511:
1113:
1100:
1505:
891:
500:
1329:
884:
737:
1644:
3986:
Orbit portraits turn out to be useful combinatorial objects in studying the connection between the dynamics and the parameter spaces of other families of maps as well. In particular, they have been used to study the patterns of all periodic dynamical rays landing on a periodic cycle of a unicritical
2152:
that every formal orbit portrait is realized by the actual orbit portrait of a periodic orbit of some quadratic one-complex-dimensional map. Orbit portraits contain dynamical information about how external rays and their landing points map in the plane, but formal orbit portraits are no more than
1095:{\displaystyle {\mathcal {P}}=\left\{\left({\frac {11}{31}},{\frac {12}{31}}\right),\left({\frac {22}{31}},{\frac {24}{31}}\right),\left({\frac {13}{31}},{\frac {17}{31}}\right),\left({\frac {26}{31}},{\frac {3}{31}}\right),\left({\frac {21}{31}},{\frac {6}{31}}\right)\right\rbrace }
1744:
1500:{\displaystyle {\mathcal {P}}=\left\{\left({\frac {74}{511}},{\frac {81}{511}},{\frac {137}{511}}\right),\left({\frac {148}{511}},{\frac {162}{511}},{\frac {274}{511}}\right),\left({\frac {296}{511}},{\frac {324}{511}},{\frac {37}{511}}\right)\right\rbrace }
1316:
1199:
747:
600:
1517:
3032:
which is strictly contained within every other critical value arc. The characteristic arc is a complete invariant of an orbit portrait, in the sense that two orbit portraits are identical if and only if they have the same characteristic arc.
593:
260:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to
Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle f(z_j) =
428:
2210:. An alternative definition is that an orbit portrait is nontrivial if it is maximal, which in this case means that there is no orbit portrait that strictly contains it (i.e. there does not exist an orbit portrait
167:
2270:
3146:. Sectors are naturally identified the complementary arcs based at the same point. The angular width of a sector is defined as the length of its corresponding complementary arc. Sectors are called
1655:
236:
1512:
For parabolic julia set c = -1.125 + 0.21650635094611*i. It is a root point between period 2 and period 6 components of
Mandelbrot set. Orbit portrait of period 2 orbit with valence 3 is :
3641:
2315:
3520:
3026:
2086:
1842:
2963:
2487:
2921:
2428:
2142:
110:
879:{\displaystyle {\mathcal {P}}=\left\{\left({\frac {4}{9}},{\frac {5}{9}}\right),\left({\frac {8}{9}},{\frac {1}{9}}\right),\left({\frac {7}{9}},{\frac {2}{9}}\right)\right\rbrace }
732:{\displaystyle {\mathcal {P}}=\left\{\left({\frac {3}{7}},{\frac {4}{7}}\right),\left({\frac {6}{7}},{\frac {1}{7}}\right),\left({\frac {5}{7}},{\frac {2}{7}}\right)\right\rbrace }
1639:{\displaystyle {\mathcal {P}}=\left\{\left({\frac {22}{63}},{\frac {25}{63}},{\frac {37}{63}}\right),\left({\frac {11}{63}},{\frac {44}{63}},{\frac {50}{63}}\right)\right\rbrace }
1240:
1957:
3700:
1123:
3975:
3911:
3578:
3419:
1233:
3752:
3665:
3602:
3487:
3445:. These two parameter rays, along with their common landing point, split the parameter space into two open components. Let the component that does not contain the point
3443:
3390:
3320:
2595:
2364:
1775:
455:
2867:
2840:
3946:
2723:
2690:
2657:
1908:
3882:
3855:
3292:
3265:
3226:
3144:
3097:
3066:
2993:
2773:
2624:
2568:
2541:
2514:
2455:
2186:
2050:
1875:
1806:
530:
485:
343:
308:
3828:
3802:
2208:
2000:
3772:
3728:
3463:
3195:
3175:
3117:
2887:
2813:
2793:
2746:
2020:
1977:
281:
256:
173:
3366:
514:
Julia set with period-two parabolic orbit. The associated orbit portrait has characteristic arc I = (22/63, 25/63) and valence v = 3 rays per orbit point.
351:
125:
1959:
are periodic under the doubling map of the circle, and all of the angles have the same exact period. This period must be a multiple of
510:
2370:. Trivial orbit portraits are pathological in some respects, and in the sequel we will refer only to nontrivial orbit portraits.
2213:
1509:
Rays for above angles land on points of that orbit . Parameter c is a center of period 9 hyperbolic component of
Mandelbrot set.
1739:{\displaystyle {\mathcal {P}}=\left\{\left({\frac {1}{15}},{\frac {2}{15}},{\frac {4}{15}},{\frac {8}{15}}\right)\right\rbrace }
182:
2317:). It is easy to see that every trivial formal orbit portrait is realized as the orbit portrait of some orbit of the map
2692:
once, except for a single arc, which it covers twice. The arc that it covers twice is called the critical value arc for
2570:
has a unique largest arc based at it, which is called its critical arc. The critical arc always has length greater than
3607:
3523:
3198:
1321:
52:
29:
3421:
be the common landing point of the two external angles in parameter space corresponding to the characteristic arc of
2275:
4162:
3492:
2998:
2055:
1811:
4114:
25:
2926:
3041:
Much as the rays landing on the orbit divide up the circle, they divide up the complex plane. For every point
2460:
2892:
2381:
2095:
1311:{\displaystyle {\mathcal {P}}=\left\{\left({\frac {1}{7}},{\frac {2}{7}},{\frac {4}{7}}\right)\right\rbrace }
60:
1194:{\displaystyle {\mathcal {R}}_{\frac {1}{7}},{\mathcal {R}}_{\frac {2}{7}},{\mathcal {R}}_{\frac {4}{7}}\,}
4058:
Milnor, John W. (1999). "Periodic Orbits, Externals Rays and the
Mandelbrot Set: An Expository Account".
2153:
combinatorial objects. Milnor's theorem states that, in truth, there is no distinction between the two.
