1411:
327:
218:
364:
759:
681:
131:
178:
1300:
482:
620:
over the entire volume. This can be used to explain many results from classical statistical mechanics, including the irreversibility of time and the increase of
418:
963:
568:
438:
384:
151:
1126:
226:
1253:
1108:
601:, with the largest eigenvalue being equal to one. For this reason, the transfer operator is sometimes called the Frobenius–Perron operator.
1084:
762:
530:. The adjoint to the transfer operator can likewise usually be interpreted as a right-shift. Particularly well studied right-shifts include the
976:
1065:
956:
909:
890:
871:
852:
1335:
980:
387:
183:
1131:
574:), the transfer operator defines how (smooth) maps evolve under iteration. Thus, transfer operators typically appear in
1187:
86:
1414:
1136:
1121:
949:
609:
1151:
335:
1396:
1156:
608:
of the transfer operator are usually fractals. When the logarithm of the transfer operator corresponds to a quantum
1440:
1350:
1274:
844:
1391:
701:
586:, where attention is focused on the time evolution of smooth functions. In turn, this has medical applications to
1207:
512:
1141:
792:
633:
1450:
1445:
1243:
1044:
527:
1116:
487:
The above definition of the transfer operator can be shown to be the point-set limit of the measure-theoretic
1340:
612:, the eigenvalues will typically be very closely spaced, and thus even a very narrow and carefully selected
104:
1371:
1315:
1279:
797:
539:
1435:
770:
692:
688:
617:
583:
51:
28:
1354:
819:
787:
587:
508:
496:
616:
of quantum states will encompass a large number of very different fractal eigenstates with non-zero
159:
1320:
1258:
972:
902:
Thermodynamic formalism: the mathematical structures of classical equilibrium statistical mechanics
488:
1345:
1212:
766:
591:
597:
It is often the case that the transfer operator is positive, has discrete positive real-valued
1325:
905:
886:
867:
848:
535:
63:
47:
920:
443:
1330:
1248:
1217:
1197:
1182:
1177:
1172:
1009:
827:
782:
613:
504:
500:
62:. In all usual cases, the largest eigenvalue is 1, and the corresponding eigenvector is the
1192:
1146:
1094:
1089:
1060:
941:
531:
1019:
393:
823:
1381:
1233:
1034:
810:
Gaspard, Pierre (1992). "r-adic one dimensional maps and the Euler summation formula".
553:
519:
423:
369:
136:
570:
naturally leads to a study of the orbits of points of X under iteration (the study of
1429:
1386:
1310:
1039:
1024:
1014:
831:
628:
605:
579:
55:
1376:
1029:
999:
684:
571:
74:
43:
1305:
1295:
1202:
1004:
523:
518:
As a general rule, the transfer operator can usually be interpreted as a (left-)
35:
322:{\displaystyle ({\mathcal {L}}\Phi )(x)=\sum _{y\,\in \,f^{-1}(x)}g(y)\Phi (y)}
17:
1238:
1078:
1074:
1070:
598:
90:
59:
621:
575:
883:
The Ruelle-Araki transfer operator in classical statistical mechanics
691:. This operator also has a continuous spectrum consisting of the
503:. The left-adjoint of the Perron–Frobenius operator is the
945:
765:. The theory of the GKW dates back to a hypothesis by Gauss on
235:
165:
864:
Time's Arrow : The origins of thermodynamic behaviour
213:{\displaystyle \{\Phi \colon X\rightarrow \mathbb {C} \}}
704:
636:
556:
446:
426:
396:
372:
338:
229:
186:
162:
139:
107:
1364:
1288:
1267:
1226:
1165:
1107:
1053:
988:
1301:Spectral theory of ordinary differential equations
753:
675:
562:
476:
432:
412:
378:
358:
321:
212:
172:
145:
125:
921:"Dynamical Zeta Functions and Transfer Operators"
359:{\displaystyle g\colon X\rightarrow \mathbb {C} }
683:is exactly solvable and is a classic example of
156:The transfer operator is defined as an operator
46:and is frequently used to study the behavior of
763:Gauss–Kuzmin–Wirsing (GKW) operator
69:The transfer operator is sometimes called the
957:
687:; the discrete eigenvalues correspond to the
101:The iterated function to be studied is a map
8:
754:{\displaystyle h(x)=1/x-\lfloor 1/x\rfloor }
748:
734:
670:
661:
207:
187:
83:Ruelle–Perron–Frobenius operator
841:Chaos, scattering and statistical mechanics
526:. The most commonly studied shifts are the
495:: in essence, the transfer operator is the
85:, in reference to the applicability of the
992:
964:
950:
942:
676:{\displaystyle b(x)=2x-\lfloor 2x\rfloor }
740:
723:
703:
635:
555:
511:. The general setting is provided by the
469:
461:
456:
445:
425:
405:
397:
395:
371:
366:is an auxiliary valuation function. When
352:
351:
337:
275:
270:
266:
262:
234:
233:
228:
203:
202:
185:
164:
163:
161:
138:
106:
1254:Group algebra of a locally compact group
698:The transfer operator of the Gauss map
126:{\displaystyle f\colon X\rightarrow X}
7:
550:Whereas the iteration of a function
538:, both of which generate systems of
937:(Provides an introductory survey).
