Knowledge (XXG)

No-wandering-domain theorem

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has wandering domains. However, the result can be generalized to many situations where the functions naturally belong to a finite-dimensional parameter space, most notably to transcendental entire and meromorphic functions with a finite number of singular values.
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Quasiconformal homeomorphisms and dynamics. I. Solution of the Fatou-Julia problem on wandering domains
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The theorem does not hold for arbitrary maps; for example, the
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Sullivan's proof of Fatou's no wandering domain conjecture
490: 350: 284: 207: 173:{\displaystyle U,f(U),f(f(U)),\dots ,f^{n}(U),\dots } 93: 395: 329: 263: 172: 510: 8: 517: 503: 349: 283: 252: 222: 212: 206: 149: 92: 183:will eventually become periodic. Here, 420:, Universitext: Tracts in Mathematics, 278:This image illustrates the dynamics of 7: 471: 469: 396:{\displaystyle f(z)=z+2\pi \sin(z)} 330:{\displaystyle f(z)=z+2\pi \sin(z)} 489:. You can help Knowledge (XXG) by 14: 473: 390: 384: 360: 354: 324: 318: 294: 288: 161: 155: 133: 130: 124: 118: 109: 103: 69:. More precisely, for every 1: 546:Theorems in dynamical systems 448:122 (1985), no. 3, 401–18. 57:) ≥ 2 does not have a 25:no-wandering-domain theorem 572: 468: 38:The theorem states that a 416:and Theodore W. Gamelin, 485:-related article is a 397: 338: 331: 265: 174: 446:Annals of Mathematics 398: 332: 277: 266: 175: 348: 282: 205: 91: 16:Mathematical theorem 49: →  556:Chaos theory stubs 424:, New York, 1993, 393: 343:transcendental map 339: 327: 261: 257: 250: 170: 498: 497: 440:Dennis Sullivan, 223: 221: 29:dynamical systems 563: 551:Complex dynamics 519: 512: 505: 477: 470: 418:Complex Dynamics 414:Lennart Carleson 402: 400: 399: 394: 336: 334: 333: 328: 270: 268: 267: 262: 256: 251: 246: 217: 216: 179: 177: 176: 171: 154: 153: 84:, the sequence 59:wandering domain 571: 570: 566: 565: 564: 562: 561: 560: 526: 525: 524: 523: 466: 422:Springer-Verlag 410: 346: 345: 280: 279: 224: 208: 203: 202: 192:-fold iteration 145: 89: 88: 33:Dennis Sullivan 27:is a result on 17: 12: 11: 5: 569: 567: 559: 558: 553: 548: 543: 538: 536:Ergodic theory 528: 527: 522: 521: 514: 507: 499: 496: 495: 478: 464: 463: 455: 438: 409: 406: 392: 389: 386: 383: 380: 377: 374: 371: 368: 365: 362: 359: 356: 353: 326: 323: 320: 317: 314: 311: 308: 305: 302: 299: 296: 293: 290: 287: 272: 271: 260: 255: 249: 245: 242: 239: 236: 233: 230: 227: 220: 215: 211: 181: 180: 169: 166: 163: 160: 157: 152: 148: 144: 141: 138: 135: 132: 129: 126: 123: 120: 117: 114: 111: 108: 105: 102: 99: 96: 67:Riemann sphere 15: 13: 10: 9: 6: 4: 3: 2: 568: 557: 554: 552: 549: 547: 544: 542: 539: 537: 534: 533: 531: 520: 515: 513: 508: 506: 501: 500: 494: 492: 488: 484: 479: 476: 472: 467: 462: 461: 456: 454: 451: 447: 443: 439: 437: 434: 431: 430:0-387-97942-5 427: 423: 419: 415: 412: 411: 407: 405: 387: 381: 378: 375: 372: 369: 366: 363: 357: 351: 344: 321: 315: 312: 309: 306: 303: 300: 297: 291: 285: 276: 258: 253: 247: 243: 240: 237: 234: 231: 228: 225: 218: 213: 209: 201: 200: 199: 197: 193: 191: 186: 167: 164: 158: 150: 146: 142: 139: 136: 127: 121: 115: 112: 106: 100: 97: 94: 87: 86: 85: 83: 79: 75: 72: 68: 64: 60: 56: 52: 48: 45: :  44: 41: 36: 34: 30: 26: 22: 491:expanding it 483:chaos theory 480: 465: 458: 441: 417: 340: 195: 189: 187:denotes the 184: 182: 81: 73: 65:denotes the 62: 54: 50: 46: 42: 40:rational map 37: 31:, proven by 24: 18: 457:S. Zakeri, 198:, that is, 21:mathematics 541:Limit sets 530:Categories 408:References 382:⁡ 376:π 316:⁡ 310:π 248:⏟ 241:∘ 238:⋯ 235:∘ 229:∘ 168:… 140:… 78:Fatou set 71:component 53:with deg( 35:in 1985. 61:, where 453:0819553 436:1230383 76:in the 428:  23:, the 481:This 487:stub 426:ISBN 379:sin 313:sin 194:of 80:of 19:In 532:: 450:MR 444:, 433:MR 518:e 511:t 504:v 493:. 391:) 388:z 385:( 373:2 370:+ 367:z 364:= 361:) 358:z 355:( 352:f 325:) 322:z 319:( 307:2 304:+ 301:z 298:= 295:) 292:z 289:( 286:f 259:. 254:n 244:f 232:f 226:f 219:= 214:n 210:f 196:f 190:n 185:f 165:, 162:) 159:U 156:( 151:n 147:f 143:, 137:, 134:) 131:) 128:U 125:( 122:f 119:( 116:f 113:, 110:) 107:U 104:( 101:f 98:, 95:U 82:f 74:U 63:Ĉ 55:f 51:Ĉ 47:Ĉ 43:f

Index

mathematics
dynamical systems
Dennis Sullivan
rational map
wandering domain
Riemann sphere
component
Fatou set
n-fold iteration
An image of the dynamical plane for f(z)=z+2\pi\sin(z).
transcendental map
Lennart Carleson
Springer-Verlag
ISBN
0-387-97942-5
MR
1230383
Annals of Mathematics
MR
0819553
Sullivan's proof of Fatou's no wandering domain conjecture
Stub icon
chaos theory
stub
expanding it
v
t
e
Categories
Ergodic theory

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