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We show that if M is a compact manifold, then there is a residual subset N of the set of homeomorphisms on M with the property that if f ∈ N , then the periodic points of f are dense in its chain recurrent set. This result was first announced in [4], but a flaw in that argument was noted in [1], where a different proof was given. It was recently noted in [5] 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 [4] to prove the result. Two fundamental results in the theory of smooth dynamical systems are the C closing lemma and the associated C general density theorem that were established by C. Pugh in [6]. The former shows that if p is a nonwandering point of a diffeomorphism f of a compact manifold M , then in any C neighborhood of f there is a diffeomorphism g for which x is a periodic point. The latter shows that C generically the nonwandering set of a diffeomorphism coincides with the closure of the set of its periodic points. This paper addresses similar (but much simpler) questions in the space of homeomorphisms of a smooth manifold M (compact and without boundary) with the C topology. In this setting the closing lemma is a ‘folk theorem’ whose proof can be found in many places; references are given below. In fact, these references show that any chain recurrent point of f is a periodic point for some g that is arbitrarily C close to f . The argument that establishes this is much simpler than any known proof of the C closing lemma. Somewhat surprisingly, however, the proof of the full C general density theorem, that for a generic homeomorphism the chain recurrent set is the same as the closure of the periodic points, has been elusive. A claim of a general density theorem (with the nonwandering set in place of the chain recurrent set) was made in [4]; a flaw in that argument was later described in [1] where a different argument was given. It was recently pointed out in [5] that the argument in [1] does not quite establish the full general density theorem; instead it shows that C generically in the C closure of the set of diffeomorphisms on M the chain recurrent set is the closure of the set of periodic points. This does not establish the full C general density theorem because in large dimensions it is known that the set of diffeomorphisms is not C dense in the set of homeomorphisms. In the current paper we show how to patch the original argument from [4], and so establish the full C general density theorem. At the end of the paper we also describe briefly how to Received by the editors October 11, 1994. 1991 Mathematics Subject Classification. Primary 58F08.

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