Abstract
It is proved that a functional law of the iterated logarithm is valid for transitiveC 2 Anosov flows on compact Riemannian manifolds when the observable belongs to a certain class of real-valued Hölder functions. The result is equally valid for semiflows over piecewise expanding interval maps that are similar to the Williams' Lorenz-attractor semiflows. Furthermore the observables need only be real-valued Hölder for these semiflows.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Billingsley, P.: Convergence of Probability Measures. New York: Wiley. 1968.
Bowen, R.: Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Lecture Notes Math. 470. Berlin-Heidelberg-New York: Springer. 1975.
Bowen, R., Ruelle, D.: The ergodic theory of Axiom-A flows. Invent. Math.18, 181–202 (1975).
Hofbauer, F., Keller, G.: Ergodic properties of invariant measures for piecewise monotonic transformations. Preprint. Universität Göttingen. 1981.
Ibragimov, I. A.: Some limit theorems for stationary processes. Th. Prob. Appl.7, 349–382 (1962).
Leonov, V. P.: On the dispersion of time-dependent means of a stationary stochastic process. Th. Prob. Appl.6, 87–93 (1961).
Ratner, M.: The central limit theorem for geodesic flows onn-dimensional manifolds of negative curvature. Israel J. Math.16, 181–197 (1973).
Ratner, M.: Bernoulli flows over maps of the interval. Israel J. Math.31, 298–314 (1978).
Reznik, M. Kh.: The law of the iterated logarithm for some classes of stationary processes. Th. Prob. Appl.13, 606–621 (1968).
Williams, R. F.: The structure of Lorenz attractors. IHES Publ. Math.50, 321–347 (1979).
Wong, S.: Two probabilistic properties of weak-Bernoulli interval maps. (To appear.)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Wong, S. Law of the iterated logarithm for transitiveC 2 Anosov flows and semiflows over maps of the interval. Monatshefte für Mathematik 94, 163–173 (1982). https://doi.org/10.1007/BF01301935
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01301935