Geodesic
On exact endomorphisms with a quasi-invariant measure
Matematičeskie zametki, Tome 21 (1977) no. 1, pp. 99-108 Cet article a éte moissonné depuis la source Math-Net.Ru

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In this paper it is proved that for any measurable partition $\xi$, $\xi\ne\varepsilon\pmod0$, of Lebesgue space with continuous measure that does not have elements of positive measure, there exists an exact endomorphism $T$ with a quasi-invariant measure for which $T^{-1}\varepsilon=\xi$. 
@article{MZM_1977_21_1_a11,
     author = {V. G. Sharapov},
     title = {On exact endomorphisms with a~quasi-invariant measure},
     journal = {Matemati\v{c}eskie zametki},
     pages = {99--108},
     year = {1977},
     volume = {21},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1977_21_1_a11/}
}
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V. G. Sharapov. On exact endomorphisms with a quasi-invariant measure. Matematičeskie zametki, Tome 21 (1977) no. 1, pp. 99-108. http://geodesic.mathdoc.fr/item/MZM_1977_21_1_a11/