Exact endomorphism

of a Lebesgue space $ ( X , \mu ) $

An endomorphism $ T $ of $ ( X , \mu ) $( cf. Metric isomorphism) such that the only measurable decomposition $ \mathop{\rm mod} 0 $ that is coarser $ \mathop{\rm mod} 0 $ than all $ T ^ {-} n \epsilon $, where $ \epsilon $ is the decomposition into points, is the trivial decomposition with as only element all of $ X $. An equivalent definition is: There is no measurable decomposition that is invariant (in older terminology — totally invariant) under $ T $( i.e. is such that $ T ^ {-} 1 \xi = \xi $ $ \mathop{\rm mod} 0 $). Examples of such endomorphisms are a one-sided Bernoulli shift and an expanding mapping.

Exact endomorphisms have strong ergodic properties analogous to those of $ K $- 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 $ K $- automorphism). Cf. $ K $- system.

References

[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)

Comments

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

The usual definition is as follows: An endomorphism $ T $ of a Lebesgue space $ ( X , \mu ) $ is said to be exact whenever $ \cap _ {n=} 0 ^ \infty T ^ {-} n {\mathcal B} = {\mathcal N} $, where $ {\mathcal B} $ is the given $ \sigma $- algebra of $ ( X , \mu ) $ and $ {\mathcal N} $ is the $ \sigma $- 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.

References

[a1]	R. Mañé, "Ergodic theory and differentiable dynamics" , Springer (1987)