22:
153:: this turns out to be an important step towards proving the more general equidistribution property. There is no universal agreement on the names of these theorems: they are variously known as the "measure rigidity theorem", the "theorem on invariant measures" and its "topological version", and so on.
118:. Ratner's theorems have guided key advances in the understanding of the dynamics of unipotent flows. Their later generalizations provide ways to both sharpen the results and extend the theory to the setting of arbitrary
920:
1134:
854:
547:
341:
1034:
806:
952:
713:
1066:
381:
774:
663:
202:
1195:
594:
408:
311:
1158:
1089:
733:
477:
253:
981:
1259:
1239:
1215:
614:
567:
517:
497:
452:
432:
277:
226:
174:
51:
859:
1497:
138:
asserts that the closures of orbits of unipotent flows on the quotient of a Lie group by a lattice are nice, geometric subsets. The
1610:
1312:
1561:; Tomanov, Georges M. (1994). "Invariant measures for actions of unipotent groups over local fields on homogeneous spaces".
1098:
811:
73:
522:
316:
44:
986:
779:
925:
671:
1039:
346:
738:
627:
1528:
Ratner, Marina (1995). "Raghunathan's conjectures for cartesian products of real and p-adic Lie groups".
205:
183:
119:
1171:
34:
1461:
Ratner, Marina (1991). "Raghunathan's topological conjecture and distributions of unipotent flows".
1275:
38:
30:
1600:
572:
386:
289:
256:
145:
55:
1357:
Ratner, Marina (1990). "Strict measure rigidity for unipotent subgroups of solvable groups".
1143:
1074:
718:
457:
149:
is the weaker statement that every ergodic invariant probability measure is homogeneous, or
1583:
1550:
1520:
1483:
1453:
1415:
1379:
1322:
231:
111:
957:
110:. The study of the dynamics of unipotent flows played a decisive role in the proof of the
8:
1605:
1441:
1331:
1298:
1244:
1224:
1218:
1200:
599:
552:
502:
482:
437:
417:
284:
262:
211:
159:
1308:
99:
1571:
1558:
1538:
1506:
1471:
1433:
1401:
1367:
1137:
115:
1542:
1475:
1579:
1546:
1516:
1479:
1449:
1411:
1375:
1318:
1092:
107:
95:
1511:
1492:
1594:
1563:
1530:
1463:
1425:
1359:
1297:. Chicago Lectures in Mathematics. Chicago, IL: University of Chicago Press.
411:
103:
142:
further asserts that each such orbit is equidistributed in its closure. The
617:
623:
123:
87:
1575:
1445:
1406:
1393:
1388:
1371:
1270:
1303:
1165:
280:
177:
1437:
915:{\displaystyle u_{t}={\begin{pmatrix}1&t\\0&1\end{pmatrix}}}
1161:
1389:"On measure rigidity of unipotent subgroups of semisimple groups"
106:
around 1990. The theorems grew out of Ratner's earlier work on
1423:
Ratner, Marina (1991). "On
Raghunathan's measure conjecture".
