2266:
2226:
2246:
2236:
2256:
1044:. There are many similarities between the complex and the nonarchimedean theories, but also many differences. A striking difference is that in the nonarchimedean setting, the Fatou set is always nonempty, but the Julia set may be empty. This is the reverse of what is true over the complex numbers. Nonarchimedean dynamics has been extended to
474:
1806:
Topics in dynamics and ergodic theory. Survey papers and mini-courses presented at the international conference and US-Ukrainian workshop on dynamical systems and ergodic theory, Katsiveli, Ukraine, August 21–30,
266:
867:
and others concerning subvarieties that contain infinitely many periodic points or that intersect an orbit in infinitely many points. These are dynamical analogues of, respectively, the
1922:
510:
977:
506:
84:
347:
1485:
1428:
Flynn, Eugene V.; Poonen, Bjorn; Schaefer, Edward F. (1997). "Cycles of quadratic polynomials and rational points on a genus-2 curve".
1348:
2249:
798:, whose second iterate is a polynomial. It turns out that this is the only way that an orbit can contain infinitely many integers.
697:
cannot have periodic points of period four, five, or six, although the result for period six is contingent on the validity of the
1949:
1884:
1780:
1739:
1685:
1273:
211:
1231:
1122:
There are many other problems of a number theoretic nature that appear in the setting of dynamical systems, including:
1914:
1826:
2300:
698:
1980:
1236:
1767:. NATO Science Series II: Mathematics, Physics and Chemistry. Vol. 237. Dordrecht: Springer Netherlands.
2290:
2239:
2026:
1197:
number-theoretic iteration problems that are not described by rational maps on varieties, for example, the
987:
that is complete with respect to a nonarchimedean absolute value. Examples of such fields are the field of
1597:
Silverman, Joseph H. (1993). "Integer points, Diophantine approximation, and iteration of rational maps".
2021:
2006:
1942:
1670:. Surveys in Differential Geometry. Vol. 10. Somerville, MA: International Press. pp. 381–430.
2295:
1810:
1546:
Poonen, Bjorn (1998). "The classification of rational preperiodic points of quadratic polynomials over
69:. A fundamental goal is to describe arithmetic properties in terms of underlying geometric structures.
1212:
gives an extensive list of articles and books covering a wide range of arithmetical dynamical topics.
1048:, which is a compact connected space that contains the totally disconnected non-locally compact field
883:. The following conjectures illustrate the general theory in the case that the subvariety is a curve.
2201:
2160:
2039:
1716:. Mathematical Surveys and Monographs. Vol. 159. Providence, RI: American Mathematical Society.
1599:
1554:
1430:
2045:
868:
2265:
1988:
1972:
2229:
2049:
1998:
1935:
624:, and the general Uniform Boundedness Conjecture says that the number of preperiodic points in
130:
The following table describes a rough correspondence between
Diophantine equations, especially
2269:
1869:
1204:
symbolic codings of dynamical systems based on explicit arithmetic expansions of real numbers.
2206:
2135:
1299:
1180:
876:
1804:
Sidorov, Nikita (2003). "Arithmetic dynamics". In
Bezuglyi, Sergey; Kolyada, Sergiy (eds.).
2196:
2031:
1836:
1790:
1749:
1695:
1620:
1575:
1524:
1504:
1461:
1414:
1371:
1328:
1283:
76:
2259:
1852:
8:
2165:
2069:
2063:
2055:
2016:
1918:
1226:
1221:
566:
43:
2255:
1666:
Zhang, Shou-Wu (2006). "Distributions in algebraic dynamics". In Yau, Shing Tung (ed.).
1508:
2211:
2155:
2059:
1976:
1840:
1717:
1579:
1528:
1494:
1465:
1439:
1316:
1111:
2125:
1880:
1822:
1776:
1735:
1681:
1583:
1483:
Stoll, Michael (2008). "Rational 6-cycles under iteration of quadratic polynomials".
