Hayashi’s connecting lemma

Let $f\colon M\to M$ be a $C^{1}$ diffeomorphism of the compact smooth manifold $M$ , and let $p,q\in M$ be such that there exists a nonperiodic point in $\omega(p,f)\cap\alpha(q,f)$ (the intersection of the alpha limit set of $q$ with the omega limit set of $p$ ). Then there exists a diffeomorphism $g$ , arbitrarily close to $f$ in the $\mathcal{C}^{1}$ topology of $\operatorname{Diff}^{1}(M)$ , such that $q$ is in the forward orbit of $p$ through $g$ , i.e. such that $g^{n}(p)=q$ for some $n>0$ .

References

1 Wen, L., Xia, Z., $\mathcal{C}^{1}$ connecting lemmas, Trans. Amer. Math. Soc. 352 (2000), no. 11, 5213-5230.

Title	Hayashi’s connecting lemma
Canonical name	HayashisConnectingLemma
Date of creation	2013-03-22 14:07:16
Last modified on	2013-03-22 14:07:16
Owner	Koro (127)
Last modified by	Koro (127)
Numerical id	6
Author	Koro (127)
Entry type	Theorem
Classification	msc 37C05
Classification	msc 37C25
Synonym	connecting lemma