33:
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497:, the existence of periodic orbits is a major unsolved problem. For instance, it is unknown if every triangular shaped table has a periodic billiard path. More progress has been made for outer billiards, though the situation is far from well understood. As mentioned above, all the orbits are periodic when the system is defined relative to a convex rational polygon in the
381:
has some aperiodic orbits, thus clarifying the distinction between the dynamics in the rational and regular cases. In 1996, Philip
Boyland showed that outer billiards relative to some shapes can have orbits which accumulate on the shape. In 2005, Daniel Genin showed that all orbits are bounded when
352:
gave the full proof of this result. A big advance in the polygonal case came over a period of several years when three teams of authors, Vivaldi-Shaidenko, Kolodziej, and Gutkin-Simanyi, each using different methods, showed that outer billiards relative to a
485:. (See the reference for the definition.) Filiz Doǧru and Samuel Otten then extended this work in 2011 by specifying the conditions under which a regular polygonal table in the hyperbolic plane have all orbits unbounded, that is, are large.
200:
of the point, namely ... x0 ↔ x1 ↔ x2 ↔ x3 ... That is, start at x0 and iteratively apply both the outer billiards map and the backwards outer billiards map. When P is a strictly convex shape, such as an
311:
informally posed the question as to whether or not one can have unbounded orbits in an outer billiards system, and Moser put it in writing in 1973. Sometimes this basic question has been called
209:, some points might not have well-defined orbits, on account of the potential ambiguity of choosing the midpoint of the relevant tangent line. Nevertheless, in the polygonal case,
535:
orbit periodic. The cases of the equilateral triangle and the square are trivial, and
Tabachnikov answered this for the regular pentagon. These are the only cases known.
62:
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multiples of each other. In 2008, Dmitry
Dolgopyat and Bassam Fayad showed that outer billiards defined relative to the semidisk has unbounded orbits. The
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shape in the plane. Given a point x0 outside P, there is typically a unique point x1 (also outside P) so that the line segment connecting x0 to x1 is
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Outer billiards is a subject still in its beginning phase. Most problems are still unsolved. Here are some open problems in the area.
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in half. The proof of
Dolgopyat-Fayad is robust, and also works for regions obtained by cutting a disk nearly in half, when the word
1936:
84:
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understand the structure of periodic orbits relative to simple shapes in the hyperbolic plane, such as small equilateral triangles.
168:(or backwards) outer billiards map is also defined, as the map x1 -> x0. One gets the inverse map simply by replacing the word
263:
Defining an outer billiards system in a higher-dimensional space is beyond the scope of this article. Unlike the case of ordinary
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501:. Moreover, it is a recent theorem of Chris Culter (written up by Sergei Tabachnikov) that outer billiards relative to any
421:. Subsequently, Schwartz showed that outer billiards has unbounded orbits when defined relative to any irrational kite. An
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polygon has all orbits bounded. The notion of quasirational is technical (see references) but it includes the class of
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to rotate 90 degrees. These distinguished tangent lines can be used to define the outer billiards map roughly as above.
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and a person walking from x0 to x1 would see P on the right. (See Figure.) The map F: x0 -> x1 is called the
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in the late 1950s, though it seems that a few people cite an earlier construction in 1945, due to M. Day.
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In 2003, Filiz Doǧru and Sergei
Tabachnikov showed that all orbits are unbounded for a certain class of
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Kite, thus answering the original Moser-Neumann question in the affirmative. The
Penrose kite is the
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more broadly, characterize the structure of the set of periodic orbits relative to the typical
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or in other spaces that suitably generalize the plane. Outer billiards differs from a usual
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body at each point. One obtains these tangents by starting with the normals and using the
1926:
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1159:"A proof of Culter's theorem on existence of periodic orbits in polygonal outer billiards"
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Schwartz, Richard E. (2009). "outer billiards on kites". Annals of
Mathematics Studies.
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coordinates. In the case of rational polygons, all the orbits are periodic. In 1995,
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showed that outer billiards has some unbounded orbits when defined relative to the
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has periodic orbits—in fact a periodic orbit outside of any given bounded region.
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1941:
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267:, the definition is not straightforward. One natural setting for the map is a
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Boyland, Philip (1996). "Dual billiards, twist maps, and impact oscillators".
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686:. Annals of Mathematics Studies. Vol. 77. Princeton University Press.
205:, every point in the exterior of P has a well defined orbit. When P is a
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in the definition given above. The figure shows the situation in the
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and solved only recently, has been a guiding problem in the field.
