Homeomorphisms of linear and planar sets of the first category into microscopic sets

Abstract J.C. Oxtoby and S. Ulam proved that each set of the first category in r -dimensional Euclidean space can be transformed into a set of Lebesgue measure zero by some automorphism, and such homeomorphisms constitute a residual set in the set of all automorphisms. We have improved this result changing the sets of measure zero by microscopic sets on the real line and on the plane.