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Conformal dimension

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447: 390: 165: 256: 317: 81: 488: 431: 97: 522: 481: 189: 532: 474: 527: 351: 424: 517: 512: 176: 417: 262: 293: 32: 62: 274: 28: 40: 507: 261:
The second inequality is true by definition. The first one is deduced from the fact that the
454: 324: 458: 446: 401: 501: 337:
is of null conformal dimension. However, there is no metric space quasisymmetric to
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be the collection of all metric spaces that are quasisymmetric to 
160:{\displaystyle \mathrm {Cdim} X=\inf _{Y\in {\mathcal {G}}}\dim _{H}Y} 251:{\displaystyle \dim _{T}X\leq \mathrm {Cdim} X\leq \dim _{H}X} 134: 68: 374:, University Lecture Series, Vol. 54, 2010, Rhodes Island 462: 405: 296: 192: 100: 65: 323:, since the topological and Hausdorff dimensions of 311: 250: 159: 75: 372:Conformal Dimension : Theory and Application 122: 482: 425: 8: 489: 475: 432: 418: 39:, that is, the class of all metric spaces 303: 298: 295: 236: 212: 197: 191: 145: 133: 132: 125: 101: 99: 67: 66: 64: 363: 277:over all spaces homeomorphic to  7: 443: 441: 386: 384: 222: 219: 216: 213: 111: 108: 105: 102: 14: 370:John M. Mackay, Jeremy T. Tyson, 269:, and thus can be defined as the 445: 388: 312:{\displaystyle \mathbf {R} ^{N}} 299: 76:{\displaystyle {\mathcal {G}}} 1: 341:with a 0 Hausdorff dimension. 87:. The conformal dimension of 461:. You can help Knowledge by 404:. You can help Knowledge by 523:Mathematical analysis stubs 352:Anomalous scaling dimension 290:The conformal dimension of 549: 440: 383: 179:, for a metric space  457:-related article is a 400:–related article is a 313: 252: 175:We have the following 161: 77: 59:be a metric space and 27:is the infimum of the 533:Metric geometry stubs 314: 263:topological dimension 253: 162: 78: 294: 190: 98: 63: 16:In mathematics, the 275:Hausdorff dimension 91:is defined as such 29:Hausdorff dimension 18:conformal dimension 528:Chaos theory stubs 309: 265:T is invariant by 248: 157: 140: 73: 470: 469: 413: 412: 121: 51:Formal definition 540: 518:Dimension theory 491: 484: 477: 449: 442: 434: 427: 420: 392: 385: 375: 368: 325:Euclidean spaces 318: 316: 315: 310: 308: 307: 302: 257: 255: 254: 249: 241: 240: 225: 202: 201: 166: 164: 163: 158: 150: 149: 139: 138: 137: 114: 82: 80: 79: 74: 72: 71: 548: 547: 543: 542: 541: 539: 538: 537: 513:Metric geometry 498: 497: 496: 495: 455:metric geometry 439: 438: 381: 379: 378: 369: 365: 360: 348: 297: 292: 291: 287: 232: 193: 188: 187: 173: 141: 96: 95: 61: 60: 53: 33:conformal gauge 12: 11: 5: 546: 544: 536: 535: 530: 525: 520: 515: 510: 500: 499: 494: 493: 486: 479: 471: 468: 467: 450: 437: 436: 429: 422: 414: 411: 410: 393: 377: 376: 362: 361: 359: 356: 355: 354: 347: 344: 343: 342: 328: 306: 301: 286: 283: 259: 258: 247: 244: 239: 235: 231: 228: 224: 221: 218: 215: 211: 208: 205: 200: 196: 172: 169: 168: 167: 156: 153: 148: 144: 136: 131: 128: 124: 120: 117: 113: 110: 107: 104: 70: 52: 49: 41:quasisymmetric 13: 10: 9: 6: 4: 3: 2: 545: 534: 531: 529: 526: 524: 521: 519: 516: 514: 511: 509: 506: 505: 503: 492: 487: 485: 480: 478: 473: 472: 466: 464: 460: 456: 451: 448: 444: 435: 430: 428: 423: 421: 416: 415: 409: 407: 403: 399: 394: 391: 387: 382: 373: 367: 364: 357: 353: 350: 349: 345: 340: 336: 333: 329: 326: 322: 304: 289: 288: 284: 282: 280: 276: 272: 268: 267:homeomorphism 264: 245: 242: 237: 233: 229: 226: 209: 206: 203: 198: 194: 186: 185: 184: 182: 178: 170: 154: 151: 146: 142: 129: 126: 118: 115: 94: 93: 92: 90: 86: 58: 50: 48: 46: 42: 38: 34: 30: 26: 23: 19: 463:expanding it 452: 406:expanding it 395: 380: 371: 366: 338: 334: 320: 278: 260: 180: 177:inequalities 174: 88: 84: 56: 54: 44: 36: 24: 22:metric space 17: 15: 502:Categories 358:References 332:Cantor set 171:Properties 243:⁡ 230:≤ 210:≤ 204:⁡ 152:⁡ 130:∈ 31:over the 508:Fractals 346:See also 285:Examples 43:to  398:fractal 273:of the 271:infimum 327:agree. 453:This 396:This 20:of a 459:stub 402:stub 330:The 55:Let 319:is 234:dim 195:dim 143:dim 123:inf 35:of 504:: 281:. 183:: 47:. 490:e 483:t 476:v 465:. 433:e 426:t 419:v 408:. 339:K 335:K 321:N 305:N 300:R 279:X 246:X 238:H 227:X 223:m 220:i 217:d 214:C 207:X 199:T 181:X 155:Y 147:H 135:G 127:Y 119:= 116:X 112:m 109:i 106:d 103:C 89:X 85:X 69:G 57:X 45:X 37:X 25:X

Index

metric space
Hausdorff dimension
conformal gauge
quasisymmetric
inequalities
topological dimension
homeomorphism
infimum
Hausdorff dimension
Euclidean spaces
Cantor set
Anomalous scaling dimension
Stub icon
fractal
stub
expanding it
v
t
e
Stub icon
metric geometry
stub
expanding it
v
t
e
Categories
Fractals
Metric geometry
Dimension theory

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