1269:
533:
131:
66:
35:
23:
1844:
1532:
1469:
2130:
2115:
1848:
1814:
1768:
1727:
1671:
1608:
1563:
1512:
1449:
1400:
1391:
1357:
1308:
1261:
1176:
1017:
and the standard definition of equicontinuity leads to the usual definition of the
31:
1612:
1453:
727:
The orbit of a rational map may contain infinitely many integers. For example, if
2144:
2120:
2035:
1876:
1832:
1818:
1786:
1745:
1691:
1616:
1571:
1520:
1457:
1410:
1367:
1324:
1279:
1198:
1191:
1045:
2181:
2100:
1984:
1676:
1297:
Northcott, Douglas
Geoffrey (1950). "Periodic points on an algebraic variety".
872:
784:-th entry in the orbit is an integer. An example of this phenomenon is the map
719:
cannot have rational periodic points of any period strictly larger than three.
656:
The
Uniform Boundedness Conjecture is not known even for quadratic polynomials
161:
87:, is an analogue of complex dynamics in which one replaces the complex numbers
51:
1896:
1772:
1516:
1362:
1343:
1265:
2284:
2140:
1992:
1958:
1709:
1088:-adic completions. Another natural generalization is to replace self-maps of
880:
824:
be a rational function of degree at least two, and assume that no iterate of
469:{\displaystyle O_{F}(P)=\left\{P,F(P),F^{(2)}(P),F^{(3)}(P),\cdots \right\}.}
55:
47:
39:
27:
1904:
1405:
1386:
1209:
2186:
2110:
2010:
1170:
1156:
1134:
1127:
1117:
864:
702:
46:. Arithmetic dynamics is the study of the number-theoretic properties of
2191:
2150:
2002:
1731:
1567:
1320:
62:
1909:
1722:
1444:
1163:
1022:
1018:
124:
120:
1879:, Boris Hasselblatt, A. B. Katok, Cambridge University Press, 2003,
1312:
1260:. Graduate Texts in Mathematics. Vol. 241. New York: Springer.
500:
858:
526:
be a rational function of degree at least two with coefficients in
1877:
A first course in dynamics: with a panorama of recent developments
1499:
1067:
There are natural generalizations of arithmetic dynamics in which
170:
1656:
is a polynomial, then already the second iterate is a polynomial.
914:
be an irreducible algebraic curve. Suppose that there is a point
605:
be a morphism of degree at least two defined over a number field
1927:
2090:
1714:
Potential theory and dynamics on the
Berkovich projective line
1809:. Lond. Math. Soc. Lect. Note Ser. Vol. 310. Cambridge:
1387:"Arithmetic properties of periodic points of quadratic maps"
584:
is bounded by a constant that depends only on the degree of
983:
is the study of classical dynamical questions over a field
1870:
Lecture Notes on
Arithmetic Dynamics Arizona Winter School
1668:
Differential
Geometry: A Tribute to Professor S.-S. Chern
1118:
Other areas in which number theory and dynamics interact
1921:'s "The Arithmetic of Dynamical Systems", reviewed by
1910:
Analysis and dynamics on the
Berkovich projective line
780:
is a polynomial with integer coefficients, then every
742:
is an integer, then it is clear that the entire orbit
61:, or algebraic points under repeated application of a
22:
is a field that amalgamates two areas of mathematics,
2105:
2095:
563:
uniform boundedness conjecture for preperiodic points
350:
261:{\displaystyle F^{(n)}=F\circ F\circ \cdots \circ F.}
214:
79:
in the setting of discrete dynamical systems, while
1427:
1765:Equidistribution in number theory, an introduction
511:Uniform boundedness conjecture for rational points
468:
260:
738:is a polynomial with integer coefficients and if
507:Uniform boundedness conjecture for torsion points
501:Number theoretic properties of preperiodic points
2282:
1762:
1344:"Rational periodic points of rational functions"
859:Dynamically defined points lying on subvarieties
1763:Granville, Andrew; Rudnick, Zeév, eds. (2007).
1341:
171:Definitions and notation from discrete dynamics
1342:Morton, Patrick; Silverman, Joseph H. (1994).