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155:
724:
R. Douady (1982). "these de 3-eme cycle". University of Paris 7.
271:. In this case, there is a natural choice of line tangent to a
108:
shape in the plane. Classically, this system is defined for the
1210:
1206:
1086:(2003). "On Polygonal Dual Billiards in the Hyperbolic Plane".
805:
Kołodziej, Rafał (1989). "The antibilliard outside a polygon".
243:) if some bounded region in the plane contains the whole orbit.
390:
condition for the system to have all orbits bounded. (Not all
26:
666:
Iota: The
Manchester University Mathematics Students' Journal
287:
Most people attribute the introduction of outer billiards to
664:
Neumann, Bernhard H. (25 Jan 1959). "Sharing Ham and Eggs".
587:. Panoramas et Synthèses. Société Mathématique de France.
758:"Global Stability of a class of discontinuous billiards"
295:
popularized the system in the 1970s as a toy model for
697:
Moser, JĂĽrgen (1978). "Is the Solar System Stable?".
315:. This question, originally posed for shapes in the
1985:
1802:
1729:
1667:
1537:
1524:
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1407:
1251:
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1041:"unbounded orbits for semicircular outer billiards"
120:in that it deals with a discrete sequence of moves
948:(2007). "unbounded orbits for outer billiards I".
611:(2002). "Dual Billiards in the Hyperbolic Plane".
299:. This system has been studied classically in the
575:
573:
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54:but its sources remain unclear because it lacks
931:Regular and chaotic dynamics of outer billiards
826:"Dual polygonal billiard and necklace dynamics"
934:(Ph.D. Thesis). Pennsylvania State University.
684:Stable and random motions in dynamical systems
386:, thus showing that quasirationality is not a
340:, that outer billiards relative to a 6-times-
160:Outer billiards defined relative to a pentagon
1222:
8:
756:Vivaldi, Franco; Shaidenko, Anna V. (1987).
112:but one can also consider the system in the
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1127:American Journal of Undergraduate Research
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1039:Dolgopyat, Dmitry; Fayad, Bassam (2009).
963:
888:
632:
429:with the following property: One of the
85:Learn how and when to remove this message
824:Gutkin, Eugene; Simanyi, Nandor (1991).
527:Show that outer billiards relative to a
469:Unbounded orbits in the hyperbolic plane
567:
398:Unbounded orbits in the Euclidean plane
1021:
1010:
831:Communications in Mathematical Physics
763:Communications in Mathematical Physics
738:
727:
517:Show that outer billiards relative to
328:Bounded orbits in the Euclidean plane
7:
1121:Doǧru, Filiz; Otten, Samuel (2011).
457:is the region one gets by cutting a
377:showed that outer billiards for the
124:the shape rather than inside of it.
1369:Measure-preserving dynamical system
25:
1937:Oleksandr Mykolayovych Sharkovsky
1123:"Sizing Up Outer Billiard Tables"
481:. The authors call such polygons
348:has all orbits bounded. In 1982,
1467:
1459:
213:point has a well-defined orbit.
31:
1108:10.1070/RD2003v008n01ABEH000226
1702:Rabinovich–Fabrikant equations
1:
1007:. Princeton University Press.
441:of equal area and the other
1088:Regular and Chaotic Dynamics
807:Bull. Polish Acad. Sci. Math
489:Existence of periodic orbits
445:divides the region into two
437:divides the region into two
180:, but the definition in the
1437:Poincaré recurrence theorem
336:sketched a proof, based on
303:, and more recently in the
2052:
1432:Poincaré–Bendixson theorem
951:Journal of Modern Dynamics
699:Mathematical Intelligencer
643:10.1088/0951-7715/15/4/305
313:the Moser-Neumann question
1784:Swinging Atwood's machine
1457:
1427:Krylov–Bogolyubov theorem
1304:
1196:10.1007/s10711-007-9196-y
1068:10.1007/s00023-009-0409-9
928:Genin, Daniel I. (2005).
907:10.1088/0951-7715/9/6/002
465:is suitably interpreted.
417:from the kites-and-darts
259:Higher-dimensional spaces
221:if it eventually repeats.
184:is essentially the same.
1692:Lotka–Volterra equations
1516:Synchronization of chaos
1319:axiom A dynamical system
363:convex rational polygons
232:) if it is not periodic.
40:This article includes a
1677:Double scroll attractor
1442:Stable manifold theorem
1349:False nearest neighbors
254:) if it is not bounded.
