846:
478:
Eremenko also asked whether every escaping point can be connected to infinity by a curve in the escaping set; it was later shown that this is not the case. Indeed, there exist entire functions whose escaping sets do not contain any curves at all.
835:
958:
311:
68:
632:
1085:
388:
123:
909:, the escaping set is much more complicated than for polynomials: in the simplest cases like the one illustrated in the picture it consists of uncountably many curves, called
899:
438:
1016:
987:
758:
237:
1055:
533:
698:
192:
579:
163:
143:
1096:
638:
Note that the final statement does not imply
Eremenko's Conjecture. (Indeed, there exist connected spaces in which the removal of a single
453:
766:
557:
For a transcendental entire function, the escaping set always intersects the Julia set. In particular, the escaping set is
487:
The following properties are known to hold for the escaping set of any non-constant and non-linear entire function. (Here
928:
921:). As mentioned above, there are examples of transcendental entire functions whose escaping set contains no curves.
245:
468:
716:
38:
596:
1060:
331:
73:
1487:
664:
852:
393:
994:
965:
906:
721:
200:
1022:
671:
of this fixed point, and hence usually referred to as the **basin of infinity**. In this case,
494:
471:
of the escaping set of a transcendental entire function is unbounded. This has become known as
475:. There are many partial results on this problem but as of 2013 the conjecture is still open.
1391:
464:
459:
The first study of the escaping set for a general transcendental entire function is due to
674:
168:
8:
543:
328:
in 1926 The escaping set occurs implicitly in his study of the explicit entire functions
1470:
1395:
1372:
Sixsmith, D.J. (2012). "Entire functions for which the escaping set is a spider's web".
1442:
1421:
1417:
1381:
1344:
1304:
1257:
564:
460:
148:
128:
1261:
29:
1358:
1399:
1354:
1314:
1228:
1192:
1176:
639:
17:
1197:
1180:
450:
Can the escaping set of a transcendental entire function have a bounded component?
925:
705:
25:
1318:
917:. In other examples the structure of the escaping set can be very different (a
660:
33:
1403:
1335:
Rippon, P. J.; Stallard, G (2011). "Boundaries of escaping Fatou components".
1481:
1212:
325:
584:
Every connected component of the closure of the escaping set is unbounded.
1466:
1295:; Schleicher, D (2011). "Dynamic rays of bounded-type entire functions".
1292:
1233:
1216:
1101:
1097:
Plotting algorithms for the
Mandelbrot set § Escape time algorithm
656:
587:
The escaping set always has at least one unbounded connected component.
551:
442:
324:
The iteration of transcendental entire functions was first studied by
709:
668:
547:
991:
962:
1447:
1426:
701:
558:
70:
belongs to the escaping set if and only if the sequence defined by
1386:
1349:
1309:
1290:
590:
The escaping set is connected or has infinitely many components.
1374:
Mathematical
Proceedings of the Cambridge Philosophical Society
760:
consists precisely of the complement of the closed unit disc:
239:, the origin belongs to the escaping set, since the sequence
1441:
Lipham, D.S. (2022). "Exponential iteration and Borel sets".
1129:
See (Eremenko, 1989), formula (1) on p. 339 and l.2 of p. 340
845:
830:{\displaystyle I(f)=\{z\in \mathbb {C} \colon |z|>1\}.}
28:ƒ consists of all points that tend to infinity under the
1217:"Sur l'itération des fonctions transcendantes Entières"
1063:
1025:
997:
968:
931:
855:
769:
724:
677:
599:
567:
497:
396:
334:
248:
203:
171:
151:
131:
76:
41:
659:of degree 2 extends to an analytic self-map of the
1079:
1049:
1010:
981:
952:
893:
840:
829:
752:
692:
642:leaves the remaining space totally disconnected.)
626:
573:
527:
432:
382:
305:
231:
186:
157:
137:
117:
62:
1479:
1420:(2020). "Escaping sets are not sigma-compact".
953:{\displaystyle F_{\sigma \delta }{\text{ set}}}
667:at infinity. The escaping set is precisely the
444:
1334:
1174:
539:The escaping set contains at least one point.
