Basis Problem for Turbulent Actions

We use modiied Tsirelson's spaces 3] to prove that there is no nite basis for turbulent Polish group actions. This answers a question of 1. Introduction One of the trends in Descriptive Set Theory during the last decade or so concerns the study of classiiability of Borel equivalence relations. Particularly well-studied are the orbit equivalence relations of Polish group actions (see 7, 10, 11, 8]). An equivalence relation is considered to be classiiable if we can associate some \sim-ple" invariants to its equivalence classes in an eeective way. The most general formalization of this notion is the one in which these simple invariants are taken to be countable structures (or equivalently, the isomorphism classes of countable structures). The rst example of a Borel equivalence relation arising from a Polish group action which is not classiiable in this way was given by H. Friedman ((6]),