187:
393:
253:
257:
271:
453:
434:
359:
56:
provides a good account of this theory with complete proofs: it also introduces a definition which make sense in any open set and dimension.
332:
230:
208:
95:
Carathéodory's principal theorem on the correspondence between boundaries under conformal mappings can be expressed as follows:
25:
376:
166:
A more precise and formal definition of the concepts of "chains of arcs" and of their equivalence classes is given in the
427:
371:
316:
41:
201:
195:
458:
276:
420:
212:
52:
in geometric terms. The theory has been generalized to more general open sets. The expository paper of
32:(i.e. a simply connected open set in the plane) by adding the boundary circle in an appropriate way.
366:
266:
83:, and conversely, many points in the boundary may correspond to a point in the prime ends of
72:
is the set of equivalence classes of chains of arcs converging to a point on the boundary of
111:
355:
328:
79:
In this way, a point in the boundary may correspond to many points in the prime ends of
346:
320:
296:
280:
60:
gives an accessible introduction to prime ends in the context of complex dynamical systems.
29:
342:
292:
350:
338:
300:
288:
249:
404:
447:
49:
45:
308:
245:
115:
17:
284:
315:, Annals of Mathematics Studies, vol. 160 (3rd ed.), Princeton, NJ:
392:
103:
324:
400:
180:
254:
Creative
Commons Attribution-ShareAlike 3.0 Unported License
408:
149:
147:
106:conformally and one-to-one onto the domain
272:Proceedings of the London Mathematical Society
428:
8:
244:This article incorporates material from the
40:The concept of prime ends was introduced by
435:
421:
231:Learn how and when to remove this message
68:The set of prime ends of the domain
194:This article includes a list of general
154:
138:
130:
53:
57:
44:to describe the boundary behavior of
7:
389:
387:
200:it lacks sufficient corresponding
14:
391:
313:Dynamics in one complex variable
185:
24:compactification is a method to
252:", which is licensed under the
167:
454:Compactification (mathematics)
1:
407:. You can help Knowledge by
269:(3 May 1981), "Prime Ends",
372:Encyclopedia of Mathematics
118:and the prime ends of
475:
386:
317:Princeton University Press
114:between the points on the
285:10.1112/plms/s3-42.3.385
277:Oxford University Press
215:more precise citations.
42:Constantin Carathéodory
325:10.1515/9781400835539
319:, pp. viii+304,
275:, s3–42 (3), Oxford:
256:but not under the
112:one-to-one mapping
416:
415:
360:978-0-691-12488-9
267:Epstein, D. B. A.
241:
240:
233:
64:Formal definition
466:
437:
430:
423:
401:topology-related
395:
388:
380:
367:"Limit elements"
353:
303:
236:
229:
225:
222:
216:
211:this article by
202:inline citations
189:
188:
181:
171:
164:
158:
151:
142:
135:
121:
109:
101:
86:
82:
75:
71:
36:Historical notes
30:topological disc
474:
473:
469:
468:
467:
465:
464:
463:
444:
443:
442:
441:
384:
365:
335:
307:
265:
237:
226:
220:
217:
207:Please help to
206:
190:
186:
179:
174:
165:
161:
152:
145:
141:, p. 385).
136:
132:
128:
119:
110:, it induces a
107:
99:
93:
84:
80:
73:
69:
66:
38:
12:
11:
5:
472:
470:
462:
461:
459:Topology stubs
456:
446:
445:
440:
439:
432:
425:
417:
414:
413:
396:
382:
381:
363:
333:
305:
239:
238:
193:
191:
184:
178:
175:
173:
172:
159:
143:
129:
127:
124:
92:
89:
65:
62:
54:Epstein (1981)
46:conformal maps
37:
34:
13:
10:
9:
6:
4:
3:
2:
471:
460:
457:
455:
452:
451:
449:
438:
433:
431:
426:
424:
419:
418:
412:
410:
406:
403:article is a
402:
397:
394:
390:
385:
378:
374:
373:
368:
364:
361:
357:
352:
348:
344:
340:
336:
334:0-691-12488-4
330:
326:
322:
318:
314:
310:
306:
302:
298:
294:
290:
286:
282:
278:
274:
273:
268:
264:
263:
262:
261:
259:
255:
251:
247:
235:
232:
224:
214:
210:
204:
203:
197:
192:
183:
182:
176:
169:
163:
160:
156:
150:
148:
144:
140:
134:
131:
125:
123:
117:
113:
105:
96:
90:
88:
77:
63:
61:
59:
58:Milnor (2006)
55:
51:
50:complex plane
47:
43:
35:
33:
31:
27:
23:
19:
409:expanding it
398:
383:
370:
312:
309:Milnor, John
270:
243:
242:
227:
218:
199:
162:
155:Epstein 1981
139:Epstein 1981
133:
97:
94:
91:Applications
78:
67:
39:
21:
15:
279:: 385–414,
246:Citizendium
213:introducing
116:unit circle
18:mathematics
448:Categories
351:1281.37001
301:0491.30027
250:Prime ends
196:references
177:References
168:references
26:compactify
377:EMS Press
311:(2006) ,
248:article "
104:unit disk
102:maps the
22:prime end
221:May 2010
379:, 2001
343:2193309
293:0614728
209:improve
48:in the
358:
349:
341:
331:
299:
291:
198:, but
170:cited.
157:, §2).
20:, the
399:This
126:Notes
405:stub
356:ISBN
329:ISBN
258:GFDL
347:Zbl
321:doi
297:Zbl
281:doi
98:If
16:In
450::
375:,
369:,
354:,
345:,
339:MR
337:,
327:,
295:,
289:MR
287:,
146:^
122:.
87:.
76:.
28:a
436:e
429:t
422:v
411:.
362:,
323::
304:.
283::
260:.
234:)
228:(
223:)
219:(
205:.
153:(
137:(
120:B
108:B
100:Ć’
85:B
81:B
74:B
70:B
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.