133:The outer billiards map
69:more precise citations.
1717:Van der Pol oscillator
1697:Mackey–Glass equations
1329:Box-counting dimension
1140:10.33697/ajur.2011.008
1046:Annales Henri Poincaré
1020:Cite journal requires
982:10.3934/jmd.2007.1.371
737:Cite journal requires
682:Moser, JĂĽrgen (1973).
323:Moser-Neumann question
161:
1867:Svetlana Jitomirskaya
1774:Multiscroll attractor
1619:Interval exchange map
1572:Dyadic transformation
1557:Complex quadratic map
1399:Topological conjugacy
1334:Correlation dimension
1309:Anosov diffeomorphism
524:has unbounded orbits.
159:
1877:Edward Norton Lorenz
946:Schwartz, Richard E.
556:Illumination problem
449:whose areas are not
394:are quasirational.)
369:whose vertices have
269:complex vector space
1837:Mitchell Feigenbaum
1779:Population dynamics
1764:Hénon–Heiles system
1624:Irrational rotation
1577:Dynamical billiards
1562:Coupled map lattice
1422:Liouville's theorem
1354:Hausdorff dimension
1339:Conservative system
1324:Bifurcation diagram
1187:2007arXiv0706.1003T
1164:Geometriae Dedicata
1100:2003RCD.....8...67D
1084:Tabachnikov, Sergei
1059:2009AnHP...10..357D
974:2007math......2073S
899:1996Nonli...9.1411B
844:1992CMaPh.143..431G
776:1987CMaPh.110..625V
625:2002Nonli..15.1051T
609:Tabachnikov, Sergei
495:polygonal billiards
297:celestial mechanics
246:An orbit is called
235:An orbit is called
224:An orbit is called
217:An orbit is called
192:An outer billiards
151:outer billiards map
2015:Santa Fe Institute
1882:Aleksandr Lyapunov
1712:Three-body problem
1599:Gingerbreadman map
1486:Bifurcation theory
1364:Lyapunov stability
1155:Tabachnikov, Serge
852:10.1007/BF02099259
784:10.1007/BF01205552
711:10.1007/BF03023062
581:Tabachnikov, Serge
375:Sergei Tabachnikov
344:shape of positive
196:is the set of all
162:
118:dynamical billiard
42:list of references
2036:Dynamical systems
2023:
2022:
1887:Benoît Mandelbrot
1852:Martin Gutzwiller
1842:Peter Grassberger
1725:
1724:
1707:Rössler attractor
1455:
1454:
1359:Invariant measure
1281:Lyapunov exponent
594:978-2-85629-030-9
277:complex structure
95:
94:
87:
16:(Redirected from
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1995:Butterfly effect
1907:Itamar Procaccia
1857:Brosl Hasslacher
1754:Elastic pendulum
1682:Duffing equation
1629:Kaplan–Yorke map
1547:Arnold's cat map
1535:
1511:Stability theory
1496:Dynamical system
1491:Control of chaos
1471:
1463:
1447:Takens's theorem
1379:Poincaré section
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883:(6): 1411–1438.
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479:hyperbolic plane
404:Richard Schwartz
379:regular pentagon
365:, namely those
359:regular polygons
309:Bernhard Neumann
305:hyperbolic plane
289:Bernhard Neumann
182:hyperbolic plane
114:hyperbolic plane
102:dynamical system
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65:this article by
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529:regular polygon
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475:convex polygons
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423:irrational kite
419:Penrose tilings
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382:the shape is a
367:convex polygons
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317:Euclidean plane
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1053:(2): 357–375.
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1828:
1827:Chen Guanrong
1825:
1823:
1820:
1818:
1815:
1813:
1812:Michael Berry
1810:
1809:
1807:
1801:
1795:
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1687:Lorenz system
1685:
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1634:Langton's ant
1632:
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1609:Horseshoe map
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1506:Quantum chaos
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1314:Arnold tongue
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452:
448:
444:
440:
436:
435:quadrilateral
432:
428:
427:quadrilateral
424:
420:
416:
415:quadrilateral
413:
409:
405:
397:
395:
393:
389:
385:
380:
376:
372:
368:
364:
360:
356:
355:quasirational
351:
347:
343:
339:
338:K.A.M. theory
335:
332:In the 70's,
327:
322:
320:
318:
314:
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306:
302:
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282:
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127:
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119:
115:
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107:
103:
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89:
86:
78:
68:
64:
58:
57:
51:
47:
43:
38:
29:
28:
19:
1957:Mary Tsingou
1922:David Ruelle
1917:Otto Rössler
1862:Michel HĂ©non
1832:Leon O. Chua
1789:Tilt-A-Whirl
1759:FPUT problem
1644:Standard map
1639:Logistic map
1581:
1464:
1238:Chaos theory
1168:
1162:
1149:
1130:
1126:
1116:
1094:(1): 67–82.
