480:
is a pseudo-Anosov map: in this case, there are two fixed points on the
Thurston boundary, one attracting and one repelling, and the homeomorphism behaves similarly to a hyperbolic automorphism of the
222:
302:
315:-coordinate, measured locally in every chart, is a geometric quantity (i.e. independent of the chart) and permits the definition of a total variation along a simple closed curve on
339:
is redefined with respect to this modified differentiable structure. With some technical modifications, these definitions extend to the case of a surface with boundary.
380:
331:≥3, are allowed. At such a singular point, the differentiable structure of the surface is modified to make the point into a conical point with the total angle
60:
500:(on the algebraic side). This leads to an analogue of Thurston classification for the case of automorphisms of free groups, developed by
513:
A. Casson, S. Bleiler, "Automorphisms of
Surfaces after Nielsen and Thurston", (London Mathematical Society Student Texts 9), (1988).
496:, the notion of a pseudo-Anosov map has been extended to self-maps of graphs (on the topological side) and outer automorphisms of
168:
523:
R. C. Penner. "A construction of pseudo-Anosov homeomorphisms", Trans. Amer. Math. Soc., 310 (1988) No 1, 179–197
582:
476:
extends to a homeomorphism of the
Thurston compactification. The dynamics of this homeomorphism is the simplest when
577:
233:
484:. A "generic" diffeomorphism of a surface of genus at least two is isotopic to a pseudo-Anosov diffeomorphism.
587:
481:
44:
446:
517:
40:
549:
353:
539:
527:
56:
561:
557:
501:
32:
90:
and a measure in the transverse direction. In some neighborhood of a regular point of
571:
36:
544:
20:
497:
553:
520:, "Travaux de Thurston sur les surfaces," Asterisque, Vols. 66 and 67 (1979).
59:, who also coined the term "pseudo-Anosov diffeomorphism" when he proved his
87:
24:
530:(1988), "On the geometry and dynamics of diffeomorphisms of surfaces",
48:
409: > 1 such that the foliations are preserved by
393:
if there exists a transverse pair of measured foliations on
311:. This assures that along a simple curve, the variation in
356:
236:
171:
217:{\displaystyle \phi _{ij}\circ \phi _{j}=\phi _{i},}
16:
Type of diffeomorphism or homeomorphism of a surface
413:and their transverse measures are multiplied by 1/
374:
296:
216:
445:Thurston constructed a compactification of the
61:classification of diffeomorphisms of a surface
532:Bulletin of the American Mathematical Society
8:
51:. Its definition relies on the notion of a
297:{\displaystyle \phi (x,y)=(f(x,y),c\pm y)}
543:
355:
235:
205:
192:
176:
170:
319:. A finite number of singularities of
43:. It is a generalization of a linear
7:
335:. The notion of a diffeomorphism of
67:Definition of a measured foliation
14:
343:Definition of a pseudo-Anosov map
460:such that the action induced on
545:10.1090/S0273-0979-1988-15685-6
405:(unstable), and a real number
366:
291:
276:
264:
258:
252:
240:
162:), with the standard property
1:
516:A. Fathi, F. Laudenbach, and
86:which consists of a singular
114:. If two such neighborhoods
82:is a geometric structure on
110:to the horizontal lines in
604:
106:which sends the leaves of
227:which must have the form
468:) by any diffeomorphism
375:{\displaystyle f:S\to S}
132:overlap then there is a
94:, there is a "flow box"
376:
298:
218:
377:
299:
219:
45:Anosov diffeomorphism
528:Thurston, William P.
492:Using the theory of
385:of a closed surface
354:
234:
169:
78:on a closed surface
482:Poincaré half-plane
134:transition function
583:Geometric topology
372:
327:-pronged saddle",
307:for some constant
294:
214:
73:measured foliation
53:measured foliation
23:, specifically in
578:Dynamical systems
447:TeichmĂĽller space
29:pseudo-Anosov map
595:
564:
547:
381:
379:
378:
373:
347:A homeomorphism
323:of the type of "
303:
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295:
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210:
209:
197:
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184:
183:
57:William Thurston
603:
602:
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593:
592:
568:
567:
526:
510:
490:
456:) of a surface
443:
352:
351:
345:
232:
231:
201:
188:
172:
167:
166:
161:
152:
143:
131:
122:
69:
31:is a type of a
17:
12:
11:
5:
601:
599:
591:
590:
588:Homeomorphisms
585:
580:
570:
569:
566:
565:
538:(2): 417–431,
534:, New Series,
524:
521:
514:
509:
506:
489:
488:Generalization
486:
442:
439:
427:stretch factor
425:is called the
383:
382:
371:
368:
365:
362:
359:
344:
341:
305:
304:
293:
290:
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281:
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225:
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213:
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200:
195:
191:
187:
182:
179:
175:
157:
148:
139:
127:
118:
68:
65:
55:introduced by
33:diffeomorphism
15:
13:
10:
9:
6:
4:
3:
2:
600:
589:
586:
584:
581:
579:
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551:
546:
541:
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533:
529:
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451:
448:
440:
438:
436:
432:
428:
424:
421:. The number
420:
416:
412:
408:
404:
401:(stable) and
400:
396:
392:
391:pseudo-Anosov
388:
369:
363:
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357:
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340:
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326:
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288:
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249:
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147:
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138:
135:
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126:
121:
117:
113:
109:
105:
101:
97:
93:
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85:
81:
77:
74:
66:
64:
62:
58:
54:
50:
46:
42:
38:
37:homeomorphism
34:
30:
26:
22:
535:
531:
504:and Handel.
494:train tracks
493:
491:
477:
473:
469:
465:
461:
457:
453:
449:
444:
441:Significance
434:
430:
426:
422:
418:
414:
410:
406:
402:
398:
394:
390:
386:
384:
346:
336:
332:
328:
324:
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316:
312:
308:
306:
226:
158:
154:
149:
145:
140:
136:
133:
128:
124:
119:
115:
111:
107:
103:
99:
95:
91:
83:
79:
75:
72:
70:
52:
28:
18:
498:free groups
144:defined on
21:mathematics
572:Categories
518:V. Poénaru
508:References
431:dilatation
389:is called
554:0002-9904
367:→
286:±
238:ϕ
203:ϕ
190:ϕ
186:∘
174:ϕ
88:foliation
502:Bestvina
25:topology
562:0956596
333:πp
47:of the
41:surface
560:
552:
423:λ
419:λ
415:λ
407:λ
146:φ
137:φ
96:φ
49:torus
39:of a
550:ISSN
417:and
123:and
27:, a
540:doi
472:of
433:of
429:or
35:or
19:In
574::
558:MR
556:,
548:,
536:19
437:.
397:,
141:ij
102:→
98::
71:A
63:.
542::
478:f
474:S
470:f
466:S
464:(
462:T
458:S
454:S
452:(
450:T
435:f
411:f
403:F
399:F
395:S
387:S
370:S
364:S
361::
358:f
337:S
329:p
325:p
321:F
317:S
313:y
309:c
292:)
289:y
283:c
280:,
277:)
274:y
271:,
268:x
265:(
262:f
259:(
256:=
253:)
250:y
247:,
244:x
241:(
212:,
207:i
199:=
194:j
181:j
178:i
159:j
155:U
153:(
150:j
129:j
125:U
120:i
116:U
112:R
108:F
104:R
100:U
92:F
84:S
80:S
76:F
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