601:
27:
1405:
139:
851:
580:
The series expansion around a fixed point and the relevant convergence properties of the solution for the resulting orbit and its analyticity properties are cogently summarized by
302:
satisfying Schröder's equation. This is one of the first steps in a long line of theorems fruitful for understanding composition operators on analytic function spaces, cf.
1266:
68:
765:
1516:
667:
It is used to analyse discrete dynamical systems by finding a new coordinate system in which the system (orbit) generated by
1688:
1486:
1440:
41:
30:
939:
determined to solve Schröder's equation: a holographic continuous interpolation of the initial discrete recursion
1937:
1932:
662:
474:-th power of a solution of Schröder's equation provides a solution of Schröder's equation with eigenvalue
20:
1640:
552:
1416:
1098:
In fact, this solution is seen to result as motion dictated by a sequence of switchback potentials,
571:
404:
1548:
1222:
1400:{\displaystyle h_{t}(x)=\Psi ^{-1}{\big (}2^{-t}\Psi (x){\big )}={\frac {x}{2^{t}+x(1-2^{t})}}.}
962:
577:
There are a good number of particular solutions dating back to Schröder's original 1870 paper.
702:
326:
183:
1865:
1804:
1718:(1904). "The principal laws of convergence of iterates and their application to analysis".
1614:
1563:
145:
49:
8:
1877:
1830:
1816:
1644:
748:
45:
1869:
1808:
1618:
1567:
1908:
1855:
1794:
1604:
1506:
698:
314:
1764:
1715:
1475:
1522:
1512:
1482:
753:
585:
1846:; Zachos, C. K. (2010). "Chaotic Maps, Hamiltonian Flows, and Holographic Methods".
600:
1900:
1873:
1843:
1812:
1778:
1753:
1697:
1659:
1622:
1592:
1571:
1540:
1470:
1452:
955:
861:
694:
303:
1737:
589:
581:
310:
295:
1904:
1683:
1626:
1544:
1926:
1782:
1526:
372:
368:
1679:
1575:
1130:, a generic feature of continuous iterates effected by Schröder's equation.
1595:; Zachos, C.K. (March 2011). "Renormalization Group Functional Equations".
1502:
1042:
were already worked out by Schröder in his original article (p. 306),
1020:
678:
More specifically, a system for which a discrete unit time step amounts to
436:
318:
1511:. Monografie Matematyczne. Warszawa: PWN – Polish Scientific Publishers.
1481:. Textbook series: Universitext: Tracts in Mathematics. Springer-Verlag.
621:, interpolated holographically through Schröder's equation. The velocity
701:) reconstructed from the solution of the above Schröder's equation, its
1912:
1758:
1741:
1701:
1664:
1456:
322:
240:
1891:
Skellam, J.G. (1951). "Random dispersal in theoretical populations".
892:
332:
An equivalent transpose form of Schröder's equation for the inverse
1645:"Recherches sur les intégrales de certaines équations fonctionelles"
527:. All solutions of Schröder's equation are related in this manner.
134:{\displaystyle \forall x\;\;\;\Psi {\big (}h(x){\big )}=s\Psi (x).}
1860:
1799:
1609:
570:, Schröder's equation is unwieldy, and had best be transformed to
26:
846:{\displaystyle h_{t}(x)=\Psi ^{-1}{\big (}s^{t}\Psi (x){\big )},}
539:
is an attracting (but not superattracting) fixed point, that is
928:
are likewise specified through the coordinate transformation
860:
real — not necessarily positive or integer. (Thus a full
604:
First five half periods of the phase-space orbit of the
480:, instead. In the same vein, for an invertible solution
313:, and have thus been extensively utilized in studies of
1785:(2009). "Evolution Profiles and Functional Equations".
1133:
A nonchaotic case he also illustrated with his method,
298:
showed in 1884 that there is an analytic (non-trivial)
33:(1841–1902) in 1870 formulated his eponymous equation.
