1190:
4839:
4730:
1287:
6291:
outside finitely many curves. Conversely, Cantat and Dupont showed that for all surface automorphisms of positive entropy except the Kummer examples, the equilibrium measure is not absolutely continuous with respect to
Lebesgue measure. In this sense, it is usual for the equilibrium measure of an
4268:
of the equilibrium measure is not too small, in the sense that its
Hausdorff dimension is always greater than zero. In that sense, an endomorphism of complex projective space with degree greater than 1 always behaves chaotically at least on part of the space. (There are examples where
5316:, a closed interval, or a circle, respectively.) Thus, outside those special cases, the equilibrium measure is highly irregular, assigning positive mass to some closed subsets of the Julia set with smaller Hausdorff dimension than the whole Julia set.
5827:
showed that every automorphism of a compact Kähler surface with positive topological entropy has simple action on cohomology. (Here an "automorphism" is complex analytic but is not assumed to preserve a Kähler metric on
435:
5398:
Gromov and Yosef Yomdin showed that the topological entropy of an endomorphism (for example, an automorphism) of a smooth complex projective variety is determined by its action on cohomology. Explicitly, for
6258:
2662:
1439:
5094:
5551:
3362:
2943:
3480:
3200:
5839:
with simple action on cohomology, some of the goals of complex dynamics have been achieved. Dinh, Sibony, and Henry de Thélin showed that there is a unique invariant probability measure
780:
5619:
3781:
6283:
to make the surface smooth. The resulting surfaces include some special K3 surfaces and rational surfaces. For the Kummer automorphisms, the equilibrium measure has support equal to
5674:
5393:
1371:
4909:
4800:
2080:
1280:
6167:
6663:
5314:
5176:
5001:
4941:
4832:
4693:
4657:
4605:
4447:
4335:
4160:
4101:
3989:
3647:
3555:
3275:
3235:
2975:
2510:
2401:
2256:
2156:
2116:
1851:
1815:
1768:
1736:
1681:
1538:
1124:
988:
923:
506:
284:
228:
176:
7858:
6834:
5256:
5437:
3104:
3032:
1242:
5136:
1336:
1180:
1096:
6955:
3608:
6880:
6356:
5940:
5790:
3700:
3438:
2478:
1040:
850:
670:
137:
6193:
6127:
6907:
6759:
6707:
6628:
6597:
6566:
6524:
6489:
6462:
6435:
6081:
6054:
6010:
5971:
5907:
5864:
5203:
4720:
4555:
4528:
4474:
4303:
4266:
4223:
4196:
4128:
4061:
4025:
3954:
3918:
3882:
3842:
3519:
3131:
3059:
2811:
2537:
2365:
2224:
710:
7004:
5282:
4376:
526:
474:
196:
3811:
3392:
2606:
1581:
1480:
6383:
5817:
5744:
5464:
2780:
2577:
2191:
877:
592:
4969:
2031:
956:
2427:
2314:
252:
5324:
More generally, complex dynamics seeks to describe the behavior of rational maps under iteration. One case that has been studied with some success is that of
7532:
6979:
4695:
whose equilibrium measure is absolutely continuous with respect to
Lebesgue measure are the Lattès examples. That is, for all non-Lattès endomorphisms,
2119:
5680:
has some chaotic behavior, in the sense that its topological entropy is greater than zero, if and only if it acts on some cohomology group with an
7907:
7469:
300:
7925:
7883:
7807:
7697:
7510:
7477:
7442:
2977:
to itself, the equilibrium measure can be much more complicated, as one sees already in complex dimension 1 from pictures of Julia sets.
1382:
8036:
8006:
7976:
7598:
7379:
7360:
7325:
6210:
2611:
1388:
5009:
1818:
6389:
has absolute value less than 1, at least one has absolute value greater than 1, and none has absolute value equal to 1. (Thus
5481:
7377:
Berteloot, François; Dupont, Christophe (2005), "Une caractérisation des endomorphismes de Lattès par leur mesure de Green",
4379:
1450:
926:
4503:
with respect to that measure, by
Fornaess and Sibony. It follows, for example, that for almost every point with respect to
7865:
7799:
7723:
4620:
1042:, the Julia set is the unit circle. For other polynomial mappings, the Julia set is often highly irregular, for example a
7901:
7463:
6974:
6914:
6717:
of a smooth projective rational surface with positive topological entropy (hence simple action on cohomology) such that
4480:
invariant measure of maximal entropy, by Briend and Duval. This is another way to say that the most chaotic behavior of
3491:
3284:
1189:
3652:
Another characterization of the equilibrium measure (due to Briend and Duval) is as follows. For each positive integer
3205:
One striking characterization of the equilibrium measure is that it describes the asymptotics of almost every point in
2823:
8122:
8028:
7998:
7950:
7502:
7426:
5684:
of absolute value greater than 1. Many projective varieties do not have such automorphisms, but (for example) many
3447:
3140:
7666:(2010), "Dynamics in several complex variables: endomorphisms of projective spaces and polynomial-like mappings",
6917:. The same holds for the subset of saddle periodic points, because both sets of periodic points grow at a rate of
958:
with complex coefficients, or more generally by a rational function.) Namely, there is always a compact subset of
8117:
8112:
4402:
532:
7009:
4838:
726:
8058:
6016:
has some chaotic behavior, and the most chaotic behavior is concentrated on the small Julia set. At least when
5563:
2368:
1704:
623:
3705:
6279:
are defined by taking the quotient space by a finite group of an abelian surface with automorphism, and then
7621:
6999:
1858:
1703:
Complex dynamics has been effectively developed in any dimension. This section focuses on the mappings from
144:
5635:
5354:
4103:. A more algebraic proof of this Zariski density was given by Najmuddin Fakhruddin. Another consequence of
1341:
7530:; Dupont, Christophe (2020), "Automorphisms of surfaces: Kummer rigidity and measure of maximal entropy",
7019:
6204:
4845:
4736:
6836:. Consider the probability measure which is evenly distributed on the isolated periodic points of period
2039:
1247:
8127:
6132:
2278:.) This measure was defined by Hans Brolin (1965) for polynomials in one variable, by Alexandre Freire,
1640:
6636:
5287:
5149:
4974:
4914:
4805:
4729:
4666:
4630:
4578:
4420:
4383:
4308:
4133:
4074:
3962:
3620:
3528:
3248:
3208:
2948:
2483:
2374:
2229:
2129:
2089:
1824:
1788:
1741:
1709:
1654:
1485:
1101:
961:
896:
479:
257:
201:
149:
7834:
4947:
In dimension 1, more is known about the "irregularity" of the equilibrium measure. Namely, define the
6791:
5210:
4500:
5216:
7721:(2010), "Super-potentials for currents on compact Kähler manifolds and dynamics of automorphisms",
7032:
6984:
6541:
and its inverse map are ergodic and, more strongly, mixing with respect to the equilibrium measure
5410:
5349:
4342:
3072:
3000:
2995:
2197:
1196:
1047:
68:
5102:
1293:
1137:
1053:
7954:
7911:
7824:
7791:
7770:
7732:
7675:
7630:
7541:
7388:
6920:
3884:, and so one gets the same limit measure by averaging only over the repelling periodic points in
3568:
1775:
6989:
6843:
6319:
5912:
5753:
5700:
3663:
3401:
2441:
2283:
1003:
813:
633:
100:
7462:(2010), "Quelques aspects des systèmes dynamiques polynomiaux: existence, exemples, rigidité",
6172:
6106:
8132:
8032:
8002:
7972:
7921:
7879:
7803:
7693:
7594:
7506:
7473:
7438:
7356:
7321:
7042:
6885:
6728:
6676:
6606:
6575:
6544:
6502:
6467:
6440:
6404:
6059:
6023:
5979:
5949:
5885:
5842:
5820:
5181:
4698:
4533:
4506:
4452:
4272:
4235:
4201:
4165:
4106:
4030:
3994:
3923:
3887:
3851:
3820:
3497:
3109:
3037:
2789:
2515:
2334:
2279:
2202:
679:
449:
64:
60:
44:
5261:
4352:
511:
459:
181:
8067:
7964:
7869:
7742:
7685:
7640:
7586:
7574:
7570:
7551:
7430:
7398:
7313:
6969:
5685:
4624:
3786:
3367:
2585:
1551:
1459:
1286:
48:
40:
8079:
8046:
8016:
7986:
7935:
7893:
7817:
7784:
7754:
7707:
7652:
7608:
7563:
7520:
7487:
7452:
7410:
7370:
7335:
6361:
5795:
5722:
5703:, which includes the case of a smooth complex projective variety. Say that an automorphism
5442:
2673:
2546:
2169:
893:
showed in the late 1910s that much of this story extends to any complex algebraic map from
855:
561:
8075:
8056:(1990), "Parabolic orbifolds and the dimension of the maximal measure for rational maps",
8042:
8012:
7982:
7931:
7889:
7813:
7780:
7750:
7714:
7703:
7671:
7659:
7648:
7616:
7604:
7582:
7559:
7516:
7483:
7448:
7418:
7406:
7366:
7352:
7331:
6710:
6288:
5467:
4954:
4608:
3238:
3066:
2317:
2287:
1867:
1446:
1183:
932:
72:
6260:, has spectral radius greater than 1, and so it gives a positive-entropy automorphism of
2406:
2293:
17:
8097:
7014:
6096:
3813:. Consider the probability measure which is evenly distributed on the points of period
3441:
1127:
627:
611:
441:
237:
97:
A simple example that shows some of the main issues in complex dynamics is the mapping
80:
6401:
with simple action on cohomology, the saddle periodic points are dense in the support
8106:
7993:
Morosawa, Shunsuke; Nishimura, Yasuichiro; Taniguchu, Masahiko; Ueda, Tetsuo (2000),
7828:
7718:
7663:
7309:
6994:
5832:. In fact, every automorphism that preserves a metric has topological entropy zero.)