1916:
3996:
3670:
2052:
are pairwise unlinked, which is to say that given any pair of them, there are two disjoint intervals of
3951:
3887:
3528:
3395:
4073:
3710:
Other than the zero portrait, there are two types of orbit portraits: primitive and satellite. If
2516:
divides the circle into a number of disjoint intervals, called complementary arcs based at the point
1204:
3733:
3646:
3583:
3468:
3424:
3371:
3301:
2573:
2320:
1756:
436:
3948:. In this case, all of the angles make up a single orbit under the doubling map. Additionally,
1112:
4126:
4063:
4040:
2845:
2818:
3919:
588:{\displaystyle {\mathcal {P}}=\left\{\left({\frac {1}{3}},{\frac {2}{3}}\right)\right\rbrace }
2695:
2662:
2629:
1880:
4136:
4032:
177:
3860:
3833:
3270:
3243:
3204:
3122:
3075:
3044:
2971:
2751:
2602:
2546:
2519:
2492:
2433:
2164:
2028:
1853:
1784:
463:
321:
286:
2366:, since every external ray of this map lands, and they all land at distinct points of the
3807:
3781:
2191:
4077:
1982:
3757:
3713:
3448:
3295:
3180:
3160:
3102:
2872:
2798:
2778:
2731:
2005:
1962:
266:
241:
4017:
3325:
3154:
when the corresponding arcs are, respectively, critical arcs and critical value arcs.
4156:
116:
4141:
4044:
423:{\displaystyle {\mathcal {P}}={\mathcal {P}}({\mathcal {O}})=\{A_{1},\ldots A_{n}\}}
3774:
is the recurrent ray period, then these two types may be characterized as follows:
3237:
3069:
1847:
1117:
315:
311:
3884:
is a pair of angles, each in a distinct orbit of the doubling map. In this case,
4101:
2149:
17:
4090:
2144:
of subsets of the circle which satisfy these four properties above is called a
4036:
2543:. The length of each interval is referred to as its angular width. Each
2367:
503:
1324:
with c= -0.03111+0.79111*i portrait of parabolic period 3 orbit is :
2188:
have only a single element are called trivial, except for orbit portrait
162:{\displaystyle f_{c}:\mathbb {\mathbb {C} } \to \mathbb {\mathbb {C} } }
2626:, except for the critical arc, maps diffeomorphically to an arc based
39:
a list of external angles for which rays land on points of that orbit
4068:
499:
4131:
1111:
509:
498:
3298:
in parameter space if and only if there exists an orbit portrait
2869:
is in every critical value arc. Also, the two inverse images of
3977:
is the base point of a parabolic bifurcation in parameter space.
3913:
is the base point of a baby
Mandelbrot set in parameter space.
3667:
is realized with a parabolic orbit only for the single value
4115:"Orbit portraits of unicritical antiholomorphic polynomials"
3961:
3897:
3739:
3686:
3652:
3627:
3620:
3589:
3506:
3499:
3474:
3430:
3405:
3377:
3307:
3012:
3005:
2302:
2254:
2230:
1762:
1661:
1523:
1335:
1246:
1174:
1152:
1130:
897:
753:
606:
536:
442:
377:
367:
357:
188:
2725:. This is not necessarily distinct from the critical arc.
2265:{\displaystyle \{A_{1}^{\prime },\ldots ,A_{n}^{\prime }\}}
487:
must have the same number of elements, which is called the
4018:"Boundaries of Bounded Fatou Components of Quadratic Maps"
3954:
3922:
3890:
3863:
3836:
3810:
3784:
3760:
3736:
3716:
3673:
3649:
3610:
3586:
3531:
3495:
3471:
3451:
3427:
3398:
3374:
3368:
as its characteristic arc. For any orbit portrait
3328:
3304:
3273:
3246:
3207:
3183:
3163:
3125:
3105:
3078:
3047:
3001:
2974:
2929:
2895:
2875:
2848:
2821:
2801:
2781:
2754:
2734:
2698:
2665:
2632:
2605:
2599:
These arcs have the property that every arc based at
2576:
2549:
2522:
2495:
2463:
2436:
2384:
2323:
2278:
2216:
2194:
2167:
2098:
2058:
2031:
2008:
1985:
1965:
1919:
1883:
1856:
1814:
1787:
1759:
1658:
1520:
1332:
1243:
1207:
1126:
894:
750:
603:
533:
466:
439:
354:
324:
289:
269:
244:
231:{\displaystyle {\mathcal {O}}=\{z_{1},\ldots z_{n}\}}
185:
128:
63:
3830:. Every ray in the portrait is mapped to itself by
2968:
Among all of the critical value arcs for all of the
2815:
has a well-defined external angle. Call this angle
4102:
Periodic orbits and external rays by Evgeny
Demidov
3969:
3940:
3905:
3876:
3849:
3822:
3796:
3766:
3746:
3722:
3694:
3659:
3636:{\displaystyle c\in {\mathcal {W}}_{\mathcal {P}}}
3635:
3596:
3572:
3514:
3481:
3457:
3437:
3413:
3384:
3360:
3314:
3286:
3259:
3220:
3189:
3169:
3138:
3111:
3091:
3060:
3020:
2995:'s, there is a unique smallest critical value arc
2987:
2957:
2915:
2881:
2861:
2834:
2807:
2787:
2767:
2740:
2717:
2684:
2651:
2618:
2589:
2562:
2535:
2508:
2481:
2449:
2422:
2358:
2309:
2264:
2202:
2180:
2136:
2080:
2044:
2014:
1994:
1971:
1951:
1902:
1869:
1836:
1800:
1769:
1738:
1638:
1499:
1310:
1227:
1193:
1094:
878:
731:
587:
479:
449:
422:
337:
302:
275:
250:
230:
161:
104:
2659:, and the critical arc covers every arc based at
4025:Journal of Difference Equations and Applications
3157:Sectors also have the interesting property that
2310:{\displaystyle A_{j}\subsetneq A_{j}^{\prime }}
1109:Valence is 3 so rays land on each orbit point.
3177:is in the critical sector of every point, and
35:In simple words one can say that it is :
3515:{\displaystyle {\mathcal {W}}_{\mathcal {P}}}
3021:{\displaystyle {\mathcal {I}}_{\mathcal {P}}}
2088:where each interval contains one of the sets.
