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Exactness of expanding mappings. (English) Zbl 0685.28006

For a suitably defined piecewise expanding \(C^ 2\)-mapping of the d- dimensional unit cube into itself the author proves that there exists a unique absolutely continuous invariant normalized measure such that the induced dynamical system is exact. The proof is based on an idea due to A. Lasota and M. C. Mackey [Probabilistic properties of deterministic systems (1986; Zbl 0606.58002)]. For related papers see G. Keller [C. R. Acad. Sci., Paris, Sér. A 289, 625-627 (1979; Zbl 0419.28007)], M. Jabłoński [Ann. Pol. Math. 42, 185-195 (1983; Zbl 0591.28014)], D. H. Mayer [Commun. Math. Phys. 95, 1-15 (1984; Zbl 0577.58022)] and D. Candeloro [Atti Semin. Mat. Fis. Univ. Modena 35, 33-42 (1987; Zbl 0656.28008)].
Reviewer: K.Krzyżewski

MSC:

28D05 Measure-preserving transformations
37A99 Ergodic theory
37D99 Dynamical systems with hyperbolic behavior
28D10 One-parameter continuous families of measure-preserving transformations
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