1091:
1087:
1077:
1050:
1044:
1034:
1013:cite journal
1004:
998:
965:math/0702073
955:
949:
940:
930:
923:
890:math/9408216
880:
876:Nonlinearity
874:
868:
835:
829:
819:
810:
806:
800:
767:
761:
751:
730:cite journal
719:
705:(2): 65–71.
702:
698:
692:
683:
665:
659:
616:
613:Nonlinearity
612:
603:
584:
533:almost every
519:almost every
512:
493:In ordinary
492:
482:
472:
462:
454:
422:
401:
387:
362:
354:
334:JĂĽrgen Moser
331:
312:
293:JĂĽrgen Moser
286:
262:
251:
247:
240:
236:
230:non-periodic
229:
225:
218:
211:almost every
191:
173:
172:by the word
169:
163:
150:
145:to P at its
136:
121:
97:
96:
81:
72:
61:Please help
53:
1942:Nina Snaith
1932:Yakov Sinai
1817:Rufus Bowen
1567:Duffing map
1552:Baker's map
1477:Theoretical
1389:SRB measure
1296:Phase space
1266:Bifurcation
137:Let P be a
128:Definitions
104:based on a
67:introducing
2000:Complexity
1897:Edward Ott
1744:Convection
1669:Continuous
1344:Ergodicity
813:: 163–168.
562:References
392:trapezoids
198:iterations
1912:Mary Rees
1872:Bryna Kra
1805:theorists
1614:Ikeda map
1604:HĂ©non map
1594:Gauss map
1276:Limit set
1261:Attractor
1178:0706.1003
1171:: 83–87.
990:119146537
860:121776396
792:111386812
651:250758250
629:CiteSeerX
585:Billiards
447:triangles
439:triangles
431:diagonals
402:In 2007,
388:necessary
384:trapezoid
346:curvature
265:billiards
248:unbounded
226:aperiodic
75:June 2013
2030:Category
1988:articles
1730:Physical
1649:Tent map
1539:Discrete
1479:branches
1409:Theorems
1245:Concepts
1157:(2007).
915:18709638
583:(1995).
550:See also
455:semidisk
451:rational
443:diagonal
371:rational
252:unstable
219:periodic
147:midpoint
1986:Related
1794:Weather
1732:systems
1525:Chaotic
1271:Fractal
1183:Bibcode
1133:: 1–8.
1096:Bibcode
1055:Bibcode
970:Bibcode
895:Bibcode
840:Bibcode
772:Bibcode
621:Bibcode
477:in the
433:of the
408:Penrose
283:History
237:bounded
207:polygon
203:ellipse
166:inverse
143:tangent
122:outside
63:improve
1892:Hee Oh
1527:maps (
1374:Mixing
988:
913:
858:
790:
649:
631:
591:
463:nearly
412:convex
273:convex
241:stable
188:Orbits
139:convex
106:convex
1803:Chaos
1582:outer
1286:Orbit
1173:arXiv
986:S2CID
960:arXiv
911:S2CID
885:arXiv
856:S2CID
788:S2CID
647:S2CID
483:large
425:is a
361:and
194:orbit
170:right
100:is a
48:, or
1529:list
1253:Core
1026:help
743:help
589:ISBN
531:has
459:disk
250:(or
239:(or
228:(or
174:left
164:The
1191:doi
1169:129
1135:doi
1104:doi
1063:doi
1005:171
978:doi
903:doi
848:doi
836:143
780:doi
768:110
707:doi
639:doi
2032::
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1181:.
1167:.
1161:.
1131:10
1129:.
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1017::
1015:}}
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728:{{
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674:^
645:.
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617:15
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570:^
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1223:t
1216:v
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1185::
1175::
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1137::
1110:.
1106::
1098::
1092:8
1071:.
1065::
1057::
1028:)
1024:(
992:.
980::
972::
962::
956:1
917:.
905::
897::
887::
881:9
862:.
850::
842::
794:.
782::
774::
745:)
741:(
713:.
709::
703:1
668:.
653:.
641::
623::
597:.
542:.
88:)
82:(
77:)
73:(
59:.
20:)
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