1842:
1777:
488:
of Schröder's equation, the (non-invertible) function
309:
Equations such as Schröder's are suitable to encoding
144:
Schröder's equation is an eigenvalue equation for the
1591:
1269:
768:
584:. Several of the solutions are furnished in terms of
71:
371:) further converts Schröder's equation to the older
1652:
1831:Evolution surfaces and Schröder functional methods
1539:
1474:
1399:
845:
133:
16:Equation for fixed point of functional composition
1742:"Regular iteration of real and complex functions"
1924:
1678:
1469:
535:Schröder's equation was solved analytically if
558:In the case of a superattracting fixed point,
1339:
1307:
835:
806:
653:. Chaos is evident in the orbit sweeping all
105:
86:
1549:"Quantum Electrodynamics at Small Distances"
954:has been constructed; in effect, the entire
917:(fractional, infinitesimal, or negative) of
880:, i.e., of all positive integer iterates of
264:
80:
79:
78:
1859:
1798:
1757:
1663:
1608:
1508:Functional equations in a single variable
1736:
1714:
1587:
1585:
1439:
599:
25:
1890:
1720:Izv. Kazan. Fiz.-Mat. Obshch. (Russian)
1639:
1925:
1773:
1771:
1501:
1435:
1433:
1431:
1582:
1443:(1870). "Ueber iterirte Functionen".
391:. Similarly, the change of variables
609: = 4 chaotic logistic map
336:of Schröder's conjugacy function is
282:is analytic on the unit disk, fixes
1768:
1428:
317:(often referred to colloquially as
13:
1325:
1293:
1019:For example, special cases of the
821:
792:
675:) looks simpler, a mere dilation.
116:
81:
72:
14:
1949:
1686:(1960). "On Analytic Iteration".
321:). It is also used in studies of
403:converts Schröder's equation to
239:does not vanish or diverge, the
1884:
1836:
1823:
1473:; Gamelin, Theodore W. (1993).
595:
1878:10.1088/1751-8113/43/44/445101
1817:10.1088/1751-8113/42/48/485208
1730:
1708:
1689:Journal d'Analyse Mathématique
1672:
1633:
1533:
1495:
1463:
1388:
1369:
1334:
1328:
1286:
1280:
830:
824:
785:
779:
743:all of its functional iterates
125:
119:
100:
94:
1:
1422:
663:Rational difference equation
530:
426:Moreover, for the velocity,
7:
1410:
10:
1954:
1917:See equations 41, 42.
1627:10.1103/PhysRevD.83.065019
660:
351:. The change of variables
18:
1905:10.1093/biomet/38.1-2.196
1119: + arcsin
1023:such as the chaotic case
899:(or Picard sequence) of
504:is also a solution, for
19:Not to be confused with
1658:(3, Supplément): 3–41.
1576:10.1103/PhysRev.95.1300
265:Functional significance
1829:Curtright, T. L.
1401:
963:functional square root
847:
756:) are provided by the
693:, can have its smooth
658:
223:. Thus, provided that
158:that sends a function
135:
52:: given the function
34:
21:Schrödinger's equation
1402:
1244:, one readily finds
848:
603:
327:renormalization group
136:
29:
1938:Mathematical physics
1933:Functional equations
1848:Journal of Physics A
1787:Journal of Physics A
1267:
766:
146:composition operator
69:
58:, find the function
50:independent variable
1870:2010JPhA...43R5101C
1809:2009JPhA...42V5208C
1619:2011PhRvD..83f5019C
1568:1954PhRv...95.1300G
1417:Böttcher's equation
1223:Beverton–Holt model
703:conjugacy equation
572:Böttcher's equation
405:Böttcher's equation
46:functional equation
38:Schröder's equation
1759:10.1007/BF02559539
1702:10.1007/BF02786856
1665:10.24033/asens.247
1457:10.1007/BF01443992
1397:
1221:Likewise, for the
1177:ln(1 − 2
961:For instance, the
843:
747:regular iteration
659:
508:periodic function
315:nonlinear dynamics
292:′(0)| < 1
131:
35:
1597:Physical Review D
1518:978-0-02-848110-4
1471:Carleson, Lennart
1392:
1211:((1 − 2
1115: − 1) (
1083:) = sin(2 arcsin
754:iterated function
586:asymptotic series
325:, as well as the
1945:
1918:
1916:
1899:(1–2): 196−218.
1888:
1882:
1881:
1863:
1844:Curtright, T. L.
1840:
1834:
1827:
1821:
1820:
1802:
1775:
1766:
1763:
1761:
1752:(3–4): 361–376.
