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.
Comments
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.
$Rufus^\infty$