715:. In this case it takes the following more explicit form; let
668:
The simplest case to which the statement above applies is
156:
The formal statement of such a result is as follows. Let
1164:
of finite volume. The theorem above implies that every
1111:
1052:
938:
881:
792:
1247:
1227:
1217:
which is either a closed curve (a horocycle around a
1203:
1174:
1146:
1101:
1077:
1042:
989:
960:
928:
862:
814:
782:
741:
721:
674:
630:
602:
575:
555:
525:
505:
485:
460:
440:
420:
389:
349:
319:
292:
265:
234:
214:
186:
162:
1129:{\displaystyle M=\Gamma \backslash \mathbb {H} ^{2}}
569:-invariant measure, and contains the closure of the
1253:
1233:
1209:
1189:
1152:
1128:
1083:
1060:
1028:
975:
946:
914:
848:
808:a closed subset which is invariant under all maps
800:
768:
727:
707:
657:
608:
588:
561:
541:
511:
491:
471:
446:
426:
402:
375:
335:
305:
271:
247:
220:
196:
168:
1493:"Raghunathan's conjectures for p-adic Lie groups"
1592:
43:but its sources remain unclear because it lacks
1557:
849:{\displaystyle \Gamma g\mapsto \Gamma (gu_{t})}
410:is homogeneous. This means that there exists a
542:{\displaystyle {\mathit {\Gamma }}\setminus G}
336:{\displaystyle {\mathit {\Gamma }}\setminus G}
1350:
1029:{\displaystyle U=\{u_{t},t\in \mathbb {R} \}}
1023:
996:
801:{\displaystyle F\subset \Gamma \backslash G}
1498:International Mathematics Research Notices
1329:
1510:
1405:
1302:
1177:
1116:
1019:
759:
698:
648:
468:
461:
74:Learn how and when to remove this message
947:{\displaystyle x\in \Gamma \backslash G}
1593:
1527:
1490:
1460:
1422:
1386:
1356:
1292:
708:{\displaystyle G=SL_{2}(\mathbb {R} )}
1061:{\displaystyle F=\Gamma \backslash G}
376:{\displaystyle \left\{xu^{t}\right\}}
1295:Ratner's Theorems on Unipotent Flows
769:{\displaystyle SL_{2}(\mathbb {R} )}
658:{\displaystyle SL_{2}(\mathbb {R} )}
129:
15:
519:under the canonical projection to
197:{\displaystyle {\mathit {\Gamma }}}
13:
1147:
1108:
1078:
1049:
935:
824:
815:
789:
722:
528:
343:. Then the closure of every orbit
322:
189:
14:
1622:
533:
454:such that the image of the orbit
327:
94:are a group of major theorems in
1190:{\displaystyle \mathbb {H} ^{2}}
20:
140:Ratner equidistribution theorem
1332:"What is... measure rigidity?"
1286:
922:. Then either there exists an
843:
827:
821:
763:
755:
702:
694:
652:
644:
283:elements, with the associated
98:concerning unipotent flows on
1:
1611:Theorems in dynamical systems
1543:10.1215/S0012-7094-95-07710-2
1476:10.1215/S0012-7094-91-06311-8
1281:
1330:Einsiedler, Manfred (2009).
136:Ratner orbit closure theorem
7:
1293:Morris, Dave Witte (2005).
1264:
120:semisimple algebraic groups
10:
1627:
1351:Selected original articles
1512:10.1155/S1073792893000145
589:{\displaystyle \phi _{t}}
499:by right translations on
403:{\displaystyle \phi _{t}}
306:{\displaystyle \phi _{t}}
1276:Equidistribution theorem
549:is closed, has a finite
29:This article includes a
1491:Ratner, Marina (1993).
1387:Ratner, Marina (1990).
1153:{\displaystyle \Gamma }
1084:{\displaystyle \Gamma }
728:{\displaystyle \Gamma }
58:more precise citations.
1255:
1235:
1211:
1191:
1154:
1130:
1085:
1062:
1030:
977:
948:
916:
850:
802:
770:
729:
709:
659:
610:
590:
563:
543:
513:
493:
473:
472:{\displaystyle \,xS\,}
448:
428:
404:
377:
337:
307:
273:
257:one-parameter subgroup
249:
222:
198:
170:
146:classification theorem
1256:
1236:
1212:
1192:
1155:
1131:
1086:
1063:
1031:
978:
949:
917:
851:
803:
771:
730:
710:
660:
611:
591:
564:
544:
514:
494:
474:
449:
429:
405:
378:
338:
308:
274:
250:
248:{\displaystyle u^{t}}
223:
199:
171:
1559:Margulis, Grigory A.