1746:Acta Mathematica
1734:
1728:
1727:
1712:
1706:
1705:
1676:
1670:
1669:
1667:
1649:
1637:
1631:
1630:
1612:
1589:
1580:
1579:
1562:(5): 1300–1312.
1553:
1537:
1531:
1530:
1499:
1493:
1492:
1480:
1477:Complex Dynamics
1467:
1461:
1460:
1437:
1406:
1404:
1403:
1398:
1393:
1391:
1387:
1386:
1362:
1361:
1348:
1343:
1342:
1324:
1323:
1311:
1310:
1304:
1303:
1279:
1278:
1259:
1254:/(1 −
1243:
1238:/(2 −
1216:
1215:) − 1)
1210:
1208:
1207:
1204:
1201:
1182:
1176:
1174:
1173:
1170:
1167:
1151:
1129:
1127:
1126:
1093:
1091:
1090:
1068:
1061:
1059:
1058:
1041:
1015:
986:
953:
938:
927:
909:
895:) is called the
890:
879:
862:continuous group
859:
852:
850:
849:
844:
839:
838:
820:
819:
810:
809:
803:
802:
778:
777:
737:
692:
656:
652:
651:
641:plotted against
640:
620:
619:
569:
550:
538:
526:
518:
503:
487:
479:
473:
464:
433:
422:
402:
390:
366:
350:
335:
304:Koenigs function
301:
293:
285:
281:
275:
260:
245:
238:
230:
222:
215:
211:
203:
189:
181:
174:
163:
157:
140:
138:
137:
132:
109:
108:
90:
89:
61:
57:
1953:
1952:
1948:
1947:
1946:
1944:
1943:
1942:
1923:
1922:
1921:
1889:
1885:
1841:
1837:
1828:
1824:
1779:Curtright, T.L.
1776:
1769:
1735:
1731:
1716:Böttcher, L. E.
1713:
1709:
1684:Jabotinsky, Eri
1677:
1673:
1647:
1638:
1634:
1593:Curtright, T.L.
1590:
1583:
1556:Physical Review
1551:
1538:
1534:
1519:
1500:
1496:
1489:
1468:
1464:
1441:Schröder, Ernst
1438:
1429:
1425:
1413:
1382:
1378:
1357:
1353:
1352:
1347:
1338:
1337:
1316:
1312:
1306:
1305:
1296:
1292:
1274:
1270:
1268:
1265:
1264:
1245:
1226:
1205:
1202:
1199:
1198:
1196:
1190:
1184:
1171:
1168:
1165:
1164:
1162:
1156:
1146:(1 −
1134:
1122:
1120:
1099:
1086:
1084:
1078:
1070:
1063:
1054:
1052:
1046:
1036:(1 −
1024:
1001:
994:
988:
972:
966:
940:
929:
918:
900:
881:
873:
865:
857:
854:
834:
833:
815:
811:
805:
804:
795:
791:
773:
769:
767:
764:
763:
731:
709:
679:
665:
657:s at all times.
654:
650:
642:
635:
623:
622:
610:
605:
598:
590:Carleman matrix
559:
553:Gabriel Koenigs
540:
536:
533:
520:
509:
489:
481:
475:
469:
442:
427:
408:
392:
376:
352:
337:
333:
311:self-similarity
299:
296:Gabriel Koenigs
287:
283:
277:
270:
267:
247:
243:
232:
224:
217:
213:
205:
191:
187:
179:
165:
159:
156:
148:
142:
104:
103:
85:
84:
70:
67:
66:
59:
53:
24:
17:
12:
11:
5:
1951:
1941:
1940:
1935:
1920:
1919:
1883:
1854:(44): 445101.
1835:
1822:
1793:(48): 485208.
1767:
1729:
1707:
1696:(1): 361–376.
1671:
1632:
1581:
1532:
1517:
1494:
1487:
1462:
1451:(2): 296–322.