2081:{\displaystyle {\mathbb {R} }/{\mathbb {Z} }}
1837:{\displaystyle {\mathbb {R} }/{\mathbb {Z} }}
8:
2417:
2385:
2259:
2217:
2131:
2099:
506:with external rays landing on period-3 orbit
417:
388:
225:
196:
259:
2958:{\displaystyle {\frac {\theta _{c}+1}{2}}}
4140:
4130:
4067:
3960:
3959:
3953:
3921:
3896:
3895:
3889:
3868:
3862:
3841:
3835:
3809:
3783:
3759:
3738:
3737:
3735:
3715:
3685:
3684:
3672:
3651:
3650:
3648:
3626:
3625:
3619:
3618:
3609:
3588:
3587:
3585:
3558:
3536:
3530:
3505:
3504:
3498:
3497:
3494:
3473:
3472:
3470:
3450:
3429:
3428:
3426:
3404:
3403:
3397:
3376:
3375:
3373:
3349:
3336:
3327:
3306:
3305:
3303:
3278:
3272:
3251:
3245:
3212:
3206:
3182:
3162:
3130:
3124:
3104:
3083:
3077:
3052:
3046:
3011:
3010:
3004:
3003:
3000:
2979:
2973:
2937:
2930:
2928:
2902:
2896:
2894:
2874:
2853:
2847:
2826:
2820:
2800:
2780:
2759:
2753:
2733:
2703:
2697:
2670:
2664:
2637:
2631:
2610:
2604:
2577:
2575:
2554:
2548:
2527:
2521:
2500:
2494:
2482:{\displaystyle \mathbb {R} /\mathbb {Z} }
2475:
2474:
2469:
2465:
2464:
2462:
2441:
2435:
2411:
2392:
2383:
2350:
2328:
2322:
2301:
2296:
2283:
2277:
2253:
2248:
2229:
2224:
2215:
2195:
2193:
2172:
2166:
2125:
2106:
2097:
2073:
2072:
2071:
2066:
2061:
2060:
2059:
2057:
2036:
2030:
2007:
1984:
1964:
1943:
1924:
1918:
1910:and preserves cyclic order of the angles.
1888:
1882:
1861:
1855:
1829:
1828:
1827:
1822:
1817:
1816:
1815:
1813:
1792:
1786:
1761:
1760:
1758:
1717:
1704:
1691:
1678:
1660:
1659:
1657:
1616:
1603:
1590:
1567:
1554:
1541:
1522:
1521:
1519:
1477:
1464:
1451:
1428:
1415:
1402:
1379:
1366:
1353:
1334:
1333:
1331:
1289:
1276:
1263:
1245:
1244:
1242:
1224:
1218:
1206:
1190:
1179:
1173:
1172:
1157:
1151:
1150:
1135:
1129:
1128:
1125:
1072:
1059:
1036:
1023:
1000:
987:
964:
951:
928:
915:
896:
895:
893:
856:
843:
820:
807:
784:
771:
752:
751:
749:
709:
696:
673:
660:
637:
624:
605:
604:
602:
566:
553:
535:
534:
532:
471:
465:
441:
440:
438:
411:
395:
376:
375:
366:
365:
356:
355:
353:
329:
323:
294:
288:
268:
243:
219:
203:
187:
186:
184:
154:
153:
152:
144:
143:
142:
133:
127:
87:
68:
62:
2916:{\displaystyle {\frac {\theta _{c}}{2}}}
432:the orbit portrait of the periodic orbit
4008:
3706:Primitive and satellite orbit portraits
2748:escapes to infinity under iteration of
2423:{\displaystyle \{A_{1},\ldots ,A_{n}\}}
2137:{\displaystyle \{A_{1},\ldots ,A_{n}\}}
263:(where subscripts are taken 1 + modulo
105:{\displaystyle f_{c}:z\mapsto z^{2}+c.}
30:one-complex dimensional quadratic maps
2161:Orbit portrait where all of the sets
1913:All of the angles in all of the sets
1850:on the circle gives a bijection from
519:Parabolic or repelling orbit portrait
7:
3730:is the valence of an orbit portrait
3604:with a repelling orbit exactly when
3228:, is in the critical value sector.
2022:is called the recurrent ray period.
1952:{\displaystyle A_{1},\ldots ,A_{n}}
3695:{\displaystyle c=r_{\mathcal {P}}}
3119:open sets called sectors based at
2965:) are both in every critical arc.
28:for understanding the behavior of
14:
4091:Chaotic 1D maps by Evgeny Demidov
2457:is a finite subset of the circle
4016:Flek, Ross; Keen, Linda (2010).
3970:{\displaystyle r_{\mathcal {P}}}
3906:{\displaystyle r_{\mathcal {P}}}
3573:{\displaystyle f_{c}(z)=z^{2}+c}
3414:{\displaystyle r_{\mathcal {P}}}
24:is a combinatorial tool used in
4142:10.1090/S1088-4173-2015-00276-3
4119:Conformal Geometry and Dynamics
3916:Satellite orbit portraits have
3778:Primitive orbit portraits have
1979:, so the period is of the form
1228:{\displaystyle z=\alpha _{c}\,}
4113:Mukherjee, Sabyasachi (2015).
3747:{\displaystyle {\mathcal {P}}}
3660:{\displaystyle {\mathcal {P}}}
3597:{\displaystyle {\mathcal {P}}}
3548:
3542:
3482:{\displaystyle {\mathcal {P}}}
3438:{\displaystyle {\mathcal {P}}}
3385:{\displaystyle {\mathcal {P}}}
3355:
3329:
3315:{\displaystyle {\mathcal {P}}}
3294:land at the same point of the
2590:{\displaystyle {\frac {1}{2}}}
2359:{\displaystyle f_{0}(z)=z^{2}}
2340:
2334:
1777:has the following properties:
1770:{\displaystyle {\mathcal {P}}}
450:{\displaystyle {\mathcal {O}}}
382:
372:
149:
80:
1:
3987:anti-holomorphic polynomial.
