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Generic expanding maps without absolutely continuous invariant \(\sigma\)-finite measure. (English) Zbl 1139.28005
The existence of type III transformations (measurable transformations of a Lebesgue space \((X,\lambda)\) with the property that there is no \(\sigma\)-finite invariant measure absolutely continuous with respect to \(\lambda\)) was conjectured by P. R. Halmos [Ann. Math. (2) 48, 735–754 (1947; Zbl 0029.35202)] and established by D. S. Ornstein [Bull. Am. Math. Soc. 66, 297–300 (1960; Zbl 0154.30502)]. In this paper a wide circle of results on particular examples and existence questions is brought to a definitive conclusion by showing that a \(C^1\)-generic expanding map of the circle is type III with respect to the Lebesgue measure. The sharpness of this result is illuminated by the fact that \(C^{1+\alpha}\) expanding maps have absolutely continuous invariant probability measures.

MSC:
28D05 Measure-preserving transformations
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