An Oxtoby and Ulam type theorem [see J. C. Oxtoby and S. M. Ulam, Ann. Math., II. Ser. 42, 874-920 (1941)] on generic ergodicity for homeomorphisms of the torus or annulus, which preserve a fixed measure and have mean rotation \(\lambda\) is obtained. It shows that a typical homeomorphism with mean rotation zero has a recurrent lift.