928:contains infinitely many points in the orbit
569:says that the number of preperiodic points of
1943:
1169:arithmetic properties of dynamically defined
613:has only finitely many preperiodic points in
550:has only finitely many preperiodic points in
158:Points of finite order on an abelian variety
1002:and the completion of its algebraic closure
1708:
722:
2245:
2235:
1950:
1936:
1486:LMS Journal of Computation and Mathematics
1349:International Mathematics Research Notices
1721:
1675:
1596:
1498:
1443:
1404:
1361:
1296:
1255:
150:Rational and integer points on a variety
1872:, March 13–17, 2010, Joseph H. Silverman
1084:are replaced by number fields and their
950:in the sense that there is some iterate
483:is preperiodic if and only if its orbit
153:Rational and integer points in an orbit
1803:
1242:
699:conjecture of Birch and Swinnerton-Dyer
75:is the study of analogues of classical
2283:
1545:
1384:
1931:
1665:
1482:
863:There are general conjectures due to
854:contains only finitely many integers.
119:and studies chaotic behavior and the
30:. Part of the inspiration comes from
758:consists of integers. Similarly, if
546:-rational preperiodic points, i.e.,
1898:The Arithmetic of Dynamical Systems
1633:An elementary theorem says that if
1258:The Arithmetic of Dynamical Systems
1232:Combinatorics and dynamical systems
769:is a rational map and some iterate
13:
1863:
1210:Arithmetic Dynamics Reference List
1062:
981:-adic (or nonarchimedean) dynamics
968:
635:may be bounded solely in terms of
14:
2312:
1957:
1890:
85:p-adic or nonarchimedean dynamics
2264:
2254:
2244:
2234:
2225:
2224:
1905:Arithmetic dynamics bibliography
683:. It is known in this case that
609:. Northcott's theorem says that
1797:
1756:
1702:
1659:
2003:analytic theory of L-functions
1981:non-abelian class field theory
1627:
1590:
1539:
1476:
1421:
1378:
1335:
1290:
1249:
1237:Arboreal Galois representation
869:Manin–Mumford conjecture
449:
443:
438:
432:
421:
415:
410:
404:
393:
387:
367:
361:
226:
220:
1:
1613:10.1215/S0012-7094-93-07129-3
1454:10.1215/S0012-7094-97-09011-6
1256:Silverman, Joseph H. (2007).
2027:Transcendental number theory
1819:10.1017/CBO9780511546716.010
7:
2250:List of recreational topics
2022:Computational number theory
2007:probabilistic number theory
1215:
1100:with self-maps (morphisms)
10:
2317:
1811:Cambridge University Press
1677:10.4310/SDG.2005.v10.n1.a9
677:over the rational numbers
504:
197:to itself. The iterate of
73:Global arithmetic dynamics
2220:
2202:Diophantine approximation
2174:
2161:Chinese remainder theorem
2083:
1965:
1773:10.1007/978-1-4020-5404-4
1712:; Baker, Matthew (2010).
1600:Duke Mathematical Journal
1555:Mathematische Zeitschrift
1552:: a refined conjecture".
1517:10.1112/S1461157000000644
1431:Duke Mathematical Journal
1363:10.1155/S1073792894000127
1266:10.1007/978-0-387-69904-2
134:, and dynamical systems:
81:local arithmetic dynamics
2046:Arithmetic combinatorics
1385:Morton, Patrick (1992).
1151:iteration of formal and
723:Integer points in orbits
2301:Algebraic number theory
2017:Geometric number theory
1973:Algebraic number theory
1652:and if some iterate of
1406:10.4064/aa-62-4-343-372
877:Mordell–Lang conjecture
540:has only finitely many
164:of a rational function
2136:Transcendental numbers
2050:additive number theory
1999:Analytic number theory
904:be a morphism and let
565:of Patrick Morton and
470:
262:
2207:Irrationality measure
2197:Diophantine equations
2040:Hodge–Arakelov theory
1300:Annals of Mathematics
828:is a polynomial. Let
705:has conjectured that
471:
324:is periodic for some
263:
142:Diophantine equations
2166:Arithmetic functions
2032:Diophantine geometry
1813:. pp. 145–189.
