3688:
3700:
3432:
3492:
3480:
2396:
138:
3724:
3610:
3664:
2008:
3709:
3598:
887:
3420:
906:
2815:
1634:
2391:{\displaystyle {\begin{aligned}{\mathcal {M}}B(z_{1})&=B(z_{2}){\;}{\mathrm {or} }{\;}{\mathcal {M}}{\;}{\mathrm {red} }\subseteq {\mathrm {green} },\\{\mathcal {M}}B(z_{2})&=B(z_{3}){\;}{\mathrm {or} }{\;}{\mathcal {M}}{\;}{\mathrm {green} }\subseteq {\mathrm {yellow} },\\{\mathcal {M}}B(z_{3})&=B(z_{1}){\;}{\mathrm {or} }{\;}{\mathcal {M}}{\;}{\mathrm {yellow} }\subseteq {\mathrm {red} }.\end{aligned}}}
726:
3739:
694:
895:
3676:
2561:
1386:
2402:
3751:
3763:
2810:{\displaystyle {\begin{aligned}z_{1}&=0.500003730675024+(6.968273875812428\times 10^{-6})i{\;}{\;}({\mathrm {red} }),\\z_{2}&=-0.138169999969259+(0.239864000061970)i{\;}{\;}({\mathrm {green} }),\\z_{3}&=-0.238618870661709-(0.264884797354373)i{\;}{\;}({\mathrm {yellow} }).\end{aligned}}}
1629:{\displaystyle {\begin{aligned}z_{1}&=0.499997032420304-(1.221880225696050\times 10^{-6})i{\;}{\;}{\mathrm {(red)} },\\z_{2}&=0.638169999974373-(0.239864000011495)i{\;}{\;}{\mathrm {(green)} },\\z_{3}&=0.799901291393262-(0.107547238170383)i{\;}{\;}{\mathrm {(yellow)} }.\end{aligned}}}
3161:
436:
2533:
1242:
2566:
2013:
1391:
3200:
2438:
1997:
1950:
1903:
3320:
3235:
2473:
2903:
808:
3155:
498:
212:
3625:
In the early 1980s, Hubbard posed the so-called twisted rabbit problem, a polynomial classification problem. The goal is to determine
Thurston equivalence types of functions of
1776:
1378:
1347:
1090:
3663:
3386:
3060:
2842:
1856:
1829:
1286:
1007:
3092:
84:
3558:
3119:
1805:
1059:
1745:
1709:
1673:
1262:
983:
963:
931:
828:
717:
663:
623:
525:
3687:
2984:
2957:
2930:
1171:
1144:
1117:
599:
572:
318:
287:
2553:
776:
749:
683:
643:
545:
3036:
3010:
2868:
1312:
1033:
3413:
945:, such as the sprouts at the one- and five-o'clock positions on the right disk or the sprouts at the seven- and eleven-o'clock positions on the left disk. When
3578:
3466:
3271:
1194:
868:
848:
260:
236:
107:
3815:
3470:
142:
3491:
343:
874:
of one another, or more specifically a similarity transformation, consisting of only scaling, rotation and translation, the filled Julia sets look
965:
is within one of these four sprouts, the associated filled Julia set in the mapping plane is said to be a Douady rabbit. For these values of
3926:
3699:
3479:
2555:
is invariant under this transformation). This rabbit is more symmetrical in the plane. The period-three fixed points then are located at
3982:
3903:
3652:. The generalization of the problem to the case where the number of post-critical points is arbitrarily large has been solved as well.
3951:"Recent Research Papers (Only since 1999) Robert L. Devaney: Rabbits, Basilicas, and Other Julia Sets Wrapped in Sierpinski Carpets"
3431:
87:
3637:
quadratic whose branch point is periodic with period three, determining which quadratic polynomial it is
Thurston equivalent to
3880:
3609:
4027:
James Belk; Justin Lanier; Dan
Margalit; Rebecca R. Winarski (2019). "Recognizing Topological Polynomials by Lifting Trees".
3723:
2481:
1199:
3597:
329:
3954:
3808:
645:) for which the associated filled Julia set is connected. The Mandelbrot set can be viewed with respect to either
3708:
3167:
3389:
2408:
1955:
1908:
1861:
3287:
3738:
3205:
2443:
262:
ranging over the plane. The resulting image can be colored by corresponding each pixel with a starting value
3750:
3920:"Twisted Three-Eared Rabbits: Identifying Topological Quadratics Up To Thurston Equivalence by Adam Chodof"
2873:
933:
as shown in the graph above. In this figure, the
Mandelbrot set superficially appears as two back-to-back
137:
3644:
of twisted rabbits, i.e. composite of the rabbit polynomial with nth powers of Dehn twists about its ears.
781:
3675:
3124:
446:
160:
1750:
1352:
1317:
1064:
898:
875:
871:
334:
118:
4090:
4006:
Laurent
Bartholdi; Volodymyr Nekrashevych (2005). "Thurston equivalence of topological polynomials".
3839:
Laurent
Bartholdi; Volodymyr Nekrashevych (2006). "Thurston equivalence of topological polynomials".
3353:
3041:
2823:
1837:
1810:
1267:
988:
4111:
3065:
886:
3919:
3762:
3510:
3419:
3097:
1092:
has three attracting fixed points. A Douady rabbit consists of the three attracting fixed points
34:
4028:
4007:
3848:
3648:
The problem was originally solved by
Laurent Bartholdi and Volodymyr Nekrashevych using iterated
1781:
3975:
1038:
905:
4106:
4052:
3900:
3641:
1714:
1678:
1642:
1247:
968:
948:
916:
813:
702:
648:
608:
510:
3858:
3715:
3649:
504:
28:
2962:
2935:
2908:
1149:
1122:
1095:
577:
550:
296:
265:
4055:
3907:
3809:"A Geometric Solution to the Twisted Rabbit Problem by Jim Belk, University of St Andrews"
2538:
761:
734:
725:
668:
628:
530:
3015:
2989:
2847:
1291:
1012:
3395:
4071:
3626:
3563:
3462:
3458:
3256:
1179:
853:
833:
602:
245:
221:
155:
114:
110:
92:
3062:
on points near the origin consists of a counterclockwise rotation about the origin of
4100:
3629:
that usually are not given by a formula (these are called topological polynomials):
913:
You can also describe the Douady rabbit utilising the
Mandelbrot set with respect to
215:
131:
128:
3669:
Gray levels indicate the speed of convergence to infinity or to the attractive cycle
141:
An example of a Douady rabbit. The colors show the number of iterations required to
3876:
3779:
3730:
3438:
3160:
2905:. The three major lobes on the left, which contain the period-three fixed points
2401:
3789:
3784:
3327:
894:
693:
4086:
4078:
3862:
3347:
3323:
3274:
3250:
431:{\displaystyle z_{n+1}={\mathcal {M}}z_{n}:=\gamma z_{n}\left(1-z_{n}\right).}
4072:"Lie Methods for Nonlinear Dynamics with Applications to Accelerator Physics"
4060:
3950:
934:
290:
1244:, a point in the five-o'clock sprout of the right disk. For this value of
3901:
Note on dynamically stable perturbations of parabolics by Tomoki
Kawahira
3634:
321:
239:
4012:
24:
4033:
3853:
125:
3877:"Period-n Rabbit Renormalization. 'Rabbit's show' by Evgeny Demidov"
4079:
Adrien Douady: La dynamique du lapin (1996) - video on the YouTube
3159:
2400:
904:
893:
885:
136:
238:
is fixed to lie in one of the two period three bulb off the main
1035:
and one other point as unstable (repelling) fixed points, and
320:
escapes a bounded region, after which it will diverge toward
3580:
ears For example, a period four bulb rabbit has three ears.
