Genericity of ergodicity for Sobolev homeomorphisms

Abstract

In this paper, we obtain a weak version of Lusin’s theorem in the Sobolev-

uniform closure of volume-preserving Lipschitz homeomorphisms on closed and connected d -dimensional manifolds,

d ≥ 2 $d \geq 2$ 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 $1\leq p<d-1$ 1 less than or equals p less than d minus 1

, the topological transitive maps are generic.