3580:realizes the orbit portrait
1322:complex quadratic polynomial
1201:, which land on fixed point
2862:{\displaystyle \theta _{c}}
2835:{\displaystyle \theta _{c}}
4179:
2795:is in the Julia set, then
4037:10.1080/10236190903205080
3941:{\displaystyle r=v\geq 2}
1120:of period 3 cycle :
2889:under the doubling map (
42:graph showing above list
2718:{\displaystyle z_{j+1}}
2685:{\displaystyle z_{j+1}}
2652:{\displaystyle z_{j+1}}
2157:Trivial orbit portraits
1903:{\displaystyle A_{j+1}}
3971:
3942:
3907:
3878:
3851:
3824:
3798:
3768:
3748:
3724:
3696:
3661:
3637:
3598:
3574:
3516:
3483:
3459:
3439:
3415:
3386:
3362:
3316:
3288:
3261:
3222:
3191:
3171:
3152:critical value sectors
3140:
3113:
3099:divide the plane into
3093:
3062:
3022:
2989:
2959:
2917:
2883:
2863:
2836:
2809:
2789:
2769:
2742:
2719:
2686:
2653:
2620:
2591:
2564:
2537:
2510:
2483:
2451:
2424:
2360:
2311:
2266:
2204:
2182:
2148:. It is a theorem of
2138:
2082:
2046:
2016:
1996:
1973:
1953:
1904:
1871:
1838:
1808:is a finite subset of
1802:
1771:
1749:Formal orbit portraits
1740:
1640:
1501:
1312:
1235:
1229:
1195:
1096:
880:
733:
589:
515:
507:
481:
451:
424:
339:
304:
277:
252:
232:
174:repelling or parabolic
163:
106:
3972:
3943:
3908:
3879:
3877:{\displaystyle A_{j}}
3852:
3850:{\displaystyle f^{n}}
3825:
3799:
3769:
3749:
3725:
3697:
3662:
3638:
3599:
3575:
3517:
3489:-wake and denoted as
3484:
3460:
3440:
3416:
3387:
3363:
3317:
3289:
3287:{\displaystyle t_{+}}
3262:
3260:{\displaystyle t_{-}}
3223:
3221:{\displaystyle f_{c}}
3192:
3172:
3141:
3139:{\displaystyle z_{j}}
3114:
3094:
3092:{\displaystyle z_{j}}
3063:
3061:{\displaystyle z_{j}}
3023:
2990:
2988:{\displaystyle A_{j}}
2960:
2918:
2884:
2864:
2837:
2810:
2790:
2770:
2768:{\displaystyle f_{c}}
2743:
2720:
2687:
2654:
2621:
2619:{\displaystyle z_{j}}
2592:
2565:
2563:{\displaystyle z_{j}}
2538:
2536:{\displaystyle z_{j}}
2511:
2509:{\displaystyle A_{j}}
2484:
2452:
2450:{\displaystyle A_{j}}
2425:
2378:In an orbit portrait
2361:
2312:
2267:
2205:
2183:
2181:{\displaystyle A_{j}}
2146:formal orbit portrait
2139:
2083:
2047:
2045:{\displaystyle A_{j}}
2017:
1997:
1974:
1954:
1905:
1872:
1870:{\displaystyle A_{j}}
1839:
1803:
1801:{\displaystyle A_{j}}
1772:
1753:Every orbit portrait
1741:
1641:
1502:
1313:
1230:
1196:
1115:
1097:
881:
734:
590:
513:
502:
482:
480:{\displaystyle A_{j}}
452:
425:
340:
338:{\displaystyle z_{j}}
305:
303:{\displaystyle A_{j}}
278:
253:
233:
164:
107:
3952:
3920:
3888:
3861:
3834:
3808:
3782:
3758:
3734:
3714:
3671:
3647:
3608:
3584:
3529:
3524:quadratic polynomial
3493:
3469:
3449:
3425:
3396:
3372:
3326:
3302:
3271:
3244:
3205:
3181:
3161:
3123:
3103:
3076:
3045:
2999:
2972:
2927:
2893:
2873:
2846:
2819:
2799:
2779:
2752:
2732:
2696:
2663:
2630:
2603:
2574:
2547:
2520:
2493:
2461:
2434:
2382:
2321:
2276:
2214:
2192:
2165:
2096:
2056:
2029:
2006:
1983:
1963:
1917:
1881:
1854:
1812:
1785:
1757:
1656:
1518:
1330:
1241:
1205:
1124:
892:
748:
601:
531:
464:
437:
352:
322:
314:whose corresponding
287:
267:
242:
183:
126:
61:
4078:1999math......5169M
3823:{\displaystyle v=2}
3797:{\displaystyle r=1}
2306:
2258:
2234:
2203:{\displaystyle {0}}
3967:
3938:
3903:
3874:
3847:
3820:
3794:
3764:
3744:
3720:
3692:
3657:
3633:
3594:
3570:
3512:
3479:
3455:
3435:
3411:
3382:
3358:
3322:with the interval
3312:
3284:
3257:
3218:
3187:
3167:
3136:
3109:
3089:
3068:of the orbit, the
3058:
3030:characteristic arc
3018:
2985:
2955:
2913:
2879:
2859:
2832:
2805:
2785:
2765:
2738:
2715:
2682:
2649:
2616:
2587:
2560:
2533:
2506:
2479:
2447:
2420:
2356:
2307:
2292:
2262:
2244:
2220:
2200:
2178:
2134:
2078:
2042:
2012:
1995:{\displaystyle rn}
1992:
1969:
1949:
1900:
1867:
1834:
1798:
1767:
1736:
1636:
1497:
1308:
1236:
1225:
1191:
1092:
876:
729:
585:
516:
508:
477:
447:
420:
335:
300:
273:
262:
248:
228:
159:
102:
4163:Dynamical systems
3767:{\displaystyle r}
3723:{\displaystyle v}
3458:{\displaystyle 0}
3190:{\displaystyle c}
3170:{\displaystyle 0}
3112:{\displaystyle v}
2953:
2911:
2882:{\displaystyle c}
2808:{\displaystyle c}
2788:{\displaystyle c}
2741:{\displaystyle c}
2585:
2015:{\displaystyle r}
1972:{\displaystyle n}
1725:
1712:
1699:
1686:
1624:
1611:
1598:
1575:
1562:
1549:
1485:
1472:
1459:
1436:
1423:
1410:
1387:
1374:
1361:
1297:
1284:
1271:
1187:
1165:
1143:
1080:
1067:
1044:
1031:
1008:
995:
972:
959:
936:
923:
864:
851:
828:
815:
792:
779:
717:
704:
681:
668:
645:
632:
574:
561:
491:of the portrait.
276:{\displaystyle n}
251:{\displaystyle f}
4170:
4147:
4146:
4144:
4134:
4110:
4104:
4099:
4093:
4088:
4082:
4081:
4071:
4055:
4049:
4048:
4031:(5–6): 555–572.