307:
240:
190:
25:
904:. Addison–Wesley, Reading.
180:acting on the space of functions
1410:
1409:
1336:Topological quantum field theory
769:and is closely related to the
714:
708:
646:
640:
470:
462:
406:
398:
348:
316:
310:
304:
298:
290:
284:
252:
246:
243:
230:
199:
173:{\displaystyle {\mathcal {L}}}
117:
1:
1132:Uniform boundedness principle
627:The transfer operator of the
42:encodes information about an
89:to the determination of the
862:Mackey, Michael C. (1992).
1467:
1275:Invariant subspace problem
845:Cambridge University Press
832:10.1088/0305-4470/25/8/017
26:
1405:
995:
881:Mayer, Dieter H. (1978).
513:Borel functional calculus
79:Perron–Frobenius operator
1244:Spectrum of a C*-algebra
839:Gaspard, Pierre (1998).
528:subshifts of finite type
87:Perron–Frobenius theorem
27:Not to be confused with
1341:Noncommutative geometry
590:, through the field of
477:{\displaystyle g=1/|J|}
440:is usually taken to be
1397:Tomita–Takesaki theory
1372:Approximation property
1316:Calculus of variations
919:Ruelle, David (2002).
900:Ruelle, David (1978).
798:Transfer-matrix method
755:
677:
564:
540:orthogonal polynomials
478:
434:
414:
380:
360:
323:
214:
174:
147:
127:
1392:Banach–Mazur distance
1355:Generalized functions
812:J. Phys. A: Math. Gen
771:Riemann zeta function
756:
693:Hurwitz zeta function
689:Bernoulli polynomials
678:
584:statistical mechanics
565:
479:
435:
415:
381:
361:
324:
215:
175:
148:
133:for an arbitrary set
128:
52:statistical mechanics
29:transfer homomorphism
1137:Kakutani fixed-point
1122:Riesz representation
793:Krein–Rutman theorem
788:Shift of finite type
702:
634:
588:rational drug design
554:
509:composition operator
497:direct image functor
444:
424:
394:
370:
336:
227:
184:
160:
137:
105:
1321:Functional calculus
1280:Mahler's conjecture
1259:Von Neumann algebra
973:Functional analysis
885:. Springer-Verlag.
866:. Springer-Verlag.
824:1992JPhA...25L.483G
767:continued fractions
685:deterministic chaos
542:via a right-shift.
499:in the category of
413:{\displaystyle |J|}
1346:Riemann hypothesis
1045:Topological vector
928:Notices of the AMS
751:
673:
592:molecular dynamics
578:problems, such as
560:
474:
430:
410:
376:
356:
319:
294:
210:
170:
143:
123:
1441:Dynamical systems
1423:
1422:
1326:Integral operator
1103:
1102:
563:{\displaystyle f}
536:Hessenberg matrix
501:measurable spaces
433:{\displaystyle g}
379:{\displaystyle f}
258:
146:{\displaystyle X}
93:of the operator.
64:invariant measure
48:dynamical systems
40:transfer operator
16:(Redirected from
1458:
1413:
1412:
1331:Jones polynomial
1249:Operator algebra
993:
966:
959:
952:
943:
935:
925:
915:
896:
877:
858:
835:
818:(8): L483–L485.
783:Bernoulli scheme
760:
758:
757:
752:
744:
727:
682:
680:
679:
674:
569:
567:
566:
561:
505:Koopman operator
483:
481:
480:
475:
473:
465:
460:
439:
437:
436:
431:
419:
417:
416:
411:
409:
401:
385:
383:
382:
377:
365:
363:
362:
357:
355:
328:
326:
325:
320:
293:
283:
282:
239:
238:
219:
217:
216:
211:
206:
179:
177:
176:
171:
169:
168:
152:
150:
149:
144:
132:
130:
129:
124:
21:
1466:
1465:
1461:
1460:
1459:
1457:
1456:
1455:
1451:Spectral theory
1446:Operator theory
1426:
1425:
1424:
1419:
1401:
1365:Advanced topics
1360:
1284:
1263:
1222:
1188:Hilbert–Schmidt
1161:
1152:Gelfand–Naimark
1099:
1049:
984:
970:
923:
918:
912:
899:
893:
880:
874:
861:
855:
838:
809:
806:
779:
700:
699:
632:
631:
552:
551:
548:
532:Jacobi operator
442:
441:
422:
421:
392:
391:
368:
367:
334:
333:
271:
225:
224:
182:
181:
158:
157:
135:
134:
103:
102:
99:
71:Ruelle operator
66:of the system.
