Genericity of ergodicity for Sobolev homeomorphisms

In this paper, we obtain a weak version of Lusin’s theorem in the Sobolev- ( 1 , p ) $(1,p)$ left parenthesis 1 comma p right parenthesis uniform closure of volume-preserving Lipschitz homeomorphisms on closed and connected d -dimensional manifolds, d ≥ 2 d ≥ 2 d greater than or equals 2 and 0 < p < 1 $0<p<1$ 0 less than p less than 1 . With this result at hand, we will be able to prove that the ergodic elements are generic. This establishes a version of the Oxtoby–Ulam theorem for this Sobolev class. We also prove that, for 1 ≤ p < d − 1 1≤ p<d-1 1 less than or equals p less than d minus 1 , the topological transitive maps are generic.