4022:
4013:
3976:
3974:
3973:
3968:
3966:
3965:
3964:
3947:
3945:
3944:
3939:
3912:
3910:
3909:
3904:
3902:
3901:
3900:
3883:
3881:
3880:
3875:
3873:
3872:
3856:
3854:
3853:
3848:
3846:
3845:
3829:
3827:
3826:
3821:
3803:
3801:
3800:
3795:
3773:
3771:
3770:
3765:
3753:
3751:
3750:
3745:
3743:
3742:
3729:
3727:
3726:
3721:
3701:
3699:
3698:
3693:
3691:
3690:
3689:
3666:
3664:
3663:
3658:
3656:
3655:
3642:
3640:
3639:
3634:
3632:
3631:
3630:
3624:
3623:
3603:
3601:
3600:
3595:
3593:
3592:
3579:
3577:
3576:
3571:
3563:
3562:
3541:
3540:
3521:
3519:
3518:
3513:
3511:
3510:
3509:
3503:
3502:
3488:
3486:
3485:
3480:
3478:
3477:
3464:
3462:
3461:
3456:
3444:
3442:
3441:
3436:
3434:
3433:
3420:
3418:
3417:
3412:
3410:
3409:
3408:
3391:
3389:
3388:
3383:
3381:
3380:
3367:
3365:
3364:
3361:{\displaystyle }
3359:
3354:
3353:
3341:
3340:
3321:
3319:
3318:
3313:
3311:
3310:
3293:
3291:
3290:
3285:
3283:
3282:
3266:
3264:
3263:
3258:
3256:
3255:
3227:
3225:
3224:
3219:
3217:
3216:
3196:
3194:
3193:
3188:
3176:
3174:
3173:
3168:
3148:critical sectors
3145:
3143:
3142:
3137:
3135:
3134:
3118:
3116:
3115:
3110:
3098:
3096:
3095:
3090:
3088:
3087:
3067:
3065:
3064:
3059:
3057:
3056:
3027:
3025:
3024:
3019:
3017:
3016:
3015:
3009:
3008:
2994:
2992:
2991:
2986:
2984:
2983:
2964:
2962:
2961:
2956:
2954:
2949:
2942:
2941:
2931:
2922:
2920:
2919:
2914:
2912:
2907:
2906:
2897:
2888:
2886:
2885:
2880:
2868:
2866:
2865:
2860:
2858:
2857:
2841:
2839:
2838:
2833:
2831:
2830:
2814:
2812:
2811:
2806:
2794:
2792:
2791:
2786:
2774:
2772:
2771:
2766:
2764:
2763:
2747:
2745:
2744:
2739:
2724:
2722:
2721:
2716:
2714:
2713:
2691:
2689:
2688:
2683:
2681:
2680:
2658:
2656:
2655:
2650:
2648:
2647:
2625:
2623:
2622:
2617:
2615:
2614:
2596:
2594:
2593:
2588:
2586:
2578:
2569:
2567:
2566:
2561:
2559:
2558:
2542:
2540:
2539:
2534:
2532:
2531:
2515:
2513:
2512:
2507:
2505:
2504:
2488:
2486:
2485:
2480:
2478:
2473:
2468:
2456:
2454:
2453:
2448:
2446:
2445:
2429:
2427:
2426:
2421:
2416:
2415:
2397:
2396:
2365:
2363:
2362:
2357:
2355:
2354:
2333:
2332:
2316:
2314:
2313:
2308:
2305:
2300:
2288:
2287:
2271:
2269:
2268:
2263:
2257:
2252:
2233:
2228:
2209:
2207:
2206:
2201:
2199:
2187:
2185:
2184:
2179:
2177:
2176:
2143:
2141:
2140:
2135:
2130:
2129:
2111:
2110:
2087:
2085:
2084:
2079:
2077:
2076:
2070:
2065:
2064:
2051:
2049:
2048:
2043:
2041:
2040:
2021:
2019:
2018:
2013:
2001:
1999:
1998:
1993:
1978:
1976:
1975:
1970:
1958:
1956:
1955:
1950:
1948:
1947:
1929:
1928:
1909:
1907:
1906:
1901:
1899:
1898:
1876:
1874:
1873:
1868:
1866:
1865:
1843:
1841:
1840:
1835:
1833:
1832:
1826:
1821:
1820:
1807:
1805:
1804:
1799:
1797:
1796:
1776:
1774:
1773:
1768:
1766:
1765:
1745:
1743:
1742:
1737:
1735:
1731:
1727:
1726:
1718:
1713:
1705:
1700:
1692:
1687:
1679:
1665:
1664:
1645:
1643:
1642:
1637:
1635:
1631:
1630:
1626:
1625:
1617:
1612:
1604:
1599:
1591:
1581:
1577:
1576:
1568:
1563:
1555:
1550:
1542:
1527:
1526:
1506:
1504:
1503:
1498:
1496:
1492:
1491:
1487:
1486:
1478:
1473:
1465:
1460:
1452:
1442:
1438:
1437:
1429:
1424:
1416:
1411:
1403:
1393:
1389:
1388:
1380:
1375:
1367:
1362:
1354:
1339:
1338:
1317:
1315:
1314:
1309:
1307:
1303:
1299:
1298:
1290:
1285:
1277:
1272:
1264:
1250:
1249:
1234:
1232:
1231:
1226:
1223:
1222:
1200:
1198:
1197:
1192:
1189:
1188:
1180:
1178:
1177:
1167:
1166:
1158:
1156:
1155:
1145:
1144:
1136:
1134:
1133:
1101:
1099:
1098:
1093:
1091:
1087:
1086:
1082:
1081:
1073:
1068:
1060:
1050:
1046:
1045:
1037:
1032:
1024:
1014:
1010:
1009:
1001:
996:
988:
978:
974:
973:
965:
960:
952:
942:
938:
937:
929:
924:
916:
901:
900:
885:
883:
882:
877:
875:
871:
870:
866:
865:
857:
852:
844:
834:
830:
829:
821:
816:
808:
798:
794:
793:
785:
780:
772:
757:
756:
738:
736:
735:
730:
728:
724:
723:
719:
718:
710:
705:
697:
687:
683:
682:
674:
669:
661:
651:
647:
646:
638:
633:
625:
610:
609:
594:
592:
591:
586:
584:
580:
576:
575:
567:
562:
554:
540:
539:
486:
484:
483:
478:
476:
475:
460:All of the sets
456:
454:
453:
448:
446:
445:
429:
427:
426:
421:
416:
415:
400:
399:
381:
380:
371:
370:
361:
360:
344:
342:
341:
336:
334:
333:
309:
307:
306:
301:
299:
298:
282:
280:
279:
274:
257:
255:
254:
249:
237:
235:
234:
229:
224:
223:
208:
207:
192:
191:
168:
