Lifting measures to Markov extensions |
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Authors: | Gerhard Keller |
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Institution: | 1. Mathematisches Institut, Universit?t Erlangen, Bismarckstrasse 1 1/2, D-8520, Erlangen, Federal Republic of Germany
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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: - 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.
- 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 iffh μ (T)=Σlogf dμ.
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