Generating Dense Subgroups of Measure Preserving Transformations

Except for a set of first category, all pairs of measure preserving transformations generate a dense subgroup of G, the group of all invertible measure preserving transformations of the unit interval when G has the weak topology. Consider G the group of invertible (Lebesgue) measure preserving transformations of the unit interval onto itself. On G, we put the weak topology (Halmos [2, pp. 61-67]). It has been asked by A. Iwanik, whether the group generated by certain pairs of measure preserving transformations is dense in G. The purpose of this note is to prove: Except for a set of first category in G X G, all pairs of measure preserving transformations generate a subgroup dense in G. Related work has been done by R. Grzaslewicz [1] who showed that for any two positive irrational numbers a and ß, less than 1, the following transformations Sa and Tß generate a dense subgroup of G, where Sa(x) = x + a (mod 1) and fx + ^modj), 0<x<|, Tß(x) i I