1426:
1424:
1421:
1420:
1419:
1412:
1409:
1408:
1407:
1396:
1390:
1385:
1381:
1377:
1374:
1371:
1368:
1365:
1360:
1356:
1351:
1346:
1341:
1336:
1333:
1330:
1327:
1322:
1319:
1315:
1309:
1302:
1299:
1295:
1291:
1288:
1285:
1282:
1277:
1273:
1219:
1218:
1188:
1096:
1095:
1074:
999:
992:
970:
869:
864:.) The set of
842:
837:
832:
829:
826:
823:
818:
814:
808:
801:
798:
794:
790:
787:
784:
781:
776:
772:
761:
729:
646:
631:
627: = d
597:
594:
532:
529:
432:) = Ψ/Ψ′
266:
263:
231:is finite and
204:, then either
152:
130:
127:
124:
121:
118:
115:
112:
107:
102:
99:
96:
93:
88:
83:
77:
74:
64:
42:Ernst Schröder
40:, named after
31:Ernst Schröder
15:
9:
6:
4:
3:
2:
1950:
1939:
1936:
1934:
1931:
1930:
1928:
1914:
1910:
1906:
1902:
1898:
1894:
1887:
1879:
1875:
1871:
1867:
1862:
1857:
1853:
1849:
1845:
1839:
1832:
1826:
1818:
1814:
1810:
1806:
1801:
1796:
1792:
1788:
1784:
1783:Zachos, C. K.
1780:
1774:
1772:
1765:
1760:
1755:
1751:
1747:
1743:
1739:
1733:
1725:
1721:
1717:
1711:
1703:
1699:
1695:
1691:
1690:
1685:
1681:
1675:
1666:
1661:
1657:
1653:
1646:
1642:
1636:
1628:
1624:
1620:
1616:
1611:
1606:
1603:(6): 065019.
1602:
1598:
1594:
1588:
1586:
1577:
1573:
1569:
1565:
1561:
1557:
1550:
1546:
1542:
1541:Gell-Mann, M.
1536:
1528:
1524:
1520:
1514:
1510:
1509:
1504:
1503:Kuczma, Marek
1498:
1490:
1488:0-387-97942-5
1484:
1479:
1478:
1472:
1466:
1458:
1454:
1450:
1446:
1442:
1436:
1434:
1432:
1427:
1418:
1415:
1414:
1394:
1383:
1379:
1375:
1372:
1366:
1363:
1358:
1354:
1349:
1344:
1331:
1320:
1317:
1313:
1300:
1297:
1289:
1283:
1275:
1271:
1263:
1262:
1261:
1257:
1253:
1249:
1241:
1237:
1233:
1229:
1224:
1214:
1194:
1187:
1180:
1160:
1155:
1154:
1153:
1149:
1145:
1141:
1137:
1131:
1125:
1118:
1114:
1110:
1106:
1102:
1089:
1082:
1077:
1073:
1066:
1057:
1050:
1045:
1044:
1043:
1039:
1035:
1031:
1027:
1022:
1017:
1016:, and so on.
1013:
1009:
1005:
998:
991:
984:
980:
976:
969:
964:
959:
957:
951:
947:
943:
936:
932:
925:
921:
916:
911:
907:
903:
898:
894:
888:
884:
877:
872:
868:
863:
853:
840:
827:
816:
812:
799:
796:
788:
782:
774:
770:
760:
759:
755:
751:
750:
744:
739:
735:
728:
724:
720:
716:
712:
706:
704:
700:
696:
690:
686:
682:
676:
674:
670:
664:
649:
645:
639:
634:
630:
626:
617:
613:
608:
602:
593:
591:
587:
583:
578:
575:
573:
567:
563:
556:
554:
548:
544:
528:
524:
516:
512:
507:
501:
497:
493:
485:
478:
472:
466:
462:
458:
454:
450:
446:
440:
438:
431:
424:
420:
416:
412:
406:
400:
396:
388:
384:
380:
374:
373:Abel equation
370:
369:Abel function
364:
360:
356:
348:
344:
340:
330:
328:
324:
320:
316:
312:
307:
305:
297:
291:
280:
273:
262:
258:
254:
250:
242:
236:
228:
220:
209:
202:
198:
194:
185:
176:
172:
168:
162:
155:
151:
147:
141:
128:
122:
113:
110:
97:
91:
75:
63:
56:
51:
47:
43:
39:
32:
28:
22:
1896:
1892:
1886:
1851:
1847:
1838:
1825:
1790:
1786:
1749:
1745:
1738:Szekeres, G.
