Hayashi’s connecting lemma
Let $f:M\to M$ 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 ${\mathrm{Diff}}^{1}(M)$, 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., ${\mathrm{C}}^{\mathrm{1}}$ 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 |