On invariants for measure preserving transformations (Q2717760)

This paper applies the theory of definable equivalence relations developed in descriptive set theory to the study of the isomorphism relation among measure preserving transformations. Two invertible measure preserving transformations on the unit interval \(\sigma_1\) and \(\sigma_2\) are isomorphic if there exists another such transformation \(\pi\) such that \(\pi \circ \sigma_1 \circ \pi^{-1} = \sigma_2\) almost everywhere. The problem of finding complete invariants for the isomorphism relation for arbitrary measure preserving transformations (as opposed to Bernoulli or discrete spectrum ones, for which there are well-known complete invariants) was stated by \textit{P. R. Halmos} [Bull. Am. Math. Soc. 55, 1015-1034 (1949; Zbl 0036.35501) and Bull. Am. Math. Soc. 67, 70-80 (1961; Zbl 0161.11401)], and the main result of the paper supplies a mathematical explanation of the failure of this search. NEWLINENEWLINENEWLINEIndeed Hjorth proves that the isomorphism relation among measure preserving transformations does not admit ``classification by countable structures'', a well-known level of classification results (see \textit{G. Hjorth} and \textit{A. S. Kechris} [Ann. Pure Appl. Logic 82, No. 3, 221-272 (1996; Zbl 0933.03056)]). This means, e.g., that there is no Borel way of associating to each measure preserving transformation \(\sigma\) a countable graph \(F(\sigma)\) as a complete invariant (i.e., \(\sigma_1\) and \(\sigma_2\) are isomorphic if and only if \(F(\sigma_1)\) and \(F(\sigma_2)\) are isomorphic as graphs). (As the author remarks this is not particular to Borel functions, but applies to any reasonable class of functions not depending on the axiom of choice.) The result applies also to rank \(2\) generalized discrete spectrum transformations, showing that discrete spectrum is indeed an upper bound for the existence of reasonable complete invariants. NEWLINENEWLINENEWLINEThe proof uses \textit{G. Hjorth}'s theory of turbulence [Classification and orbit equivalence relations, Providence, RI: Am. Math. Soc. (2000; Zbl 0942.03056)].