5329:
4567:
4068:
2325:
2321:
231:
76:
7527:
7494:
7459:
7342:
7037:
5824:
5475:
4612:
2540:
1771:
890:
886:
7746:
6091:
Some abelian varieties have an automorphism of positive entropy. For example, let
5205:
is equal to the
Hausdorff dimension of its support (the Julia set) if and only if
7862:
Laminations and foliations in dynamics, geometry and topology (Stony Brook, 1998)
3525:
goes to infinity. In more detail: only finitely many closed complex subspaces of
8092:
8053:
7942:
7689:
4660:
4226:
3617:
to be the unique largest totally invariant closed complex subspace not equal to
2123:
1628:
1612:
7874:
7761:
Fakhruddin, Najmuddin (2003), "Questions on self maps of algebraic varieties",
7346:
6725:
is conjugate to an irrational rotation. Points in that open set never approach
63:
is iterated. In geometric terms, that amounts to iterating a mapping from some
7900:
Guedj, Vincent (2010), "Propriétés ergodiques des applications rationnelles",
7644:
7590:
7317:
6280:
5689:
5681:
1374:
540:
56:
1778:
to itself. Note, however, that many varieties have no interesting self-maps.
7423:
Frontiers in complex dynamics: in celebration of John Milnor's 80th birthday
4723:
992:
619:
92:
7968:
7434:
6709:
of the equilibrium measure. For example, Eric
Bedford, Kyounghee Kim, and
6083:
assigns zero mass to all sets of sufficiently small
Hausdorff dimension.)
3959:
The equilibrium measure gives zero mass to any closed complex subspace of
3848:
goes to infinity. Moreover, most periodic points are repelling and lie in
2543:(the standard measure, scaled to have total measure 1) on the unit circle
6669:
with simple action on cohomology, there can be a nonempty open subset of
1456:
of the Fatou set is pre-periodic, meaning that there are natural numbers
1445:, the complement of the Julia set, where the dynamics is "tame". Namely,
430:{\displaystyle z,\;f(z)=z^{2},\;f(f(z))=z^{4},f(f(f(z)))=z^{8},\;\ldots }
7141:
Milnor (2006), Theorem 5.2 and problem 14-2; Fornaess (1996), Chapter 3.
8071:
7555:
4663:, François Berteloot, and Christophe Dupont, the only endomorphisms of
4496:
1043:
7186:
Zdunik (1990), Theorem 2; Berteloot & Dupont (2005), introduction.
7959:
7916:
7775:
7402:
7393:
6499:(or even just the saddle periodic points contained in the support of
1623:
is conjugate to an irrational rotation of the open unit disk; or (4)
7348:
Iteration of rational functions: complex analytic dynamical systems
4842:
A random sample from the equilibrium measure of the non-Lattès map
3237:
when followed backward in time, by Jean-Yves Briend, Julien Duval,
448:| is less than 1, then the orbit converges to 0, in fact more than
8027:, London Mathematical Society Lecture Note Series, vol. 274,
7737:
7680:
7635:
7546:
6673:
on which neither forward nor backward orbits approach the support
4837:
4728:
2818:
1285:
1188:
7619:(2012), "Dynamics of automorphisms on compact Kähler manifolds",
7497:(2014), "Dynamics of automorphisms of compact complex surfaces",
7272:
Dinh & Sibony (2010), "Super-potentials ...", Theorem 4.4.2.
6393:
is expanding in some directions and contracting at others, near
1738:
to itself, the richest source of examples. The main results for
606:
is chaotic, in various ways. For example, for almost all points
7306:
A history of complex dynamics: from Schröder to Fatou and Julia
4733:
A random sample from the equilibrium measure of the Lattès map
3817:. Then these measures also converge to the equilibrium measure
7245:
Dinh & Sibony (2010), "Super-potentials ...", section 4.4.
6772:
describes the distribution of the isolated periodic points of
6253:{\displaystyle {\begin{pmatrix}2&1\\1&1\end{pmatrix}}}
5178:
of degree greater than 1, Zdunik showed that the dimension of
2657:{\displaystyle f\colon \mathbf {CP} ^{n}\to \mathbf {CP} ^{n}}
1434:{\displaystyle f\colon \mathbf {CP} ^{1}\to \mathbf {CP} ^{1}}
1050:
is not an integer. This occurs even for mappings as simple as
4530:, its forward orbit is uniformly distributed with respect to
929:
greater than 1. (Such a mapping may be given by a polynomial
7123:
Dinh & Sibony (2010), "Dynamics ...", Proposition 1.2.3.
6768:
At least in complex dimension 2, the equilibrium measure of
4484:
is concentrated on the support of the equilibrium measure.
4162:
is that each point has zero mass. As a result, the support
456:| is greater than 1, then the orbit converges to the point
5089:{\displaystyle \dim(\mu )=\inf\{\dim _{H}(Y):\mu (Y)=1\},}
4130:
giving zero mass to closed complex subspaces not equal to
3991:
that is not the whole space. Since the periodic points in
2986:
A basic property of the equilibrium measure is that it is
2266:, that describes the most chaotic part of the dynamics of
7102:
Dinh & Sibony (2010), "Dynamics ...", Theorem 1.4.13.
7084:
Dinh & Sibony (2010), "Dynamics ...", Theorem 1.7.11.
7159:
Dinh & Sibony (2010), "Dynamics ...", Theorem 1.6.3.
7150:
Dinh & Sibony (2010), "Dynamics ...", Theorem 1.7.1.
7093:
Dinh & Sibony (2010), "Dynamics ...", Theorem 1.4.1.
5546:{\displaystyle H^{p,p}(X)\subset H^{2p}(X,\mathbf {C} )}
143:
to itself. It is helpful to view this as a map from the
6633:
A notable difference with the case of endomorphisms of
6268:
is the Haar measure (the standard
Lebesgue measure) on
6219:
3920:. There may also be repelling periodic points outside
7837:
6923:
6888:
6846:
6794:
6731:
6679:
6639:
6609:
6578:
6547:
6505:
6470:
6443:
6407:
6364:
6322:
6213:
6175:
6135:
6109:
6062:
6026:
5982:
5952:
5915:
5888:
5845:
5798:
5756:
5725:
5638:
5566:
5484:
5445:
5413:
5357:
5290:
5264:
5219:
5184:
5152:
5105:
5012:
4977:
4957:
4917:
4848:
4808:
4739:
4701:
4669:
4633:
4581:
4536:
4509:
4455:
4423:
4355:
4311:
4275:
4238:
4204:
4168:
4136:
4109:
4077:
4033:
3997:
3965:
3926:
3890:
3854:
3823:
3789:
3708:
3666:
3623:
3571:
3531:
3500:
3450:
3404:
3370:
3287:
3251:
3211:
3143:
3112:
3075:
3040:
3003:
2951:
2826:
2792:
2676:
2614:
2588:
2549:
2518:
2486:
2444:
2409:
2377:
2337:
2296:
2232:
2205:
2172:
2132:
2092:
2042:
1870:
1827:
1791:
1744:
1712:
1657:
1554:
1488:
1462:
1391:
1344:
1296:
1250:
1199:
1140:
1104:
1056:
1006:
964:
935:
899:
858:
816:
729:
682:
636:
564:
514:
482:
462:
303:
260:
240:
204:
184:
152:
103:
7903:
Quelques aspects des systèmes dynamiques polynomiaux
7465:
Quelques aspects des systèmes dynamiques polynomiaux
6780:
or an iterate, which are ignored here.) Namely, let
1540:. Therefore, to analyze the dynamics on a component
6056:has positive Hausdorff dimension. (More precisely,
7852:
6949:
6901:
6874:
6828:
6753:
6701:
6657:
6622:
6591:
6560:
6518:
6491:vanishes on closed complex subspaces not equal to
6483:
6456:
6429:
6377:
6350:
6252:
6187:
6161:
6121:
6075:
6048:
6004:
5965:
5934:
5901:
5858:
5811:
5784:
5738:
5668:
5613:
5545:
5458:
5431:
5387:
5308:
5276:
5250:
5197:
5170:
5130:
5088:
4995:
4963:
4943:, but the equilibrium measure is highly irregular.
