1764:
82:
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1410:
could be a necessary and sufficient condition for the FPP to hold. The problem was open for 20 years until the conjecture was disproved by
Kinoshita, who found an example of a compact contractible space without the FPP.
1569:(poset) to itself is the fixed point which is less than each other fixed point, according to the order of the poset. A function need not have a least fixed point, but if it does then the least fixed point is unique.
714:
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has the FPP. Compactness alone does not imply the FPP, and convexity is not even a topological property, so it makes sense to ask how to topologically characterize the FPP. In 1932
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2375:"P. Cousot & R. Cousot, Abstract interpretation: A unified lattice model for static analysis of programs by construction or approximation of fixpoints"
1996:
1781:
1738:, fixed-point logics are extensions of classical predicate logic that have been introduced to express recursion. Their development has been motivated by
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2219:
1535:
fix point." A fixed point is a point that is both a prefixpoint and a postfixpoint. Prefixpoints and postfixpoints have applications in
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1803:
1138:
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2122:
1908:
180:, a fixed point is an element that is mapped to itself by the function. Any set of fixed points of a transformation is also an
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that coincides with its least prefixpoint (and similarly its greatest fixed point coincides with its greatest postfixpoint).
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173:
31:
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1507:). The opposite usage occasionally appears. Malkis justifies the definition presented here as follows: "since
1986:
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821:
A fixed-point theorem is a result saying that at least one fixed point exists, under some general condition.
2055:
2003:
1970:
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1912:
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2294:"Renormalization Group and Critical Phenomena. I. Renormalization Group and the Kadanoff Scaling Picture"
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2159:"Multithreaded-Cartesian Abstract Interpretation of Multithreaded Recursive Programs Is Polynomial"
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998:
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2335:"Renormalization Group and Critical Phenomena. II. Phase-Space Cell Analysis of Critical Behavior"
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are fundamental concepts that can be described in terms of fixed points. Some examples follow.
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that returns a fixed point of its argument function, if one exists. Formally, if the function
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to itself must have a fixed point, but it doesn't describe how to find the fixed point.
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is a method of computing fixed points of a function. Specifically, given a function
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1713:{\displaystyle \operatorname {\mathsf {fix}} f=f(\operatorname {\mathsf {fix}} f).}
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1911:. They are also the core concept used by the generic program analysis method
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cannot have any fixed point if its domain is disjoint from its codomain. If
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828:(1922) gives a general criterion guaranteeing that, if it is satisfied,
17:
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1788: in this section. Unsourced material may be challenged and removed.
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1128:
879:
1395:
2107:
803:
The notions of attracting fixed points, repelling fixed points, and
2100:
Proceedings, 18th IEEE Symposium on
Foundations of Computer Science
1952:
is the fixed point of the one step transition probability function.
1903:
use fixed point computations for program analysis, for example in
245:
1423:, the notion and terminology of fixed points is generalized to a
1863:
for his seminal paper that won him the Nobel prize in economics.
2093:"The Category-Theoretic Solution of Recursive Domain Equations"
1757:
2051:"On Some Contractible Continua without Fixed Point Property"
2102:. SIAM Journal of Computing (volume 11). pp. 761–783.
258:
corresponds to an intersection of the curve with the line
2123:"Constructive Versions of Tarski's Fixed Point Theorems"
30:
Fixed points in mathematics are not to be confused with
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is continuous, then one can prove that the obtained
1890:, and to the mathematical explanation of the term "
807:are defined with respect to fixed-point iteration.
27:
Element mapped to itself by a mathematical function
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357:Not all functions have fixed points: for example,
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709:{\displaystyle x_{0},f(x_{0}),f(f(x_{0})),\dots }
555:{\displaystyle x_{n+1}=f(x_{n}),\,n=0,1,2,\dots }
1937:values of all web pages is the fixed point of a
1926:allows definition of recursive functions in the
172:, is a value that does not change under a given
152:(shown in red) has the fixed points 0, 1, and 2.
2091:Smyth, Michael B.; Plotkin, Gordon D. (1982).
1955:Fixed points are used to finding formulas for
2392:An Elegant Solution for Drawing a Fixed Point
8:
1197:{\displaystyle R^{f}=\{r\in R\mid f(r)=r\}.}
1188:
1155:
1074:{\displaystyle G^{f}=\{g\in G\mid f(g)=g\}.}
1065:
1032:
1997:Infinite compositions of analytic functions
427:with the same domain and codomain, a point
244:, it corresponds, in graphical terms, to a
1886:'s Nobel prize-winning work inventing the
1210:, the set of the fixed points of a set of
2358:
2317:
2067:
1804:Learn how and when to remove this message
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1406:asked whether compactness together with
619:{\displaystyle x_{0},x_{1},x_{2},\dots }
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2026:Fixed Point Theory and Its Applications
2016:
1987:Fixed points of a Möbius transformation
832:will always converge to a fixed point.