3047:
2829:
2335:
2267:
2206:
2138:
2086:
2018:
1961:
1914:
1867:
1843:
1816:
1757:
1359:
1273:
1071:
994:
368:
1380:(also called period-three fixed points) have the locations
3415:
plane. The Douady rabbit is visible in the cross-section.
4085:
This article incorporates material from Douady Rabbit on
2535:, a point in the eleven-o'clock sprout on the left disk (
2478:
As a second example, Figure 5 shows a Douady rabbit when
3338:
The Julia set has no direct analog in three dimensions.
3012:, and their counterparts on the right meet at the point
3284:
is the symmetrical image of the rabbit. Here parameter
2002:
Corresponding to these relations there are the results
2528:{\displaystyle \gamma =2-\gamma _{D}=-.55268+.959456i}
37:
3566:
3513:
3398:
3356:
3290:
3259:
3208:
3170:
3127:
3100:
3068:
3044:
3018:
2992:
2965:
2938:
2911:
2876:
2850:
2826:
2564:
2541:
2484:
2446:
2411:
2011:
1958:
1911:
1864:
1840:
1813:
1784:
1753:
1717:
1681:
1645:
1389:
1355:
1320:
1294:
1270:
1250:
1237:{\displaystyle \gamma =\gamma _{D}=2.55268-0.959456i}
1202:
1182:
1176:
For example, Figure 4 shows the Douady rabbit in the
1152:
1125:
1098:
1067:
1041:
1015:
991:
971:
951:
919:
856:
836:
816:
784:
764:
737:
705:
671:
651:
631:
611:
601:) for which the iteration remains bounded. Then, the
580:
553:
533:
513:
449:
346:
299:
268:
248:
224:
163:
95:
3976:"Polynomials, dynamics, and trees by Becca Winarski"
1639:
The red, green, and yellow points lie in the basins
1778:, respectively. The white points lie in the basin
3572:
3552:
3407:
3380:
3314:
3265:
3229:
3194:
3149:
3113:
3086:
3054:
3030:
3004:
2978:
2951:
2924:
2897:
2862:
2836:
2809:
2547:
2527:
2467:
2432:
2390:
1991:
1944:
1897:
1850:
1823:
1799:
1770:
1739:
1703:
1667:
1628:
1372:
1341:
1306:
1280:
1256:
1236:
1188:
1165:
1138:
1111:
1084:
1053:
1027:
1001:
977:
957:
925:
862:
842:
822:
802:
770:
743:
711:
677:
657:
637:
617:
593:
566:
539:
519:
492:
430:
312:
281:
254:
230:
206:
101:
78:
1061:as an attracting fixed point. Moreover, the map
503:Irrespective of the specific iteration used, the
4091:Creative Commons Attribution/Share-Alike License
3121:, followed by scaling (dilation) by a factor of
1858:on these fixed points is given by the relations
909:Representation of the dynamics inside the rabbit
154:The Douady rabbit is generated by iterating the
878:for either form of the iteration given above.
8:
3441:copy of rabbit in the center of a Julia set
3330:of their post-critical set are the rabbit.
3195:{\displaystyle \gamma =-0.55268+0.959456i}
2770:
2767:
2694:
2691:
2624:
2621:
2341:
2331:
2318:
2212:
2202:
2189:
2092:
2082:
2069:
1589:
1586:
1516:
1513:
1449:
1446:
830:has additional horizontal symmetry. Since
4032:
4011:
3852:
3565:
3512:
3397:
3355:
3289:
3258:
3207:
3169:
3136:
3128:
3126:
3105:
3099:
3067:
3046:
3045:
3043:
3017:
2991:
2970:
2964:
2943:
2937:
2916:
2910:
2875:
2849:
2828:
2827:
2825:
2776:
2775:
2769:
2766:
2732:
2700:
2699:
2693:
2690:
2656:
2630:
2629:
2623:
2620:
2605:
2573:
2565:
2563:
2540:
2501:
2483:
2445:
2433:{\displaystyle \gamma =2.55268-0.959456i}
2410:
2369:
2368:
2344:
2343:
2340:
2334:
2333:
2330:
2321:
2320:
2317:
2308:
2282:
2266:
2265:
2237:
2236:
2215:
2214:
2211:
2205:
2204:
2201:
2192:
2191:
2188:
2179:
2153:
2137:
2136:
2111:
2110:
2095:
2094:
2091:
2085:
2084:
2081:
2072:
2071:
2068:
2059:
2033:
2017:
2016:
2012:
2010:
1992:{\displaystyle {\mathcal {M}}z_{3}=z_{1}}
1983:
1970:
1960:
1959:
1957:
1945:{\displaystyle {\mathcal {M}}z_{2}=z_{3}}
1936:
1923:
1913:
1912:
1910:
1898:{\displaystyle {\mathcal {M}}z_{1}=z_{2}}
1889:
1876:
1866:
1865:
1863:
1842:
1841:
1839:
1815:
1814:
1812:
1783:
1762:
1756:
1755:
1752:
1728:
1716:
1692:
1680:
1656:
1644:
1592:
1591:
1588:
1585:
1554:
1519:
1518:
1515:
1512:
1481:
1452:
1451:
1448:
1445:
1430:
1398:
1390:
1388:
1364:
1358:
1357:
1354:
1319:
1293:
1272:
1271:
1269:
1249:
1213:
1201:
1181:
1157:
1151:
1130:
1124:
1103:
1097:
1076:
1070:
1069:
1066:
1040:
1014:
993:
992:
990:
970:
950:
918:
855:
835:
815:
783:
763:
736:
704:
670:
650:
630:
610:
585:
579:
558:
552:
532:
512:
478:
473:
454:
448:
414:
393:
377:
367:
366:
351:
345:
304:
298:
273:
267:
247:
223:
192:
187:
168:
162:
94:
64:
42:
36:
3315:{\displaystyle c\approx -0.1226-0.7449i}
3800:
3659:
3590:
3560:th bulb of the main cardioid will have
3475:
3230:{\displaystyle \mu =0.122565-0.744864i}
2468:{\displaystyle \mu =0.122565-0.744864i}
1349:. The three attracting fixed points of
3038:. It can be shown that the effect of
890:Douady rabbit in an exponential family
2898:{\displaystyle z=1.450795+0.7825835i}
810:, the Mandelbrot set with respect to
16:Fractal related to the mandelbrot set
7:
803:{\displaystyle \gamma \to 2-\gamma }
778:is invariant under the substitution
3756:A Douady rabbit on a red background
3150:{\displaystyle |\gamma |=1.1072538}
493:{\displaystyle w_{n+1}=w_{n}^{2}+c}
327:It can also be described using the
207:{\displaystyle z_{n+1}=z_{n}^{2}+c}
2792:
2789:
2786:
2783:
2780:
2777:
2713:
2710:
2707:
2704:
2701:
2637:
2634:
2631:
2376:
2373:
2370:
2360:
2357:
2354:
2351:
2348:
2345:
2325:
2322:
2253:
2250:
2247:
2244:
2241:
2238:
2228:
2225:
2222:
2219:
2216:
2196:
2193:
2124:
2121:
2118:
2115:
2112:
2102:
2099:
2096:
2076:
2073:
1791:
1771:{\displaystyle {\mathcal {M}}^{3}}
1611:
1608:
1605:
1602:
1599:
1596:
1535:
1532:
1529:
1526:
1523:
1462:
1459:
1456:
1373:{\displaystyle {\mathcal {M}}^{3}}
1342:{\displaystyle z=.656747-.129015i}
1085:{\displaystyle {\mathcal {M}}^{3}}
1048:
547:) consists of all starting points
14:
507:associated with a given value of
109:is near the center of one of the
3761:
3749:
3737:
3722:
3707:
3698:
3686:
3674:
3662:
3608:
3596:
3490:
3478:
3430:
3418:
1173:and their basins of attraction.
724:
692:
3988:from the original on 2022-11-01
3957:from the original on 2019-10-23
3932:from the original on 2022-05-03
3883:from the original on 2022-05-03
3821:from the original on 2022-11-01
3507:In general, the rabbit for the
3381:{\displaystyle c=-0.123+0.745i}
3249:is the composition of a rabbit
1288:has the repelling fixed points
4089:, which is licensed under the
3547:
3535:
3137:
3129:
3081:
3075:
3055:{\displaystyle {\mathcal {M}}}
2837:{\displaystyle {\mathcal {M}}}
2820:The repelling fixed points of
2797:
2772:
2760:
2754:
2718:
2696:
2684:
2678:
2642:
2626:
2614:
2592:
2314:
2301:
2288:
2275:
2185:
2172:
2159:
2146:
2065:
2052:
2039:
2026:
1851:{\displaystyle {\mathcal {M}}}
1824:{\displaystyle {\mathcal {M}}}
1794:
1788:
1734:
1721:
1698:
1685:
1662:
1649:
1614:
1593:
1579:
1573:
1538:
1520:
1506:
1500:
1465:
1453:
1439:
1417:
1281:{\displaystyle {\mathcal {M}}}
1002:{\displaystyle {\mathcal {M}}}
788:
289:and calculating the amount of
54:
48:
1:
3457:has c at the root of the 1/3-
3087:{\displaystyle \arg(\gamma )}
293:required before the value of
79:{\textstyle f_{c}(z)=z^{2}+c}
3553:{\displaystyle period-(n+1)}
3164:Figure 5: Douady rabbit for
3114:{\displaystyle 120^{\circ }}
2405:Figure 4: Douady rabbit for
605:consists of those values of
4128:
3350:Julia set with parameters
2986:, meet at the fixed point
1800:{\displaystyle B(\infty )}
731:The Mandelbrot set in the
699:The Mandelbrot set in the
3863:10.1007/s11511-006-0007-3
3768:A chain of Douady rabbits
3744:Multibrot-4 Douady rabbit
1054:{\displaystyle z=\infty }
3906:October 2, 2006, at the
3681:Boundaries of level sets
1740:{\displaystyle B(z_{3})}
1704:{\displaystyle B(z_{2})}
1668:{\displaystyle B(z_{1})}
4056:"Douady Rabbit Fractal"
3322:. It is one of 2 other
1257:{\displaystyle \gamma }
985:, it can be shown that
978:{\displaystyle \gamma }
958:{\displaystyle \gamma }
926:{\displaystyle \gamma }
901:of the rabbit Julia set
823:{\displaystyle \gamma }
712:{\displaystyle \gamma }
658:{\displaystyle \gamma }
618:{\displaystyle \gamma }
520:{\displaystyle \gamma }
441:which is equivalent to
3621:Twisted rabbit problem
3574:
3554:
3409:
3382:
3316:
3267:
3237:
3231:
3196:
3151:
3115:
3088:
3056:
3032:
3006:
2980:
2953:
2926:
2899:
2864:
2844:itself are located at
2838:
2811:
2549:
2529:
2475:
2469:
2434:
2392:
1993:
1946:
1899:
1852:
1825:
1801:
1772:
1741:
1705:
1669:
1630:
1374:
1343:
1308:
1282:
1258:
1238:
1190:
1167:
1140:
1113:
1086:
1055:
1029:
1003:
979:
959:
927:
910:
902:
891:
872:affine transformations
864:
844:
824:
804:
772:
745:
713:
679:
659:
639:
619:
595:
568:
541:
521:
494:
432:
314:
283:
256:
232:
208:
146:
103:
80:
3615:Perturbed rabbit zoom
3575:
3555:
3467:parabolic fixed point
3410:
3383:
3317:
3268:
3232:
3197:
3163:
3152:
3116:
3089:
3057:
3033:
3007:
2981:
2979:{\displaystyle z_{3}}
2954:
2952:{\displaystyle z_{2}}
2927:
2925:{\displaystyle z_{1}}
2900:
2865:
2839:
2812:
2550:
2530:
2470:
2435:
2404:
2393:
1994:
1947:
1900:
1853:
1826:
1802:
1773:
1742:
1706:
1670:
1631:
1375:
1344:
1309:
1283:
1259:
1239:
1191:
1168:
1166:{\displaystyle z_{3}}
1141:
1139:{\displaystyle z_{2}}
1114:
1112:{\displaystyle z_{1}}
1087:
1056:
1030:
1004:
980:
960:
928:
908:
897:
889:
865:
845:
825:
805:
773:
746:
714:
680:
660:
640:
620:
596:
594:{\displaystyle w_{0}}
569:
567:{\displaystyle z_{0}}
542:
522:
495:
433:
335:complex quadratic map
315:
313:{\displaystyle z_{n}}
284:
282:{\displaystyle z_{0}}
257:
233:
209:
140:
119:complex quadratic map
104:
81:
3693:Binary decomposition
3564:
3511:
3497:Parabolic chessboard
3396:
3354:
3288:
3257:
3206:
3168:
3125:
3098:
3066:
3042:
3016:
2990:
2963:
2936:
2909:
2874:
2848:
2824:
2562:
2548:{\displaystyle \mu }
2539:
2482:
2444:
2409:
2009:
1956:
1909:
1862:
1838:
1811:
1782:
1751:
1715:
1679:
1643:
1387:
1353:
1318:
1292:
1268:
1248:
1200:
1180:
1150:
1123:
1096:
1065:
1039:
1013:
989:
969:
949:
917:
882:Detailed description
854:
834:
814:
782:
771:{\displaystyle \mu }
762:
744:{\displaystyle \mu }
735:
703:
678:{\displaystyle \mu }
669:
649:
638:{\displaystyle \mu }
629:
609:
578:
551:
540:{\displaystyle \mu }
531:
511:
447:
344:
297:
266:
246:
222:
161:
93:
35:
3031:{\displaystyle z=1}
3005:{\displaystyle z=0}
2863:{\displaystyle z=0}
1307:{\displaystyle z=0}
1028:{\displaystyle z=0}
483:
197:
4053:Weisstein, Eric W.
3570:
3550:
3408:{\displaystyle xy}
3405:
3378:
3326:inducing the same
3312:
3263:
3238:
3227:
3192:
3147:
3111:
3084:
3052:
3028:
3002:
2976:
2949:
2922:
2895:
2860:
2834:
2807:
2805:
2545:
2525:
2476:
2465:
2430:
2388:
2386:
1989:
1942:
1895:
1848:
1821:
1797:
1768:
1737:
1701:
1665:
1626:
1624:
1370:
1339:
1304:
1278:
1254:
1234:
1186:
1163:
1136:
1109:
1082:
1051:
1025:
999:
975:
955:
923:
911:
903:
892:
860:
840:
820:
800:
768:
741:
709:
675:
655:
635:
615:
591:
564:
537:
517:
490:
469:
428:
310:
279:
252:
228:
218:, where parameter
204:
183:
156:Mandelbrot set map
147:
124:It is named after
111:period three bulbs
99:
76:
3650:monodromic groups
3642:equivalence class
3588:Perturbed rabbit
3573:{\displaystyle n}
3437:A small embedded
3266:{\displaystyle n}
3094:, or very nearly
2758:0.264884797354373
2749:0.238618870661709
2682:0.239864000061970
2673:0.138169999969259
2596:6.968273875812428
2587:0.500003730675024
1577:0.107547238170383
1568:0.799901291393262
1504:0.239864000011495
1495:0.638169999974373
1421:1.221880225696050
1412:0.499997032420304
1189:{\displaystyle z}
863:{\displaystyle w}
843:{\displaystyle z}
330:logistic map form
255:{\displaystyle z}
231:{\displaystyle c}
102:{\displaystyle c}
27:derived from the
4119:
4075:
4066:
4065:
4039:
4038:
4036:
4024:
4018:
4017:
4015:
4003:
3997:
3996:
3994:
3993:
3987:
3980:
3972:
3966:
3965:
3963:
3962:
3947:
3941:
3940:
3938:
3937:
3931:
3924:
3916:
3910:
3898:
3892:
3891:
3889:
3888:
3873:
3867:
3866:
3856:
3841:Acta Mathematica
3836:
3830:
3829:
3827:
3826:
3820:
3813:
3805:
3765:
3753:
3741:
3726:
3711:
3702:
3690:
3678:
3666:
3640:determining the
3612:
3603:Perturbed rabbit
3600:
3592:Perturbed rabbit
3579:
3577:
3576:
3571:
3559:
3557:
3556:
3551:
3494:
3482:
3434:
3422:
3414:
3412:
3411:
3406:
3387:
3385:
3384:
3379:
3321:
3319:
3318:
3313:
3277:about its ears.