4935:
4903:
4826:
4794:
4714:
4687:
4651:
4599:
4549:
4522:
4468:
4441:
4370:
4329:
4297:
4260:
4217:
4190:
4154:
4122:
4095:
4055:
4019:
3983:
3948:
3912:
3876:
3836:
3805:
3775:
3694:
3641:
3602:
3549:
3513:
3474:
3432:
3386:
3357:{\displaystyle (1/d^{rn})(f^{r})^{*}(\delta _{z})}
3356:
3269:
3229:
3194:
3125:
3098:
3053:
3026:
2969:
2937:
2805:
2774:
2656:
2600:
2571:
2531:
2504:
2472:
2421:
2395:
2359:
2308:
2250:
2218:
2185:
2162:is greater than 1; then the degree of the mapping
2150:
2110:
2074:
2025:
1845:
1809:
1762:
1730:
1675:
1639:is conjugate to an irrational rotation of an open
1575:
1532:
1474:
1433:
1365:
1330:
1274:
1236:
1174:
1118:
1090:
1034:
982:
950:
917:
871:
844:
774:
704:
664:
586:
520:
508:, again more than exponentially fast. (Here 0 and
500:
468:
429:
278:
246:
222:
190:
170:
131:
7670:, Lecture Notes in Mathematics, vol. 1998,
6913:goes to infinity, by Eric Bedford, Lyubich, and
5628:is also the logarithm of the spectral radius of
5583:
5284:. (In the latter cases, the Julia set is all of
5031:
4487:Finally, one can say more about the dynamics of
6784:be an automorphism of a compact Kähler surface
6603:are both uniformly distributed with respect to
6358:, at least one eigenvalue of the derivative of
6207:has absolute value greater than 2, for example
5138:denotes the Hausdorff dimension of a Borel set
2938:{\displaystyle \{:|z_{1}|=\cdots =|z_{n}|=1\}.}
5344:to itself. The case of main interest is where
4349:is always greater than zero, in fact equal to
626:on the circle. There are also infinitely many
551:fixed point means one where the derivative of
7763:Journal of the Ramanujan Mathematical Society
6776:. (There may also be complex curves fixed by
3475:{\displaystyle E\subsetneq \mathbf {CP} ^{n}}
3195:{\displaystyle f^{*}\mu _{f}=\deg(f)\mu _{f}}
1643:. (Note that the "backward orbit" of a point
790:can be considered chaotic, since points near
440:behave, qualitatively? The answer is: if the
51:mapping. This article focuses on the case of
8:
7533:Journal of the European Mathematical Society
5792:has only one eigenvalue with absolute value
5080:
5034:
5003:(or more generally on a smooth manifold) by
2982:Characterizations of the equilibrium measure
2929:
2827:
7499:Frontiers in complex dynamics (Banff, 2011)
6980:Infinite compositions of analytic functions
4615:. In this case, the equilibrium measure of
4491:on the support of the equilibrium measure:
2945:For more general holomorphic mappings from
7254:Cantat & Dupont (2020), section 1.2.1.
7168:Berteloot & Dupont (2005), Théorème 1.
7119:
7117:
7111:Fornaess & Sibony (2001), Theorem 4.3.
7005:Carathéodory's theorem (conformal mapping)
6721:has a Siegel disk, on which the action of
6312:periodic point if, for a positive integer
3656:, the number of periodic points of period
1699:The equilibrium measure of an endomorphism
1603:approach a fixed point in the boundary of
775:{\displaystyle f(f(\cdots (f(z))\cdots ))}
423:
339:
310:
7958:
7915:
7873:
7844:
7839:
7836:
7774:
7736:
7679:
7634:
7545:
7392:
7263:Cantat & Dupont (2020), Main Theorem.
7231:De Thélin & Dinh (2012), Theorem 1.2.
7080:
7078:
6941:
6931:
6922:
6893:
6887:
6882:). Then this measure converges weakly to
6851:
6845:
6820:
6793:
6736:
6730:
6684:
6678:
6649:
6641:
6638:
6614:
6608:
6583:
6577:
6568:. It follows that for almost every point
6552:
6546:
6510:
6504:
6495:. It follows that the periodic points of
6475:
6469:
6448:
6442:
6412:
6406:
6369:
6363:
6327:
6321:
6214:
6212:
6174:
6151:
6134:
6108:
6067:
6061:
6031:
6025:
5987:
5981:
5957:
5951:
5926:
5914:
5893:
5887:
5850:
5844:
5803:
5797:
5761:
5755:
5730:
5724:
5658:
5643:
5637:
5614:{\displaystyle h(f)=\max _{p}\log d_{p}.}
5602:
5586:
5565:
5535:
5517:
5489:
5483:
5450:
5444:
5412:
5377:
5362:
5356:
5300:
5292:
5289:
5263:
5239:
5218:
5189:
5183:
5162:
5154:
5151:
5110:
5104:
5041:
5011:
4987:
4979:
4976:
4956:
4927:
4919:
4916:
4895:
4886:
4880:
4847:
4818:
4810:
4807:
4786:
4777:
4771:
4738:
4706:
4700:
4679:
4671:
4668:
4643:
4635:
4632:
4591:
4583:
4580:
4541:
4535:
4514:
4508:
4460:
4454:
4433:
4425:
4422:
4354:
4321:
4313:
4310:
4280:
4274:
4243:
4237:
4209:
4203:
4173:
4167:
4146:
4138:
4135:
4114:
4108:
4087:
4079:
4076:
4063:, it follows that the periodic points of
4038:
4032:
4002:
3996:
3975:
3967:
3964:
3931:
3925:
3895:
3889:
3859:
3853:
3828:
3822:
3794:
3788:
3758:
3746:
3716:
3707:
3671:
3665:
3633:
3625:
3622:
3576:
3570:
3541:
3533:
3530:
3505:
3499:
3466:
3458:
3449:
3409:
3403:
3375:
3369:
3345:
3332:
3322:
3303:
3294:
3286:
3261:
3253:
3250:
3221:
3213:
3210:
3186:
3158:
3148:
3142:
3117:
3111:
3090:
3080:
3074:
3045:
3039:
3018:
3008:
3002:
2961:
2953:
2950:
2918:
2912:
2903:
2889:
2883:
2874:
2862:
2843:
2825:
2797:
2791:
2760:
2755:
2736:
2731:
2709:
2690:
2675:
2648:
2640:
2630:
2622:
2613:
2587:
2558:
2550:
2548:
2523:
2517:
2496:
2488:
2485:
2464:
2443:
2408:
2387:
2379:
2376:
2342:
2336:
2295:
2242:
2234:
2231:
2210:
2204:
2177:
2171:
2142:
2134:
2131:
2102:
2094:
2091:
2066:
2047:
2041:
2011:
1992:
1979:
1957:
1938:
1925:
1903:
1884:
1869:
1837:
1829:
1826:
1801:
1793:
1790:
1754:
1746:
1743:
1722:
1714:
1711:
1667:
1659:
1656:
1553:
1515:
1493:
1487:
1461:
1425:
1417:
1407:
1399:
1390:
1343:
1316:
1295:
1249:
1219:
1198:
1160:
1139:
1111:
1103:
1076:
1055:
1026:
1005:
974:
966:
963:
934:
909:
901:
898:
863:
857:
821:
815:
728:
687:
681:
641:
635:
573:
565:
563:
513:
492:
484:
481:
461:
414:
368:
330:
302:
270:
262:
259:
239:
214:
206:
203:
183:
162:
154:
151:
123:
102:
8025:The Mandelbrot set, theme and variations
7241:
7239:
7237:
3776:{\displaystyle (d^{r(n+1)}-1)/(d^{r}-1)}
234:.) The basic question is: given a point
7056:
6292:automorphism to be somewhat irregular.
5746:takes its maximum value, the action of
4225:has no isolated points, and so it is a
1587:contains an attracting fixed point for
1383:classification of the possible dynamics
4341:has some chaotic behavior is that the
8093:Gallery of dynamics (Curtis McMullen)
7796:Dynamics in several complex variables
6599:, the forward and backward orbits of
5669:{\displaystyle H^{*}(X,\mathbf {C} )}
5388:{\displaystyle H^{*}(X,\mathbf {Z} )}
5320:Automorphisms of projective varieties
1366:{\displaystyle c\doteq 0.383-0.0745i}
7:
4904:{\displaystyle f(z)=(z-2)^{4}/z^{4}}
4795:{\displaystyle f(z)=(z-2)^{2}/z^{2}}
4607:obtained from an endomorphism of an
4337:.) Another way to make precise that
2403:; this is simply the Julia set when
2328:in any dimension (around 1994). The
883:has absolute value greater than 1.)
3364:which is evenly distributed on the
3281:, consider the probability measure
2075:{\displaystyle f_{0},\ldots ,f_{n}}
1275:{\displaystyle a\doteq -0.5+0.866i}
630:on the circle, meaning points with
602:. At these points, the dynamics of
7421:; Sutherland, Scott, eds. (2014),
6788:with positive topological entropy
6537:with simple action on cohomology,
6162:{\displaystyle GL(2,\mathbf {Z} )}
5553:. Then the topological entropy of
3241:, and Sibony. Namely, for a point
1853:to itself, for a positive integer
515:
463:
185:
25:
7380:Commentarii Mathematici Helvetici
7132:Fakhruddin (2003), Corollary 5.3.
