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It is well known that a transverse homoclinic point of a hyperbolic periodic point of a $C^1$-diffeomorphism of a compact manifold $M$ persists under small $C^1$ perturbations. This follows easily from the general continuity properties of hyperbolic sets and the fact that the union of orbits of the periodic point and the homoclinic point form a hyperbolic set.

Does the same hold for non-compact manifolds (with the usual Whitney topology on the set of all $C^1$ diffeomorphisms)?

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  • $\begingroup$ The periodic orbit and the orbit of some transverse homoclinic intersection belong to a compact set, so same argument works. $\endgroup$
    – rpotrie
    Jun 17, 2021 at 0:23
  • $\begingroup$ Could you please elaborate a bit on how the argument can be adapted to this setting? The above arguments seem to crucially depend on working with diffeomorphisms of compact manifolds. $\endgroup$
    – Leon Staresinic
    Jun 18, 2021 at 9:40
  • $\begingroup$ To be more specific, it seems implicit in your response that results for compact manifolds continue to work in the non-compact setting, as long as the property we are interested in (such as the continuity of a homoclinic point) "lives" in a compact set. Could you elaborate more on why this is so? $\endgroup$
    – Leon Staresinic
    Jun 18, 2021 at 10:55
  • $\begingroup$ What I say is that all you need to control does not escape a compact set, so, the same proof applies. Say that $y$ is a transverse homoclinic point of a periodic point $p$. So, the forward orbit of $y$ belongs to a compact submanifold $S$ of the stable manifold of the orbit of $p$ (which is forward invariant) and the backward orbit to a compact submanifold $U$ of the unstable manifold of the orbit of $p$ (backward invariant). There is a neighborhood $V$ of $f$ in $C^1$ topology such that if $g \in V$ then $g$ and its derivative are close to $f$ in a nbhdd of $U \cup S$. $\endgroup$
    – rpotrie
    Jun 18, 2021 at 23:21

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