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


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.


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


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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

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