Topological Rokhlin's Theorem.
For diffeomorphisms of a compact manifold, see the topological tower theorem in
C. Bonatti and S. Crovisier, Récurrence et généricité. Inventiones Mathematicae 158 (2004), 33–104.
This has been generalized to homeomorphisms on a space with finite dimension in
Y. Gutman, Mean dimension and Jaworski-type theorems. Proc. Lond. Math. Soc. 111 (2015), 831–850.