Hostname: page-component-848d4c4894-ttngx Total loading time: 0 Render date: 2024-05-23T15:52:37.028Z Has data issue: false hasContentIssue false
  • English
  • Français

Estimating invariant measures and Lyapunov exponents

Published online by Cambridge University Press:  19 September 2008

Brian R. Hunt
Brian R. Hunt
Affiliation:
Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742, USA (e-mail: bhunt@ipst.umd.edu)
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper describes a method for obtaining rigorous numerical bounds on time averages for a class of one-dimensional expanding maps. The idea is to directly estimate the absolutely continuous invariant measure for these maps, without computing trajectories. The main theoretical result is a bound on the convergence rate of the Frobenius—Perron operator for such maps. The method is applied to estimate the Lyapunov exponents for a planar map of recent interest.

Type
Survey Article
Information
Ergodic Theory and Dynamical Systems , Volume 16 , Issue 4 , August 1996 , pp. 735 - 749
Copyright
Copyright © Cambridge University Press 1996

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Alexander, J. C., Hunt, B. R., Kan, I. and Yorke, J. A.. Intermingled basins for the triangle map. Ergod. Th. & Dynam. Sys. pp. 651–662 of this issue.Google Scholar
[2]Alexander, J. C., Kan, I., Yorke, J. A. and You, Z-P.. Riddled Basins. Int. J. Bif. Chaos 2 (1992), 795–813.CrossRefGoogle Scholar
[3]Birkhoff, G. D.. Proof of the ergodic theorem. Proc. Nat. Acad. Sci. 17 (1931), 656–660.CrossRefGoogle ScholarPubMed
[4]Blank, M. L.. A bound for the rate of decrease of correlation in one-dimensional dynamical systems. Functional Anal. Appl. 18 (1984), 50–52.CrossRefGoogle Scholar
[5]Chiu, C., Du, Q. and Li, T. Y.. Error estimates of the Markov finite approximation of the Frobenius—Perron operator. Nonlin. Anal. Th. Meth. Appl. 19 (1992), 291–308.CrossRefGoogle Scholar
[6]Eckmann, J-P. and Ruelle, D.. Ergodic theory of chaos and strange attractors. Rev. Mod. Phys. 57 (1985), 617–656.CrossRefGoogle Scholar
[7]Hofbauer, F. and Keller, G.. Ergodic properties of invariant measures for piecewise monotonic transformations. Math. Z. 180 (1982), 119–140.CrossRefGoogle Scholar
[8]Hunt, F. Y.. A Monte Carlo approach to the approximation of invariant measures. National Institute of Standard and Technology Internal Report 4980 (1993).Google Scholar
[9]Keller, G.. On the rate of convergence to equilibrium in one-dimensional systems. Comm. Math. Phys. 96 (1984), 181–193.CrossRefGoogle Scholar
[10]Lasota, A. and Yorke, J. A.. On the existence of invariant measures for piecewise monotonic transformations. Trans. Amer. Math. Soc. 186 (1973), 481–488.CrossRefGoogle Scholar
[11]Li, T. Y.. Finite approximation for the Frobenius—Perron operator, a solution to Ulam's conjecture. J. Approx. Theory 17 (1976), 177–186.CrossRefGoogle Scholar
[12]Lopes, A.. Dynamics of real polynomials on the plane and triple point phase transition. Math. Comp. Mod. 13 (1990), 17–32.CrossRefGoogle Scholar
[13]Miller, W. M.. Stability and approximation of invariant measures for a class of non-expanding transformations. Nonlin. Anal. Th. Meth. Appl. 23 (1994), 1013–1025.CrossRefGoogle Scholar
[14]Oseledec, V. I.. A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc. 19 (1968), 197–231.Google Scholar
[15]Ruelle, D.. Applications conservant une mesure absolument continue par rapport à dx sur [0, 1]. Comm. Math. Phys. 55 (1977), 47–51.CrossRefGoogle Scholar
[16]Rychlik, M.. Bounded variation and invariant measures. Studia Math. 76 (1983), 69–80.CrossRefGoogle Scholar
[17]Rychlik, M.. Regularity of the metric entropy for expanding maps. Trans. Amer. Math. Soc. 315 (1989), 833–847.CrossRefGoogle Scholar
[18]Rychlik, M. and Sorets, E.. Regularity and other properties of absolutely continuous invariant measures for the quadratic family. Comm. Math. Phys. 150 (1992), 217–236.CrossRefGoogle Scholar
[19]. Hyperbolicity is dense in the real quadratic family. Preprint #1992/10, SUNY Stony Brook IMS, revised 1994.Google Scholar
[20]Varga, Richard S.. Matrix Iterative Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1962.Google Scholar
[21]Young, L-S.. Decay of correlations for certain quadratic maps. Comm. Math. Phys. 146 (1992), 123–138.CrossRefGoogle Scholar
[A]Liverani, C.. Decay of correlations. Ann. Math. 142 (1995), 239–301.CrossRefGoogle Scholar
[B]Liverani, C.. Decay of correlations for piecewise expanding maps. J. Stat. Phys. 78 (1995), 1111–1129.CrossRefGoogle Scholar