1732:
1723:
1719:
1710:
1693:
1687:
1674:
1655:
1651:
1635:
1600:
1596:
1559:
1555:
1535:
1507:
1497:
1476:
1465:
1448:
1444:
1255:
1251:
1247:
1239:
1235:
1231:
1227:
1220:
1212:
1192:
1185:
1183:, and hence
1178:
1158:
1147:
1143:
1139:
1135:
1132:
1123:
1116:
1112:
1108:
1104:
1100:
1097:
1087:
1080:
1075:
1071:
1069:, and hence
1064:
1055:
1051:) = (arcsin
1048:
1037:
1033:
1029:
1025:
1021:logistic map
1018:
1011:
1007:
1003:
996:
989:
982:
978:
974:
967:
960:
949:
945:
941:
934:
930:
923:
919:
915:all iterates
914:
912:
905:
901:
896:
886:
882:
875:
870:
866:
855:
762:
757:
746:
742:
741:In general,
740:
733:
726:
722:
718:
714:
710:
707:
688:
684:
680:
677:
672:
668:
666:
647:
643:
637:
632:
628:
624:
615:
611:
606:
596:Applications
579:
576:
565:
561:
557:
546:
542:
534:
522:
519:with period
514:
510:
505:
499:
495:
491:
483:
476:
470:
467:
460:
456:
452:
448:
444:
435:
429:
425:
418:
414:
410:
398:
394:
386:
382:
378:
362:
358:
354:
346:
342:
338:
331:
319:chaos theory
308:
289:
278:
271:
268:
256:
252:
248:
246:is given by
234:
226:
218:
207:
200:
196:
192:
177:
170:
166:
160:
153:
149:
143:
65:
54:
37:
36:
1680:Erdős, Paul
1641:Koenigs, G.
1152:, yields
441:,
439:'s equation
184:fixed point
1927:Categories
1893:Biometrika
1726:: 155–234.
1423:References
1260:, so that
987:, so that
661:See also:
434:,
397:) = log(φ(
357:) = log(Ψ(
323:turbulence
241:eigenvalue
190:, meaning
62:such that
1861:1002.0104
1800:0909.2424
1610:1010.5174
1545:Low, F.E.
1527:489667432
1445:Math. Ann
1376:−
1326:Ψ
1318:−
1298:−
1294:Ψ
913:However,
893:semigroup
822:Ψ
797:−
793:Ψ
708:That is,
549:)| < 1
531:Solutions
465:, holds.
233:Ψ′(
117:Ψ
82:Ψ
73:∀
48:with one
1740:(1958).
1643:(1884).
1547:(1954).
1505:(1968).
1411:See also
897:splinter
582:Szekeres
564:′(
555:(1884).
545:′(
541:0 < |
455:′(
417:)) = (φ(
288:0 < |
255:′(
1913:2332328
1866:Bibcode
1805:Bibcode
1615:Bibcode
1564:Bibcode
1209:
1197:
1175:
1163:
1121:√
1085:√
1053:√
498:(log Ψ(
385:)) = α(
361:))/log(
345:)) = Φ(
294:, then
44:, is a
1911:
1525:
1515:
1485:
977:) = Ψ(
752:, see
717:) = Ψ(
588:, cf.
568:)| = 0
286:, and
1909:JSTOR
1856:arXiv
1795:arXiv
1648:(PDF)
1605:arXiv
1552:(PDF)
1250:) =
1195:) = −
1161:) = −
1142:) = 2
1032:) = 4
1006:)) =
956:orbit
758:orbit
749:group
745:(its
725:)) ≡
695:orbit
451:)) =
437:Julia
389:) + 1
367:(the
334:Φ = Ψ
276:, if
216:) or
210:) = 0
182:is a
1523:OCLC
1513:ISBN
1483:ISBN
1234:) =
1107:) ∝
856:for
699:flow
697:(or
521:log(
468:The
269:For
212:(or
199:) =
173:(.))
1901:doi
1874:doi
1813:doi
1754:doi
1750:100
1698:doi
1660:doi
1623:doi
1572:doi
1453:doi
1067:= 4
1000:1/2
993:1/2
971:1/2
965:is
551:by
506:any
459:)β(
341:(Φ(
274:= 0
221:= 1
186:of
178:If
164:to
1929::
1907:.
