On Borel Non–reducibility of P-ideals

V. Kanovei

Let E be the least class of equivalence relations on Polish spaces which contains all equalities and is closed under the operations of countable power E^{∞} and Fubini product Fin ⨂_{n} E_{n} over the ideal Fin of all finite sets.

Theorem 1. Any equivalence relation generated by a Borel P-ideal I other than Fin or 0 × Fin, is \underline{not} Borel reducible to an equivalence relation in E.

(Equivalence relations induced by Fin and 0 × Fin clearly belong to E.)

The subclass T ⊆ E obtained by the closure under countable powers only, includes only equivalence relations induced by continuous actions of S_{∞}, the symmetric group, so that for T instead of E Theorem 1 follows from Hjorth’s theory of turbulence. (A particular case of I = the density-0 ideal here, was considered by H. Friedman.) Note that E contains many remarkable relations which do not belong to T , for instance, Frechet ideals Fin_{α}, obtained from Fin by iteration of Fubini products.

Our technical approach is based on the notion of generic ergodicity, due to Hjorth. An equivalence relation E is generically F-ergodic, if any Borel map ϑ with x E y ⟹ ϑ(x) F ϑ(y) is F-constant on a comeager set. We prove that a certain stronger version of the ergodicity is preserved under the same transformations of F as above. Let a Polish group G act continuously on a Polish space X. Say that the induced equivalence E_{X}^{G} is hereditarily generically (hg, for brevity) F-ergodic if the local relation ∼_{Y}^{U} is F-ergodic whenever U ⊆ G is open, symmetric, and contains 1_{G}, and ∅ ≠ Y ⊆ X is open, and the local orbit O(Y, U, x) is dense in Y for comeager-many x ∈ Y.

Theorem 2. Suppose that the action is generically turbulent. Let F be a family of analytic equivalence relations such that E_{X}^{G} is hg F-ergodic for each F in F. Let F′ be the closure of F under the operations of countable power and Fubini product over Fin. Then E_{X}^{G} is hg F’-ergodic for each F′ in F′.

Address: MCCME, Bol. Vlasevski Per. 11, Moscow 121002, Russia

E-mail: kanovei@math.uni-wuppertal.de