# 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 HayashisConnectingLemma 2013-03-22 14:07:16 2013-03-22 14:07:16 Koro (127) Koro (127) 6 Koro (127) Theorem msc 37C05 msc 37C25 connecting lemma