Timeline for How to find representatives of $SL(2,\mathbb{R})/SL(2, \mathbb{Z})$
Current License: CC BY-SA 3.0
24 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Apr 13, 2017 at 12:19 | history | edited | CommunityBot |
replaced http://math.stackexchange.com/ with https://math.stackexchange.com/
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| May 19, 2015 at 20:47 | comment | added | GH from MO | @BarbaraSchapira: Thanks for your comment. I think the OP's question concerned the geodesic flow on the unit tangent bundle of the upper half-plane, and how it projects on the unit tangent bundle of the modular surface. Here, certain distances blow up exponentially, which explains the OP's numerical instability and difficulties. Or at least this is what I meant in my first comment (on May 13 at 17:33). | |
| May 19, 2015 at 20:01 | comment | added | Barbara Schapira | @GH. There is some confusion about what is an Anosov flow/diffeo. The geodesic flow is Anosov on the unit tangent bundle of a compact hyperbolic manifold. However, the distance between two orbits does NOT grow exponentially, by triangular inequality: $d(g^t v, g^t w)\le 2t+d(v,w)$. The exponential growth/decay holds infinitesimally, for the differential. | |
| May 14, 2015 at 20:34 | vote | accept | john mangual | ||
| May 14, 2015 at 18:44 | comment | added | john mangual | @Asaf Perhaps you can put an answer below explaining why my choice of words is ambiguous? | |
| May 14, 2015 at 18:43 | history | edited | john mangual | CC BY-SA 3.0 |
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| May 14, 2015 at 18:37 | history | edited | john mangual | CC BY-SA 3.0 |
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| May 14, 2015 at 18:05 | comment | added | Asaf | @johnmangual , I don't understand your terminology (what is "to compute" or what is "generic" in your terms). In Ergoidc Theory, generic points have a definite meaning and one can show that Lebesgue a.e. point is generic. About the non-generic points, in those cases, there are non-generic points (actually, the exceptional set is of full dimension) that can have rather strange behavior (their orbit can be contained inside a fractal say, or have a dense orbit which is not equidistributed). | |
| May 14, 2015 at 17:57 | comment | added | john mangual | @Asaf The only time you can compute to arbitrary precision is when on points which are not generic (and not all generic points are trivial). I start to wonder if these "generic" points exist at all. | |
| May 14, 2015 at 17:08 | comment | added | Asaf | @coudy, I've meant to compute explicitly the iterates of a certain point to arbitrary precision, because of Bernoullicity of the system (which is basically the problem with the geodesic flow as-well). | |
| May 14, 2015 at 15:04 | comment | added | coudy | @Asaf The powers are explicitely given by ${\tiny \pmatrix{1 & 1 \cr 1 & 0 \cr}}^n = {\tiny \pmatrix{F_{n+1} & F_n \cr F_n & F_{n-1}\cr}}$ where $(F_n)$ is the Fibonacci sequence. I hope there is no typo this time. | |
| May 14, 2015 at 14:56 | comment | added | coudy | @Asaf. Yes I am sorry. I wanted to write $x\mapsto {\tiny \pmatrix{1&1\cr 1&0\cr}}x$, which is the square root of the cat map and is Anosov, mixing, etc. Its iterates can be expressed in term of the Fibonacci numbers. The short window left for editing comments is annoying. | |
| May 14, 2015 at 11:00 | comment | added | Asaf | @coudy , the map you've written is not the cat map, the cat map is hyperbolic. The iterates can't be computed easily for that map, for the same reason (it is Bernoulli). The map you've defined is just a simple shearing. | |
| May 14, 2015 at 9:37 | comment | added | GH from MO | @coudy: Thanks for this example. I am sure the geodesic flow is more complex in many ways, so perhaps my meta claims can still be justified, but I am not qualified to do so! | |
| May 14, 2015 at 9:34 | comment | added | GH from MO | @Asaf: Thanks for the correction, I was typing too quickly. | |
| May 14, 2015 at 7:06 | comment | added | coudy | @GH Translations of the circle are not mixing nor Anosov indeed, I was thinking of the map $x\mapsto 2x$ on the circle. Let me give an example of an Anosov mixing map for which the iterates can be computed: the cat map $x\mapsto {\tiny \pmatrix{1&1\cr 0& 1\cr}}x$ on ${\bf R}^2/{\bf Z}^2$. The coefficients of its powers are given by the Fibonacci numbers. | |
| May 14, 2015 at 5:33 | comment | added | Asaf | @coudy, the Kronecker system (rotation) is an almost periodic map (or periodic in the rational case)!, certainly not mixing/expansive in any way. It's computation (or more specifically, cutting sequence and Poincare section) is related to the continued fraction expansion of $\sqrt{2}$. GH, the image of the iterates stay in the same distance throughout the orbit (hence also $0$ entropy). | |
| May 13, 2015 at 20:51 | comment | added | john mangual | @GHfromMO I am being contractictory. Numerically, due to the chaotic nature of this map, this problem is intractible! Unless I pick directions - no longer generic - where this map can be exactly computed. | |
| May 13, 2015 at 20:41 | comment | added | GH from MO | @coudy: But it is not Anosov, is it? If you start with two $x$'s at distance $\epsilon$ apart, the $n$-th iterates are at distance at most $n\epsilon$ apart. Unlike in the case of the geodesic flow on $\mathrm{SL}_2(\mathbb{R})/\mathrm{SL}_2(\mathbb{Z})$ where the distance grows exponentially. At any rate, this is not my field of expertise, I just thought the OP observed something (namely unstability of the orbits) that has a theoretical basis. | |
| May 13, 2015 at 18:55 | answer | added | coudy | timeline score: 8 | |
| May 13, 2015 at 17:33 | comment | added | GH from MO | I doubt there is a good way to compute this, because the geodesic flow you are looking at is strongly chaotic (ergodic, mixing, Anosov etc.). | |
| May 13, 2015 at 16:54 | history | edited | john mangual | CC BY-SA 3.0 |
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| May 13, 2015 at 15:19 | history | asked | john mangual | CC BY-SA 3.0 |