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:
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(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.
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(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 iffh μ (T)=Σlogf dμ.
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Keller, G. Lifting measures to Markov extensions. Monatshefte für Mathematik 108, 183–200 (1989). https://doi.org/10.1007/BF01308670
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DOI: https://doi.org/10.1007/BF01308670