166:
165:
160:
158:
157:
148:
147:
138:
137:
111:
109:
108:
103:
92:
91:
73:
72:
26:complex dynamics
4178:
4177:
4173:
4172:
4171:
4169:
4168:
4167:
4153:
4152:
4151:
4150:
4112:
4111:
4107:
4100:
4096:
4089:
4085:
4057:
4056:
4052:
4020:
4015:
4014:
4010:
4005:
3993:
3984:
3982:Generalizations
3955:
3950:
3949:
3918:
3917:
3891:
3886:
3885:
3864:
3859:
3858:
3837:
3832:
3831:
3806:
3805:
3780:
3779:
3756:
3755:
3732:
3731:
3712:
3711:
3708:
3680:
3669:
3668:
3645:
3644:
3617:
3606:
3605:
3582:
3581:
3554:
3532:
3527:
3526:
3496:
3491:
3490:
3467:
3466:
3447:
3446:
3423:
3422:
3399:
3394:
3393:
3370:
3369:
3345:
3332:
3324:
3323:
3300:
3299:
3274:
3269:
3268:
3247:
3242:
3241:
3234:
3232:Parameter wakes
3208:
3203:
3202:
3179:
3178:
3159:
3158:
3126:
3121:
3120:
3101:
3100:
3079:
3074:
3073:
3048:
3043:
3042:
3039:
3002:
2997:
2996:
2975:
2970:
2969:
2933:
2932:
2925:
2924:
2898:
2891:
2890:
2871:
2870:
2849:
2844:
2843:
2822:
2817:
2816:
2797:
2796:
2777:
2776:
2755:
2750:
2749:
2730:
2729:
2699:
2694:
2693:
2666:
2661:
2660:
2633:
2628:
2627:
2606:
2601:
2600:
2572:
2571:
2550:
2545:
2544:
2523:
2518:
2517:
2496:
2491:
2490:
2459:
2458:
2437:
2432:
2431:
2407:
2388:
2380:
2379:
2376:
2346:
2324:
2319:
2318:
2279:
2274:
2273:
2212:
2211:
2190:
2189:
2168:
2163:
2162:
2159:
2121:
2102:
2094:
2093:
2092:Any collection
2054:
2053:
2032:
2027:
2026:
2004:
2003:
1981:
1980:
1961:
1960:
1939:
1920:
1915:
1914:
1884:
1879:
1878:
1857:
1852:
1851:
1810:
1809:
1788:
1783:
1782:
1755:
1754:
1751:
1677:
1673:
1669:
1654:
1653:
1651:
1589:
1585:
1540:
1536:
1535:
1531:
1516:
1515:
1450:
1446:
1401:
1397:
1352:
1348:
1347:
1343:
1328:
1327:
1262:
1258:
1254:
1239:
1238:
1214:
1203:
1202:
1171:
1149:
1127:
1122:
1121:
1107:
1058:
1054:
1022:
1018:
986:
982:
950:
946:
914:
910:
909:
905:
890:
889:
888:
842:
838:
806:
802:
770:
766:
765:
761:
746:
745:
744:
741:
695:
691:
659:
655:
623:
619:
618:
614:
599:
598:
597:
552:
548:
544:
529:
528:
526:
521:
497:
467:
462:
461:
435:
434:
407:
391:
350:
349:
325:
320:
319:
290:
285:
284:
265:
264:
240:
239:
215:
199:
181:
180:
129:
124:
123:
83:
64:
59:
58:
49:
12:
11:
5:
4176:
4174:
4166:
4165:
4155:
4154:
4149:
4148:
4105:
4094:
4083:
4050:
4007:
4006:
4004:
4001:
4000:
3999:
3992:
3989:
3983:
3980:
3979:
3978:
3963:
3958:
3937:
3934:
3931:
3928:
3925:
3914:
3899:
3894:
3871:
3867:
3844:
3840:
3819:
3816:
3813:
3793:
3790:
3787:
3763:
3741:
3719:
3707:
3704:
3688:
3683:
3679:
3676:
3654:
3629:
3622:
3616:
3613:
3591:
3569:
3566:
3561:
3557:
3553:
3550:
3547:
3544:
3539:
3535:
3508:
3501:
3476:
3465:be called the
3454:
3432:
3407:
3402:
3379:
3357:
3352:
3348:
3344:
3339:
3335:
3331:
3309:
3296:Mandelbrot set
3281:
3277:
3254:
3250:
3238:parameter rays
3233:
3230:
3215:
3211:
3199:critical value
3186:
3166:
3133:
3129:
3108:
3086:
3082:
3055:
3051:
3038:
3035:
3014:
3007:
2982:
2978:
2952:
2948:
2945:
2940:
2936:
2910:
2905:
2901:
2878:
2856:
2852:
2829:
2825:
2804:
2784:
2762:
2758:
2737:
2712:
2709:
2706:
2702:
2679:
2676:
2673:
2669:
2646:
2643:
2640:
2636:
2613:
2609:
2584:
2581:
2557:
2553:
2530:
2526:
2503:
2499:
2477:
2472:
2467:
2444:
2440:
2419:
2414:
2410:
2406:
2403:
2400:
2395:
2391:
2387:
2375:
2372:
2353:
2349:
2345:
2342:
2339:
2336:
2331:
2327:
2304:
2299:
2295:
2291:
2286:
2282:
2261:
2256:
2251:
2247:
2243:
2240:
2237:
2232:
2227:
2223:
2219:
2198:
2175:
2171:
2158:
2155:
2133:
2128:
2124:
2120:
2117:
2114:
2109:
2105:
2101:
2090:
2089:
2075:
2069:
2063:
2039:
2035:
2023:
2011:
1991:
1988:
1968:
1946:
1942:
1938:
1935:
1932:
1927:
1923:
1911:
1897:
1894:
1891:
1887:
1864:
1860:
1844:
1831:
1825:
1819:
1795:
1791:
1764:
1750:
1747:
1734:
1730:
1724:
1721:
1716:
1711:
1708:
1703:
1698:
1695:
1690:
1685:
1682:
1676:
1672:
1668:
1663:
1650:
1647:
1634:
1629:
1623:
1620:
1615:
1610:
1607:
1602:
1597:
1594:
1588:
1584:
1580:
1574:
1571:
1566:
1561:
1558:
1553:
1548:
1545:
1539:
1534:
1530:
1525:
1495:
1490:
1484:
1481:
1476:
1471:
1468:
1463:
1458:
1455:
1449:
1445:
1441:
1435:
1432:
1427:
1422:
1419:
1414:
1409:
1406:
1400:
1396:
1392:
1386:
1383:
1378:
1373:
1370:
1365:
1360:
1357:
1351:
1346:
1342:
1337:
1306:
1302:
1296:
1293:
1288:
1283:
1280:
1275:
1270:
1267:
