Abstract
The essential decorrelation rate of a hyperbolic dynamical system is the decay rate of time-correlations one expects to see stably for typical observables once resonances are projected out. We define and illustrate these notions and study the conjecture that for observables in \(\mathcal{C}^1\), the essential decorrelation rate is never faster than what is dictated by the smallest unstable Liapunov multiplier.
Article PDF
Similar content being viewed by others
REFERENCES
V. Baladi, unpublished (1989).
V. Baladi, Positive transfer operators and decay of correlations, in Advanced Series in Nonlinear Dynamics, Vol. 16 (World Scientific, River Edge, NJ, 2000).
R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Springer Lect. Notes in Math., Vol. 470 (Berlin-Heidelberg-New York, 1975).
P. Collet and S. Isola, On the essential spectrum of the transfer operator for expanding Markov maps, Comm. Math. Phys. 139:551–557 (1991).
D. E. Edmunds and W. D. Evans, Spectral Theory and Differential Operators (Oxford University Press, New York, 1987).
V. Gundlach and Y. Latushkin, A sharp formula for the essential spectral radius of Ruelle's operator on smooth and Hölder spaces, Ergod. Theor. Dyn. Syst. 23:175–191 (2003).
M. Hirsch, C. Pugh, and M. Shub, Invariant manifolds, Springer Lect. Notes in Math., Vol. 583 (Berlin-New York, 1977).
T. Kato, Perturbation Theory for Linear Operators (Springer-Verlag, Berlin, 1984), Second corrected printing of the second edition.
A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications, Vol. 54 (Cambridge University Press, Cambridge, 1995), with a supplementary chapter by A. Katok and L. Mendoza.
G. Keller, Ergodicité et mesures invariantes pour les transformations dilatantes par morceaux d'une région bornée du plan, C. R. Acad. Sci. Paris Sér. A-B. 289:A625-A627 (1979).
G. Keller, On the rate of convergence to equilibrium in one-dimensional systems, Comm. Math. Phys. 96:181–193 (1984).
P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, Cambridge Studies in Advanced Mathematics, Vol. 44 (Cambridge University Press, Cambridge, 1995), Fractals and rectifiability.
J. Rousseau-Egele, Un théorème de la limite locale pour une classe de transformations dilatantes et monotones par morceaux, Ann. Probab. 11:772–788 (1983).
D. Ruelle, Thermodynamic Formalism (Addison-Wesley, Reading, MA, 1978).
D. Ruelle, Resonances of chaotic dynamical systems, Phys. Rev. Lett. 56:405–407 (1986).
D. Ruelle, Resonances for Axiom A flows, J. Differential Geom. 25:99–116 (1987).
L.-S. Young, Recurrence times and rates of mixing, Israel. J. Math. 110:153–188 (1999).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Collet, P., Eckmann, JP. Liapunov Multipliers and Decay of Correlations in Dynamical Systems. Journal of Statistical Physics 115, 217–254 (2004). https://doi.org/10.1023/B:JOSS.0000019817.71073.61
Issue Date:
DOI: https://doi.org/10.1023/B:JOSS.0000019817.71073.61