Timeline for Under reasonable assumptions, is a closed invariant graph with only negative Lyapunov exponents necessarily stable?
Current License: CC BY-SA 4.0
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| Sep 1, 2021 at 4:47 | comment | added | Julian Newman | I've realised that this is probably trivial. Cover graph$A$ by finitely many charts. Fix sufficiently large $n$ that max log norm $\partial_x(\Theta^n)(\omega,A(\omega))$ over all the charts is strictly negative; then take a nbhd $O$ of graph$A$ contained in the union of the charts such that log norm $\partial_x(\Theta^n)$ has strictly negative supremum over all the charts intersected with $O$. Then $\Theta^n$ is uniformly Lipschitz w.r.t. its second input across the whole of $O$, and hence $\Theta^{tn}(O)\to$ graph$A$ as $t\to\infty$, and therefore $\Theta^t(O)\to$ graph$A$ as $t\to\infty$. | |
| Aug 31, 2021 at 21:44 | history | edited | Julian Newman | CC BY-SA 4.0 |
put Sturman & Stark result in a remark at the end
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| Aug 31, 2021 at 21:36 | history | asked | Julian Newman | CC BY-SA 4.0 |