1243:Notes and references
1112:projective varieties
643:, and the degree of
591:More generally, let
348:
212:
97:-adic field such as
77:diophantine geometry
38:of self-maps of the
16:Field of mathematics
2291:Arithmetic dynamics
2212:Continued fractions
2075:Arithmetic dynamics
2070:Arithmetic topology
2064:P-adic Hodge theory
2056:Arithmetic geometry
1989:Iwasawa–Tate theory
1923:Robert L. Benedetto
1919:Joseph H. Silverman
1509:2008arXiv0803.2836S
1227:Arithmetic topology
1222:Arithmetic geometry
1110:of other affine or
138:
44:algebraic varieties
34:, the study of the
20:Arithmetic dynamics
2156:Modular arithmetic
2126:Irrational numbers
2060:anabelian geometry
1977:class field theory
1568:10.1007/PL00004405
1025:of a rational map
466:
258:
162:Preperiodic points
145:Dynamical systems
137:
2296:Dynamical systems
2278:
2277:
2175:Advanced concepts
2131:Algebraic numbers
2116:Composite numbers
1885:978-0-521-58750-1
1782:978-1-4020-5403-7
1741:978-0-8218-4924-8
1687:978-1-57146-116-2
1275:978-0-387-69903-5
838:. Then the orbit
534:Douglas Northcott
205:times is denoted
179:be a set and let
168:
167:
132:abelian varieties
67:rational function
42:or other complex
24:dynamical systems
2308:
2268:
2258:
2248:
2247:
2238:
2237:
2228:
2227:
2121:Rational numbers
1952:
1945:
1938:
1929:
1928:
1857:
1856:
1801:
1795:
1794:
1760:
1754:
1753:
1732:10.1090/surv/159
1725:
1706:
1700:
1699:
1679:
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1655:
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1625:
1624:
1594:
1588:
1587:
1551:
1543:
1537:
1536:
1502:
1480:
1474:
1473:
1447:
1425:
1419:
1418:
1408:
1392:Acta Arithmetica
1382:
1376:
1375:
1365:
1339:
1333:
1332:
1294:
1288:
1287:
1253:
1192:Drinfeld modules
1186:
1183:, especially on
1177:equidistribution
1154:
1147:
1109:
1099:
1093:
1087:
1083:
1072:
1058:
1043:
1016:
1013:. The metric on
1012:
1001:
991:-adic rationals
990:
986:
980:
963:
959:
955:
949:
946:is periodic for
945:
941:
927:
923:
913:
903:
853:
837:
827:
823:
797:
783:
779:
768:
757:
741:
737:
718:
696:
682:
676:
652:
646:
642:
639:, the degree of
638:
634:
623:
612:
608:
604:
587:
583:
572:
567:Joseph Silverman
560:
549:
545:
539:
531:
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200:
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139:
136:
118:
107:
96:
92:
58:
32:complex dynamics
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2280:
2279:
2274:
2216:
2182:Quadratic forms
2170:
2145:P-adic analysis
2101:Natural numbers
2079:
2036:Arakelov theory
1961:
1956:
1893:
1866:
1864:Further reading
1861:
1860:
1829:
1802:
1798:
1783:
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1336:
1313:10.2307/1969504
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1291:
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1245:
1218:
1199:Collatz problem
1184:
1152:
1138:
1135:function fields
1120:
1101:
1095:
1089:
1085:
1082:
1074:
1068:
1065:
1063:Generalizations
1057:
1049:
1046:Berkovich space
1026:
1014:
1011:
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1000:
992:
988:
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978:
974:
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551:
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541:
537:
532:. A theorem of
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117:
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88:
56:
17:
12:
11:
5:
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2304:
2303:
2298:
2293:
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2275:
2273:
2272:
2262:
2252:
2242:
2240:List of topics
2232:
2221:
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2214:
2209:
2204:
2199:
2194:
2189:
2184:
2178:
2176:
2172:
2171:
2169:
2168:
2163:
2158:
2153:
2148:
2141:P-adic numbers
2138:
2133:
2128:
2123:
2118:
2113:
2108:
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2098:
2093:
2087:
2085:
2081:
2080:
2078:
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2067:
2053:
2043:
2029:
2024:
2019:
2014:
1996:
1985:Iwasawa theory
1969:
1967:
1963:
1962:
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1940:
1932:
1926:
1925:
1912:
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1902:
1892:
1891:External links
1889:
1888:
1887:
1875:Chapter 15 of
1873:
1865:
1862:
1859:
1858:
1827:
1796:
1781:
1755:
1740:
1710:Rumely, Robert
1701:
1686:
1658:
1626:
1607:(3): 793–829.
