Timeline for Periodic billiard paths in hyperbolic triangles
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 3, 2017 at 4:12 | comment | added | Takahiro Waki | At first, it should be about the case of spherical tringle. | |
| Sep 20, 2016 at 22:56 | answer | added | Igor Rivin | timeline score: 4 | |
| Sep 20, 2016 at 13:47 | history | edited | Joseph O'Rourke | CC BY-SA 3.0 |
Removed misplaced italics.
|
| Sep 20, 2016 at 12:29 | comment | added | Benoît Kloeckner | @Neal: I can't speak for Anton Petrunin, but take a periodic orbit in a Euclidean triangle and consider its combinatorics (the word of the successively sides it hits, up to circular permutation) ; I would guess that this orbit is a (at least local) length minimizer in the class of closed curves that touch the boundary with the same combinatorics. Then a very small hyperbolic triangle close to that Euclidean triangle would have even better stability properties for the corresponding minimizing problem (it costs more to move a point across a vertex), and a minimizer must be a periodic orbit. | |
| Sep 20, 2016 at 12:14 | comment | added | Ian Morris | Theorem 3 of "Expansive billiard flows" (arxiv.org/pdf/1207.3116) asserts that a billiard flow in a polygonal subset of the Poincare disc has a dense set of periodic orbits, but this paper seems to be unpublished and uncited. | |
| Sep 20, 2016 at 11:33 | comment | added | Joseph O'Rourke | @GerryMyerson: One quote from the reference you cite: "the billiard map of a triangle on the hyperbolic disk has non-vanishing Lyapunov exponents." However, I am uncertain of the consequences of this fact. | |
| Sep 20, 2016 at 9:45 | comment | added | Ian Morris | This is only a wild guess, but my suspicion is that billiards in hyperbolic triangles are chaotic in the sense of Devaney and should have infinitely many periodic orbits irrespective of their angles. | |
| Sep 20, 2016 at 3:24 | comment | added | Neal | @AntonPetrunin Why do you believe the sentence you have marked with a (?)? (Also, shouldn't the orthic triangle work for all acute hyperbolic triangles?) | |
| Sep 20, 2016 at 2:56 | comment | added | Gerry Myerson | The review of Alfonso Artigue, Expansive flows of the three-sphere, Differential Geom. Appl. 41 (2015) 91-101, MR3353741 refers to "the billiard in a geodesic triangle in the hyperbolic disc", so it might be good to have a look there. | |
| Sep 20, 2016 at 0:46 | comment | added | Joseph O'Rourke | @AntonPetrunin: "all hyperbolic triangles have to have a periodic trajectory": Remarkable! | |
| Sep 20, 2016 at 0:31 | comment | added | Anton Petrunin | Consider one parameter family of hyperbolic triangles such that the angles decrease. It seems that if at the beginning you had a closed orbit then it survives in the family (?). If this is true then you may start with any Euclidean triangle for which you know that a trajectory exists and get a trajectory for any hyperbolic triangle with smaller angles. It seems that the set of Euclidean triangles with periodic trajectories is dense therefore all hyperbolic triangles have to have a periodic trajectory. | |
| Sep 20, 2016 at 0:29 | history | edited | Joseph O'Rourke | CC BY-SA 3.0 |
deleted 2 characters in body
|
| Sep 19, 2016 at 23:50 | history | asked | Joseph O'Rourke | CC BY-SA 3.0 |