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Like the closing lemma, the connecting lemma is of fundamental importance in dynamical systems. Hayashi recently proved the C 1 connecting lemma for stable and unstable manifolds of a hyperbolic invariant set. In this paper, we prove several very general C 1 connecting lemmas. We simplify Hayashi's proof and extend the results to more general cases.doi:10.1090/s0002-9947-00-02553-8 fatcat:wejdlc32kfgtlbmscslv2pjwwi