Geb. 30, 253–254 (1974)
Kifer, Y.: Ergodic theory of random transformations. Progress in probability and statistics. Basel Boston: Birkhäuser 1986
Katok, A., Strelcyn, J.M.: Smooth maps with singularities: invariant manifolds, entropy and billiards. Lect. Notes Math., vol. 1222. Berlin Heidelberg New York: Springer 1986
Kunita, H.: Stochastic differential equations and stochastic flow of diffeomorphisms. In: Hennequin, P.L. (ed.) Ecole d'Eté de Probabilités de Saint-Flour XII, 1982 (Lect. Notes Math., vol. 1097, pp. 144–303) Berlin Heidelberg New York: Springer 1984
Ledrappier, F.: Propriétés ergodiques des mesures de Sinai. Publ. Math., Inst. Hautes Etud. Sci. 59, 163–188 (1984)
Ledrappier, F.: Dimension of invariant measures. Proceedings of the conference ergodic theory and related topics II. Georgenthal 1986. Teubner-Texte zur Mathematik 94, 116–124 (1987)
Ledrappier, F.: Quelques propriétés des exposants caractéristiques. Lect. Notes Math., vol. 1097, pp. 305–396. Berlin Heidelberg New York: Springer 1984
Le Jan, Y.: On isotropic Brownian motion. Z. Wahrscheinlichkeitstheor. Verw. Geb. 70, 609–620 (1985)
Ledrappier, F., Strelcyn, J.-M.: A proof of the estimation from below in Pesin entropy formula. Ergodic Theory Dyn. Syst. 2, 203–219 (1982)
Ledrappier, F., Young, L.-S.: The metric entropy of diffeomorphisms. Ann. Math. 122, 509–574 (1985)
Mañé, R.: A proof of Pesin's formula. Ergodic Theory Dyn. Syst. 1, 95–102 (1981)
Parry, W.: Entropy and generators in ergodic theory. Math. Lecture Notes Series. New York: Benjamin 1969
Pesin, Ia.B.: Lyapunov characteristic exponents and smooth ergodic theory. Russ. Math. Surv. 32, 55–114 (1977)
Rohlin, V.A.: On the fundamental ideas of measure theory. Transl., I. Ser., Am. Math. Soc. 10, 1–54 (1962)
Rudolph, D.: If a finite extension of a Bernoulli shift has no finite rotation factors, it is Bernoulli. Isr. J. Math. 30, 193–206 (1978)
Ruelle, D.: Ergodic theory of differentiable dynamical systems. Publ. Math., Inst. Hautes Etud. Sci. 50, 27–58 (1979)
Sinai, Ya.: Classical dynamical systems with countable Lebesgue spectrum II. Izv. Akad. Nauk Arm. SSR, Mat. 30, 15–68 (1966); Transl., II. Ser., Am. Math. Soc. 68, 34–88 (1968)
Thouvenot, J.P.: Remarques sur les systèmes dynamiques donnés avec plusieurs facteurs. Isr. J. Math. 21, 215–232 (1975)
Young, L.-S.: Stochastic stability of Hyperbolic Attractors. Ergodic Theory Dyn. Syst. 6, 31–319 (1986)
Author information
Authors and Affiliations
Additional information
The research of this author is partially supported by the National Science Foundation and the Sloan Foundation.
Rights and permissions
About this article
Cite this article
Ledrappier, F., Young, L.S. Entropy formula for random transformations. Probab. Th. Rel. Fields 80, 217–240 (1988). https://doi.org/10.1007/BF00356103
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00356103