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Some Dynamical Properties of Certain Differentiable Mappings of an Interval

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

Abstract

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:
$$ {f_r}\left( x \right) = r \cdot x\left( {1 - x} \right){\text{}}\left\langle {0,1} \right\rangle {\hbox{\ \hbox{$\mid$}\kern -1em \lower .5em \hbox{$\leftarrow$}}} ,\;0 \leqslant r \leqslant 4.\;\left[ R \right] $$
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Keywords

Lebesgue Measure Invariant Measure Differentiable Mapping Pairwise Disjoint Periodic Point 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  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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