1245:
1225:
1201:
1172:
1144:
1099:
1075:
1040:
987:
976:{\displaystyle F=xU}
958:
926:
860:
812:
780:
739:
719:
672:
628:
600:
573:
553:
523:
503:
483:
458:
438:
418:
387:
347:
317:
290:
263:
232:
212:
184:
160:
112:Oppenheim conjecture
1071:In geometric terms
1576:10.1007/BF01231565
1407:10.1007/BF02391906
1372:10.1007/BF01231511
1339:Notices of the AMS
1251:
1231:
1207:
1187:
1150:
1126:
1095:, so the quotient
1081:
1058:
1026:
973:
944:
912:
906:
846:
798:
766:
725:
705:
655:
606:
586:
559:
539:
509:
489:
479:for the action of
469:
444:
424:
414:, closed subgroup
400:
373:
333:
303:
269:
245:
218:
194:
166:
100:homogeneous spaces
31:list of references
1314:978-0-226-53984-3
1254:{\displaystyle M}
1234:{\displaystyle M}
1210:{\displaystyle M}
609:{\displaystyle x}
562:{\displaystyle S}
512:{\displaystyle G}
492:{\displaystyle S}
447:{\displaystyle G}
427:{\displaystyle S}
272:{\displaystyle G}
221:{\displaystyle G}
169:{\displaystyle G}
130:Short description
92:Ratner's theorems
84:
83:
76:
1618:
1587:
1554:
1524:
1514:
1487:
1457:
1419:
1409:
1383:
1346:
1336:
1326:
1306:
1260:
1258:
1257:
1252:
1240:
1238:
1237:
1232:
1216:
1214:
1213:
1208:
1197:has an image in
1196:
1194:
1193:
1188:
1186:
1185:
1180:
1160:is a hyperbolic
1159:
1157:
1156:
1151:
1138:hyperbolic plane
1135:
1133:
1132:
1127:
1125:
1124:
1119:
1090:
1088:
1087:
1082:
1067:
1065:
1064:
1059:
1035:
1033:
1032:
1027:
1022:
1008:
1007:
982:
980:
979:
974:
953:
951:
950:
945:
921:
919:
918:
913:
911:
910:
872:
871:
855:
853:
852:
847:
842:
841:
807:
805:
804:
799:
775:
773:
772:
767:
762:
754:
753:
735:be a lattice in
734:
732:
731:
726:
714:
712:
711:
706:
701:
693:
692:
664:
662:
661:
656:
651:
643:
642:
615:
613:
612:
607:
595:
593:
592:
587:
585:
584:
568:
566:
565:
560:
548:
546:
545:
540:
532:
531:
518:
516:
515:
510:
498:
496:
495:
490:
478:
476:
475:
470:
453:
451:
450:
445:
433:
431:
430:
425:
409:
407:
406:
401:
399:
398:
382:
380:
379:
374:
372:
368:
367:
366:
342:
340:
339:
334:
326:
325:
312:
310:
309:
304:
302:
301:
278:
276:
275:
270:
254:
252:
251:
246:
244:
243:
227:
225:
224:
219:
203:
201:
200:
195:
193:
192:
175:
173:
172:
167:
116:Grigory Margulis
79:
72:
68:
65:
59:
54:this article by
45:inline citations
24:
23:
16:
1626:
1625:
1621:
1620:
1619:
1617:
1616:
1615:
1591:
1590:
1438:10.2307/2944357
1353:
1334:
1315:
1289:
1284:
1267:
1246:
1243:
1242:
1226:
1223:
1222:
1202:
1199:
1198:
1181:
1176:
1175:
1173:
1170:
1169:
1145:
1142:
1141:
1120:
1115:
1114:
1100:
1097:
1096:
1076:
1073:
1072:
1041:
1038:
1037:
1018:
1003:
999:
988:
985:
984:
959:
956:
955:
927:
924:
923:
905:
904:
899:
893:
892:
887:
877:
876:
867:
863:
861:
858:
857:
837:
833:
813:
810:
809:
781:
778:
777:
758:
749:
745:
740:
737:
736:
720:
717:
716:
697:
688:
684:
673:
670:
669:
666:
647:
638:
634:
629:
626:
625:
601:
598:
597:
580:
576:
574:
571:
570:
554:
551:
550:
527:
526:
524:
521:
520:
504:
501:
500:
484:
481:
480:
459:
456:
455:
439:
436:
435:
419:
416:
415:
394:
390:
388:
385:
384:
362:
358:
354:
350:
348:
345:
344:
321:
320:
318:
315:
314:
297:
293:
291:
288:
287:
264:
261:
260:
239:
235:
233:
230:
229:
213:
210:
209:
188:
187:
185:
182:
181:
161:
158:
157:
144:Ratner measure
132:
108:horocycle flows
80:
69:
63:
60:
49:
35:related reading
25:
21:
12:
11:
5:
1624:
1614:
1613:
1608:
1603:
1601:Ergodic theory
1589:
1588:
1570:(1): 347–392.
