Exact endomorphism

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of a Lebesgue space

An endomorphism of (cf. Metric isomorphism) such that the only measurable decomposition that is coarser than all , where is the decomposition into points, is the trivial decomposition with as only element all of . An equivalent definition is: There is no measurable decomposition that is invariant (in older terminology — totally invariant) under (i.e. is such that ). Examples of such endomorphisms are a one-sided Bernoulli shift and an expanding mapping.

Exact endomorphisms have strong ergodic properties analogous to those of -systems (to which they are related: there is a construction associating an automorphism to some endomorphism — its natural extension; for an exact endomorphism the latter is a -automorphism). Cf. -system.


[1] V.A. Rokhlin, "Exact endomorphisms of a Lebesgue space" Izv. Akad. Nauk SSSR Ser. Mat. , 25 : 4 (1961) pp. 499–530 (In Russian)
[2] I.P. [I.P. Kornfel'd] Cornfel'd, S.V. Fomin, Ya.G. Sinai, "Ergodic theory" , Springer (1982) (Translated from Russian)


Instead of "(measurable) decomposition" one also uses (measurable) partition.

The usual definition is as follows: An endomorphism of a Lebesgue space is said to be exact whenever , where is the given -algebra of and is the -algebra of subsets of measure 0 or 1. For a proof that expanding mappings are exact with respect to some measure, see e.g. [a1], Sect. III.1.


[a1] R. Mañé, "Ergodic theory and differentiable dynamics" , Springer (1987)
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Exact endomorphism. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article