6658:{\displaystyle \mathbf {CP} ^{n}}
6464:. On the other hand, the measure
5309:{\displaystyle \mathbf {CP} ^{1}}
5171:{\displaystyle \mathbf {CP} ^{1}}
4996:{\displaystyle \mathbf {CP} ^{1}}
4936:{\displaystyle \mathbf {CP} ^{1}}
4827:{\displaystyle \mathbf {CP} ^{1}}
4688:{\displaystyle \mathbf {CP} ^{n}}
4652:{\displaystyle \mathbf {CP} ^{n}}
4600:{\displaystyle \mathbf {CP} ^{n}}
4442:{\displaystyle \mathbf {CP} ^{n}}
4413:. For a holomorphic endomorphism
4330:{\displaystyle \mathbf {CP} ^{n}}
4155:{\displaystyle \mathbf {CP} ^{n}}
4096:{\displaystyle \mathbf {CP} ^{n}}
3984:{\displaystyle \mathbf {CP} ^{n}}
3702:), counted with multiplicity, is
3642:{\displaystyle \mathbf {CP} ^{n}}
3550:{\displaystyle \mathbf {CP} ^{n}}
3270:{\displaystyle \mathbf {CP} ^{n}}
3230:{\displaystyle \mathbf {CP} ^{n}}
2970:{\displaystyle \mathbf {CP} ^{n}}
2505:{\displaystyle \mathbf {CP} ^{1}}
2396:{\displaystyle \mathbf {CP} ^{n}}
2251:{\displaystyle \mathbf {CP} ^{n}}
2151:{\displaystyle \mathbf {CP} ^{n}}
2111:{\displaystyle \mathbf {CP} ^{n}}
2036:for some homogeneous polynomials
1846:{\displaystyle \mathbf {CP} ^{n}}
1810:{\displaystyle \mathbf {CP} ^{n}}
1770:have been extended to a class of
1763:{\displaystyle \mathbf {CP} ^{n}}
1731:{\displaystyle \mathbf {CP} ^{n}}
1676:{\displaystyle \mathbf {CP} ^{1}}
1544:, one can assume after replacing
1533:{\displaystyle f^{a}(U)=f^{b}(U)}
1119:{\displaystyle c\in \mathbf {C} }
983:{\displaystyle \mathbf {CP} ^{1}}
918:{\displaystyle \mathbf {CP} ^{1}}
555:has absolute value less than 1.)
501:{\displaystyle \mathbf {CP} ^{1}}
279:{\displaystyle \mathbf {CP} ^{1}}
223:{\displaystyle \mathbf {CP} ^{1}}
171:{\displaystyle \mathbf {CP} ^{1}}
67:to itself. The related theory of
7947:Dynamics in one complex variable
7853:{\displaystyle \mathbf {P} ^{2}}
7840:
6966:Dynamics in complex dimension 1
6645:
6642:
6152:
5659:
5536:
5378:
5296:
5293:
5209:is conjugate to a Lattès map, a
5158:
5155:
4983:
4980:
4923:
4920:
4814:
4811:
4722:assigns its full mass 1 to some
4675:
4672:
4639:
4636:
4587:
4584:
4429:
4426:
4389:For any continuous endomorphism
4317:
4314:
4142:
4139:
4083:
4080:
3971:
3968:
3629:
3626:
3537:
3534:
3462:
3459:
3257:
3254:
3217:
3214:
2957:
2954:
2644:
2641:
2626:
2623:
2492:
2489:
2383:
2380:
2238:
2235:
2193:, which is also greater than 1.
2138:
2135:
2098:
2095:
1833:
1830:
1797:
1794:
1750:
1747:
1718:
1715:
1663:
1660:
1599:in the sense that all points in
1421:
1418:
1403:
1400:
1290:The Julia set of the polynomial
1193:The Julia set of the polynomial
1112:
970:
967:
905:
902:
794:diverge exponentially fast from
558:On the other hand, suppose that
488:
485:
266:
263:
210:
207:
158:
155:
7213:Cantat (2010), sections 7 to 9.
6829:{\displaystyle h(f)=\log d_{1}}
4401:is equal to the maximum of the
2582:More generally, for an integer
2270:. (It has also been called the
1819:morphism of algebraic varieties
786:on the circle, the dynamics of
87:Dynamics in complex dimension 1
7908:Société Mathématique de France
7470:Société Mathématique de France
6938:
6924:
6863:
6857:
6804:
6798:
6748:
6742:
6696:
6690:
6424:
6418:
6339:
6333:
6156:
6142:
6043:
6037:
5999:
5993:
5779:
5773:
5663:
5649:
5576:
5570:
5540:
5526:
5507:
5501:
5382:
5368:
5251:{\displaystyle f(z)=z^{\pm d}}
5229:
5223:
5125:
5119:
5071:
5065:
5056:
5050:
5025:
5019:
4877:
4864:
4858:
4852:
4768:
4755:
4749:
4743:
4292:
4286:
4255:
4249:
4185:
4179:
4050:
4044:
4014:
4008:
3943:
3937:
3907:
3901:
3871:
3865:
3770:
3751:
3743:
3732:
3720:
3709:
3683:
3677:
3591:
3585:
3421:
3415:
3351:
3338:
3329:
3315:
3312:
3288:
3179:
3173:
2919:
2904:
2890:
2875:
2868:
2830:
2766:
2724:
2718:
2715:
2683:
2680:
2636:
2559:
2551:
2454:
2448:
2371:of the equilibrium measure in
2354:
2348:
2122:, this is the same thing as a
2020:
2017:
1985:
1963:
1931:
1918:
1912:
1909:
1877:
1874:
1564:
1558:
1527:
1521:
1505:
1499:
1413:
1306:
1300:
1209:
1203:
1150:
1144:
1130:is the set of complex numbers
1066:
1060:
1016:
1010:
945:
939:
833:
827:
769:
766:
760:
757:
751:
745:
739:
733:
699:
693:
653:
647:
574:
566:
404:
401:
398:
392:
386:
380:
358:
355:
349:
343:
320:
314:
113:
107:
1:
7866:American Mathematical Society
7800:American Mathematical Society
7747:10.1090/S1056-3911-10-00549-7
7724:Journal of Algebraic Geometry
7668:Holomorphic dynamical systems
7222:Cantat (2014), section 2.4.3.
6264:. The equilibrium measure of
5715:if: there is only one number
5432:{\displaystyle 0\leq p\leq n}
5213:(up to sign), or a power map
4405:(or "metric entropy") of all
4397:, the topological entropy of
3099:{\displaystyle f^{*}\mu _{f}}
3027:{\displaystyle f_{*}\mu _{f}}
2786:Then the equilibrium measure
2086:that have no common zeros in
1857:. Such a mapping is given in
1631:, meaning that the action of
1615:, meaning that the action of
1237:{\displaystyle f(z)=z^{2}+az}
712:means the result of applying
178:to itself, by adding a point
8098:Surveys in Dynamical Systems
7281:Cantat (2010), Théorème 9.8.
7204:Cantat (2000), Théorème 2.2.
7177:Milnor (2006), problem 14-2.
6975:Complex quadratic polynomial
6665:is that for an automorphism
5692:do have such automorphisms.
5624:(The topological entropy of
5131:{\displaystyle \dim _{H}(Y)}
3490:, the measures just defined
1331:{\displaystyle f(z)=z^{2}+c}
1175:{\displaystyle f(z)=z^{2}+c}
1091:{\displaystyle f(z)=z^{2}+c}
1000:is chaotic. For the mapping
547:is zero at those points. An
7690:10.1007/978-3-642-13171-4_4
7290:Cantat (2014), Theorem 8.2.
7195:Milnor (2006), problem 5-3.
6950:{\displaystyle (d_{1})^{r}}
6437:of the equilibrium measure
5713:simple action on cohomology
5332:complex projective variety
3603:{\displaystyle f^{-1}(S)=S}
3494:to the equilibrium measure
2813:is the Haar measure on the
1691:, need not be contained in
1381:There is a rather complete
1134:such that the Julia set of
996:, on which the dynamics of
782:.) Even at periodic points
622:in the circle, and in fact
230:has the advantage of being
71:studies iteration over the
8149:
8029:Cambridge University Press
7999:Cambridge University Press
7951:Princeton University Press
7503:Princeton University Press
7427:Princeton University Press
7304:Alexander, Daniel (1994),
7063:Milnor (2006), section 13.
7029:Related areas of dynamics
6875:{\displaystyle f^{r}(z)=z}
6713:constructed automorphisms
6351:{\displaystyle f^{r}(z)=z}
5935:{\displaystyle \log d_{p}}
5880:measure of maximal entropy
5785:{\displaystyle H^{p,p}(X)}
5474:acting by pullback on the
4911:. The Julia set is all of
4802:. The Julia set is all of
4449:, the equilibrium measure
4393:of a compact metric space
3695:{\displaystyle f^{r}(z)=z}
3433:{\displaystyle f^{r}(w)=z}
2512:, the equilibrium measure
2473:{\displaystyle f(z)=z^{2}}
2276:measure of maximal entropy
1035:{\displaystyle f(z)=z^{2}}
845:{\displaystyle f^{r}(z)=z}
802:. (The periodic points of
672:for some positive integer
665:{\displaystyle f^{r}(z)=z}
610:on the circle in terms of
132:{\displaystyle f(z)=z^{2}}
90:
7645:10.1016/j.aim.2012.01.014
7591:10.1007/978-1-4612-4364-9
7318:10.1007/978-3-663-09197-4
6195:integer matrices acts on
6188:{\displaystyle 2\times 2}
6122:{\displaystyle E\times E}
5348:acts nontrivially on the
4951:of a probability measure
4403:measure-theoretic entropy
4386:, and Feliks Przytycki.
3482:such that for all points
598:is on the unit circle in
198:to the complex numbers. (
139:from the complex numbers
18:Complex analytic dynamics
8059:Inventiones Mathematicae
7072:Guedj (2010), Theorem B.