1555:
1537:(2): 275–382.
1525:
1505:(5): 141–146.
1488:
1470:(1): 235–280.
1458:
1432:(3): 545–607.
1420:
1400:(1): 229–309.
1384:
1366:(2): 449–482.
1352:
1349:
1348:
1347:
1327:
1313:
1288:
1285:
1283:
1280:
1279:
1278:
1273:
1266:
1263:
1250:
1241:) or dense in
1230:
1206:
1184:
1179:
1149:
1123:
1118:
1113:
1110:
1107:
1104:
1093:Fuchsian group
1091:is a cofinite
1080:
1057:
1054:
1051:
1048:
1045:
1025:
1021:
1017:
1014:
1011:
1006:
1002:
998:
995:
992:
972:
969:
966:
963:
943:
940:
937:
934:
931:
909:
903:
900:
898:
895:
894:
891:
888:
886:
883:
882:
880:
875:
870:
866:
845:
840:
836:
832:
829:
826:
823:
820:
817:
797:
794:
791:
788:
785:
765:
761:
757:
752:
748:
744:
724:
704:
700:
696:
691:
687:
683:
680:
677:
665:
654:
650:
646:
641:
637:
633:
622:
605:
583:
579:
558:
538:
535:
530:
508:
488:
467:
464:
443:
423:
397:
393:
371:
365:
361:
357:
353:
332:
329:
324:
300:
296:
279:consisting of
268:
242:
238:
217:
191:
165:
131:
128:
96:ergodic theory
82:
81:
64:September 2019
39:external links
28:
26:
19:
9:
6:
4:
3:
2:
1623:
1612:
1609:
1607:
1604:
1602:
1599:
1598:
1596:
1585:
1581:
1577:
1573:
1569:
1566:
1565:
1564:Invent. Math.
1560:
1556:
1552:
1548:
1544:
1540:
1536:
1533:
1532:
1531:Duke Math. J.
1526:
1522:
1518:
1513:
1508:
1504:
1500:
1499:
1494:
1489:
1485:
1481:
1477:
1473:
1469:
1466:
1465:
1464:Duke Math. J.
1459:
1455:
1451:
1447:
1443:
1439:
1435:
1431:
1428:
1427:
1426:Ann. of Math.
1421:
1417:
1413:
1408:
1403:
1399:
1396:
1395:
1390:
1385:
1381:
1377:
1373:
1369:
1365:
1362:
1361:
1360:Invent. Math.
1355:
1354:
1345:(5): 600–601.
