On proofs of the $C^0$ general density theorem

Abstract:

We show that if $M$ is a compact manifold, then there is a residual subset $\mathcal {N}$ of the set of homeomorphisms on $M$ with the property that if $f\in \mathcal {N}$, then the periodic points of $f$ are dense in its chain recurrent set. This result was first announced in [J. Palis, C. Pugh, M. Shub, M. Sullivan, Genericity theorems in topological dynamics, Dynamical Systems – Warwick 1974 (Springer Lect. Notes in Math. #468), Springer-Verlag, New York, 1975, pp. 241–250], but a flaw in that argument was noted in [E.M. Coven, J. Madden, Z. Nitecki, A note on generic properties of continuous maps, Ergodic Theory and Dynamical Systems II, Boston, Birkhäuser, 1982, pp. 97–101], where a different proof was given. It was recently noted in [S.Y. Pilyugin, The Space of Dynamical Systems with the $C^0$ Topology, (Springer Lect. Notes in Math #1571), Springer-Verlag, New York, 1994.] that this new argument only serves to show that the density of periodic points in the chain recurrent set is generic in the closure of the set of diffeomorphisms. We show how to patch the original argument from [J. Palis, C. Pugh, M. Shub, M. Sullivan, Genericity theorems in topological dynamics, Dynamical Systems – Warwick 1974 (Springer Lect. Notes in Math. #468), Springer-Verlag, New York, 1975, pp. 241–250] to prove the result.