1589:
1538:
1475:
1438:(3): 435–463.
1420:
1399:(4): 343–372.
1377:
1334:
1307:(1): 167–177.
1289:
1274:
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1240:
1239:
1234:
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1217:
1214:
1206:
1205:
1202:
1195:
1188:
1179:and invariant
1174:
1167:
1160:
1149:
1133:dynamics over
1131:
1126:dynamics over
1119:
1116:
1078:
1064:
1061:
1053:
1007:
996:
973:
972:-adic dynamics
967:
966:
965:
932:
873:Michel Raynaud
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83:, also called
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2187:Modular forms
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2112:
2111:Prime numbers
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1993:Kummer theory
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1959:Number theory
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1850:
1846:
1842:
1838:
1834:
1830:
1828:0-521-53365-1
1824:
1820:
1816:
1812:
1808:
1800:
1792:
1788:
1784:
1778:
1774:
1770:
1766:
1759:
1751:
1747:
1743:
1737:
1733:
1729:
1724:
1719:
1715:
1711:
1705:
1697:
1693:
1689:
1683:
1678:
1673:
1669:
1662:
1649:
1645:
1641:
1637:
1630:
1622:
1618:
1614:
1610:
1606:
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1593:
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1577:
1573:
1569:
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1518:
1514:
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1501:
1496:
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1471:
1467:
1463:
1459:
1455:
1451:
1446:
1441:
1437:
1433:
1432:
1424:
1416:
1412:
1407:
1402:
1398:
1394:
1393:
1388:
1381:
1373:
1369:
1364:
1359:
1356:(2): 97–110.
1355:
1351:
1350:
1345:
1338:
1330:
1326:
1322:
1318:
1314:
1310:
1306:
1302:
1301:
1293:
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1281:
1277:
1271:
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1263:
1259:
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1238:
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1233:
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1225:
1223:
1220:
1219:
1213:
1211:
1203:
1200:
1196:
1193:
1189:
1187:-adic spaces.
1182:
1178:
1175:
1172:
1171:moduli spaces
1168:
1165:
1161:
1158:
1150:
1145:
1141:
1136:
1132:
1129:
1128:finite fields
1125:
1124:
1123:
1115:
1113:
1108:
1104:
1098:
1092:
1081:
1077:
1071:
1060:
1056:
1052:
1047:
1041:
1037:
1033:
1029:
1024:
1020:
1010:
1006:
999:
995:
982:
976:The field of
971:
954:
939:
935:
922:
918:
912:
908:
902:
898:
894:
889:
886:
885:
884:
882:
881:Gerd Faltings
878:
874:
870:
866:
851:
846:
842:
836:
832:
821:
817:
813:
809:
804:
801:
800:
799:
796:
792:
788:
777:
773:
766:
762:
755:
750:
746:
735:
731:
720:
716:
712:
704:
700:
694:
690:
681:
675:
671:
667:
663:
654:
651:
632:
628:
621:
617:
603:
599:
595:
589:
581:
577:
568:
564:
558:
554:
544:
535:
530:
523:
519:
512:
508:
498:
494:
490:
463:
459:
455:
452:
446:
435:
428:
424:
418:
407:
400:
396:
390:
384:
381:
378:
374:
370:
364:
356:
352:
344:
343:
342:
337:
332:
328:
321:
317:
312:
309:The point is
307:
303:
297:
293:
289:
284:
279:
275:
255:
252:
249:
246:
243:
240:
237:
234:
231:
223:
216:
208:
207:
206:
191:
187:
183:
163:
160:
157:
156:
152:
149:
148:
144:
141:
140:
135:
133:
128:
126:
122:
116:
112:
105:
101:
91:
86:
82:
78:
74:
70:
68:
64:
60:
53:
49:
45:
41:
40:complex plane
37:
33:
29:
28:number theory
25:
21:
2084:Key concepts
2074:
2011:sieve theory
1897:
1805:
1799:
1764:
1758:
1723:math/0407433
1713:
1704:
1667:
1661:
1647:
1643:
1639:
1635:
1629:
1604:
1598:
1592:
1562:(1): 11–29.
