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