1897:38
1895:.
1872:.
1864:.
1852:43
1850:.
1811:.
1803:.
1791:42
1789:.
1781:;
1770:^
1748:.
1744:.
1724:14
1722:.
1692:.
1682:;
1654:.
1650:.
1621:.
1613:.
1601:83
1599:.
1584:^
1570:.
1560:95
1558:.
1554:.
1543:;
1521:.
1447:.
1430:^
1246:Ψ(
1225:,
1157:Ψ(
1117:nπ
1062:,
1047:Ψ(
985:))
981:Ψ(
958:.
944:→
910:.
738:.
721:Ψ(
705:.
683:→
636:/d
592:.
574:.
502:))
494:)
490:Ψ(
482:Ψ(
443:β(
428:β(
423:.
421:))
409:φ(
407:,
401:))
393:Ψ(
377:α(
375:,
353:α(
347:sy
329:.
306:.
261:.
251:=
225:Ψ(
206:Ψ(
175:.
1915:.
1903::
1880:.
1876::
1868::
1858::
1833:.
1819:.
1815::
1807::
1797::
1762:.
1756::
1704:.
1700::
1694:8
1668:.
1662::
1656:1
1629:.
1625::
1617::
1607::
1578:.
1574::
1566::
1529:.
1491:.
1459:.
1455::
1449:3
1395:.
1389:)
1384:t
1380:2
1373:1
1370:(
1367:x
1364:+
1359:t
1355:2
1350:x
1345:=
1340:)
1335:)
1332:x
1329:(
1321:t
1314:2
1308:(
1301:1
1290:=
1287:)
1284:x
1281:(
1276:t
1272:h
1258:)
1256:x
1252:x
1248:x
1242:)
1240:x
1236:x
1232:x
1230:(
1228:h
1217:.
1213:x
1206:2
1203:/
1200:1
1193:x
1191:(
1189:t
1186:h
1181:)
1179:x
1172:2
1169:/
1166:1
1159:x
1150:)
1148:x
1144:x
1140:x
1138:(
1136:h
1128:)
1124:x
1113:x
1111:(
1109:x
1105:x
1103:(
1101:V
1094:.
1092:)
1088:x
1081:x
1079:(
1076:t
1072:h
1065:s
1060:)
1056:x
1049:x
1040:)
1038:x
1034:x
1030:x
1028:(
1026:h
1014:)
1012:x
1010:(
1008:h
1004:x
1002:(
997:h
995:(
990:h
983:x
979:s
975:x
973:(
968:h
952:)
950:x
948:(
946:h
942:x
937:)
935:x
933:(
931:Ψ
926:)
924:x
922:(
920:h
908:)
906:x
904:(
902:h
891:(
889:)
887:x
885:(
883:h
878:)
876:x
874:(
871:n
867:h
858:t
841:,
836:)
831:)
828:x
825:(
817:t
813:s
807:(
800:1
789:=
786:)
783:x
780:(
775:t
771:h
736:)
734:x
732:(
730:1
727:h
723:x
719:s
715:x
713:(
711:h
691:)
689:x
687:(
685:h
681:x
673:x
671:(
669:h
655:x
648:t
644:h
638:t
633:t
629:h
625:v
618:)
616:x
614:(
612:h
607:s
566:a
562:h
560:|
547:a
543:h
537:a
525:)
523:s
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515:x
513:(
511:k
500:x
496:k
492:x
486:)
484:x
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471:n
463:)
461:x
457:x
453:f
449:x
447:(
445:f
430:x
419:x
415:x
413:(
411:h
399:x
395:x
387:x
383:x
381:(
379:h
365:)
363:s
359:x
355:x
349:)
343:y
339:h
300:Ψ
290:h
284:0
279:h
272:a
259:)
257:a
253:h
249:s
244:s
237:)
235:a
229:)
227:a
219:s
214:∞
208:a
201:a
197:a
195:(
193:h
188:h
180:a
171:h
169:(
167:f
161:f
154:h
150:C
129:.
126:)
123:x
120:(
114:s
111:=
106:)
101:)
98:x
95:(
92:h
87:(
76:x
60:Ψ
55:h
23:.
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