Pugh's closure lemma

In the theory of dynamic systems, Pugh's closure lemma states that a dynamic system with non-moving points in the topology can be approximated arbitrarily well by dynamic systems with periodic orbites . It was proven by Charles C. Pugh . ${\ displaystyle C ^ {1}}$

It is an open question whether this also applies to the topology (10th Smale's problem ). Before Pugh, René Thom published an incorrect proof ( Mauricio Peixoto found the error ). ${\ displaystyle C ^ {\ infty}}$

10. Smale's problem

Let be a - diffeomorphism of a compact manifold and a non-migrating point of . ${\ displaystyle f \ colon M \ to M}$ ${\ displaystyle C ^ {\ infty}}$ ${\ displaystyle M}$ ${\ displaystyle p \ in M}$ ${\ displaystyle f}$

One of the problems Smaleschen asks whether in the topology arbitrarily close to lying -Diffeomorphismen are, for a periodic point is. ${\ displaystyle C ^ {\ infty}}$ ${\ displaystyle f}$ ${\ displaystyle C ^ {\ infty}}$ ${\ displaystyle g \ colon M \ to M}$ ${\ displaystyle p}$

So there should be a -diffeomorphism for each , so that ${\ displaystyle \ epsilon> 0}$ ${\ displaystyle C ^ {\ infty}}$ ${\ displaystyle g \ colon M \ to M}$

{\ displaystyle d (f (x), g (x)) <\ epsilon}

and

{\ displaystyle \ parallel D_ {x} ^ {k} f-D_ {x} ^ {k} g \ parallel <\ epsilon \ \ forall k \ in \ mathbb {N} \ \ forall x \ in M}

(for an arbitrarily chosen Riemannian metric ) as well as for a is. ${\ displaystyle g ^ {n} (p) = p}$ ${\ displaystyle n \ in \ mathbb {N}}$

This question is an open problem. Only the following closure lemma from Pugh is proven, which only guarantees approximability in the topology. ${\ displaystyle C ^ {1}}$