6902:{\displaystyle \mu _{f}}
6754:{\displaystyle J^{*}(f)}
6702:{\displaystyle J^{*}(f)}
6623:{\displaystyle \mu _{f}}
6592:{\displaystyle \mu _{f}}
6561:{\displaystyle \mu _{f}}
6519:{\displaystyle \mu _{f}}
6484:{\displaystyle \mu _{f}}
6457:{\displaystyle \mu _{f}}
6430:{\displaystyle J^{*}(f)}
6385:on the tangent space at
6076:{\displaystyle \mu _{f}}
6049:{\displaystyle J^{*}(f)}
6005:{\displaystyle J^{*}(f)}
5966:{\displaystyle \mu _{f}}
5902:{\displaystyle \mu _{f}}
5859:{\displaystyle \mu _{f}}
5632:on the whole cohomology
5198:{\displaystyle \mu _{f}}
4715:{\displaystyle \mu _{f}}
4550:{\displaystyle \mu _{f}}
4523:{\displaystyle \mu _{f}}
4469:{\displaystyle \mu _{f}}
4298:{\displaystyle J^{*}(f)}
4261:{\displaystyle J^{*}(f)}
4218:{\displaystyle \mu _{f}}
4191:{\displaystyle J^{*}(f)}
4123:{\displaystyle \mu _{f}}
4056:{\displaystyle J^{*}(f)}
4020:{\displaystyle J^{*}(f)}
3949:{\displaystyle J^{*}(f)}
3913:{\displaystyle J^{*}(f)}
3877:{\displaystyle J^{*}(f)}
3837:{\displaystyle \mu _{f}}
3610:), and one can take the
3514:{\displaystyle \mu _{f}}
3126:{\displaystyle \mu _{f}}
3054:{\displaystyle \mu _{f}}
2994:, in the sense that the
2806:{\displaystyle \mu _{f}}
2532:{\displaystyle \mu _{f}}
2360:{\displaystyle J^{*}(f)}
2219:{\displaystyle \mu _{f}}
2158:to itself.) Assume that
1705:complex projective space
705:{\displaystyle f^{r}(z)}
7622:Advances in Mathematics
7000:Riemann mapping theorem
6526:) are Zariski dense in
6397:.) For an automorphism
6103:be the abelian surface
5866:of maximal entropy for
5336:, meaning isomorphisms
5277:{\displaystyle d\geq 2}
4726:of Lebesgue measure 0.
4409:-invariant measures on
4371:{\displaystyle n\log d}
3277:and a positive integer
3069:, the pullback measure
2196:Then there is a unique
1859:homogeneous coordinates
1651:, the set of points in
1385:of a rational function
806:on the unit circle are
614:, the forward orbit of
521:{\displaystyle \infty }
469:{\displaystyle \infty }
191:{\displaystyle \infty }
145:complex projective line
8023:Tan, Lei, ed. (2000),
7875:10.1090/conm/269/04329
7854:
6951:
6903:
6876:
6830:
6755:
6703:
6659:
6624:
6593:
6562:
6520:
6485:
6458:
6431:
6379:
6352:
6296:Saddle periodic points
6254:
6189:
6163:
6123:
6077:
6050:
6006:
5967:
5936:
5903:
5860:
5813:
5786:
5740:
5670:
5615:
5547:
5460:
5433:
5389:
5310:
5278:
5252:
5199:
5172:
5142:. For an endomorphism
5132:
5090:
4997:
4965:
4944:
4937:
4905:
4835:
4828:
4796:
4716:
4689:
4653:
4601:
4551:
4524:
4470:
4443:
4372:
4331:
4299:
4262:
4219:
4192:
4156:
4124:
4097:
4057:
4021:
3985:
3950:
3914:
3878:
3838:
3807:
3806:{\displaystyle d^{rn}}
3777:
3696:
3643:
3604:
3551:
3515:
3476:
3434:
3388:
3387:{\displaystyle d^{rn}}
3358:
3271:
3231:
3196:
3127:
3100:
3055:
3028:
2971:
2939:
2807:
2776:
2658:
2602:
2601:{\displaystyle d>1}
2573:
2533:
2506:
2474:
2423:
2397:
2361:
2316:(around 1983), and by
2310:
2252:
2220:
2187:
2152:
2112:
2076:
2027:
1847:
1811:
1785:be an endomorphism of
1764:
1732:
1687:under some iterate of
1677:
1577:
1576:{\displaystyle f(U)=U}
1534:
1476:
1475:{\displaystyle a<b}
1435:
1378:
1367:
1332:
1283:
1276:
1238:
1176:
1120:
1092:
1046:in the sense that its
1036:
984:
952:
919:
873:
846:
776:
706:
666:
588:
522:
502:
470:
431:
280:
248:
224:
192:
172:
133:
7969:10.1515/9781400835539
7855:
7831:(2001), "Dynamics of
7435:10.1515/9781400851317
6952:
6904:
6877:
6831:
6756:
6704:
6660:
6625:
6594:
6563:
6521:
6486:
6459:
6432:
6380:
6378:{\displaystyle f^{r}}
6353:
6255:
6190:
6164:
6124:
6078:
6051:
6007:
5968:
5937:
5904:
5861:
5814:
5812:{\displaystyle d_{p}}
5787:
5741:
5739:{\displaystyle d_{p}}
5671:
5616:
5548:
5461:
5459:{\displaystyle d_{p}}
5434:
5403:of complex dimension
5390:
5311:
5279:
5253:
5200:
5173:
5133:
5091:
4998:
4966:
4938:
4906:
4841:
4829:
4797:
4732:
4717:
4690:
4654:
4621:absolutely continuous
4602:
4552:
4525:
4471:
4444:
4373:
4332:
4300:
4263:
4220:
4193:
4157:
4125:
4098:
4058:
4022:
3986:
3951:
3915:
3879:
3839:
3808:
3778:
3697:
3644:
3605:
3552:
3516:
3477:
3435:
3389:
3359:
3272:
3232:
3197:
3128:
3106:is also defined, and
3101:
3056:
3029:
2972:
2940:
2808:
2777:
2775:{\displaystyle f()=.}
2659:
2603:
2574:
2572:{\displaystyle |z|=1}
2534:
2507:
2475:
2424:
2398:
2362:
2311:
2253:
2221:
2188:
2186:{\displaystyle d^{n}}
2153:
2113:
2077:
2028:
1848:
1812:
1765:
1733:
1678:
1578:
1535:
1477:
1436:
1368:
1333:
1289:
1277:
1239:
1192:
1177:
1121:
1093:
1037:
985:
953:
920:
874:
872:{\displaystyle f^{r}}
847:
777:
707:
667:
624:uniformly distributed
589:
587:{\displaystyle |z|=1}
523:
503:
471:
432:
281:
249:
225:
193:
173:
134:
29:Branch of mathematics
7995:Holomorphic dynamics
7835:
7674:, pp. 165–294,
7505:, pp. 463–514,
6921:
6886:
6844:
6792:
6761:under the action of
6729:
6677:
6637:
6607:
6576:
6545:
6533:For an automorphism
6503:
6468:
6441:
6405:
6362:
6320:
6277:Kummer automorphisms
6211:
6199:. Any group element
6173:
6133:
6107:
6087:Kummer automorphisms
6060:
6024:
5980:
5950:
5913:
5886:
5843:
5835:For an automorphism
5796:
5754:
5723:
5636:
5564:
5482:
5443:
5411:
5355:
5288:
5262:
5217:
5211:Chebyshev polynomial
5182:
5150:
5103:
5010:
4975:
4964:{\displaystyle \mu }
4955:
4915:
4846:
4806:
4737:
4699:
4667:
4631:
4579:
4534:
4507:
4499:and, more strongly,
4453:
4421:
4353:
4309:
4273:
4236:
4202:
4166:
4134:
4107:
4075:
4031:
3995:
3963:
3924:
3888:
3852:
3821:
3787:
3706:
3664:
3621:
3569:
3529:
3498:
3448:
3402:
3368:
3285:
3249:
3209:
3141:
3110:
3073:
3038:
3001:
2949:
2824:
2790:
2674:
2612:
2586:
2547:
2516:
2484:
2442:
2407:
2375:
2335:
2294:
2230:
2203:
2170:
2130:
2090:
2040:
2026:{\displaystyle f()=}
1868:
1825:
1789:
1742:
1710:
1655:
1552:
1486:
1460:
1389:
1342:
1294:
1248:
1197:
1138:
1102:
1054:
1004:
962:
951:{\displaystyle f(z)}
933:
897:
856:
852:, the derivative of
814:
727:
680:
634:
562:
512:
480:
460:
301:
258:
238:
202:
182:
150:
101:
37:holomorphic dynamics
7910:, pp. 97–202,
7825:Fornaess, John Erik
7792:Fornaess, John Erik
7417:Bonifant, Araceli;
7033:Arithmetic dynamics
7010:Böttcher's equation
