On invariants for measure preserving transformations

Abstract

Abstract. The classification problem for measure preserving transformations is strictly more complicated than that of graph isomorphism. Preamble.We consider the group M ∞ of all invertible measure preserving transformations either on the unit interval or any other reasonable measure space. It seems natural to say that two of these transformations, σ 1 , σ 2 , are equivalent or isomorphic if there is a third, π, so thatTo what extent can this equivalence relation be considered classifiable? In specific cases-for instance σ 1 , σ 2 both Bernoulli or discrete spectrum-there are well accepted systems of complete invariants. However, in the completely general context of arbitrary measure preserving transformations there is no known satisfactory system of complete invariants nor even a clear statement of what this would entail.For instance, Halmos in [7] despairs of precisely formulating the problem but at page 1029 suggests that its solution should fulfill the "the vague task of finding a complete set of invariants. . . " At page 75 of [8] he proposes that the central problem is to "find usable necessary and sufficient conditions for the conjugacy of two measure preserving transformations." Some time later Weiss at page 670 of [23] raises the problem of finding "a set of invariants large enough so that if all invariants agree for two m.p.t. one can conclude that the m.p.t. are isomorphic".This article considers attempts to make this precise and ask abstractly whether the problem of classifiability could in principle have a positive solu-2000 Mathematics Subject Classification: Primary 03A15. Key words and phrases: classification, measure preserving transformation, Polish group action.The author gratefully acknowledges support from NSF grants DMS 99-70403 and DMS 96-22977.[51] 52 G. Hjorth tion. We present a clearly identifiable lower bound on the classification difficulty of the isomorphism relation for measure preserving transformations.One precise formulation of the problem would be to understand a classifiable equivalence relation on a Polish space to be one for which we may find a Borel assignment of reals or points in some other standard Borel space as complete invariants. This is the notion of classifiable suggested by the Glimm-Effros dichotomy of [10]. Indeed, Feldman in [4] takes exactly that position. Appealing to [18] he observes that the isomorphism relation for Bernoulli shifts allows real numbers to be assigned in a Borel manner as a complete invariant and uses [19] to remark that such an assignment is already impossible for the class of measure preserving transformations having the property of K.A more generous notion of classification, closer to the kinds we consider below, is already implicit in sources [7], [8], [23]. In each case the results of [9] are accepted as providing a complete classification for the discrete spectrum measure preserving transformations. Here the invariants are not real numbers or single points in a standard Borel space, but rather countable sets of complex numbers. The sig...

The conjugacy problem in ergodic theory

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