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Lifting measures to Markov extensions. (English) Zbl 0712.28008
Generalizing a theorem of F. Hofbauer [Isr. J. Math. 34, 213-237 (1979; Zbl 0422.28015)] we give conditions under which invariant measures for piecewise invertible dynamical systems can be lifted to Markov extensions. In contrast to Hofbauer’s proof we do not use coding techniques to construct a pointwise near-isomorphism between the inverse limits of the dynamical system and its Markov extension. Instead we use rather straightforward entropy arguments proving that an ergodic invariant measure of the original system can be lifted to an invariant measure of the extension provided its entropy is greater than the “topological entropy at infinity” of the extension. In the special case of twice continuously differentiable, piecewise monotone interval maps with nondegenerate critical points it is proved that each nonatomic ergodic invariant measure with positive Lyapunov exponent can be lifted to the canonical Markov extension.
Reviewer: G.Keller

MSC:
28D05 Measure-preserving transformations
28D20 Entropy and other invariants
37A99 Ergodic theory
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