Some Dynamical Properties of Certain Differentiable Mappings of an Interval

Part II
Part of the Progress in Mathematics book series (PM, volume 21)


This paper is a continuation of the paper [S]. We present here a sufficient condition for the existence of invariant measure (absolutely continuous with respect to Lebesgue measure) for differentiable mappings of an interval. The condition is equivalent to the one presented in [S] but its formulation is very close to Ruelle’s conjecture on absolutely continuous invariant measures for the parabolic mappings:
$$ {f_r}\left( x \right) = r \cdot x\left( {1 - x} \right){\text{}}\left\langle {0,1} \right\rangle {\hbox{\ \hbox{$\mid$}\kern -1em \lower .5em \hbox{$\leftarrow$}}} ,\;0 \leqslant r \leqslant 4.\;\left[ R \right] $$


Lebesgue Measure Invariant Measure Differentiable Mapping Pairwise Disjoint Periodic Point 
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  1. [M]
    M. Misiurewicz, Absolutely continuous measures for certain maps of an interval, Preprint IHES/M/79/293.Google Scholar
  2. [E]
    D. Ruelle, Applications conservant une measure absolu-ment continue par rapport à dx sur <0, 1>, Comm. Math. Phys. vol. 55 (1977), 47–51.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [S]
    W. Szlenk, Some dynamical properties of certain differentiable mappings of an interval, Part I. Bol. Soc. Mat. Mex., to appear.Google Scholar
  4. [O]
    A. Ognev, Metric Properties of a class of Maps of the Interval, Mat. Zametki (1981) (to appear).Google Scholar

Copyright information

© Springer Science+Business Media New York 1982

Authors and Affiliations

  1. 1.Institute of Applied MathematicsWarsaw Agricultural UniversityWarsawPoland

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