420:
196:
243:
471:
706:
141:
302:
567:
614:
535:
637:
509:
927:
893:
340:
1002:
908:
958:
153:
212:
1007:
441:
903:
754:, and the order of the points on each such orbit coincides with the order of the points for a rotation by
843:
is a unique minimal set and the orbits of all points both in forward and backward direction converge to
652:
673:
727:
for topological classification of homeomorphisms of the circle. There are two distinct possibilities.
100:
46:
898:
810:
640:
251:
20:
944:
540:
203:
575:
1012:
780:
425:
327:
66:
925:[On the Differentiable Conjugation of Diffeomorphisms from the Circle to Rotations].
936:
514:
476:
736:
86:
78:
772:, but the limiting periodic orbits in forward and backward directions may be different.
747:
622:
494:
82:
31:
27:
972:
436:
is unique modulo integers, therefore the rotation number is a well-defined element of
996:
986:
948:
724:
207:
144:
50:
428:
proved that the limit exists and is independent of the choice of the starting point
977:
148:
923:"Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations"
922:
790:
has no periodic orbits (this follows immediately by considering a periodic point
38:
888:
829:
74:
70:
818:
940:
415:{\displaystyle \omega (f)=\lim _{n\to \infty }{\frac {F^{n}(x)-x}{n}}.}
54:
81:. Poincaré later proved a theorem characterizing the existence of
821:. Denjoy proved that this possibility is always realized when
715:
of the circle into itself (not necessarily homeomorphic) then
475:
Intuitively, it measures the average rotation angle along the
26:"Map winding number" redirects here. Not to be confused with
873:
when viewed as a map from the group of homeomorphisms (with
723:
have the same rotation numbers. It was used by
Poincaré and
859:
of degree 1 is constant on components of the complement of
676:
625:
578:
543:
517:
497:
444:
343:
254:
215:
156:
103:
762:
converges to a periodic orbit. The same is true for
700:
631:
608:
561:
529:
503:
465:
414:
296:
237:
190:
135:
360:
191:{\displaystyle S^{1}=\mathbb {R} /\mathbb {Z} .}
851:is semiconjugate to the irrational rotation by
238:{\displaystyle F:\mathbb {R} \to \mathbb {R} }
8:
466:{\displaystyle \mathbb {R} /\mathbb {Z} .}
989:. From MathWorld--A Wolfram Web Resource.
879:topology) of the circle into the circle.
805:There exists a dense orbit. In this case
675:
667:are two homeomorphisms of the circle and
624:
577:
542:
516:
496:
456:
455:
450:
446:
445:
443:
382:
375:
363:
342:
253:
231:
230:
223:
222:
214:
181:
180:
175:
171:
170:
161:
155:
127:
114:
102:
16:Invariant of homeomorphisms of the circle
921:Herman, Michael Robert (December 1979).
766:orbits, corresponding to iterations of
651:The rotation number is invariant under
825:is twice continuously differentiable.
7:
928:Publications Mathématiques de l'IHÉS
758:. Moreover, every forward orbit of
809:is topologically conjugate to the
750:, every periodic orbit has period
370:
14:
701:{\displaystyle h\circ f=g\circ h}
655:, and even monotone topological
136:{\displaystyle f:S^{1}\to S^{1}}
855:, and the semiconjugating map
711:for a monotone continuous map
588:
582:
394:
388:
367:
353:
347:
285:
279:
270:
258:
227:
120:
1:
297:{\displaystyle F(x+m)=F(x)+m}
245:of the real line, satisfying
143:is an orientation-preserving
742:(in the lowest terms). Then
69:in 1885, in relation to the
619:and its rotation number is
562:{\displaystyle 0<N<1}
326:is defined in terms of the
1029:
1003:Fixed points (mathematics)
971:Michał Misiurewicz (ed.).
909:Poincaré–Bendixson theorem
798:). There are two subcases.
25:
18:
957:for smaller file size in
609:{\displaystyle F(x)=x+N,}
89:of the rotation number.
65:It was first defined by
19:Not to be confused with
869:The rotation number is
775:The rotation number of
731:The rotation number of
702:
633:
610:
563:
531:
530:{\displaystyle 2\pi N}
505:
467:
416:
307:for every real number
298:
239:
192:
137:
894:Denjoy diffeomorphism
703:
653:topological conjugacy
634:
611:
564:
532:
506:
468:
417:
299:
240:
193:
138:
987:"Map Winding Number"
674:
623:
576:
541:
515:
495:
442:
341:
252:
213:
154:
101:
985:Weisstein, Eric W.
