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.

单词	HayashisConnectingLemma
释义	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.
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