821:
785:
621:
615:
306:{\displaystyle 0,1,e,e^{e},e^{e^{e}},\dots }
550:. In particular, the escaping set is never
491:means that the function is not of the form
1446:
1425:
1385:
1348:
1308:
1232:
1196:
1019:. For functions in the exponential class
795:
56:
1371:
1269:Banach Center Publications, Warsawa, PWN
1256:
844:
1465:
454:(more unsolved problems in mathematics)
1480:
1440:
1286:
1284:
1282:
1262:"On the iteration of entire functions"
924:By definition, the escaping set is an
1416:
1330:
1328:
1211:
715:For instance the escaping set of the
708:subset of the complex plane, and the
63:{\displaystyle z_{0}\in \mathbb {C} }
1205:
1181:"On questions of Fatou and Eremenko"
1170:
1168:
1166:
1164:
627:{\displaystyle I(f)\cup \{\infty \}}
1279:
1252:
1250:
1248:
1246:
1244:
546:of the escaping set is exactly the
13:
1325:
1080:{\displaystyle G_{\delta \sigma }}
618:
14:
1499:
1459:
1161:
383:{\displaystyle f(z)=z+1+\exp(-z)}
118:{\displaystyle z_{n+1}:=f(z_{n})}
16:In mathematics, and particularly
1241:
145:gets large. The escaping set of
1471:"A poem on Eremenko conjecture"
1434:
1410:
1365:
1359:10.1090/s0002-9939-2011-10842-6
907:transcendental entire functions
841:Transcendental entire functions
712:is the boundary of this basin.
445:Unsolved problem in mathematics
1141:
1132:
1123:
1114:
1038:
1032:
880:
871:
865:
856:
811:
803:
779:
773:
734:
728:
687:
681:
650:
609:
603:
507:
501:
427:
421:
406:
400:
377:
368:
344:
338:
213:
207:
181:
175:
112:
99:
1:
1198:10.1090/s0002-9939-04-07805-0
1154:
1147:Theorem 3 of (Eremenko, 1989)
1138:Theorem 2 of (Eremenko, 1989)
1120:Theorem 1 of (Eremenko, 1989)
894:{\displaystyle (\exp(z)-1)/2}
482:
433:{\displaystyle f(z)=c\sin(z)}
1291:Rottenfußer, G; Rückert, J;
717:complex quadratic polynomial
665:super-attracting fixed point
467:. He conjectured that every
7:
1319:10.4007/annals.2010.173.1.3
1090:
1011:{\displaystyle F_{\sigma }}
982:{\displaystyle G_{\delta }}
645:
10:
1504:
1057:, the escaping set is not
753:{\displaystyle f(z)=z^{2}}
319:
232:{\displaystyle f(z)=e^{z}}
1404:10.1017/S0305004111000582
1050:{\displaystyle \exp(z)+a}
528:{\displaystyle f(z)=az+b}
125:converges to infinity as
1107:
1081:
1051:
1012:
983:
954:
902:
895:
831:
754:
694:
628:
575:
529:
434:
384:
307:
233:
188:
159:
139:
119:
64:
1337:Proc. Amer. Math. Soc
1185:Proc. Amer. Math. Soc
1082:
1052:
1013:
984:
955:
896:
848:
832:
755:
695:
629:
576:
530:
473:Eremenko's conjecture
435:
385:
308:
234:
189:
160:
140:
120:
65:
1061:
1023:
995:
966:
929:
853:
767:
722:
693:{\displaystyle I(f)}
675:
597:
565:
495:
465:Wiman-Valiron theory
394:
332:
246:
201:
187:{\displaystyle I(f)}
169:
149:
129:
74:
39:
30:repeated application
1396:2011MPCPS.151..551S
669:basin of attraction
469:connected component
316:tends to infinity.
1234:10.1007/bf02559517
1077:
1047:
1008:
979:
950:
903:
891:
827:
750:
690:
624:
571:
525:
461:Alexandre Eremenko
430:
380:
303:
229:
184:
155:
135:
115:
60:
948:
574:{\displaystyle f}
197:For example, for
158:{\displaystyle f}
138:{\displaystyle n}
32:of ƒ. That is, a
1495:
1488:Complex analysis
1474:
1453:
1452:
1450:
1438:
1432:
1431:
1429:
1414:
1408:
1407:
1389:
1369:
1363:
1362:
1352:
1343:(8): 2807–2820.