1344:
1340:
1333:
1328:
1324:
1320:
1316:
1310:
1305:
1300:
1296:
1291:
1290:
1277:
1274:
1272:
1269:
1268:
1262:
1248:
1228:
1220:
1204:
1182:
1167:
1163:
1139:
1121:
1105:
1102:
1094:
1069:
1055:
1046:
1043:
1015:
1012:
1009:
1004:
1000:
993:
990:
970:
967:
964:
961:
941:
932:
929:
907:
901:
896:
889:
884:
878:
873:
868:
864:
838:
834:
830:
818:
795:
786:
783:
750:
746:
742:
689:
685:
681:
678:
675:
639:
635:
631:
621:
619:
603:
581:
577:
556:
536:
506:
486:
465:
462:
441:
421:
413:
395:
391:
369:
363:
359:
355:
351:
330:
298:
294:
286:
282:
266:
258:
240:
236:
215:
207:
179:
163:
154:
152:
148:
147:
141:
137:
127:
125:
121:
117:
113:
109:
105:
104:Marina Ratner
101:
97:
93:
89:
78:
75:
67:
57:
53:
47:
46:
40:
36:
32:
27:
18:
17:
1567:
1562:
1534:
1529:
1502:
1496:
1467:
1462:
1429:
1424:
1397:
1392:
1363:
1358:
1342:
1338:
1304:math/0310402
1294:
1070:
667:
618:dense subset
155:
150:
143:
139:
135:
133:
91:
85:
70:
61:
50:Please help
42:
1287:Expositions
124:local field
88:mathematics
56:introducing
1606:Lie groups
1595:Categories
1394:Acta Math.
1282:References
1271:Danzer set
954:such that
596:-orbit of
102:proved by
1166:horocycle
1148:Γ
1112:∖
1109:Γ
1079:Γ
1053:∖
1050:Γ
1016:∈
939:∖
936:Γ
933:∈
825:Γ
822:↦
816:Γ
793:∖
790:Γ
787:⊂
723:Γ
624:Example:
578:ϕ
534:∖
529:Γ
412:connected
392:ϕ
328:∖
323:Γ
295:ϕ
281:unipotent
190:Γ
178:Lie group
151:algebraic
1265:See also
1162:orbifold
1584:1253197
1551:1321062
1521:1219864
1484:1106945
1454:1135878
1446:2944357
1416:1075042
1380:1062971
1323:2158954
1136:of the
983:(where
206:lattice
122:over a
52:improve
1582:
1549:
1519:
1482:
1452:
1444:
1414:
1378:
1321:
1311:
856:where
228:, and
1442:JSTOR
1335:(PDF)
1299:arXiv
1036:) or
616:as a
176:be a
37:, or
1503:1993
1309:ISBN
1219:cusp
776:and
285:flow
134:The
1572:doi
1568:116
1539:doi
1507:doi
1472:doi
1434:doi
1430:134
1402:doi
1398:165
1368:doi
1364:101
1221:of
1168:of
1140:by
1068:.
434:of
383:of
313:on
259:of
208:in
114:by
86:In
1597::
1580:MR
1578:.
1547:MR
1545:.
1535:77
1517:MR
1515:.
1501:.
1495:.
1480:MR
1478:.
1468:63
1450:MR
1448:.
1440:.
1412:MR
1410:.
1391:.
1376:MR
1374:.
1343:56
1341:.
1337:.
1319:MR
1317:.
1307:.
1261:.
620:.
255:a
204:a
180:,
126:.
90:,
41:,
33:,
1586:.
1574::
1553:.
1541::
1523:.
1509::
1486:.
1474::
1456:.
1436::
1418:.
1404::
1382:.
1370::
1325:.
1301::
1249:M
1229:M
1205:M
1183:2
1178:H
1122:2
1117:H
1106:=
1103:M
1056:G
1047:=
1044:F
1024:}
1020:R
1013:t
1010:,
1005:t
1001:u
997:{
994:=
991:U
971:U
968:x
965:=
962:F
942:G
930:x
908:)
902:1
897:0
890:t
885:1
879:(
874:=
869:t
865:u
844:)
839:t
835:u
831:g
828:(
819:g
796:G
784:F
764:)
760:R
756:(
751:2
747:L
743:S
703:)
699:R
695:(
690:2
686:L
682:S
679:=
676:G
653:)
649:R
645:(
640:2
636:L
632:S
604:x
582:t
557:S
537:G
507:G
487:S
466:S
463:x
442:G
422:S
396:t
370:}
364:t
360:u
356:x
352:{
331:G
299:t
267:G
241:t
237:u
216:G
164:G
77:)
71:(
66:)
62:(
48:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.