904:Poincaré recurrence
817:and all orbits are
811:irrational rotation
641:irrational rotation
21:Rotation (quantity)
941:10.1007/BF02684798
698:
629:
606:
559:
527:
501:
463:
412:
374:
311:and every integer
294:
235:
188:
133:
1008:Dynamical systems
973:"Rotation theory"
781:irrational number
632:{\displaystyle N}
511:is a rotation by
504:{\displaystyle f}
407:
359:
1020:
982:
952:
899:Poincaré section
878:
862:
858:
854:
850:
847:. In this case,
846:
842:
838:
835:invariant under
834:
824:
816:
808:
797:
793:
789:
785:
778:
771:
761:
757:
753:
745:
741:
734:
722:
718:
714:
707:
705:
704:
699:
666:
662:
638:
636:
635:
630:
615:
613:
612:
607:
568:
566:
565:
560:
536:
534:
533:
528:
510:
508:
507:
502:
482:
474:
472:
470:
469:
464:
459:
454:
449:
435:
431:
421:
419:
418:
413:
408:
403:
387:
386:
376:
373:
333:
325:
314:
310:
303:
301:
300:
295:
244:
242:
241:
236:
234:
226:
201:
197:
195:
194:
189:
184:
179:
174:
166:
165:
142:
140:
139:
134:
132:
131:
119:
118:
1028:
1027:
1023:
1022:
1021:
1019:
1018:
1017:
993:
992:
970:
967:
920:
917:
885:
874:
860:
856:
852:
848:
844:
840:
836:
832:
828:There exists a
822:
814:
806:
795:
791:
787:
783:
776:
767:
759:
755:
751:
743:
739:
737:rational number
732:
720:
716:
712:
672:
671:
664:
660:
649:
621:
620:
574:
573:
539:
538:
513:
512:
493:
492:
489:
480:
440:
439:
437:
433:
429:
378:
377:
339:
338:
331:
323:
320:rotation number
312:
308:
250:
249:
211:
210:
199:
157:
152:
151:
123:
110:
99:
98:
95:
83:periodic orbits
79:planetary orbit
63:
43:rotation number
35:
24:
17:
12:
11:
5:
1026:
1024:
1016:
1015:
1010:
1005:
995:
994:
991:
990:
983:
966:
965:External links
963:
962:
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901:
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891:
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881:
867:
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864:
826:
800:
799:
773:
748:periodic orbit
709:
708:
697:
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691:
688:
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682:
679:
648:
645:
628:
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605:
602:
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555:
552:
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546:
526:
523:
520:
500:
488:
485:
462:
458:
453:
448:
426:Henri Poincaré
423:
422:
411:
406:
402:
399:
396:
393:
390:
385:
381:
372:
369:
366:
362:
358:
355:
352:
349:
346:
305:
304:
293:
290:
287:
284:
281:
278:
275:
272:
269:
266:
263:
260:
257:
233:
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221:
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187:
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169:
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130:
126:
122:
117:
113:
109:
106:
94:
91:
67:Henri Poincaré
62:
59:
51:homeomorphisms
32:Turning number
28:Winding number
15:
13:
10:
9:
6:
4:
3:
2:
1025:
1014:
1011:
1009:
1006:
1004:
1001:
1000:
998:
988:
984:
980:
979:
974:
969:
968:
964:
960:
956:
950:
946:
942:
938:
934:
931:(in French).
930:
929:
924:
919:
918:
914:
910:
907:
905:
902:
900:
897:
895:
892:
890:
887:
886:
882:
880:
877:
872:
831:
827:
820:
813:by the angle
812:
804:
803:
802:
801:
782:
774:
770:
765:
749:
738:
730:
729:
728:
726:
725:Arnaud Denjoy
695:
692:
689:
686:
683:
680:
677:
670:
669:
668:
658:
657:semiconjugacy
654:
646:
644:
642:
626:
603:
600:
597:
594:
591:
585:
579:
572:
571:
570:
556:
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550:
547:
544:
524:
521:
518:
498:
486:
484:
478:
460:
451:
427:
409:
404:
400:
397:
391:
383:
379:
364:
356:
350:
344:
337:
336:
335:
329:
321:
316:
291:
288:
282:
276:
273:
267:
264:
261:
255:
248:
247:
246:
219:
216:
209:
208:homeomorphism
205:
185:
176:
167:
162:
158:
150:
146:
145:homeomorphism
128:
124:
115:
111:
107:
104:
97:Suppose that
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90:
88:
84:
80:
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72:
68:
60:
58:
56:
52:
48:
44:
40:
33:
29:
22:
978:Scholarpedia
976:
954:
932:
926:
875:
870:
868:
768:
763:
710:
656:
650:
618:
490:
424:
319:
317:
306:
96:
85:in terms of
64:
42:
36:
959:pdf ver 1.3
432:. The lift
87:rationality
39:mathematics
997:Categories
915:References
889:Circle map
871:continuous
830:Cantor set
647:Properties
93:Definition
75:perihelion
71:precession
949:118356096
935:: 5–233.
693:∘
681:∘
522:π
398:−
371:∞
368:→
345:ω
228:→
121:→
47:invariant
1013:Rotation
955:SciSpace
883:See also
764:backward
569:), then
328:iterates
953:, also
839:. Then
786:. Then
537:(where
487:Example
473:
438:
202:may be
147:of the
73:of the
61:History
53:of the
947:
779:is an
746:has a
477:orbits
204:lifted
149:circle
55:circle
45:is an
41:, the
945:S2CID
819:dense
735:is a
659:: if
639:(cf.
206:to a
198:Then
77:of a
719:and
663:and
554:<
548:<
318:The
937:doi
794:of
756:p/q
740:p/q
643:).
491:If
479:of
361:lim
330:of
322:of
49:of
37:In
30:or
999::
975:.
943:.
933:49
483:.
334::
315:.
57:.
981:.
951:.
939::
876:C
863:.
861:C
857:h
853:θ
849:f
845:C
841:C
837:f
833:C
823:f
815:θ
807:f
796:f
792:x
788:f
784:θ
777:f
769:f
760:f
752:q
744:f
733:f
721:g
717:f
713:h
696:h
690:g
687:=
684:f
678:h
665:g
661:f
627:N
604:,
601:N
598:+
595:x
592:=
589:)
586:x
583:(
580:F
557:1
551:N
545:0
525:N
519:2
499:f
481:f
461:.
457:Z
452:/
447:R
434:F
430:x
410:.
405:n
401:x
395:)
392:x
389:(
384:n
380:F
365:n
357:=
354:)
351:f
348:(
332:F
324:f
313:m
309:x
292:m
289:+
286:)
283:x
280:(
277:F
274:=
271:)
268:m
265:+
262:x
259:(
256:F
232:R
224:R
220::
217:F
200:f
186:.
182:Z
177:/
172:R
168:=
163:1
159:S
129:1
125:S
116:1
112:S
108::
105:f
34:.
23:.
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