Monatshefte für Mathematik

, Volume 108, Issue 2–3, pp 183–200

Lifting measures to Markov extensions

  • Gerhard Keller
Article

DOI: 10.1007/BF01308670

Cite this article as:
Keller, G. Monatshefte für Mathematik (1989) 108: 183. doi:10.1007/BF01308670

Abstract

Generalizing a theorem ofHofbauer (1979), we give conditions under which invariant measures for piecewise invertible dynamical systems can be lifted to Markov extensions. Using these results we prove:
  1. (1)

    IfT is anS-unimodal map with an attracting invariant Cantor set, then ∫log|T′|dμ=0 for the unique invariant measure μ on the Cantor set.

     
  2. (2)

    IfT is piecewise invertible, iff is the Radon-Nikodym derivative ofT with respect to a σ-finite measurem, if logf has bounded distortion underT, and if μ is an ergodicT-invariant measure satisfying a certain lower estimate for its entropy, then μ≪m iffhμ (T)=Σlogf dμ.

     

Copyright information

© Springer-Verlag 1989

Authors and Affiliations

  • Gerhard Keller
    • 1
  1. 1.Mathematisches InstitutUniversität ErlangenErlangenFederal Republic of Germany

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