Advertisement

Journal of Statistical Physics

, Volume 146, Issue 4, pp 850–863 | Cite as

Harmonic Averages and New Explicit Constants for Invariant Densities of Piecewise Expanding Maps of the Interval

  • Paweł Góra
  • Zhenyang Li
  • Abraham Boyarsky
  • Harald Proppe
Article

Abstract

The statistical behavior of families of maps is important in studying the stability properties of chaotic maps. For a piecewise expanding map τ whose slope >2 in magnitude, much is known about the stability of the associated invariant density. However, when the map has slope magnitude ≤2 many different behaviors can occur as shown in (Keller in Monatsh. Math. 94(4): 313–333, 1982) for W maps. The main results of this note use a harmonic average of slopes condition to obtain new explicit constants for the upper and lower bounds of the invariant probability density function associated with the map, as well as a bound for the speed of convergence to the density. Since these constants are determined explicitly the results can be extended to families of approximating maps.

Keywords

Absolutely continuous invariant measures Piecewise expanding maps of interval Lower bound for invariant density Explicit constants for rate of convergence Harmonic average of slopes 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Baladi, V.: Positive Transfer Operators and Decay of Correlations. Advanced Series in Nonlinear Dynamics, vol. 16, p. x+314. World Scientific, River Edge (2000). ISBN:981-02-3328-0 CrossRefGoogle Scholar
  2. 2.
    Boyarsky, A., Góra, P.: Laws of Chaos. Invariant Measures and Dynamical Systems in One Dimension. Probability and Its Applications. Birkhäuser, Boston (1997) zbMATHGoogle Scholar
  3. 3.
    Eslami, P., Góra, P.: Stronger Lasota-Yorke inequality for piecewise monotonic transformations. Preprint Google Scholar
  4. 4.
    Eslami, P., Misiurewicz, M.: Singular limits of absolutely continuous invariant measures for families of transitive map. J. Differ. Equ. Appl. (2011). doi: 10.1080/10236198.2011.590480 Google Scholar
  5. 5.
    Keller, G.: Stochastic stability in some chaotic dynamical systems. Monatsh. Math. 94(4), 313–333 (1982) MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Keller, G.: Piecewise monotonic transformations and exactness. Collection: Seminar on Probability, Rennes 1978, Exp. No. 6, 32 pp. Univ. Rennes, Rennes (in French) Google Scholar
  7. 7.
    Keller, G.: Interval maps with strictly contracting Perron-Frobenius operators. Int. J. Bifurc. Chaos Appl. Sci. Eng. 9(9), 1777–1783 (1999) zbMATHCrossRefGoogle Scholar
  8. 8.
    Keller, G., Liverani, C.: Stability of the spectrum for transfer operators. Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 28(1), 141–152 (1999) MathSciNetzbMATHGoogle Scholar
  9. 9.
    Kowalski, Z.S.: Invariant measures for piecewise monotonic transformation has a positive lower bound on its support. Bull. Acad. Pol. Sci., Sér. Sci. Math. XXVII(1), 53–57 (1979) Google Scholar
  10. 10.
    Liverani, C.: Decay of correlations for piecewise expanding maps. J. Stat. Phys. 78(3–4), 1111–1129 (1995) MathSciNetADSzbMATHCrossRefGoogle Scholar
  11. 11.
    Liverani, C.: Decay of correlations. Ann. Math. (2) 142(2), 239–301 (1995) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Góra, P., Li, Zh., Boyarsky, A.: Harmonic average of slopes and the stability of ACIM. Preprint Google Scholar
  13. 13.
    Li, Zh.: W-like maps with various instabilities of ACIM’s. Available at http://arxiv.org/abs/1109.5199
  14. 14.
    Li, Zh., Góra, P., Boyarsky, A., Proppe, H., Eslami, P.: A family of piecewise expanding maps having singular measure as a limit of ACIM’s. Ergod. Theory Dyn. Syst. (accepted) Google Scholar
  15. 15.
    Schmitt, B.: Contributions à l’étude de systèmes dynamiques unidimensionnels en théorie ergodique. Ph.D. Thesis, University of Bourgogne (1986) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Paweł Góra
    • 1
  • Zhenyang Li
    • 1
  • Abraham Boyarsky
    • 1
  • Harald Proppe
    • 1
  1. 1.Department of Mathematics and StatisticsConcordia UniversityMontrealCanada

Personalised recommendations