2121:Patrick Cousot; Radhia Cousot (1979).
1693:
1690:
1687:
1665:
1662:
1659:
1628:
1625:
1622:
1427:. Let ≤ be a partial order over a set
2168:. Lecture Notes in Computer Science.
7:
1786:adding citations to reliable sources
145:{\displaystyle f(x)=x^{3}-3x^{2}+3x}
1907:, which is often required for code
1647:has one or more fixed points, then
1376:. The FPP is also preserved by any
870:give a way to count fixed points.
25:
2028:. American Mathematical Society.
1948:The stationary distribution of a
2236:Lectures on the Modal μ-calculus
2217:Lectures on the Modal μ-calculus
1762:
1610:, a fixed-point combinator is a
1515:the inequality sign in the term
1226:Topological fixed point property
334:{\displaystyle f(x)=x^{2}-3x+4,}
1851:is a fixed point of the game's
1773:needs additional citations for
1636:{\displaystyle {\mathsf {fix}}}
1372:, i.e., it is preserved by any
921:is said to be a fixed point of
474:, the fixed-point iteration is
196:is a fixed point of a function
2131:Pacific Journal of Mathematics
1816:In many fields, equilibria or
1704:
1682:
1415:Fixed points of partial orders
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1340:
1294:{\displaystyle f\colon X\to X}
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2280:Synthetic Projective Geometry
2049:Kinoshita, Shin'ichi (1953).
1740:descriptive complexity theory
1222:of the set of automorphisms.
874:Fixed point of a group action
860:Lefschetz fixed-point theorem
2178:10.1007/978-3-319-24537-9_11
1861:Kakutani fixed-point theorem
1853:best response correspondence
1537:theoretical computer science
372:has no fixed points because
2333:Wilson, Kenneth G. (1971).
2292:Wilson, Kenneth G. (1971).
2263:University of Toronto Press
1872:theory of phase transitions
1385:Brouwer fixed-point theorem
864:Nielsen fixed-point theorem
837:Brouwer fixed-point theorem
341:then 2 is a fixed point of
32:other uses of "fixed point"
2428:
2407:Fixed points (mathematics)
2157:Malkis, Alexander (2015).
2024:Brown, R. F., ed. (1988).
1742:and their relationship to
1727:
1595:
1546:
1229:
950:{\displaystyle g\cdot x=x}
826:Banach fixed-point theorem
814:
392:
29:
1455:, sometimes shortened to
188:Fixed point of a function
1882:fixed point has led to
1870:, more precisely in the
1744:database query languages
564:which gives rise to the
164:(sometimes shortened to
2360:10.1103/PhysRevB.4.3184
2319:10.1103/PhysRevB.4.3174
2241:March 21, 2012, at the
2222:March 21, 2012, at the
2004:Invariant (mathematics)
1971:Cycles and fixed points
1928:untyped lambda calculus
1913:abstract interpretation
1572:One way to express the
1131:of the fixed points of
252:, and each fixed-point
71:{\displaystyle f'(x)=0}
2259:Non-Euclidean Geometry
2144:10.2140/pjm.1979.82.43
1924:fixed-point combinator
1714:
1637:
1598:Fixed point combinator
1592:Fixed-point combinator
1574:Knaster–Tarski theorem
1359:
1358:{\displaystyle f(x)=x}
1324:
1323:{\displaystyle x\in X}
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906:{\displaystyle \cdot }
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2166:Reachability Problems
2069:10.4064/fm-40-1-96-98
1939:linear transformation
1888:renormalization group
1828:, a fixed point of a
1715:
1638:
1612:higher-order function
1567:partially ordered set
1370:topological invariant
1360:
1325:
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1254:
1199:
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1106:{\displaystyle R^{f}}
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983:{\displaystyle G^{f}}
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839:(1911) says that any
830:fixed-point iteration
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621:
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447:{\displaystyle x_{0}}
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405:fixed-point iteration
395:Fixed-point iteration
389:Fixed point iteration
385:for any real number.
