Timeline for Quantitive and computational improvement of the Oseledets multiplicative ergodic theorem for irrational rotation
Current License: CC BY-SA 4.0
19 events
| when toggle format | what | by | license | comment | |
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| Jul 25, 2019 at 15:08 | comment | added | Adam | @Jairo Bochi. Thanks. I'll check it out. | |
| Jul 25, 2019 at 10:33 | comment | added | Jairo Bochi | Here is a sketch on how to show that if the matrices are entrywise positive then there is a continuous invariant 1-dimensional subbundle (without appealing to general facts about domination): Consider any continuous field of lines $L(x)$ contained in the positive orthant, where $x$ is in the circle. Now let $L'_x$ be the new field of lines obtained by applying the dynamics (a la graph transform): $L'(x) := g(T^{-1}x) L(T^{-1} x)$. The transform $L \mapsto L'$ is a contraction with respect to the Hilbert projective metric. So it has a fixed point, which is the desired subbundle. | |
| Jul 25, 2019 at 10:18 | comment | added | Jairo Bochi | @RR Unfortunately there exists no comprehensive reference on dominated splittings. The ICTP lecture notes by Crovisier and Potrie is a good start. There are two caveats about it: (1) it's written for the derivative cocycle of a diffeomorphism (and not general linear cocycles); (2) the cones in the cone criterion are "ice-cream cones" (and so we can't apply it directly to the positive orthant in order to say something about positive matrices). Nevertheless the arguments extend to the more general situations. | |
| Jul 24, 2019 at 23:20 | comment | added | Adam | @Jairo Bochi. Could you please introduce a reference for it? | |
| Jul 21, 2019 at 22:18 | comment | added | Jairo Bochi | @r-r If the matrices are entrywise positive then the bundle $V_1$ is one-dimensional and dominating (in particular continuous). | |
| Jul 19, 2019 at 15:19 | comment | added | Adam | @AlekseiKulikov You can use topological presuree that is analytic. | |
| Jul 19, 2019 at 13:30 | comment | added | Aleksei Kulikov | @JairoBochi even worse, my coefficients are in general complex (though we of course can reduce $GL(n, \mathbb{C})$ to $GL(2n, \mathbb{R})$, but then we will definitely have negative coefficients). | |
| Jul 19, 2019 at 13:09 | comment | added | Adam | @AlekseiKulikov I know some papers by Backes who proved continuity of the subbundle for periodic point. It might help you. | |
| Jul 19, 2019 at 13:04 | comment | added | Adam | @JairoBochi : I have a question about your last comment. What you wanna get by positive entries? Hyperbolicity? As far as i know matrices have positive entries implies hyperbolicity in 2D. It does not work for higher dimention( Do you know contour example?). | |
| Jul 18, 2019 at 21:05 | comment | added | Jairo Bochi | Hmm, your situation seems complicated. Do your matrices have positive entries? This could help a lot. | |
| Jul 18, 2019 at 19:46 | comment | added | Aleksei Kulikov | @JairoBochi part about lack of continuity looks even more disappointing. But actually in my problem setup is a bit different then what I wrote in my post: my function $g$ is analytic on $[0, 1]$ but $g(0)$ and $g(1)$ may be not equal. May be something like what I want can be proved in analytic category or even in my setting with potential discontinuity? There are some things about analyticity in the papers which Dan and R R suggested but I'm still reading through them. | |
| Jul 18, 2019 at 19:41 | comment | added | Aleksei Kulikov | @JairoBochi papers of Morris and Furman looks interesting, thanks but I unfortunately has no idea how to prove continuity of the subbundle since I do not know how to calculate anything about it. | |
| Jul 18, 2019 at 19:18 | comment | added | Jairo Bochi | Part (1) is extremely delicate: you may check this paper by Wang and You to have an idea of the difficulties that are involved... | |
| Jul 18, 2019 at 19:10 | comment | added | Jairo Bochi | Property (2) is true for $\ell=1$, as a consequence of Theorem 1 of this old paper by Furman, or as a consequence of the more general Theorem A.3 of this paper by Morris. More generally, property (2) holds for any value of $\ell$ such that $V_\ell$ is a continuous subbundle (e.g. the dominated subbundle of a dominated splitting): just restrict the cocycle to this subbundle and use Morris' theorem. | |
| Jul 18, 2019 at 14:51 | comment | added | Aleksei Kulikov | @RR thanks, I'll take a look into this! | |
| Jul 17, 2019 at 13:44 | comment | added | Adam | Check this paper. You need domination for your questions. This kind of cocycle is called "quasi periodic cocycle". As @DanRust mentioned, Duarte and Klein's book explain it. | |
| Jul 17, 2019 at 9:58 | comment | added | Anthony Quas | Interesting question. A couple of comments: (1) the MET. actually gives you something stronger than you state- $\mathbb R^n$ may be expressed as a direct sum of subspaces with expansion rates $\lambda_1$,...,$\lambda_k$. (2) I guess you want none of the $\lambda_i$ to be 0? (although of course you can multiply the generator by a constant, adding a constant to the $\lambda_i$’s.). | |
| Jul 16, 2019 at 21:42 | comment | added | Dan Rust | Have you tried checking out Chapter 6 of Duarte and Klein's book? It deals with Cocycles whose base dynamics are irrational rotations satisfying diophantine approximation, so you should be able to find a partial answer there (sorry, I don't know the book myself, a colleague recommended it after I showed him your question). | |
| Jul 16, 2019 at 16:19 | history | asked | Aleksei Kulikov | CC BY-SA 4.0 |