1559:
1553:
1548:
1541:
1490:
1484:
1478:
1445:math/9508211
1435:
1429:
1423:
1396:
1390:
1380:
1353:
1347:
1337:
1304:
1298:
1292:
1257:
1251:
1207:
1190:dynamics on
1162:dynamics on
1157:power series
1143:
1139:
1121:
1106:
1102:
1096:
1090:
1079:
1075:
1069:
1066:
1054:
1050:
1039:
1035:
1031:
1027:
1008:
1004:
997:
993:
975:
969:
952:
937:
930:
920:
916:
910:
906:
900:
896:
892:
887:
879:, proven by
871:, proven by
865:Shouwu Zhang
862:
849:
844:
840:
834:
830:
819:
815:
811:
807:
802:
794:
790:
786:
775:
771:
764:
760:
753:
748:
744:
733:
729:
726:
714:
707:
703:Bjorn Poonen
692:
685:
679:
673:
669:
665:
658:
655:
649:
630:
626:
619:
615:
601:
597:
593:
590:
579:
575:
562:
556:
552:
542:
528:
521:
517:
514:
492:
485:
478:
335:
333:
326:
319:
315:
310:
308:
301:
295:
291:
287:
282:
277:
273:
270:
201:with itself
189:
185:
181:
174:
129:
114:
110:
103:
99:
89:
80:
72:
71:
19:
18:
2270:Wikiversity
2192:L-functions
1915:Book review
1493:: 367–380.
888:Conjecture.
497:is finite.
341:is the set
311:preperiodic
2285:Categories
2151:Arithmetic
1853:1051.37007
1164:Lie groups
1023:Julia sets
964:to itself.
960:that maps
924:such that
875:, and the
536:says that
505:See also:
125:Julia sets
63:polynomial
1900:home page
1584:118160396
1500:0803.2836
456:⋯
299:for some
250:∘
247:⋯
244:∘
238:∘
36:iteration
2260:Wikibook
2230:Category
1845:15482676
1533:14082110
1470:15169450
1216:See also
1181:measures
1137:such as
942:. Then
895: :
803:Theorem.
596: :
336:orbit of
283:periodic
271:A point
184: :
52:rational
2091:Numbers
1837:2052279
1791:2290490
1750:2599526
1696:2408228
1621:1240603
1576:1617987
1525:2465796
1505:Bibcode
1462:1480542
1415:1199627
1372:1264933
1329:0034607
1321:1969504
1284:2316407
48:integer
1966:Fields
1883:
1851:
1843:
1835:
1825:
1789:
1779:
1748:
1738:
1694:
1684:
1619:
1582:
1574:
1531:
1523:
1468:
1460:
1413:
1370:
1327:
1319:
1282:
1272:
1155:-adic
561:. The
2106:Unity
1841:S2CID
1718:arXiv
1580:S2CID
1529:S2CID
1495:arXiv
1466:S2CID
1440:arXiv
1317:JSTOR
1019:Fatou
647:over
479:Thus
121:Fatou
93:by a
59:-adic
1881:ISBN
1823:ISBN
1807:2000
1777:ISBN
1736:ISBN
1682:ISBN
1642:) ∈
1354:1994
1270:ISBN
1208:The
1073:and
1034:) ∈
1021:and
890:Let
814:) ∈
805:Let
793:) =
668:) =
515:Let
509:and
294:) =
175:Let
123:and
26:and
1917:of
1849:Zbl
1815:doi
1769:doi
1728:doi
1672:doi
1609:doi
1564:doi
1560:228
1513:doi
1450:doi
1401:doi
1358:doi
1309:doi
1262:doi
1094:or
956:of
573:in
329:≥ 1
313:if
304:≥ 1
285:if
281:is
108:or
65:or
2287::
2062:,
2038:,
2009:,
2005:,
1991:,
1987:,
1983:,
1979:,
1847:.
