# K-flows reference

The following paper is about how a K-flow is produced from a K-induced map, but it is written in Russian. Does someone know where to find its English version? Do some textbooks include this topic?

B. Gurevič, Certain conditions for the existence of K-decompositions for special flows, Tr. Mosk. Mat. Obs., 1967, Volume 17, Pages 89–116.

• I don't read/understand Russian, but just skimming through the paper it seems very clear (i.e. p. 93) they use a suspension construction. Are you asking whether if one $T$ is a K-map, then all other powers (and the suspension map) are K-maps? this is indeed correct...
– Asaf
Commented Dec 26, 2021 at 7:45
• yes, but want to understand its proof. Do you know which textbooks/notes talk about it? Thanks! Commented Dec 26, 2021 at 14:31
• I guess the quickest way would be to either show that the conditions holds for powers (shouldn't be hard, given a generating partition) and then play a bit with the definition of the suspension, or otherwise go through the equivalence to uniform mixing, then the theorem is clear for powers, probably suspension is not hard as well...
– Asaf
Commented Dec 26, 2021 at 21:44
• Its not true that all suspensions of K-induced map are K-flows, its true if the roof function is not "essentially-constant". You can show this by (extending/understandng) the proof in: Totoki, Haruo. On a class of special flows. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 15 (1970), 157–167. Commented Dec 27, 2021 at 8:27
• @user103342 great, this is very helpful. But it is strange, Totoki's paper cited Gurevič's paper, but in Mathscient, Totoki's paper is not in the list of citations of Gurevič's paper. Commented Dec 27, 2021 at 18:55

The Gurevich paper you cite was translated as "Some Existence Conditions for K-Decompositions for Special Flows" in Trans. Moscow Math. Soc. 17 (1967), 99-128 (if you don't have access to a university library I can send a copy to you; you can find my university email address by searching my name). There is also the Totoki paper as mentioned in the comments. Totoki also has a book titled Ergodic Theory, which includes the argument in the paper without much change it seems. Ornstein and Smorodinsky's paper "Ergodic flows of positive entropy can be time changed to become $$K$$-flows" has an argument very similar to Totoki's argument.
Sinai's two papers "Dynamical systems with countable Lebesgue spectrum " I and II are the standard references for $$K$$-flows.
Katok has a related paper titled "Smooth Non-Bernoulli $$K$$-Automorphisms" where he interprets the $$K$$-property for skew products as a cohomological equation. (This paper uses smooth structures; instead of being purely ergodic theoretical.)
Finally, most ergodic theory books don't include flows at all; one book that does is Cornfeld, Fomin, Sinai's Ergodic Theory with a special emphasis on the connection between $$K$$-property and spectral properties (there is also a chapter on special flows; but the connection between $$K$$-property for special flows and the $$K$$-property for the base automorphism is not developed). Nadkarni's Basic Ergodic Theory also has a chapter on flows; but it does not have anything on the $$K$$-property.