1261:
1257:
1253:
1248:
1221:
1217:
1213:
1210:
1186:
1183:
1176:
1170:
1164:
1161:
1154:
1148:
1142:
1139:
1132:
1106:
1103:
1090:
1085:
1079:
1076:
1071:
1066:
1063:
1057:
1053:
1049:
1043:
1040:
1035:
1030:
1027:
1021:
1017:
1013:
1007:
1004:
999:
994:
991:
985:
981:
977:
971:
968:
963:
958:
955:
949:
945:
941:
935:
932:
927:
922:
919:
913:
908:
904:
899:
874:
869:
863:
860:
855:
850:
847:
841:
837:
833:
827:
824:
819:
814:
811:
805:
801:
797:
791:
788:
783:
778:
775:
769:
764:
760:
755:
727:
722:
716:
713:
708:
703:
700:
694:
690:
686:
680:
677:
672:
667:
664:
658:
654:
650:
644:
641:
636:
631:
628:
622:
617:
613:
608:
583:
579:
573:
570:
565:
560:
557:
551:
547:
543:
538:
525:
522:
520:
517:
496:
493:
474:
470:
444:
419:
414:
410:
406:
403:
398:
394:
390:
387:
384:
379:
374:
369:
364:
359:
332:
328:
310:be the set of
297:
293:
272:
247:
227:
222:
218:
214:
211:
206:
202:
198:
195:
190:
170:
169:
156:
151:
146:
141:
136:
132:
113:
112:
101:
98:
95:
90:
86:
82:
79:
76:
71:
67:
48:
45:
44:
43:
40:
22:orbit portrait
13:
10:
9:
6:
4:
3:
2:
4175:
4164:
4161:
4160:
4158:
4143:
4138:
4133:
4128:
4124:
4120:
4116:
4109:
4106:
4103:
4098:
4095:
4092:
4087:
4084:
4079:
4075:
4070:
4065:
4061:
4054:
4051:
4046:
4042:
4038:
4034:
4030:
4026:
4019:
4012:
4009:
4002:
3998:
3995:
3994:
3990:
3988:
3981:
3956:
3935:
3932:
3929:
3926:
3923:
3915:
3892:
3869:
3865:
3842:
3838:
3817:
3814:
3811:
3791:
3788:
3785:
3777:
3776:
3775:
3761:
3717:
3705:
3703:
3681:
3677:
3674:
3614:
3611:
3567:
3564:
3559:
3555:
3551:
3545:
3537:
3533:
3525:
3452:
3400:
3350:
3346:
3342:
3337:
3333:
3297:
3279:
3275:
3252:
3248:
3239:
3231:
3229:
3213:
3209:
3200:
3184:
3164:
3155:
3153:
3149:
3131:
3127:
3106:
3084:
3080:
3071:
3070:external rays
3053:
3049:
3036:
3034:
3031:
3028:, called the
2980:
2976:
2966:
2950:
2946:
2943:
2938:
2934:
2908:
2903:
2899:
2876:
2854:
2850:
2827:
2823:
2802:
2782:
2760:
2756:
2735:
2726:
2710:
2707:
2704:
2700:
2677:
2674:
2671:
2667:
2644:
2641:
2638:
2634:
2611:
2607:
2597:
2582:
2579:
2555:
2551:
2528:
2524:
2501:
2497:
2470:
2442:
2438:
2412:
2408:
2404:
2401:
2398:
2393:
2389:
2373:
2371:
2369:
2351:
2347:
2343:
2337:
2329:
2325:
2297:
2293:
2289:
2284:
2280:
2249:
2245:
2241:
2238:
2235:
2225:
2221:
2196:
2173:
2169:
2156:
2154:
2151:
2147:
2126:
2122:
2118:
2115:
2112:
2107:
2103:
2067:
2037:
2033:
2024:
2009:
1989:
1986:
1966:
1944:
1940:
1936:
1933:
1930:
1925:
1921:
1912:
1895:
1892:
1889:
1885:
1862:
1858:
1849:
1845:
1823:
1793:
1789:
1780:
1779:
1778:
1748:
1746:
1732:
1728:
1722:
1719:
1714:
1709:
1706:
1701:
1696:
1693:
1688:
1683:
1680:
1674:
1670:
1666:
1648:
1646:
1632:
1627:
1621:
1618:
1613:
1608:
1605:
1600:
1595:
1592:
1586:
1582:
1578:
1572:
1569:
1564:
1559:
1556:
1551:
1546:
1543:
1537:
1532:
1528:
1513:
1510:
1507:
1493:
1488:
1482:
1479:
1474:
1469:
1466:
1461:
1456:
1453:
1447:
1443:
1439:
1433:
1430:
1425:
1420:
1417:
1412:
1407:
1404:
1398:
1394:
1390:
1384:
1381:
1376:
1371:
1368:
1363:
1358:
1355:
1349:
1344:
1340:
1325:
1323:
1318:
1304:
1300:
1294:
1291:
1286:
1281:
1278:
1273:
1268:
1265:
1259:
1255:
1251:
1219:
1215:
1211:
1208:
1184:
1181:
1168:
1162:
1159:
1146:
1140:
1137:
1119:
1118:external rays
1114:
1110:
1104:
1102:
1088:
1083:
1077:
1074:
1069:
1064:
1061:
1055:
1051:
1047:
1041:
1038:
1033:
1028:
1025:
1019:
1015:
1011:
1005:
1002:
997:
992:
989:
983:
979:
975:
969:
966:
961:
956:
953:
947:
943:
939:
933:
930:
925:
920:
917:
911:
906:
902:
886:
872:
867:
861:
858:
853:
848:
845:
839:
835:
831:
825:
822:
817:
812:
809:
803:
799:
795:
789:
786:
781:
776:
773:
767:
762:
758:
742:
739:
725:
720:
714:
711:
706:
701:
698:
692:
688:
684:
678:
675:
670:
665:
662:
656:
652:
648:
642:
639:
634:
629:
626:
620:
615:
611:
595:
581:
577:
571:
568:
563:
558:
555:
549:
545:
541:
523:
518:
512:
505:
501:
494:
492:
490:
472:
468:
458:
433:
412:
408:
404:
401:
396:
392:
385:
362:
348:Then the set
346:
330:
326:
317:
316:external rays
313:
295:
291:
270:
245:
220:
216:
212:
209:
204:
200:
193:
179:
175:
139:
134:
130:
122:
121:
120:
118:
117:complex plane
99:
96:
93:
88:
84:
77:
74:
69:
65:
57:
56:
55:
54:
53:quadratic map
46:
41:
38:
37:
36:
33:
31:
27:
23:
19:
4125:(3): 35–50.
