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Invariant measures for interval maps with critical points and singularities. (English) Zbl 1184.37032
The paper explores conditions which allow to establish the existence of absolutely continuous invariant measures for interval maps which have both, singularities and critical points (in which the dervative vanishes). The authors find conditions on the singular points, outside from the neighborhoods of critical points (expansion conditions) and on the critical points which allow guarantee that, for a piecewise \(C^2\) interval map, the set of absolutely continuous invariant measures is finite and their basis covers the interval up to a set of Lebesgue zero measure.

MSC:
37E05 Dynamical systems involving maps of the interval
37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems
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