3272:
3270:
3269:
3264:
3236:
3234:
3233:
3228:
3201:
3199:
3198:
3193:
3156:
3154:
3153:
3148:
3140:
3132:
3120:
3118:
3117:
3112:
3110:
3109:
3093:
3091:
3090:
3085:
3061:
3059:
3058:
3053:
3051:
3050:
3037:
3035:
3034:
3029:
3011:
3009:
3008:
3003:
2985:
2983:
2982:
2977:
2975:
2974:
2958:
2956:
2955:
2950:
2948:
2947:
2931:
2929:
2928:
2923:
2921:
2920:
2904:
2902:
2901:
2896:
2869:
2867:
2866:
2861:
2843:
2841:
2840:
2835:
2833:
2832:
2816:
2814:
2813:
2808:
2806:
2796:
2795:
2771:
2768:
2737:
2736:
2717:
2716:
2695:
2692:
2661:
2660:
2641:
2640:
2625:
2622:
2613:
2612:
2578:
2577:
2554:
2552:
2551:
2546:
2534:
2532:
2531:
2526:
2506:
2505:
2474:
2472:
2471:
2466:
2439:
2437:
2436:
2431:
2397:
2395:
2394:
2389:
2387:
2380:
2379:
2364:
2363:
2342:
2339:
2338:
2332:
2329:
2328:
2319:
2313:
2312:
2287:
2286:
2271:
2270:
2257:
2256:
2232:
2231:
2213:
2210:
2209:
2203:
2200:
2199:
2190:
2184:
2183:
2158:
2157:
2142:
2141:
2128:
2127:
2106:
2105:
2093:
2090:
2089:
2083:
2080:
2079:
2070:
2064:
2063:
2038:
2037:
2022:
2021:
1998:
1996:
1995:
1990:
1988:
1987:
1975:
1974:
1965:
1964:
1951:
1949:
1948:
1943:
1941:
1940:
1928:
1927:
1918:
1917:
1904:
1902:
1901:
1896:
1894:
1893:
1881:
1880:
1871:
1870:
1857:
1855:
1854:
1849:
1847:
1846:
1830:
1828:
1827:
1822:
1820:
1819:
1806:
1804:
1803:
1798:
1777:
1775:
1774:
1769:
1767:
1766:
1761:
1760:
1746:
1744:
1743:
1738:
1733:
1732:
1710:
1708:
1707:
1702:
1697:
1696:
1674:
1672:
1671:
1666:
1661:
1660:
1635:
1633:
1632:
1627:
1625:
1618:
1617:
1590:
1587:
1559:
1558:
1542:
1541:
1517:
1514:
1486:
1485:
1469:
1468:
1450:
1447:
1438:
1437:
1403:
1402:
1379:
1377:
1376:
1371:
1369:
1368:
1363:
1362:
1348:
1346:
1345:
1340:
1313:
1311:
1310:
1305:
1287:
1285:
1284:
1279:
1277:
1276:
1263:
1261:
1260:
1255:
1243:
1241:
1240:
1235:
1218:
1217:
1195:
1193:
1192:
1187:
1172:
1170:
1169:
1164:
1162:
1161:
1145:
1143:
1142:
1137:
1135:
1134:
1118:
1116:
1115:
1110:
1108:
1107:
1091:
1089:
1088:
1083:
1081:
1080:
1075:
1074:
1060:
1058:
1057:
1052:
1034:
1032:
1031:
1026:
1008:
1006:
1005:
1000:
998:
997:
984:
982:
981:
976:
964:
962:
961:
956:
932:
930:
929:
924:
869:
867:
866:
861:
849:
847:
846:
841:
829:
827:
826:
821:
809:
807:
806:
801:
777:
775:
774:
769:
750:
748:
747:
742:
728:
718:
716:
715:
710:
696:
684:
682:
681:
676:
664:
662:
661:
656:
644:
642:
641:
636:
624:
622:
621:
616:
600:
598:
597:
592:
590:
589:
573:
571:
570:
565:
563:
562:
546:
544:
543:
538:
526:
524:
523:
518:
505:filled Julia set
499:
497:
496:
491:
482:
477:
465:
464:
437:
435:
434:
429:
424:
420:
419:
418:
398:
397:
382:
381:
372:
371:
362:
361:
319:
317:
316:
311:
309:
308:
288:
286:
285:
280:
278:
277:
261:
259:
258:
253:
237:
235:
234:
229:
213:
211:
210:
205:
196:
191:
179:
178:
108:
106:
105:
100:
85:
83:
82:
77:
69:
68:
47:
46:
31:of the function
4127:
4126:
4122:
4121:
4120:
4118:
4117:
4116:
4097:
4096:
4069:
4051:
4050:
4047:
4042:
4026:
4025:
4021:
4005:
4004:
4000:
3991:
3989:
3985:
3978:
3974:
3973:
3969:
3960:
3958:
3949:
3948:
3944:
3935:
3933:
3929:
3922:
3918:
3917:
3913:
3908:Wayback Machine
3899:
3895:
3886:
3884:
3875:
3874:
3870:
3838:
3837:
3833:
3824:
3822:
3818:
3811:
3807:
3806:
3802:
3798:
3776:
3769:
3766:
3757:
3754:
3745:
3742:
3733:
3727:
3718:
3712:
3703:
3694:
3691:
3682:
3679:
3670:
3667:
3658:
3627:complex numbers
3623:
3616:
3613:
3604:
3601:
3586:
3562:
3561:
3509:
3508:
3505:
3498:
3495:
3486:
3483:
3447:
3428:
3394:
3393:
3352:
3351:
3344:
3336:
3286:
3285:
3255:
3254:
3243:
3204:
3203:
3166:
3165:
3123:
3122:
3101:
3096:
3095:
3064:
3063:
3040:
3039:
3014:
3013:
2988:
2987:
2966:
2961:
2960:
2939:
2934:
2933:
2912:
2907:
2906:
2872:
2871:
2846:
2845:
2822:
2821:
2804:
2803:
2738:
2728:
2725:
2724:
2662:
2652:
2649:
2648:
2601:
2579:
2569:
2560:
2559:
2537:
2536:
2497:
2480:
2479:
2442:
2441:
2407:
2406:
2385:
2384:
2304:
2291:
2278:
2262:
2261:
2175:
2162:
2149:
2133:
2132:
2055:
2042:
2029:
2007:
2006:
1979:
1966:
1954:
1953:
1932:
1919:
1907:
1906:
1885:
1872:
1860:
1859:
1836:
1835:
1809:
1808:
1780:
1779:
1754:
1749:
1748:
1724:
1713:
1712:
1688:
1677:
1676:
1652:
1641:
1640:
1623:
1622:
1560:
1550:
1547:
1546:
1487:
1477:
1474:
1473:
1426:
1404:
1394:
1385:
1384:
1356:
1351:
1350:
1316:
1315:
1290:
1289:
1266:
1265:
1246:
1245:
1209:
1198:
1197:
1178:
1177:
1153:
1148:
1147:
1126:
1121:
1120:
1099:
1094:
1093:
1068:
1063:
1062:
1037:
1036:
1011:
1010:
987:
986:
967:
966:
947:
946:
915:
914:
884:
852:
851:
832:
831:
812:
811:
780:
779:
760:
759:
756:
755:
754:
753:
752:
733:
732:
729:
721:
720:
701:
700:
697:
667:
666:
647:
646:
627:
626:
607:
606:
581:
576:
575:
554:
549:
548:
529:
528:
509:
508:
450:
445:
444:
410:
403:
399:
389:
373:
347:
342:
341:
337:, specifically
300:
295:
294:
269:
264:
263:
244:
243:
220:
219:
164:
159:
158:
152:
91:
90:
60:
38:
33:
32:
17:
12:
11:
5:
4125:
4123:
4115:
4114:
4109:
4099:
4098:
4082:
4081:
4076:
4067:
4046:
4045:External links
4043:
4041:
4040:
4019:
4013:math/0510082v3
3998:
3967:
3942:
3911:
3893:
3868:
3831:
3799:
3797:
3794:
3793:
3792:
3787:
3782:
3775:
3772:
3771:
3770:
3767:
3760:
3758:
3755:
3748:
3746:
3743:
3736:
3734:
3728:
3721:
3719:
3713:
3706:
3704:
3697:
3695:
3692:
3685:
3683:
3680:
3673:
3671:
3668:
3661:
3657:
3654:
3646:
3645:
3638:
3622:
3619:
3618:
3617:
3614:
3607:
3605:
3602:
3595:
3593:
3585:
3582:
3569:
3549:
3546:
3543:
3540:
3537:
3534:
3531:
3528:
3525:
3522:
3519:
3516:
3504:
3501:
3500:
3499:
3496:
3489:
3487:
3484:
3477:
3463:Mandelbrot set
3446:
3443:
3427:
3424:
3404:
3401:
3377:
3374:
3371:
3368:
3365:
3362:
3359:
3343:
3340:
3335:
3332:
3311:
3308:
3305:
3302:
3299:
3296:
3293:
3262:
3247:twisted rabbit
3242:
3239:
3226:
3223:
3220:
3217:
3214:
3211:
3191:
3188:
3185:
3182:
3179:
3176:
3173:
3146:
3143:
3139:
3135:
3131:
3108:
3104:
3083:
3080:
3077:
3074:
3071:
3049:
3027:
3024:
3021:
3001:
2998:
2995:
2973:
2969:
2946:
2942:
2919:
2915:
2894:
2891:
2888:
2885:
2882:
2879:
2859:
2856:
2853:
2831:
2818:
2817:
2802:
2799:
2794:
2791:
2788:
2785:
2782:
2779:
2774:
2765:
2762:
2759:
2756:
2753:
2750:
2747:
2744:
2741:
2739:
2735:
2731:
2727:
2726:
2723:
2720:
2715:
2712:
2709:
2706:
2703:
2698:
2689:
2686:
2683:
2680:
2677:
2674:
2671:
2668:
2665:
2663:
2659:
2655:
2651:
2650:
2647:
2644:
2639:
2636:
2633:
2628:
2619:
2616:
2611:
2608:
2604:
2600:
2597:
2594:
2591:
2588:
2585:
2582:
2580:
2576:
2572:
2568:
2567:
2544:
2524:
2521:
2518:
2515:
2512:
2509:
2504:
2500:
2496:
2493:
2490:
2487:
2464:
2461:
2458:
2455:
2452:
2449:
2429:
2426:
2423:
2420:
2417:
2414:
2399:
2398:
2383:
2378:
2375:
2372:
2367:
2362:
2359:
2356:
2353:
2350:
2347:
2337:
2327:
2324:
2316:
2311:
2307:
2303:
2300:
2297:
2294:
2292:
2290:
2285:
2281:
2277:
2274:
2269:
2264:
2263:
2260:
2255:
2252:
2249:
2246:
2243:
2240:
2235:
2230:
2227:
2224:
2221:
2218:
2208:
2198:
2195:
2187:
2182:
2178:
2174:
2171:
2168:
2165:
2163:
2161:
2156:
2152:
2148:
2145:
2140:
2135:
2134:
2131:
2126:
2123:
2120:
2117:
2114:
2109:
2104:
2101:
2098:
2088:
2078:
2075:
2067:
2062:
2058:
2054:
2051:
2048:
2045:
2043:
2041:
2036:
2032:
2028:
2025:
2020:
2015:
2014:
1986:
1982:
1978:
1973:
1969:
1963:
1939:
1935:
1931:
1926:
1922:
1916:
1892:
1888:
1884:
1879:
1875:
1869:
1845:
1834:The action of
1818:
1796:
1793:
1790:
1787:
1765:
1759:
1736:
1731:
1727:
1723:
1720:
1700:
1695:
1691:
1687:
1684:
1664:
1659:
1655:
1651:
1648:
1637:
1636:
1621:
1616:
1613:
1610:
1607:
1604:
1601:
1598:
1595:
1584:
1581:
1578:
1575:
1572:
1569:
1566:
1563:
1561:
1557:
1553:
1549:
1548:
1545:
1540:
1537:
1534:
1531:
1528:
1525:
1522:
1511:
1508:
1505:
1502:
1499:
1496:
1493:
1490:
1488:
1484:
1480:
1476:
1475:
1472:
1467:
1464:
1461:
1458:
1455:
1444:
1441:
1436:
1433:
1429:
1425:
1422:
1419:
1416:
1413:
1410:
1407:
1405:
1401:
1397:
1393:
1392:
1367:
1361:
1338:
1335:
1332:
1329:
1326:
1323:
1303:
1300:
1297:
1275:
1253:
1233:
1230:
1227:
1224:
1221:
1216:
1212:
1208:
1205:
1185:
1160:
1156:
1133:
1129:
1106:
1102:
1079:
1073:
1050:
1047:
1044:
1024:
1021:
1018:
996:
974:
954:
922:
883:
880:
859:
839:
819:
799:
796:
793:
790:
787:
767:
740:
730:
723:
722:
708:
698:
691:
690:
689:
688:
687:
674:
654:
634:
614:
603:Mandelbrot set
588:
584:
561:
557:
536:
516:
489:
486:
481:
476:
472:
468:
463:
460:
457:
453:
439:
438:
427:
423:
417:
413:
409:
406:
402:
396:
392:
388:
385:
380:
376:
370:
365:
360:
357:
354:
350:
307:
303:
276:
272:
251:
227:
203:
200:
195:
190:
186:
182:
177:
174:
171:
167:
151:
148:
115:Mandelbrot set
98:
75:
72:
67:
63:
59:
56:
53:
50:
45:
41:
15:
13:
10:
9:
6:
4:
3:
2:
4124:
4113:
4110:
4108:
4105:
4104:
4102:
4095:
4094:
4092:
4088:
4080:
4077:
4073:
4068:
4063:
4062:
4057:
4054:
4049:
4048:
4044:
4035:
4030:
4023:
4020:
4014:
4009:
4002:
3999:
3984:
3977:
3971:
3968:
3956:
3952:
3946:
3943:
3928:
3921:
3915:
3912:
3909:
3905:
3902:
3897:
3894:
3882:
3878:
3872:
3869:
3864:
3860:
3855:
3850:
3846:
3842:
3835:
3832:
3817:
3810:
3804:
3801:
3795:
3791:
3788:
3786:
3783:
3781:
3778:
3777:
3773:
3764:
3759:
3752:
3747:
3740:
3735:
3732:
3731:external rays
3725:
3720:
3717:
3710:
3705:
3701:
3696:
3689:
3684:
3677:
3672:
3665:
3660:
3655:
3653:
3651:
3643:
3639:
3636:
3632:
3631:
3630:
3628:
3620:
3611:
3606:
3599:
3594:
3591:
3589:
3583:
3581:
3567:
3544:
3541:
3538:
3532:
3529:
3526:
3523:
3520:
3517:
3514:
3502:
3493:
3488:
3481:
3476:
3474:
3472:
3468:
3464:
3460:
3456:
3455:chubby rabbit
3452:
3444:
3442:
3440:
3435:
3433:
3425:
3423:
3421:
3416:
3402:
3399:
3391:
3390:cross-section
3375:
3372:
3369:
3366:
3363:
3360:
3357:
3349:
3341:
3339:
3333:
3331:
3329:
3325:
3309:
3306:
3303:
3300:
3297:
3294:
3291:
3283:
3278:
3276:
3260:
3252:
3248:
3240:
3224:
3221:
3218:
3215:
3212:
3209:
3189:
3186:
3183:
3180:
3177:
3174:
3171:
3162:
3158:
3144:
3141:
3133:
3106:
3102:
3078:
3072:
3069:
3025:
3022:
3019:
2999:
2996:
2993:
2971:
2967:
2944:
2940:
2917:
2913:
2892:
2889:
2886:
2883:
2880:
2877:
2857:
2854:
2851:
2800:
2763:
2757:
2751:
2748:
2745:
2742:
2740:
2733:
2729:
2721:
2687:
2681:
2675:
2672:
2669:
2666:
2664:
2657:
2653:
2645:
2617:
2609:
2606:
2602:
2598:
2595:
2589:
2586:
2583:
2581:
2574:
2570:
2558:
2557:
2556:
2542:
2522:
2519:
2516:
2513:
2510:
2507:
2502:
2498:
2494:
2491:
2488:
2485:
2462:
2459:
2456:
2453:
2450:
2447:
2427:
2424:
2421:
2418:
2415:
2412:
2403:
2381:
2365:
2309:
2305:
2298:
2295:
2293:
2283:
2279:
2272:
2258:
2233:
2180:
2176:
2169:
2166:
2164:
2154:
2150:
2143:
2129:
2107:
2060:
2056:
2049:
2046:
2044:
2034:
2030:
2023:
2005:
2004:
2003:
2000:
1984:
1980:
1976:
1971:
1967:
1937:
1933:
1929:
1924:
1920:
1890:
1886:
1882:
1877:
1873:
1832:
1785:
1763:
1729:
1725:
1718:
1693:
1689:
1682:
1657:
1653:
1646:
1619:
1582:
1576:
1570:
1567:
1564:
1562:
1555:
1551:
1543:
1509:
1503:
1497:
1494:
1491:
1489:
1482:
1478:
1470:
1442:
1434:
1431:
1427:
1423:
1420:
1414:
1411:
1408:
1406:
1399:
1395:
1383:
1382:
1381:
1365:
1336:
1333:
1330:
1327:
1324:
1321:
1301:
1298:
1295:
1251:
1231:
1228:
1225:
1222:
1219:
1214:
1210:
1206:
1203:
1183:
1174:
1158:
1154:
1131:
1127:
1104:
1100:
1077:
1045:
1042:
1022:
1019:
1016:
972:
952:
944:
940:
936:
920:
907:
900:
896:
888:
881:
879:
877:
873:
857:
837:
817:
797:
794:
791:
785:
765:
738:
727:
706:
695:
686:
672:
652:
632:
612:
604:
586:
582:
559:
555:
534:
514:
506:
501:
487:
484:
479:
474:
470:
466:
461:
458:
455:
451:
442:
425:
421:
415:
411:
407:
404:
400:
394:
390:
386:
383:
378:
374:
363:
358:
355:
352:
348:
340:
339:
338:
336:
332:
331:
325:
323:
305:
301:
292:
274:
270:
249:
241:
225:
217:
216:complex plane
201:
198:
193:
188:
184:
180:
175:
172:
169:
165:
157:
149:
144:
139:
135:
133:
132:Adrien Douady
130:
129:mathematician
127:
122:
120:
116:
112:
96:
89:
73:
70:
65:
61:
57:
51:
43:
39:
30:
26:
22:
21:Douady rabbit
4084:
4083:
4059:
4034:1906.07680v1
4022:
4001:
3990:. Retrieved
3970:
3959:. Retrieved
3945:
3934:. Retrieved
3914:
3896:
3885:. Retrieved
3871:
3854:math/0510082
3844:
3840:
3834:
3823:. Retrieved
3803:
3780:Dragon curve
3647:
3624:
3587:
3506:
3454:
3450:
3448:
3439:homeomorphic
3436:
3429:
3417:
3345:
3337:
3281:
3279:
3246:
3244:
2819:
2477:
2001:
1833:
1638:
1175:
942:
938:
912:
758:Noting that
757:
502:
443:
440:
328:
326:
153:
123:
20:
18:
3790:Siegel disc
3785:Herman ring
3635:topological
3465:. It has a
3328:permutation
3324:polynomials
3275:Dehn twists
1196:plane when
4112:Limit sets
4101:Categories
4087:PlanetMath
4070:Dragt, A.
3992:2022-05-08
3961:2020-04-07
3936:2022-05-03
3887:2022-05-03
3825:2022-05-03
3796:References
3503:n-th eared
3485:Fat rabbit
3451:fat rabbit
3348:quaternion
3273:powers of
3251:polynomial
1264:, the map
935:unit disks
899:Lamination
291:iterations
150:Background
4061:MathWorld
3584:Perturbed
3533:−
3364:−
3304:−
3298:−
3295:≈
3219:−
3210:μ
3178:−
3172:γ
3145:1.1072538
3134:γ
3107:∘
3079:γ
3073:
2890:0.7825835
2752:−
2746:−
2670:−
2607:−
2599:×
2543:μ
2511:−
2499:γ
2495:−
2486:γ
2457:−
2448:μ
2422:−
2413:γ
2366:⊆
2234:⊆
2108:⊆
1792:∞
1571:−
1498:−
1432:−
1424:×
1415:−
1331:−
1252:γ
1226:−
1211:γ
1204:γ
1049:∞
973:γ
953:γ
921:γ
818:γ
798:γ
795:−
789:→
786:γ
766:μ
739:μ
707:γ
673:μ
653:γ
633:μ
613:γ
535:μ
515:γ
408:−
387:γ
88:parameter
29:Julia set
4107:Fractals
3983:Archived
3955:Archived
3927:Archived
3904:Archived
3881:Archived
3847:: 1–51.