336:
147:
84:
73:
1898:Programming language
1782:improve this article
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1617:
1334:
1308:
1273:
1261:fixed point property
1259:is said to have the
1243:
1232:Fixed-point property
1139:
1090:
1016:
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962:fixed-point subgroup
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817:Fixed-point theorems
811:Fixed-point theorems
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780:is a fixed point of
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204:belongs to both the
176:. Specifically, for
168:), also known as an
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2351:1971PhRvB...4.3184W
2310:1971PhRvB...4.3174W
1892:critical phenomenon
1826:projective geometry
1746:, in particular to
1443:be a function over
1265:continuous function
1212:field automorphisms
1085:fixed-point subring
841:continuous function
2234:Yde Venema (2008)
2215:Yde Venema (2008)
1957:iterated functions
1945:'s link structure.
1905:data-flow analysis
1832:has been called a
1736:mathematical logic
1724:Fixed-point logics
1710:
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868:algebraic topology
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401:numerical analysis
378:is never equal to
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277:is defined on the
240:is defined on the
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2339:Physical Review B
2298:Physical Review B
2255:Coxeter, H. S. M.
2187:978-3-319-24536-2
1941:derived from the
1814:
1813:
1806:
1730:Fixed-point logic
1604:combinatory logic
1586:least fixed point
1578:monotone function
1576:is to say that a
1559:least fixed point
1549:Least fixed point
1543:Least fixed point
1483:. Analogously, a
1383:According to the
1263:(FPP) if for any
1252:{\displaystyle X}
1238:topological space
824:For example, the
793:{\displaystyle f}
773:{\displaystyle x}
753:{\displaystyle f}
733:{\displaystyle x}
628:iterated function
467:{\displaystyle f}
454:in the domain of
420:{\displaystyle f}
230:. In particular,
36:stationary points
16:(Redirected from
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1582:complete lattice
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1973:of permutations
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1485:postfixed point
1453:pre-fixed point
1417:
1408:contractibility
1400:Euclidean space
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1083:Similarly, the
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853:Euclidean space
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2386:External links
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2205:on 2022-08-10.
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1449:prefixed point
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302:
299:
296:
293:
290:
189:
186:
174:transformation
141:
138:
135:
130:
126:
122:
119:
114:
110:
106:
103:
100:
97:
94:
67:
64:
61:
58:
55:
51:
48:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2424:
2413:
2410:
2408:
2405:
2404:
2402:
2393:
2390:
2389:
2385:
2376:
2370:
2367:
2361:
2356:
2352:
2348:
2344:
2340:
2336:
2329:
2326:
2320:
2315:
2311:
2307:
2303:
2299:
2295:
2288:
2285:
2281:
2277:
2276:G. B. Halsted
2272:
2269:
2265:. p. 36.
2264:
2260:
2256:
2250:
2247:
2244:
2240:
2237:
2231:
2228:
2225:
2221:
2218:
2212:
2209:
2201:
2197:
2193:
2189:
2183:
2179:
2175:
2171:
2167:
2160:
2153:
2150:
2145:
2141:
2137:
2133:
2132:
2124:
2117:
2114:
2109:
2105:
2101:
2094:
2087:
2084:
2079:
2075:
2070:
2065:
2061:
2058:
2057:
2052:
2045:
2042:
2037:
2035:0-8218-5080-6
2031:
2027:
2020:
2017:
2010:
2005:
2002:
2001:
1998:
1995:
1993:
1990:
1988:
1985:
1983:
1980:
1978:
1975:
1972:
1969:
1968:
1963:
1958:
1954:
1951:
1947:
1944:
1940:
1936:
1932:
1929:
1925:
1921:
1917:
1914:
1910:
1906:
1902:
1899:
1896:
1893:
1889:
1885:
1881:
1877:
1876:linearization
1873:
1869:
1865:
1862:
1858:
1854:
1850:
1846:
1842:
1838:
1835:
1831:
1827:
1823:
1822:
1821:
1819:
1808:
1805:
1797:
1787:
1783:
1777:
1776:
1771:This section
1769:
1765:
1760:
1759:
1753:
1751:
1749:
1745:
1741:
1737:
1731:
1723:
1707:
1701:
1698:
1679:
1676:
1673:
1670:
1650:
1649:
1648:
1646:
1613:
1609:
1605:
1599:
1591:
1589:
1587:
1583:
1579:
1575:
1570:
1568:
1564:
1560:
1556:
1550:
1542:
1540:
1538:
1532:
1530:
1526:
1522:
1518:
1512:
1510:
1506:
1502:
1498:
1494:
1490:
1486:
1482:
1478:
1474:
1470:
1466:
1462:
1458:
1454:
1450:
1446:
1442:
1438:
1434:
1430:
1426:
1425:partial order
1422:
1421:domain theory
1414:
1412:
1409:
1405:
1401:
1397:
1394:
1390:
1386:
1381:
1379:
1375:
1374:homeomorphism
1371:
1368:The FPP is a
1366:
1352:
1349:
1343:
1337:
1317:
1314:
1311:
1304:there exists
1288:
1282:
1279:
1276:
1269:
1268:
1267:
1266:
1262:
1246:
1239:
1233:
1225:
1223:
1221:
1217:
1213:
1209:
1208:Galois theory
1204:
1191:
1185:
1182:
1176:
1170:
1167:
1164:
1161:
1158:
1152:
1147:
1143:
1134:
1130:
1126:
1123:
1119:
1116:
1098:
1094:
1086:
1081:
1068:
1062:
1059:
1053:
1047:
1044:
1041:
1038:
1035:
1029:
1024:
1020:
1011:
1007:
1003:
1000:
996:
993:
975:
971:
963:
958:
944:
941:
938:
935:
932:
924:
920:
916:
900:
893:
889:
885:
881:
873:
871:
869:
865:
861:
856:
854:
851:-dimensional
850:
846:
842:
838:
833:
831:
827:
822:
818:
810:
808:
806:
801:
787:
767:
747:
727:
719:
703:
700:
689:
685:
678:
672:
669:
661:
657:
650:
647:
642:
638:
630:applications
629:
613:
610:
605:
601:
597:
592:
588:
584:
579:
575:
567:
562:
549:
546:
543:
540:
537:
534:
531:
528:
525:
521:
513:
509:
502:
499:
494:
491:
488:
484:
475:
461:
439:
435:
414:
406:
402:
396:
388:
386:
382:
376:
369:
365:
361:
355:
351:
345:
328:
325:
322:
319:
316:
313:
308:
304:
300:
294:
288:
280:
275:
269:
266:
262:
256:
251:
247:
243:
238:
228:
224:
220:
211:
207:
187:
185:
183:
182:invariant set
179:
175:
171:
167:
163:
159:
139:
136:
133:
128:
124:
120:
117:
112:
108:
104:
98:
92:
85:The function
83:
65:
62:
56:
49:
46:
37:
33:
19:
2369:
2342:
2338:
2328:
2301:
2297:
2287:
2279:
2271:
2258:
2249:
2230:
2211:
2200:the original
2169:
2165:
2152:
2138:(1): 43–57.
2135:
2129:
2116:
2099:
2086:
2062:(1): 96–98.
2059:
2054:
2044:
2025:
2019:
1950:Markov chain
1909:optimization
1879:
1875:
1834:double point
1833:
1830:projectivity
1815:
1800:
1791:
1780:Please help
1775:verification
1772:
1754:Applications
1733:
1644:
1601:
1571:
1555:order theory
1552:
1531:is called a
1528:
1524:
1520:
1516:
1508:
1504:
1500:
1496:
1492:
1488:
1484:
1480:
1476:
1472:
1468:
1464:
1461:pre-fixpoint
1460:
1456:
1452:
1448:
1444:
1440:
1436:
1432:
1428:
1418:
1382:
1367:
1303:
1235:
1205:
1132:
1124:
1117:
1115:automorphism
1082:
1009:
1001:
994:
992:automorphism
959:
922:
918:
914:
892:group action
887:
883:
877:
857:
848:
834:
823:
820:
802:
563:
476:
404:
398:
380:
374:
367:
363:
359:
356:
349:
343:
279:real numbers
273:
270:
264:
260:
254:
242:real numbers
236:
226:
222:
218:
191:
169:
165:
161:
155:
2412:Game theory
2172:: 114–127.
2056:Fund. Math.
1992:Idempotence
1982:Equilibrium
1977:Eigenvector
1920:type theory
1457:prefixpoint
1220:fixed field
1218:called the
1135:, that is,
720:to a point
347:, because
162:fixed point
158:mathematics
2401:Categories
1495:such that
1471:such that
1378:retraction
1330:such that
192:Formally,
2282:, page 27
2078:0016-2736
1901:compilers
1857:John Nash
1841:economics
1818:stability
1794:July 2018
1699:
1671:
1447:. Then a
1315:∈
1286:→
1280::
1168:∣
1162:∈
1045:∣
1039:∈
936:⋅
901:⋅
862:(and the
845:unit ball
704:…
614:…
550:…
314:−
178:functions
118:−
2257:(1942).