1332:
1323:
1322:
1312:
1288:
1277:
1276:
1266:
1254:
1239:
1238:
1236:
1209:
1203:
1202:
1200:
1191:(4): 1119–1126.
1172:
1148:
1145:
1139:
1136:
1130:
1127:
1121:
1118:
1086:
1084:
1083:
1078:
1076:
1075:
1056:
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1017:
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1007:
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988:
986:
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980:
978:
977:
961:. It is neither
959:
957:
956:
951:
949:
946:
944:
943:
900:
898:
897:
892:
887:
849:Escaping set of
836:
834:
833:
828:
814:
806:
798:
759:
757:
756:
751:
749:
748:
699:
697:
696:
691:
640:dispersion point
633:
631:
630:
625:
581:is a polynomial.
580:
578:
577:
572:
534:
532:
531:
526:
446:
439:
437:
436:
431:
389:
387:
386:
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312:
310:
309:
304:
296:
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294:
293:
276:
275:
238:
236:
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193:
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185:
164:
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161:
156:
144:
142:
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136:
124:
122:
121:
116:
111:
110:
92:
91:
69:
67:
66:
61:
59:
51:
50:
18:complex dynamics
1503:
1502:
1498:
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1493:
1492:
1478:
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1462:
1457:
1456:
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1370:
1366:
1333:
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1289:
1280:
1264:
1255:
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1210:
1206:
1175:Rippon, P. J.;
1173:
1162:
1157:
1152:
1151:
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1137:
1133:
1128:
1124:
1119:
1115:
1110:
1093:
1068:
1064:
1062:
1059:
1058:
1024:
1021:
1020:
1002:
998:
996:
993:
992:
973:
969:
967:
964:
963:
945:
936:
932:
930:
927:
926:
883:
854:
851:
850:
843:
810:
802:
794:
768:
765:
764:
744:
740:
723:
720:
719:
676:
673:
672:
653:
648:
598:
595:
594:
566:
563:
562:
561:if and only if
496:
493:
492:
485:
457:
456:
451:
448:
395:
392:
391:
333:
330:
329:
322:
289:
285:
284:
280:
271:
267:
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244:
243:
223:
219:
202:
199:
198:
170:
167:
166:
150:
147:
146:
130:
127:
126:
106:
102:
81:
77:
75:
72:
71:
55:
46:
42:
40:
37:
36:
26:entire function
12:
11:
5:
1501:
1491:
1490:
1476:
1475:
1461:
1460:External links
1458:
1455:
1454:
1433:
1409:
1380:(3): 551–571.
1364:
1324:
1278:
1240:
1227:(4): 337–370.
1204:
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1111:
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775:
772:
747:
743:
739:
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733:
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689:
686:
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661:Riemann sphere
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635:
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165:is denoted by
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134:
114:
109:
105:
101:
98:
95:
90:
87:
84:
80:
58:
54:
49:
45:
34:complex number
9:
6:
4:
3:
2:
1500:
1489:
1486:
1485:
1483:
1472:
1468:
1464:
1463:
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1397:
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1320:
1316:
1311:
1306:
1302:
1298:
1294:
1287:
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1283:
1274:
1270:
1263:
1259:
1253:
1251:
1249:
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1230:
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1218:
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1199:
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1178:
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1169:
1167:
1165:
1160:
1144:
1135:
1126:
1117:
1113:
1103:
1100:
1098:
1095:
1094:
1088:
1072:
1069:
1065:
1044:
1041:
1035:
1029:
1026:
1018:
1003:
999:
989:
974:
970:
960:
940:
937:
933:
922:
920:
916:
912:
908:
888:
884:
877:
874:
868:
862:
859:
847:
824:
818:
815:
807:
799:
791:
788:
782:
776:
770:
763:
762:
761:
745:
741:
737:
731:
725:
718:
713:
711:
707:
703:
684:
678:
670:
666:
662:
658:
643:
641:
634:is connected.
