Metric properties of measure preserving homeomorphisms

We study “typical” metric (ergodic) properties of measure preserving homeomorphisms of regularly connected cellular polyhedra and of some other spaces. In 1941 Oxtoby and Ulam proved (for a narrower class of spaces) that ergodicity is such a property. Using a modification of their construction and the method of approximating metric automorphisms by periodic ones, we prove in this paper that almost all properties that are “typical' for the metric automorphisms of the Lebesgue spaces are also "typical” for the situation under discussion.