32:
23:
22:
18:Ruelle operator
15:
12:
11:
5:
1464:
1462:
1454:
1453:
1448:
1443:
1438:
1428:
1427:
1421:
1420:
1418:
1417:
1406:
1403:
1402:
1400:
1399:
1394:
1389:
1384:
1382:Choquet theory
1379:
1374:
1368:
1366:
1362:
1361:
1359:
1358:
1348:
1343:
1338:
1333:
1328:
1323:
1318:
1313:
1308:
1303:
1298:
1292:
1290:
1286:
1285:
1283:
1282:
1277:
1271:
1269:
1265:
1264:
1262:
1261:
1256:
1251:
1246:
1241:
1236:
1234:Banach algebra
1230:
1228:
1224:
1223:
1221:
1220:
1215:
1210:
1205:
1200:
1195:
1190:
1185:
1180:
1175:
1169:
1167:
1163:
1162:
1160:
1159:
1157:Banach–Alaoglu
1154:
1149:
1144:
1139:
1134:
1129:
1124:
1119:
1113:
1111:
1105:
1104:
1101:
1100:
1098:
1097:
1092:
1087:
1085:Locally convex
1082:
1068:
1063:
1057:
1055:
1051:
1050:
1048:
1047:
1042:
1037:
1032:
1027:
1022:
1017:
1012:
1007:
1002:
996:
990:
986:
985:
971:
969:
968:
961:
954:
946:
940:
939:
916:
910:
897:
891:
878:
872:
859:
853:
836:
805:
802:
801:
800:
795:
790:
785:
778:
775:
761:is called the
750:
747:
743:
739:
736:
733:
730:
726:
722:
719:
716:
713:
710:
707:
672:
669:
666:
663:
660:
657:
654:
651:
648:
645:
642:
639:
606:eigenfunctions
572:point dynamics
559:
547:
544:
520:shift operator
472:
468:
464:
459:
455:
452:
449:
429:
408:
404:
400:
375:
354:
350:
347:
344:
341:
330:
329:
318:
315:
312:
309:
306:
303:
300:
297:
292:
289:
286:
281:
278:
274:
269:
265:
261:
257:
254:
251:
248:
245:
242:
237:
232:
209:
205:
201:
198:
195:
192:
189:
167:
142:
122:
119:
116:
113:
110:
98:
95:
24:
14:
13:
10:
9:
6:
4:
3:
2:
1463:
1452:
1449:
1447:
1444:
1442:
1439:
1437:
1434:
1433:
1431:
1416:
1408:
1407:
1404:
1398:
1395:
1393:
1390:
1388:
1387:Weak topology
1385:
1383:
1380:
1378:
1375:
1373:
1370:
1369:
1367:
1363:
1356:
1352:
1349:
1347:
1344:
1342:
1339:
1337:
1334:
1332:
1329:
1327:
1324:
1322:
1319:
1317:
1314:
1312:
1311:Index theorem
1309:
1307:
1304:
1302:
1299:
1297:
1294:
1293:
1291:
1287:
1281:
1278:
1276:
1273:
1272:
1270:
1268:Open problems
1266:
1260:
1257:
1255:
1252:
1250:
1247:
1245:
1242:
1240:
1237:
1235:
1232:
1231:
1229:
1225:
1219:
1216:
1214:
1211:
1209:
1206:
1204:
1201:
1199:
1196:
1194:
1191:
1189:
1186:
1184:
1181:
1179:
1176:
1174:
1171:
1170:
1168:
1164:
1158:
1155:
1153:
1150:
1148:
1145:
1143:
1140:
1138:
1135:
1133:
1130:
1128:
1125:
1123:
1120:
1118:
1115:
1114:
1112:
1110:
1106:
1096:
1093:
1091:
1088:
1086:
1083:
1080:
1076:
1072:
1069:
1067:
1064:
1062:
1059:
1058:
1056:
1052:
1046:
1043:
1041:
1038:
1036:
1033:
1031:
1028:
1026:
1023:
1021:
1018:
1016:
1013:
1011:
1008:
1006:
1003:
1001:
998:
997:
994:
991:
987:
982:
978:
974:
967:
962:
960:
955:
953:
948:
947:
944:
938:
934:(8): 887–895.