612:
606:
600:
592:
589:
586:
583:
568:
560:
556:
553:
549:
545:
541:
538:
537:
536:
522:
519:
516:
513:
510:
504:
498:
490:
480:
476:
474:
470:
466:
462:
455:
441:
424:
418:
415:
412:
409:
403:
397:
374:
371:
365:
362:
359:
356:
353:
350:
347:
341:
335:
327:
317:
300:
297:
290:
286:
281:
277:
272:
268:
264:
261:
258:
255:
252:
249:
242:
241:
240:
224:
220:
216:
210:
204:
195:
178:
172:
152:
132:
107:
103:
96:
93:
88:
85:
82:
78:
52:
47:
43:
35:
31:
27:
23:
19:
1436:
1418:Rempe, Lasse
1412:
1377:
1373:
1367:
1340:
1336:
1300:
1297:Ann. of Math
1296:
1272:
1268:
1224:
1220:
1207:
1188:
1184:
1143:
1134:
1125:
1116:
923:
919:spider's web
918:
914:
910:
904:
714:
654:
637:
488:
486:
477:
472:
458:
326:Pierre Fatou
323:
315:
196:
22:escaping set
21:
15:
1467:Lasse Rempe
1258:Eremenko, A
1177:Stallard, G
663:, having a
651:Polynomials
1448:2010.13876
1427:2006.16946
1303:: 77–125.
1275:: 339–345.
1155:References
1102:target set
657:polynomial
483:Properties
1387:1012.1303
1350:1009.4450
1310:0704.3213
1221:Acta Math
1213:Fatou, P.
1073:σ
1070:δ
1030:
1004:σ
975:δ
947: set
941:δ
938:σ
875:−
863:
800::
792:∈
710:Julia set
706:connected
619:∞
613:∪
548:Julia set
489:nonlinear
463:who used
419:
372:−
366:
301:…
53:∈
1482:Category
1293:Rempe, L
1260:(1989).
1215:(1926).
1179:(2005).
1091:See also
646:Examples
593:The set
544:boundary
1392:Bibcode
320:History
700:is an
552:closed
24:of an
20:, the
1443:arXiv
1422:arXiv
1382:arXiv
1345:arXiv
1305:arXiv
1265:(PDF)
1108:Notes
911:hairs
990:nor
915:rays
905:For
816:>
704:and
702:open
559:open
542:The
390:and
1400:doi
1378:151
1355:doi
1341:139
1315:doi
1301:173
1229:doi
1193:doi
1189:133
1027:exp
913:or
860:exp
535:.)
416:sin
363:exp
1484::
1469:.
1398:.
1390:.
1376:.
1353:.
1339:.
1327:^
1313:.
1299:.
1281:^
1273:23
1271:.
1267:.
1243:^
1225:47
1223:.
1219:.
1187:.
1183:.
1163:^
1087:.
655:A
440:.
194:.
94::=
1473:.
1451:.
1445::
1430:.
1424::
1406:.
1402::
1394::
1384::
1361:.
1357::
1347::
1321:.
1317::
1307::
1237:.
1231::
1201:.
1195::
1066:G
1045:a
1042:+
1039:)
1036:z
1033:(
1000:F
971:G
934:F
901:.
889:2
885:/
881:)
878:1
872:)
869:z
866:(
857:(
825:.
822:}
819:1
812:|
808:z
804:|
796:C
789:z
786:{
783:=
780:)
777:f
774:(
771:I
746:2
742:z
738:=
735:)
732:z
729:(
726:f
688:)
685:f
682:(
679:I
622:}
616:{
610:)
607:f
604:(
601:I
569:f
554:.
523:b
520:+
517:z
514:a
511:=
508:)
505:z
502:(
499:f
447::
428:)
425:z
422:(
413:c
410:=
407:)
404:z
401:(
398:f
378:)
375:z
369:(
360:+
357:1
354:+
351:z
348:=
345:)
342:z
339:(
336:f
298:,
291:e
287:e
282:e
278:,
273:e
269:e
265:,
262:e
259:,
256:1
253:,
250:0
225:z
221:e
217:=
214:)
211:z
208:(
205:f
182:)
179:f
176:(
173:I
153:f
133:n
113:)
108:n
104:z
100:(
97:f
89:1
86:+
83:n
79:z
57:C
48:0
44:z
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
