Combinatorial and topological aspects of measure preserving homomorphisms
by
Alfio Giarlotta
University of Illinois at Urbana-Champaign
Coauthors: Vittorino Pata (University of Brescia, Italy), Pietro Ursino (University of Catania, Italy)

We study measure preserving homomorphisms between Lebesgue measurable subsets of the real line. This analysis is carried out using particular bijections of the interval [0, 1), called shifts. Within the set of all shifts, it is possible to identify a countable subfamily, formed by the so called ``rational" shifts. Then, the group S generated by rational shifts witnesses the separability of the space of homomorphisms. S can be viewed as a combinatorial object, since it is a directed colimit of a system of symmetric groups. This allows us both to express every measure preserving homomorphism as a countable sequence of finite permutations of natural numbers, and to endow S with some natural combinatorial metrics. Next, some topological properties of the sets E and A of endomorphisms and automorphisms of [0, 1) are analyzed. In particular, we show that E, endowed with the metric of convergence in measure, is a complete separable metric space, and that the algebraic and topological structures of A are compatible.

Date received: January 30, 2000