933:
929:
922:
917:
913:
911:0-201-13504-3
907:
903:
898:
894:
892:0-387-09990-5
888:
884:
879:
875:
873:0-387-94093-6
869:
865:
860:
856:
854:0-521-39511-9
850:
846:
842:
837:
833:
829:
825:
821:
817:
813:
808:
807:
803:
799:
796:
794:
791:
789:
786:
784:
781:
780:
776:
774:
772:
768:
764:
745:
741:
737:
731:
728:
724:
720:
717:
711:
705:
696:
694:
690:
686:
667:
664:
658:
655:
652:
649:
643:
637:
630:
629:Bernoulli map
625:
623:
619:
615:
611:
607:
602:
600:
595:
593:
589:
585:
581:
580:quantum chaos
577:
573:
557:
545:
543:
541:
537:
533:
529:
525:
521:
516:
514:
510:
506:
502:
498:
494:
490:
485:
466:
457:
453:
450:
447:
427:
402:
389:
373:
345:
342:
339:
313:
301:
295:
287:
279:
276:
272:
267:
263:
259:
255:
249:
223:
222:
221:
196:
193:
154:
140:
120:
114:
111:
108:
96:
94:
92:
88:
84:
80:
76:
72:
67:
65:
61:
57:
56:quantum chaos
53:
49:
45:
41:
37:
30:
19:
1436:Chaos theory
1377:Balanced set
1351:Distribution
1289:Applications
1142:Krein–Milman
1127:Closed graph
936:
931:
927:
901:
882:
863:
840:
815:
811:
697:
626:
603:
596:
549:
546:Applications
522:acting on a
517:
492:
486:
390:determinant
331:
155:
100:
82:
78:
75:David Ruelle
70:
68:
44:iterated map
39:
33:
1306:Heat kernel
1296:Hardy space
1203:Trace class
1117:Hahn–Banach
1079:Topological
610:Hamiltonian
599:eigenvalues
524:shift space
489:pushforward
91:eigenvalues
36:mathematics
1430:Categories
1239:C*-algebra
1054:Properties
804:References
97:Definition
1213:Unbounded
1208:Transpose
1166:Operators
1095:Separable
1090:Reflexive
1075:Algebraic
1061:Barrelled
749:⌋
735:⌊
732:−
671:⌋
662:⌊
659:−
349:→
343::
308:Φ
277:−
268:∈
260:∑
241:Φ
200:→
194::
191:Φ
118:→
112::
77:, or the
1415:Category
1227:Algebras
1109:Theorems
1066:Complete
1035:Schwartz
981:glossary
777:See also
614:ensemble
534:and the
388:Jacobian
73:, after
60:fractals
1218:Unitary
1198:Nuclear
1183:Compact
1178:Bounded
1173:Adjoint
1147:Min–max
1040:Sobolev
1025:Nuclear
1015:Hilbert
1010:Fréchet
975: (
820:Bibcode
622:entropy
618:support
576:physics
420:, then
1193:Normal
1030:Orlicz
1020:Hölder
1000:Banach
989:Spaces
977:topics
908:
889:
870:
851:
386:has a
332:where
38:, the
1005:Besov
924:(PDF)
1353:(or
1071:Dual
906:ISBN
887:ISBN
868:ISBN
849:ISBN
604:The
582:and
58:and
828:doi
507:or
491:of
220:as
153:.
81:or
34:In
1432::
979:–
932:49
930:.
926:.
847:.
843:.
826:.
816:25
814:.
773:.
695:.
624:.
594:.
515:.
484:.
54:,
50:,
1357:)
1081:)
1077:/
1073:(
983:)
965:e
958:t
951:v
914:.
895:.
876:.
857:.
834:.
830::
822::
746:x
742:/
738:1
729:x
725:/
721:1
718:=
715:)
712:x
709:(
706:h
668:x
665:2
656:x
653:2
650:=
647:)
644:x
641:(
638:b
558:f
493:g
471:|
467:J
463:|
458:/
454:1
451:=
448:g
428:g
407:|
403:J
399:|
374:f
353:C
346:X
340:g
317:)
314:y
311:(
305:)
302:y
299:(
296:g
291:)
288:x
285:(
280:1
273:f
264:y
256:=
253:)
250:x
247:(
244:)
236:L
231:(
208:}
204:C
197:X
188:{
166:L
141:X
121:X
115:X
109:f
31:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.