275:
475:
403:
has wandering domains. However, the result can be generalized to many situations where the functions naturally belong to a finite-dimensional parameter space, most notably to transcendental entire and meromorphic functions with a finite number of singular values.
269:
178:
401:
335:
274:
545:
204:
516:
429:
337:; the Fatou set (consisting entirely of wandering domains) is shown in white, while the Julia set is shown in tones of gray.
90:
555:
550:
509:
535:
342:
347:
281:
445:
442:
Quasiconformal homeomorphisms and dynamics. I. Solution of the Fatou-Julia problem on wandering domains
540:
502:
188:
425:
39:
486:
413:
28:
452:
435:
449:
432:
421:
70:
32:
66:
529:
58:
474:
482:
459:
20:
77:
264:{\displaystyle f^{n}=\underbrace {f\circ f\circ \cdots \circ f} _{n}.}
273:
341:
The theorem does not hold for arbitrary maps; for example, the
460:
Sullivan's proof of Fatou's no wandering domain conjecture
490:
350:
284:
207:
173:{\displaystyle U,f(U),f(f(U)),\dots ,f^{n}(U),\dots }
93:
395:
329:
263:
172:
510:
8:
517:
503:
349:
283:
252:
222:
212:
206:
149:
92:
183:will eventually become periodic. Here,
420:, Universitext: Tracts in Mathematics,
278:This image illustrates the dynamics of
7:
471:
469:
396:{\displaystyle f(z)=z+2\pi \sin(z)}
330:{\displaystyle f(z)=z+2\pi \sin(z)}
489:. You can help Knowledge (XXG) by
14:
473:
390:
384:
360:
354:
324:
318:
294:
288:
161:
155:
133:
130:
124:
118:
109:
103:
69:. More precisely, for every
1:
546:Theorems in dynamical systems
448:122 (1985), no. 3, 401–18.
57:) ≥ 2 does not have a
25:no-wandering-domain theorem
572:
468:
38:The theorem states that a
416:and Theodore W. Gamelin,
485:-related article is a
397:
338:
331:
265:
174:
446:Annals of Mathematics
398:
332:
277:
266:
175:
348:
282:
205:
91:
16:Mathematical theorem
49: →
556:Chaos theory stubs
424:, New York, 1993,
393:
343:transcendental map
339:
327:
261:
257:
250:
170:
498:
497:
440:Dennis Sullivan,
223:
221:
29:dynamical systems
563:
551:Complex dynamics
519:
512:
505:
477:
470:
418:Complex Dynamics
414:Lennart Carleson
402:
400:
399:
394:
336:
334:
333:
328:
270:
268:
267:
262:
256:
251:
246:
217:
216:
179:
177:
176:
171:
154:
153:
84:, the sequence
59:wandering domain
571:
570:
566:
565:
564:
562:
561:
560:
526:
525:
524:
523:
466:
422:Springer-Verlag
410:
346:
345:
280:
279:
224:
208:
203:
202:
192:-fold iteration
145:
89:
88:
33:Dennis Sullivan
27:is a result on
17:
12:
11:
5:
569:
567:
559:
558:
553:
548:
543:
538:
536:Ergodic theory
528:
527:
522:
521:
514:
507:
499:
496:
495:
478:
464:
463:
455:
438:
409:
406:
392:
389:
386:
383:
380:
377:
374:
371:
368:
365:
362:
359:
356:
353:
326:
323:
320:
317:
314:
311:
308:
305:
302:
299:
296:
293:
290:
287:
272:
271:
260:
255:
249:
245:
242:
239:
236:
233:
230:
227:
220:
215:
211:
181:
180:
169:
166:
163:
160:
157:
152:
148:
144:
141:
138:
135:
132:
129:
126:
123:
120:
117:
114:
111:
108:
105:
102:
99:
96:
67:Riemann sphere
15:
13:
10:
9:
6:
4:
3:
2:
568:
557:
554:
552:
549:
547:
544:
542:
539:
537:
534:
533:
531:
520:
515:
513:
508:
506:
501:
500:
494:
492:
488:
484:
479:
476:
472:
467:
462:
461:
456:
454:
451:
447:
443:
439:
437:
434:
431:
430:0-387-97942-5
427:
423:
419:
415:
412:
411:
407:
405:
387:
381:
378:
375:
372:
369:
366:
363:
357:
351:
344:
321:
315:
312:
309:
306:
303:
300:
297:
291:
285:
276:
258:
253:
247:
243:
240:
237:
234:
231:
228:
225:
218:
213:
209:
201:
200:
199:
197:
193:
191:
186:
167:
164:
158:
150:
146:
142:
139:
136:
127:
121:
115:
112:
106:
100:
97:
94:
87:
86:
85:
83:
79:
75:
72:
68:
64:
60:
56:
52:
48:
45: :
44:
41:
36:
34:
30:
26:
22:
491:expanding it
483:chaos theory
480:
465:
458:
441:
417:
340:
195:
189:
187:denotes the
184:
182:
81:
73:
65:denotes the
62:
54:
50:
46:
42:
40:rational map
37:
31:, proven by
24:
18:
457:S. Zakeri,
198:, that is,
21:mathematics
541:Limit sets
530:Categories
408:References
382:
376:π
316:
310:π
248:⏟
241:∘
238:⋯
235:∘
229:∘
168:…
140:…
78:Fatou set
71:component
53:with deg(
35:in 1985.
61:, where
453:0819553
436:1230383
76:in the
428:
23:, the
481:This
487:stub
426:ISBN
379:sin
313:sin
194:of
80:of
19:In
532::
450:MR
444:,
433:MR
518:e
511:t
504:v
493:.
391:)
388:z
385:(
373:2
370:+
367:z
364:=
361:)
358:z
355:(
352:f
325:)
322:z
319:(
307:2
304:+
301:z
298:=
295:)
292:z
289:(
286:f
259:.
254:n
244:f
232:f
226:f
219:=
214:n
210:f
196:f
190:n
185:f
165:,
162:)
159:U
156:(
151:n
147:f
143:,
137:,
134:)
131:)
128:U
125:(
122:f
119:(
116:f
113:,
110:)
107:U
104:(
101:f
98:,
95:U
82:f
74:U
63:Ĉ
55:f
51:Ĉ
47:Ĉ
43:f
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.