# 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)

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.