# Special flow

constructed from an automorphism $S$ of a measure space $(X,\nu)$ and a measurable function $f$ (defined on $X$ and taking positive values)

A measurable flow in a certain new measure space $(M,\mu)$ constructed in the following way. The points of $M$ are the pairs $(x,s)$, where $x\in X$ and $0\leq s<f(x)$, and $\mu$ is the restriction to $M$, regarded as a subset of $X\times[0,\infty)$, of the direct product of the measure $\nu$ on $X$ and the Lebesgue measure on $[0,\infty)$. If $\mu(M)=\int_Xfd\nu<\infty$, then one usually normalizes this measure. The motion takes place in such a way that the second coordinate of $(x,s)$ increases with unit speed until it reaches the value $f(x)$, and then the point jumps into the position $(Sx,0)$.

In ergodic theory, special flows play a role similar to that of sections and the Poincaré return map in the study of smooth dynamical systems: $X$ plays the role of the section and $S$ that of the map. But in the topological theory one can usually construct a section in the form of a manifold only locally. In ergodic theory, the construction of a global section is possible under very general conditions, because here there are no restrictions connected with the topology. (Even if the initial flow is smooth, one nevertheless allows for the section to be discontinuous.) Therefore, under very general conditions a measurable flow is metrically isomorphic to some special flow, even with some extra conditions on $f$ (see [1]). A related concept is that of a special automorphism.

#### References

 [1] I.P. [I.P. Kornfel'd] Cornfel'd, S.V. Fomin, Ya.G. Sinai, "Ergodic theory" , Springer (1982) (Translated from Russian)

Each measurable flow on a Lebesgue space without fixed points is metrically isomorphic to a special flow; see [1], Chapt. 11, §2. For such a "special representation" with extra conditions on $f$ (Rudolph's theorem), see [1], Chapt. 11, §4. Special representations are used e.g. for the analysis of the spectrum of $K$-flows (cf. $K$-system).