2239:Archived
2220:Archived
2196:17640585
1964:See also
1935:PageRank
1880:unstable
1878:near an
1563:function
1431:and let
1387:, every
1006:subgroup
718:converge
566:sequence
210:codomain
208:and the
166:fixpoint
50:′
18:Fixpoint
2347:Bibcode
2306:Bibcode
2278:(1906)
1868:physics
1748:Datalog
1565:from a
1527:, such
1491:is any
1467:is any
1389:compact
1129:subring
1127:is the
1004:is the
890:with a
880:algebra
866:) from
352:(2) = 2
248:in the
2194:
2184:
2076:
2032:
1922:, the
1884:Wilson
1584:has a
1557:, the
1513:before
1404:Borsuk
1396:subset
1393:convex
1113:of an
990:of an
216:, and
206:domain
38:where
2203:(PDF)
2192:S2CID
2162:(PDF)
2126:(PDF)
2096:(PDF)
2011:Notes
1847:of a
1580:on a
1561:of a
1463:) of
1398:of a
1216:field
1214:is a
1120:of a
999:group
997:of a
740:. If
246:curve
34:, or
2182:ISBN
2170:9328
2074:ISSN
2030:ISBN
1849:game
1843:, a
1606:for
1523:) ≤
1479:) ≤
1391:and
1122:ring
960:The
858:The
835:The
366:) =
225:) =
160:, a
2355:doi
2314:doi
2174:doi
2140:doi
2104:doi
2064:doi
1918:In
1866:In
1839:In
1824:In
1784:by
1734:In
1602:In
1553:In
1533:pre
1511:is
1487:of
1459:or
1419:In
1206:In
1008:of
925:if
917:in
878:In
847:in
626:of
399:In
383:+ 1
370:+ 1
281:by
212:of
200:if
156:In
2403::
2353:.
2341:.
2337:.
2312:.
2300:.
2296:.
2261:.
2190:.
2180:.
2164:.
2136:82
2134:.
2128:.
2098:.
2072:.
2060:40
2053:.
1894:."
1874:,
1855:.
1750:.
1539:.
1499:≤
1439:→
1435::
1380:.
1365:.
1236:A
1012::
957:.
913:,
800:.
403:,
354:.
184:.
2377:.
2363:.
2357::
2349::
2343:4
2322:.
2316::
2308::
2302:4
2176::
2146:.
2142::
2110:.
2106::
2080:.
2066::
2038:.
1959:.
1930:.
1915:.
1836:.
1807:)
1801:(
1796:)
1792:(
1778:.
1708:.
1705:)
1702:f
1694:x
1691:i
1688:f
1683:(
1680:f
1677:=
1674:f
1666:x
1663:i
1660:f
1645:f
1629:x
1626:i
1623:f
1529:x
1525:x
1521:x
1519:(
1517:f
1509:f
1505:p
1503:(
1501:f
1497:p
1493:p
1489:f
1481:p
1477:p
1475:(
1473:f
1469:p
1465:f
1445:X
1441:X
1437:X
1433:f
1429:X
1353:x
1350:=
1347:)
1344:x
1341:(
1338:f
1318:X
1312:x
1289:X
1283:X
1277:f
1247:X
1192:.
1189:}
1186:r
1183:=
1180:)
1177:r
1174:(
1171:f
1165:R
1159:r
1156:{
1153:=
1148:f
1144:R
1133:f
1125:R
1118:f
1099:f
1095:R
1069:.
1066:}
1063:g
1060:=
1057:)
1054:g
1051:(
1048:f
1042:G
1036:g
1033:{
1030:=
1025:f
1021:G
1010:G
1002:G
995:f
976:f
972:G
945:x
942:=
939:x
933:g
923:g
919:X
915:x
888:X
884:G
849:n
788:f
768:x
748:f
728:x
701:,
698:)
695:)
690:0
686:x
682:(
679:f
676:(
673:f
670:,
667:)
662:0
658:x
654:(
651:f
648:,
643:0
639:x
611:,
606:2
602:x
598:,
593:1
589:x
585:,
580:0
576:x
547:,
544:2
541:,
538:1
535:,
532:0
529:=
526:n
522:,
519:)
514:n
510:x
506:(
503:f
500:=
495:1
492:+
489:n
485:x
462:f
440:0
436:x
415:f
381:x
375:x
368:x
364:x
362:(
360:f
350:f
344:f
329:,
326:4
323:+
320:x
317:3
309:2
305:x
301:=
298:)
295:x
292:(
289:f
274:f
265:x
261:y
255:c
237:f
232:f
227:c
223:c
221:(
219:f
214:f
202:c
198:f
194:c
140:x
137:3
134:+
129:2
125:x
121:3
113:3
109:x
105:=
102:)
99:x
96:(
93:f
78:.
66:0
63:=
60:)
57:x
54:(
47:f
20:)
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