1839:.
1833:MR
1831:.
1821:.
1787:MR
1785:.
1775:.
1746:MR
1744:.
1734:.
1726:.
1692:MR
1690:.
1680:.
1617:MR
1615:.
1605:71
1603:.
1578:.
1572:MR
1570:.
1558:.
1527:.
1521:MR
1519:.
1511:.
1503:.
1491:11
1489:.
1464:.
1458:MR
1456:.
1448:.
1436:90
1434:.
1411:MR
1409:.
1397:62
1395:.
1389:.
1368:MR
1366:.
1352:.
1346:.
1325:MR
1323:.
1315:.
1305:51
1303:.
1280:MR
1278:.
1268:.
1114:.
1105:→
1059:.
919:∈
909:⊂
899:→
833:∈
701:.
672:+
653:.
600:→
588:.
331:.
306:.
276:∈
188:→
127:.
54:,
50:,
2147:)
2143:(
2096:0
2066:)
2058:(
2052:)
2048:(
2042:)
2034:(
2013:)
2001:(
1995:)
1975:(
1951:e
1944:t
1937:v
1855:.
1817::
1793:.
1771::
1752:.
1730::
1720::
1698:.
1674::
1654:F
1650:)
1648:x
1646:(
1644:C
1640:x
1638:(
1636:F
1623:.
1611::
1586:.
1566::
1549:Q
1535:.
1515::
1507::
1497::
1472:.
1452::
1442::
1417:.
1403::
1374:.
1360::
1331:.
1311::
1286:.
1264::
1201:.
1194:.
1185:p
1173:.
1166:.
1159:.
1153:p
1148:.
1146:)
1144:x
1142:(
1140:C
1130:.
1107:V
1103:V
1097:P
1091:P
1086:p
1080:p
1076:Q
1070:Q
1055:p
1051:C
1042:)
1040:x
1038:(
1036:K
1032:x
1030:(
1028:F
1015:K
1009:p
1005:C
998:p
994:Q
989:p
985:K
979:p
970:p
962:C
958:F
953:F
948:F
944:C
940:)
938:P
936:(
933:F
931:O
926:C
921:P
917:P
911:P
907:C
901:P
897:P
893:F
852:)
850:a
848:(
845:F
841:O
835:Q
831:a
826:F
822:)
820:x
818:(
816:Q
812:x
810:(
808:F
795:x
791:x
789:(
787:F
782:n
778:)
776:x
774:(
772:F
767:)
765:x
763:(
761:F
756:)
754:a
752:(
749:F
745:O
740:a
736:)
734:x
732:(
730:F
717:)
715:x
713:(
710:c
708:F
695:)
693:x
691:(
688:c
686:F
680:Q
674:c
670:x
666:x
664:(
661:c
659:F
650:Q
645:K
641:F
637:N
633:)
631:K
629:(
627:P
622:)
620:K
618:(
616:P
611:F
607:K
602:P
598:P
594:F
586:F
582:)
580:Q
578:(
576:P
571:F
559:)
557:Q
555:(
553:P
548:F
543:Q
538:F
529:Q
524:)
522:x
520:(
518:F
495:)
493:P
491:(
488:F
486:O
481:P
464:.
460:}
453:,
450:)
447:P
444:(
439:)
436:3
433:(
429:F
425:,
422:)
419:P
416:(
411:)
408:2
405:(
401:F
397:,
394:)
391:P
388:(
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382:,
379:P
375:{
371:=
368:)
365:P
362:(
357:F
353:O
339:P
327:k
322:)
320:P
318:(
316:F
302:n
296:P
292:P
290:(
288:F
278:S
274:P
256:.
253:F
241:F
235:F
232:=
227:)
224:n
221:(
217:F
203:n
199:F
195:S
190:S
186:S
182:F
177:S
115:p
111:C
104:p
100:Q
95:p
90:C
57:p
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