Communications in Mathematical Physics

, Volume 127, Issue 2, pp 319–337 | Cite as

Quadratic maps without asymptotic measure

  • Franz Hofbauer
  • Gerhard Keller


An interval map is said to have an asymptotic measure if the time averages of the iterates of Lebesgue measure converge weakly. We construct quadratic maps which have no asymptotic measure. We also find examples of quadratic maps which have an asymptotic measure with very unexpected properties, e.g. a map with the point mass on an unstable fix point as asymptotic measure. The key to our construction is a new characterization of kneading sequences.


Neural Network Statistical Physic Complex System Nonlinear Dynamics Lebesgue Measure 
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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Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • Franz Hofbauer
    • 1
  • Gerhard Keller
    • 2
  1. 1.Institut für MathematikUniversität WienWienAustria
  2. 2.Mathematisches InstitutUniversität ErlangenErlangenFederal Republic of Germany

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