3816:Archived
3774:See also
3633:given a
3426:Embedded
3282:corabbit
3241:Variants
3222:0.744864
3216:0.122565
3187:0.959456
2884:1.450795
2460:0.744864
2454:0.122565
2425:0.959456
1229:0.959456
322:infinity
240:cardioid
3656:Gallery
3469:with 3
3461:of the
3392:in the
3181:0.55268
2520:.959456
2419:2.55268
1334:.129015
1328:.656747
1223:2.55268
939:sprouts
876:similar
333:of the
214:on the
113:of the
86:, when
25:fractal
3471:petals
3388:and a
3307:0.7449
3301:0.1226
2959:, and
2514:.55268
1952:, and
1711:, and
1146:, and
143:escape
126:French
117:for a
4029:arXiv
4008:arXiv
3986:(PDF)
3979:(PDF)
3930:(PDF)
3923:(PDF)
3849:arXiv
3819:(PDF)
3812:(PDF)
3729:With
3716:spine
3714:With
3373:0.745
3367:0.123
3253:with
937:with
751:plane
719:plane
23:is a
3459:limb
3449:The
3280:The
2870:and
1314:and
1009:has
943:buds
870:are
850:and
625:(or
574:(or
527:(or
242:and
3859:doi
3845:197
3453:or
3445:Fat
3202:or
3103:120
3070:arg
2440:or
1807:of
1747:of
941:or
665:or
121:.
4103::
4058:.
3981:.
3953:.
3925:.
3879:.
3857:.
3843:.
3814:.
3473:.
3346:A
3342:4D
3334:3D
3245:A
3157:.
2603:10
1999:.
1905:,
1831:.
1675:,
1428:10
1119:,
685:.
500:.
384::=
324:.
134:.
19:A
4093:.
4074:.
4064:.
4037:.
4031::
4016:.
4010::
3995:.
3964:.
3939:.
3890:.
3865:.
3861::
3851::
3828:.
3568:n
3548:)
3545:1
3542:+
3539:n
3536:(
3530:d
3527:o
3524:i
3521:r
3518:e
3515:p
3403:y
3400:x
3376:i
3370:+
3361:=
3358:c
3310:i
3292:c
3261:n
3225:i
3213:=
3190:i
3184:+
3175:=
3142:=
3138:|
3130:|
3082:)
3076:(
3048:M
3026:1
3023:=
3020:z
3000:0
2997:=
2994:z
2972:3
2968:z
2945:2
2941:z
2932:,
2918:1
2914:z
2893:i
2887:+
2881:=
2878:z
2858:0
2855:=
2852:z
2830:M
2801:.
2798:)
2793:w
2790:o
2787:l
2784:l
2781:e
2778:y
2773:(
2764:i
2761:)
2755:(
2743:=
2734:3
2730:z
2722:,
2719:)
2714:n
2711:e
2708:e
2705:r
2702:g
2697:(
2688:i
2685:)
2679:(
2676:+
2667:=
2658:2
2654:z
2646:,
2643:)
2638:d
2635:e
2632:r
2627:(
2618:i
2615:)
2610:6
2593:(
2590:+
2584:=
2575:1
2571:z
2523:i
2517:+
2508:=
2503:D
2492:2
2489:=
2463:i
2451:=
2428:i
2416:=
2382:.
2377:d
2374:e
2371:r
2361:w
2358:o
2355:l
2352:l
2349:e
2346:y
2336:M
2326:r
2323:o
2315:)
2310:1
2306:z
2302:(
2299:B
2296:=
2289:)
2284:3
2280:z
2276:(
2273:B
2268:M
2259:,
2254:w
2251:o
2248:l
2245:l
2242:e
2239:y
2229:n
2226:e
2223:e
2220:r
2217:g
2207:M
2197:r
2194:o
2186:)
2181:3
2177:z
2173:(
2170:B
2167:=
2160:)
2155:2
2151:z
2147:(
2144:B
2139:M
2130:,
2125:n
2122:e
2119:e
2116:r
2113:g
2103:d
2100:e
2097:r
2087:M
2077:r
2074:o
2066:)
2061:2
2057:z
2053:(
2050:B
2047:=
2040:)
2035:1
2031:z
2027:(
2024:B
2019:M
1985:1
1981:z
1977:=
1972:3
1968:z
1962:M
1938:3
1934:z
1930:=
1925:2
1921:z
1915:M
1891:2
1887:z
1883:=
1878:1
1874:z
1868:M
1844:M
1817:M
1795:)
1789:(
1786:B
1764:3
1758:M
1735:)
1730:3
1726:z
1722:(
1719:B
1699:)
1694:2
1690:z
1686:(
1683:B
1663:)
1658:1
1654:z
1650:(
1647:B
1620:.
1615:)
1612:w
1609:o
1606:l
1603:l
1600:e
1597:y
1594:(
1583:i
1580:)
1574:(
1565:=
1556:3
1552:z
1544:,
1539:)
1536:n
1533:e
1530:e
1527:r
1524:g
1521:(
1510:i
1507:)
1501:(
1492:=
1483:2
1479:z
1471:,
1466:)
1463:d
1460:e
1457:r
1454:(
1443:i
1440:)
1435:6
1418:(
1409:=
1400:1
1396:z
1366:3
1360:M
1337:i
1325:=
1322:z
1302:0
1299:=
1296:z
1274:M
1232:i
1220:=
1215:D
1207:=
1184:z
1159:3
1155:z
1132:2
1128:z
1105:1
1101:z
1078:3
1072:M
1046:=
1043:z
1023:0
1020:=
1017:z
995:M
858:w
838:z
792:2
587:0
583:w
560:0
556:z
488:c
485:+
480:2
475:n
471:w
467:=
462:1
459:+
456:n
452:w
426:.
422:)
416:n
412:z
405:1
401:(
395:n
391:z
379:n
375:z
369:M
364:=
359:1
356:+
353:n
349:z
306:n
302:z
275:0
271:z
250:z
226:c
202:c
199:+
194:2
189:n
185:z
181:=
176:1
173:+
170:n
166:z
145:.
97:c
74:c
71:+
66:2
62:z
58:=
55:)
52:z
49:(
44:c
40:f
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