4122:
4118:
4108:
4097:
4086:
4069:math/9905169
4059:
4053:
4028:
4024:
4011:
3985:
3709:
3240:with angles
3235:
3156:
3151:
3147:
3040:
3029:
2967:
2727:
2598:
2377:
2160:
2145:
2091:
1848:doubling map
1752:
1652:
1514:
1511:
1508:
1326:
1319:
1237:
1108:
887:
743:
740:
596:
527:
488:
459:
431:
347:
171:
114:
50:
34:
21:
15:
3072:landing at
2150:John Milnor
119:to itself
18:mathematics
4003:References
3997:Lamination
3702:for about
2775:, or when
2489:, so each
2272:such that
430:is called
258:, so that
47:Definition
4132:1404.7193
3933:≥
3615:∈
3338:−
3253:−
2935:θ
2900:θ
2851:θ
2824:θ
2402:…
2368:Julia set
2303:′
2290:⊊
2255:′
2239:…
2231:′
2116:…
2025:The sets
1934:…
1649:valence 4
1216:α
1105:valence 3
524:valence 2
504:Julia set
405:…
213:…
176:periodic
150:→
115:from the
81:↦
4157:Category
4060:Preprint
4045:54997658
3991:See also
3857:. Each
2002:, where
495:Examples
318:land at
261:z_{j+1}}
51:Given a
4074:Bibcode
3037:Sectors
2430:, each
489:valence
283:), let
4043:
3197:, the
312:angles
172:and a
4127:arXiv
4064:arXiv
4041:S2CID
4021:(PDF)
3522:. A
2728:When
1781:Each
178:orbit
20:, an
3804:and
3754:and
3392:let
3267:and
3236:Two
3150:or
2923:and
2374:Arcs
1846:The
1320:For
4137:doi
4033:doi
3643:.
3201:of
2842:.
1877:to
1483:511
1470:511
1467:324
1457:511
1454:296
1434:511
1431:274
1421:511
1418:162
1408:511
1405:148
1385:511
1382:137
1372:511
1359:511
238:of
16:In
4159::
4135:.
4123:19
4121:.
4117:.
4072:.
4062:.
4039:.
4029:16
4027:.
4023:.
1723:15
1710:15
1697:15
1684:15
1622:63
1619:50
1609:63
1606:44
1596:63
1593:11
1573:63
1570:37
1560:63
1557:25
1547:63
1544:22
1480:37
1369:81
1356:74
1116:3
1078:31
1065:31
1062:21
1042:31
1029:31
1026:26
1006:31
1003:17
993:31
990:13
970:31
967:24
957:31
954:22
934:31
931:12
921:31
918:11
457:.
345:.
32:.
4145:.
4139::
4129::
4080:.
4076::
4066::
4047:.
4035::
3962:P
3957:r
3936:2
3930:v
3927:=
3924:r
3898:P
3893:r
3870:j
3866:A
3843:n
3839:f
3818:2
3815:=
3812:v
3792:1
3789:=
3786:r
3762:r
3740:P
3718:v
3687:P
3682:r
3678:=
3675:c
3653:P
3628:P
3621:W
3612:c
3590:P
3568:c
3565:+
3560:2
3556:z
3552:=
3549:)
3546:z
3543:(
3538:c
3534:f
3507:P
3500:W
3475:P
3453:0
3431:P
3406:P
3401:r
3378:P
3356:]
3351:+
3347:t
3343:,
3334:t
3330:[
3308:P
3280:+
3276:t
3249:t
3214:c
3210:f
3185:c
3165:0
3132:j
3128:z
3107:v
3085:j
3081:z
3054:j
3050:z
3013:P
3006:I
2981:j
2977:A
2951:2
2947:1
2944:+
2939:c
2909:2
2904:c
2877:c
2855:c
2828:c
2803:c
2783:c
2761:c
2757:f
2736:c
2711:1
2708:+
2705:j
2701:z
2678:1
2675:+
2672:j
2668:z
2645:1
2642:+
2639:j
2635:z
2612:j
2608:z
2583:2
2580:1
2556:j
2552:z
2529:j
2525:z
2502:j
2498:A
2476:Z
2471:/
2466:R
2443:j
2439:A
2418:}
2413:n
2409:A
2405:,
2399:,
2394:1
2390:A
2386:{
2352:2
2348:z
2344:=
2341:)
2338:z
2335:(
2330:0
2326:f
2298:j
2294:A
2285:j
2281:A
2260:}
2250:n
2246:A
2242:,
2236:,
2226:1
2222:A
2218:{
2197:0
2174:j
2170:A
2132:}
2127:n
2123:A
2119:,
2113:,
2108:1
2104:A
2100:{
2074:Z
2068:/
2062:R
2038:j
2034:A
2010:r
1990:n
1987:r
1967:n
1945:n
1941:A
1937:,
1931:,
1926:1
1922:A
1896:1
1893:+
1890:j
1886:A
1863:j
1859:A
1830:Z
1824:/
1818:R
1794:j
1790:A
1763:P
1733:}
1729:)
1720:8
1715:,
1707:4
1702:,
1694:2
1689:,
1681:1
1675:(
1671:{
1667:=
1662:P
1633:}
1628:)
1614:,
1601:,
1587:(
1583:,
1579:)
1565:,
1552:,
1538:(
1533:{
1529:=
1524:P
1494:}
1489:)
1475:,
1462:,
1448:(
1444:,
1440:)
1426:,
1413:,
1399:(
1395:,
1391:)
1377:,
1364:,
1350:(
1345:{
1341:=
1336:P
1305:}
1301:)
1295:7
1292:4
1287:,
1282:7
1279:2
1274:,
1269:7
1266:1
1260:(
1256:{
1252:=
1247:P
1220:c
1212:=
1209:z
1185:7
1182:4
1175:R
1169:,
1163:7
1160:2
1153:R
1147:,
1141:7
1138:1
1131:R
1089:}
1084:)
1075:6
1070:,
1056:(
1052:,
1048:)
1039:3
1034:,
1020:(
1016:,
1012:)
998:,
984:(
980:,
976:)
962:,
948:(
944:,
940:)
926:,
912:(
907:{
903:=
898:P
873:}
868:)
862:9
859:2
854:,
849:9
846:7
840:(
836:,
832:)
826:9
823:1
818:,
813:9
810:8
804:(
800:,
796:)
790:9
787:5
782:,
777:9
774:4
768:(
763:{
759:=
754:P
726:}
721:)
715:7
712:2
707:,
702:7
699:5
693:(
689:,
685:)
679:7
676:1
671:,
666:7
663:6
657:(
653:,
649:)
643:7
640:4
635:,
630:7
627:3
621:(
616:{
612:=
607:P
582:}
578:)
572:3
569:2
564:,
559:3
556:1
550:(
546:{
542:=
537:P
473:j
469:A
443:O
418:}
413:n
409:A
402:,
397:1
393:A
389:{
386:=
383:)
378:O
373:(
368:P
363:=
358:P
331:j
327:z
296:j
292:A
271:n
246:f
226:}
221:n
217:z
210:,
205:1
201:z
197:{
194:=
189:O
155:C
145:C
140::
135:c
131:f
100:.
97:c
94:+
89:2
85:z
78:z
75::
70:c
66:f
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
