In 1978 Lind and Thouvenot proved that every finite entropy ergodic transformation is isomorphic to a Lebesgue measure-preserving homeomorphism of T^2. During their proof they also arrive at an isomorphism with a measure preserving diffeomorphism of T^2, however it does not necessarily preserve Lebesgue measure. The diffeomorphism preserves an Oxtoby-Ulam measure. Joseph Rosenblatt and I are considering continuous and smooth representations of weakly-mixing, rigid transformations on T^2.