5882:). (In particular,
5872:equilibrium measure
5350:singular cohomology
4949:Hausdorff dimension
4571:is an endomorphism
4343:topological entropy
3783:, which is roughly
2996:pushforward measure
2765:
2741:
2422:{\displaystyle n=1}
2320:, Peter Papadopol,
2309:{\displaystyle n=1}
2260:equilibrium measure
2198:probability measure
2082:of the same degree
1548:by an iterate that
1451:connected component
1048:Hausdorff dimension
539:, meaning that the
69:arithmetic dynamics
8072:10.1007/BF01234434
7868:, pp. 47–85,
7850:
7615:de Thélin, Henry;
7472:, pp. 13–95,
6947:
6899:
6872:
6826:
6751:
6699:
6655:
6620:
6589:
6558:
6516:
6481:
6454:
6427:
6375:
6348:
6250:
6244:
6185:
6159:
6119:
6073:
6046:
6002:
5963:
5946:.) The support of
5932:
5899:
5856:
5809:
5782:
5736:
5666:
5611:
5591:
5543:
5456:
5429:
5385:
5306:
5274:
5248:
5195:
5168:
5128:
5086:
4993:
4961:
4945:
4933:
4901:
4836:
4824:
4792:
4712:
4685:
4649:
4597:
4547:
4520:
4466:
4439:
4384:Michał Misiurewicz
4368:
4327:
4295:
4258:
4215:
4188:
4152:
4120:
4093:
4053:
4017:
3981:
3946:
3910:
3874:
3834:
3803:
3773:
3692:
3639:
3600:
3547:
3511:
3472:
3440:. Then there is a
3430:
3384:
3354:
3267:
3227:
3192:
3137:in the sense that
3123:
3096:
3051:
3024:
2967:
2935:
2803:
2772:
2751:
2727:
2654:
2598:
2569:
2529:
2502:
2470:
2419:
2393:
2357:
2306:
2248:
2216:
2183:
2148:
2108:
2072:
2023:
1843:
1807:
1776:projective variety
1760:
1728:
1673:
1583:. Then either (1)
1573:
1530:
1472:
1431:
1379:
1363:
1328:
1284:
1272:
1234:
1172:
1116:
1088:
1032:
980:
948:
915:
869:
842:
772:
702:
662:
584:
518:
498:
466:
427:
276:
244:
220:
188:
168:
129:
53:algebraic dynamics
39:, is the study of
8123:Dynamical systems
7927:978-2-85629-338-6
7885:978-0-8218-1985-2
7809:978-0-8218-0317-2
7699:978-3-642-13170-7
7575:Gamelin, Theodore
7571:Carleson, Lennart
7512:978-0-691-15929-4
7479:978-2-85629-338-6
7444:978-0-691-15929-4
7043:Symbolic dynamics
6300:A periodic point
6129:. Then the group
5821:simple eigenvalue
5686:rational surfaces
5582:
4659:. Conversely, by
4611:by dividing by a
3559:totally invariant
3135:totally invariant
1449:showed that each
247:{\displaystyle z}
65:algebraic variety
61:rational function
41:dynamical systems
16:(Redirected from
8140:
8118:Complex analysis
8113:Complex dynamics
8082:
8049:
8019:
7989:
7962:
7949:(3rd ed.),
7938:
7919:
7896:
7877:
7859:
7857:
7856:
7851:
7849:
7848:
7843:
7820:
7787:
7778:
7757:
7740:
7715:Dinh, Tien-Cuong
7710:
7683:
7660:Dinh, Tien-Cuong
7655:
7638:
7629:(5): 2640–2655,
7617:Dinh, Tien-Cuong
7611:
7579:Complex dynamics
7566:
7556:10.4171/JEMS/946
7549:
7540:(4): 1289–1351,
7523:
7490:
7455:
7419:Lyubich, Mikhail
7413:
7396:
7373:
7338:
7291:
7288:
7282:
7279:
7273:
7270:
7264:
7261:
7255:
7252:
7246:
7243:
7232:
7229:
7223:
7220:
7214:
7211:
7205:
7202:
7196:
7193:
7187:
7184:
7178:
7175:
7169:
7166:
7160:
7157:
7151:
7148:
7142:
7139:
7133:
7130:
7124:
7121:
7112:
7109:
7103:
7100:
7094:
7091:
7085:
7082:
7073:
7070:
7064:
7061:
6985:Montel's theorem
6970:Complex analysis
6956:
6954:
6953:
6948:
6946:
6945:
6936:
6935:
6908:
6906:
6905:
6900:
6898:
6897:
6881:
6879:
6878:
6873:
6856:
6855:
6835:
6833:
6832:
6827:
6825:
6824:
6765:or its inverse.
6760:
6758:
6757:
6752:
6741:
6740:
6708:
6706:
6705:
6700:
6689:
6688:
6664:
6662:
6661:
6656:
6654:
6653:
6648:
6629:
6627:
6626:
6621:
6619:
6618:
6598:
6596:
6595:
6590:
6588:
6587:
6572:with respect to
6567:
6565:
6564:
6559:
6557:
6556:
6525:
6523:
6522:
6517:
6515:
6514:
6490:
6488:
6487:
6482:
6480:
6479:
6463:
6461:
6460:
6455:
6453:
6452:
6436:
6434:
6433:
6428:
6417:
6416:
6384:
6382:
6381:
6376:
6374:
6373:
6357:
6355:
6354:
6349:
6332:
6331:
6259:
6257:
6256:
6251:
6249:
6248:
6194:
6192:
6191:
6186:
6168:
6166:
6165:
6160:
6155:
6128:
6126:
6125:
6120:
6082:
6080:
6079:
6074:
6072:
6071:
6055:
6053:
6052:
6047:
6036:
6035:
6011:
6009:
6008:
6003:
5992:
5991:
5972:
5970:
5969:
5964:
5962:
5961:
5942:with respect to
5941:
5939:
5938:
5933:
5931:
5930:
5908:
5906:
5905:
5900:
5898:
5897:
5865:
5863:
5862:
5857:
5855:
5854:
5819:, and this is a
5818:
5816:
5815:
5810:
5808:
5807:
5791:
5789:
5788:
5783:
5772:
5771:
5745:
5743:
5742:
5737:
5735:
5734:
5675:
5673:
5672:
5667:
5662:
5648:
5647:
5620:
5618:
5617:
5612:
5607:
5606:
5590:
5552:
5550:
5549:
5544:
5539:
5525:
5524:
5500:
5499:
5476:Hodge cohomology
5465:
5463:
5462:
5457:
5455:
5454:
5438:
5436:
5435:
5430:
5394:
5392:
5391:
5386:
5381:
5367:
5366:
5315:
5313:
5312:
5307:
5305:
5304:
5299:
5283:
5281:
5280:
5275:
5257:
5255:
5254:
5249:
5247:
5246:
5204:
5202:
5201:
5196:
5194:
5193:
5177:
5175:
5174:
5169:
5167:
5166:
5161:
5137:
5135:
5134:
5129:
5115:
5114:
5095:
5093:
5092:
5087:
5046:
5045:
5002:
5000:
4999:
4994:
4992:
4991:
4986:
4970:
4968:
4967:
4962:
4942:
4940:
4939:
4934:
4932:
4931:
4926:
4910:
4908:
4907:
4902:
4900:
4899:
4890:
4885:
4884:
4833:
4831:
4830:
4825:
4823:
4822:
4817:
4801:
4799:
4798:
4793:
4791:
4790:
4781:
4776:
4775:
4721:
4719:
4718:
4713:
4711:
4710:
4694:
4692:
4691:
4686:
4684:
4683:
4678:
4658:
4656:
4655:
4650:
4648:
4647:
4642:
4625:Lebesgue measure
4623:with respect to
4606:
4604:
4603:
4598:
4596:
4595:
4590:
4556:
4554:
4553:
4548:
4546:
4545:
4529:
4527:
4526:
4521:
4519:
4518:
4475:
4473:
4472:
4467:
4465:
4464:
4448:
4446:
4445:
4440:
4438:
4437:
4432:
4377:
4375:
4374:
4369:
4336:
4334:
4333:
4328:
4326:
4325:
4320:
4304:
4302:
4301:
4296:
4285:
4284:
4267:
4265:
4264:
4259:
4248:
4247:
4224:
4222:
4221:
4216:
4214:
4213:
4197:
4195:
4194:
4189:
4178:
4177:
4161:
4159:
4158:
4153:
4151:
4150:
4145:
4129:
4127:
4126:
4121:
4119:
4118:
4102:
4100:
4099:
4094:
4092:
4091:
4086:
4062:
4060:
4059:
4054:
4043:
4042:
4026:
4024:
4023:
4018:
4007:
4006:
3990:
3988:
3987:
3982:
3980:
3979:
3974:
3955:
3953:
3952:
3947:
3936:
3935:
3919:
3917:
3916:
3911:
3900:
3899:
3883:
3881:
3880:
3875:
3864:
3863:
3843:
3841:
3840:
3835:
3833:
3832:
3812:
3810:
3809:
3804:
3802:
3801:
3782:
3780:
3779:
3774:
3763:
3762:
3750:
3736:
3735:
3701:
3699:
3698:
3693:
3676:
3675:
3648:
3646:
3645:
3640:
3638:
3637:
3632:
3609:
3607:
3606:
3601:
3584:
3583:
3556:
3554:
3553:
3548:
3546:
3545:
3540:
3520:
3518:
3517:
3512:
3510:
3509:
3481:
3479:
3478:
3473:
3471:
3470:
3465:
3439:
3437:
3436:
3431:
3414:
3413:
3393:
3391:
3390:
3385:
3383:
3382:
3363:
3361:
3360:
3355:
3350:
3349:
3337:
3336:
3327:
3326:
3311:
3310:
3298:
3276:
3274:
3273:
3268:
3266:
3265:
3260:
3236:
3234:
3233:
3228:
3226:
3225:
3220:
3201:
3199:
3198:
3193:
3191:
3190:
3163:
3162:
3153:
3152:
3132:
3130:
3129:
3124:
3122:
3121:
3105:
3103:
3102:
3097:
3095:
3094:
3085:
3084:
3060:
3058:
3057:
3052:
3050:
3049:
3033:
3031:
3030:
3025:
3023:
3022:
3013:
3012:
2976:
2974:
2973:
2968:
2966:
2965:
2960:
2944:
2942:
2941:
2936:
2922:
2917:
2916:
2907:
2893:
2888:
2887:
2878:
2867:
2866:
2848:
2847:
2812:
2810:
2809:
2804:
2802:
2801:
2781:
2779:
2778:
2773:
2764:
2759:
2740:
2735:
2714:
2713:
2695:
2694:
2663:
2661:
2660:
2655:
2653:
2652:
2647:
2635:
2634:
2629:
2607:
2605:
2604:
2599:
2578:
2576:
2575:
2570:
2562:
2554:
2538:
2536:
2535:
2530:
2528:
2527:
2511:
2509:
2508:
2503:
2501:
2500:
2495:
2479:
2477:
2476:
2471:
2469:
2468:
2438:For the mapping
2428:
2426:
2425:
2420:
2402:
2400:
2399:
2394:
2392:
2391:
2386:
2366:
2364:
2363:
2358:
2347:
2346:
2315:
2313:
2312:
2307:
2257:
2255:
2254:
2249:
2247:
2246:
2241:
2225:
2223:
2222:
2217:
2215:
2214:
2192:
2190:
2189:
2184:
2182:
2181:
2157:
2155:
2154:
2149:
2147:
2146:
2141:
2117:
2115:
2114:
2109:
2107:
2106:
2101:
2081:
2079:
2078:
2073:
2071:
2070:
2052:
2051:
2032:
2030:
2029:
2024:
2016:
2015:
1997:
1996:
1984:
1983:
1962:
1961:
1943:
1942:
1930:
1929:
1908:
1907:
1889:
1888:
1852:
1850:
1849:
1844:
1842:
1841:
1836:
1816:
1814:
1813:
1808:
1806:
1805:
1800:
1769:
1767:
1766:
1761:
1759:
1758:
1753:
1737:
1735:
1734:
1729:
1727:
1726:
1721:
1682:
1680:
1679:
1674:
1672:
1671:
1666:
1582:
1580:
1579:
1574:
1539:
1537:
1536:
1531:
1520:
1519:
1498:
1497:
1481:
1479:
1478:
1473:
1440:
1438:
1437:
1432:
1430:
1429:
1424:
1412:
1411:
1406:
1372:
1370:
1369:
1364:
1337:
1335:
1334:
1329:
1321:
1320:
1281:
1279:
1278:
1273:
1243:
1241:
1240:
1235:
1224:
1223:
1181:
1179:
1178:
1173:
1165:
1164:
1125:
1123:
1122:
1117:
1115:
1097:
1095:
1094:
1089:
1081:
1080:
1041:
1039:
1038:
1033:
1031:
1030:
989:
987:
986:
981:
979:
978:
973:
957:
955:
954:
949:
924:
922:
921:
916:
914:
913:
908:
878:
876:
875:
870:
868:
867:
851:
849:
848:
843:
826:
825:
781:
779:
778:
773:
711:
709:
708:
703:
692:
691:
671:
669:
668:
663:
646:
645:
593:
591:
590:
585:
577:
569:
527:
525:
524:
519:
507:
505:
504:
499:
497:
496:
491:
475:
473:
472:
467:
436:
434:
433:
428:
419:
418:
373:
372:
335:
334:
285:
283:
282:
277:
275:
274:
269:
253:
251:
250:
245:
229:
227:
226:
221:
219:
218:
213:
197:
195:
194:
189:
177:
175:
174:
169:
167:
166:
161:
138:
136:
135:
130:
128:
127:
73:rational numbers
49:complex analytic
33:Complex dynamics
21:
8148:
8147:
8143:
8142:
8141:
8139:
8138:
8137:
8103:
8102:
8089:
8052:
8039:
8022:
8009:
7992:
7979:
7941:
7928:
7899:
7886:
7838:
7833:
7832:
7823:
7810:
7790:
7760:
7713:
7700:
7672:Springer-Verlag
7658:
7614:
7601:
7583:Springer-Verlag
7569:
7526:
7513:
7493:
7480:
7458:
7445:
7416:
7376:
7363:
7353:Springer-Verlag
7341:
7328:
7303:
7300:
7295:
7294:
7289:
7285:
7280:
7276:
7271:
7267:
7262:
7258:
7253:
7249:
7244:
7235:
7230:
7226:
7221:
7217:
7212:
7208:
7203:
7199:
7194:
7190:
7185:
7181:
7176:
7172:
7167:
7163:
7158:
7154:
7149:
7145:
7140:
7136:
7131:
7127:
7122:
7115:
7110:
7106:
7101:
7097:
7092:
7088:
7083:
7076:
7071:
7067:
7062:
7058:
7053:
7015:Orbit portraits
6990:Poincaré metric
6963:
6937:
6927:
6919:
6918:
6889:
6884:
6883:
6847:
6842:
6841:
6816:
6790:
6789:
6732:
6727:
6726:
6711:Curtis McMullen
6680:
6675:
6674:
6640:
6635:
6634:
6610:
6605:
6604:
6579:
6574:
6573:
6548:
6543:
6542:
6506:
6501:
6500:
6471:
6466:
6465:
6444:
6439:
6438:
6408:
6403:
6402:
6365:
6360:
6359:
6323:
6318:
6317:
6298:
6243:
6242:
6237:
6231:
6230:
6225:
6215:
6209:
6208:
6171:
6170:
6131:
6130:
6105:
6104:
6089:
6063:
6058:
6057:
6027:
6022:
6021:
6020:is projective,
5983:
5978:
5977:
5975:small Julia set
5953:
5948:
5947:
5922:
5911:
5910:
5889:
5884:
5883:
5846:
5841:
5840:
5823:. For example,
5799:
5794:
5793:
5757:
5752:
5751:
5726:
5721:
5720:
5701:Kähler manifold
5639:
5634:
5633:
5598:
5562:
5561:
5513:
5485:
5480:
5479:
5468:spectral radius
5446:
5441:
5440:
5409:
5408:
5358:
5353:
5352:
5322:
5291:
5286:
5285:
5260:
5259:
5235:
5215:
5214:
5185:
5180:
5179:
5153:
5148:
5147:
5106:
5101:
5100:
5037:
5008:
5007:
4978:
4973:
4972:
4953:
4952:
4918:
4913:
4912:
4891:
4876:
4844:
4843:
4809:
4804:
4803:
4782:
4767:
4735:
4734:
4702:
4697:
4696:
4670:
4665:
4664:
4634:
4629:
4628:
4609:abelian variety
4582:
4577:
4576:
4563:
4537:
4532:
4531:
4510:
4505:
4504:
4456:
4451:
4450:
4424:
4419:
4418:
4351:
4350:
4312:
4307:
4306:
4276:
4271:
4270:
4239:
4234:
4233:
4205:
4200:
4199:
4169:
4164:
4163:
4137:
4132:
4131:
4110:
4105:
4104:
4078:
4073:
4072:
4034:
4029:
4028:
3998:
3993:
3992:
3966:
3961:
3960:
3927:
3922:
3921:
3891:
3886:
3885:
3855:
3850:
3849:
3824:
3819:
3818:
3790:
3785:
3784:
3754:
3712:
3704:
3703:
3667:
3662:
3661:
3624:
3619:
3618:
3612:exceptional set
3572:
3567:
3566:
3532:
3527:
3526:
3501:
3496:
3495:
3492:converge weakly
3457:
3446:
3445:
3405:
3400:
3399:
3371:
3366:
3365:
3341:
3328:
3318:
3299:
3283:
3282:
3252:
3247:
3246:
3239:Tien-Cuong Dinh
3212:
3207:
3206:
3182:
3154:
3144:
3139:
3138:
3113:
3108:
3107:
3086:
3076:
3071:
3070:
3067:finite morphism
3041:
3036:
3035:
3014:
3004:
2999:
2998:
2984:
2952:
2947:
2946:
2908:
2879:
2858:
2839:
2822:
2821:
2793:
2788:
2787:
2705:
2686:
2672:
2671:
2639:
2621:
2610:
2609:
2584:
2583:
2545:
2544:
2519:
2514:
2513:
2487:
2482:
2481:
2460:
2440:
2439:
2435:
2405:
2404:
2378:
2373:
2372:
2338:
2333:
2332:
2330:small Julia set
2292:
2291:
2288:Mikhail Lyubich
2233:
2228:
2227:
2206:
2201:
2200:
2173:
2168:
2167:
2133:
2128:
2127:
2093:
2088:
2087:
2062:
2043:
2038:
2037:
2007:
1988:
1975:
1953:
1934:
1921:
1899:
1880:
1866:
1865:
1828:
1823:
1822:
1792:
1787:
1786:
1745:
1740:
1739:
1713:
1708:
1707:
1701:
1658:
1653:
1652:
1550:
1549:
1511:
1489:
1484:
1483:
1458:
1457:
1447:Dennis Sullivan
1416:
1398:
1387:
1386:
1340:
1339:
1312:
1292:
1291:
1246:
1245:
1215:
1195:
1194:
1156:
1136:
1135:
1100:
1099:
1098:for a constant
1072:
1052:
1051:
1022:
1002:
1001:
965:
960:
959:
931:
930:
900:
895:
894:
859:
854:
853:
817:
812:
811:
798:upon iterating
725:
724:
683:
678:
677:
637:
632:
631:
628:periodic points
594:, meaning that
560:
559:
530:superattracting
510:
509:
483:
478:
477:
458:
457:
410:
364:
326:
299:
298:
286:, how does its
261:
256:
255:
236:
235:
205:
200:
199:
180:
179:
153:
148:
147:
119:
99:
98:
95:
89:
81:complex numbers
79:instead of the
30:
23:
22:
15:
12:
11:
5:
8146:
8144:
8136:
8135:
8130:
8125:
8120:
8115:
8105:
8104:
8101:
8100:
8095:
8088:
8087:External links
8085:
8084:
8083:
8066:(3): 627–649,
8050:
8037:
8020:
8007:
7990:
7977:
7939:
7926:
7897:
7884:
7847:
7842:
7829:Sibony, Nessim
7821:
7808:
7788:
7769:(2): 109–122,
7758:
7731:(3): 473–529,
7719:Sibony, Nessim
7711:
7698:
7664:Sibony, Nessim
7656:
7612:
7599:
7567:
7524:
7511:
7491:
7478:
7456:
7443:
7414:
7403:10.4171/CMH/21
7387:(2): 433–454,
7374:
7361:
7339:
7326:
7299:
7296:
7293:
7292:
7283:
7274:
7265:
7256:
7247:
7233:
7224:
7215:
7206:
7197:
7188:
7179:
7170:
7161:
7152:
7143:
7134:
7125:
7113:
7104:
7095:
7086:
7074:
7065:
7055:
7054:
7052:
7049:
7048:
7047:
7046:
7045:
7040:
7035:
7026:
7025:
7024:
7023:
7017:
7012:
7007:
7002:
6997:
6992:
6987:
6982:
6977:
6972:
6962:
6959:
6944:
6940:
6934:
6930:
6926:
6896:
6892:
6871:
6868:
6865:
6862:
6859:
6854:
6850:
6840:(meaning that
6823:
6819:
6815:
6812:
6809:
6806:
6803:
6800:
6797:
6750:
6747:
6744:
6739:
6735:
6698:
6695:
6692:
6687:
6683:
6652:
6647:
6644:
6617:
6613:
6586:
6582:
6555:
6551:
6513:
6509:
6478:
6474:
6451:
6447:
6426:
6423:
6420:
6415:
6411:
6372:
6368:
6347:
6344:
6341:
6338:
6335:
6330:
6326:
6297:
6294:
6247:
6241:
6238:
6236:
6233:
6232:
6229:
6226:
6224:
6221:
6220:
6218:
6184:
6181:
6178:
6169:of invertible
6158:
6154:
6150:
6147:
6144:
6141:
6138:
6118:
6115:
6112:
6097:elliptic curve
6088:
6085:
6070:
6066:
6045:
6042:
6039:
6034:
6030:
6012:. Informally:
6001:
5998:
5995:
5990:
5986:
5973:is called the
5960:
5956:
5929:
5925:
5921:
5918:
5896:
5892:
5853:
5849:
5806:
5802:
5781:
5778:
5775:
5770:
5767:
5764:
5760:
5733:
5729:
5665:
5661:
5657:
5654:
5651:
5646:
5642:
5622:
5621:
5610:
5605:
5601:
5597:
5594:
5589:
5585:
5581:
5578:
5575:
5572:
5569:
5542:
5538:
5534:
5531:
5528:
5523:
5520:
5516:
5512:
5509:
5506:
5503:
5498:
5495:
5492:
5488:
5453:
5449:
5428:
5425:
5422:
5419:
5416:
5384:
5380:
5376:
5373:
5370:
5365:
5361:
5321:
5318:
5303:
5298:
5295:
5273:
5270:
5267:
5245:
5242:
5238:
5234:
5231:
5228:
5225:
5222:
5192:
5188:
5165:
5160:
5157:
5127:
5124:
5121:
5118:
5113:
5109:
5097:
5096:
5085:
5082:
5079:
5076:
5073:
5070:
5067:
5064:
5061:
5058:
5055:
5052:
5049:
5044:
5040:
5036:
5033:
5030:
5027:
5024:
5021:
5018:
5015:
4990:
4985:
4982:
4960:
4930:
4925:
4922:
4898:
4894:
4889:
4883:
4879:
4875:
4872:
4869:
4866:
4863:
4860:
4857:
4854:
4851:
4821:
4816:
4813:
4789:
4785:
4780:
4774:
4770:
4766:
4763:
4760:
4757:
4754:
4751:
4748:
4745:
4742:
4709:
4705:
4682:
4677:
4674:
4646:
4641:
4638:
4594:
4589:
4586:
4562:
4559:
4544:
4540:
4517:
4513:
4463:
4459:
4436:
4431:
4428:
4380:Mikhail Gromov
4367:
4364:
4361:
4358:
4324:
4319:
4316:
4294:
4291:
4288:
4283:
4279:
4257:
4254:
4251:
4246:
4242:
4212:
4208:
4187:
4184:
4181:
4176:
4172:
4149:
4144:
4141:
4117:
4113:
4090:
4085:
4082:
4052:
4049:
4046:
4041:
4037:
4016:
4013:
4010:
4005:
4001:
3978:
3973:
3970:
3945:
3942:
3939:
3934:
3930:
3909:
3906:
3903:
3898:
3894:
3873:
3870:
3867:
3862:
3858:
3831:
3827:
3800:
3797:
3793:
3772:
3769:
3766:
3761:
3757:
3753:
3749:
3745:
3742:
3739:
3734:
3731:
3728:
3725:
3722:
3719:
3715:
3711:
3691:
3688:
3685:
3682:
3679:
3674:
3670:
3660:(meaning that
3636:
3631:
3628:
3599:
3596:
3593:
3590:
3587:
3582:
3579:
3575:
3565:(meaning that
3544:
3539:
3536:
3508:
3504:
3469:
3464:
3461:
3456:
3453:
3442:Zariski closed
3429:
3426:
3423:
3420:
3417:
3412:
3408:
3381:
3378:
3374:
3353:
3348:
3344:
3340:
3335:
3331:
3325:
3321:
3317:
3314:
3309:
3306:
3302:
3297:
3293:
3290:
3264:
3259:
3256:
3224:
3219:
3216:
3189:
3185:
3181:
3178:
3175:
3172:
3169:
3166:
3161:
3157:
3151:
3147:
3120:
3116:
3093:
3089:
3083:
3079:
3048:
3044:
3021:
3017:
3011:
3007:
2983:
2980:
2979:
2978:
2964:
2959:
2956:
2934:
2931:
2928:
2925:
2921:
2915:
2911:
2906:
2902:
2899:
2896:
2892:
2886:
2882:
2877:
2873:
2870:
2865:
2861:
2857:
2854:
2851:
2846:
2842:
2838:
2835:
2832:
2829:
2800:
2796:
2784:
2783:
2782:
2771:
2768:
2763:
2758:
2754:
2750:
2747:
2744:
2739:
2734:
2730:
2726:
2723:
2720:
2717:
2712:
2708:
2704:
2701:
2698:
2693:
2689:
2685:
2682:
2679:
2666:
2665:
2664:be the mapping
2651:
2646:
2643:
2638:
2633:
2628:
2625:
2620:
2617:
2597:
2594:
2591:
2580:
2568:
2565:
2561:
2557:
2553:
2526:
2522:
2499:
2494:
2491:
2467:
2463:
2459:
2456:
2453:
2450:
2447:
2434:
2431:
2418:
2415:
2412:
2390:
2385:
2382:
2356:
2353:
2350:
2345:
2341:
2305:
2302:
2299:
2245:
2240:
2237:
2213:
2209:
2180:
2176:
2145:
2140:
2137:
2120:Chow's theorem
2105:
2100:
2097:
2069:
2065:
2061:
2058:
2055:
2050:
2046:
2034:
2033:
2022:
2019:
2014:
2010:
2006:
2003:
2000:
1995:
1991:
1987:
1982:
1978:
1974:
1971:
1968:
1965:
1960:
1956:
1952:
1949:
1946:
1941:
1937:
1933:
1928:
1924:
1920:
1917:
1914:
1911:
1906:
1902:
1898:
1895:
1892:
1887:
1883:
1879:
1876:
1873:
1840:
1835:
1832:
1804:
1799:
1796:
1757:
1752:
1749:
1725:
1720:
1717:
1700:
1697:
1670:
1665:
1662:
1572:
1569:
1566:
1563:
1560:
1557:
1529:
1526:
1523:
1518:
1514:
1510:
1507:
1504:
1501:
1496:
1492:
1471:
1468:
1465:
1428:
1423:
1420:
1415:
1410:
1405:
1402:
1397:
1394:
1362:
1359:
1356:
1353:
1350:
1347:
1327:
1324:
1319:
1315:
1311:
1308:
1305:
1302:
1299:
1271:
1268:
1265:
1262:
1259:
1256:
1253:
1233:
1230:
1227:
1222:
1218:
1214:
1211:
1208:
1205:
1202:
1171:
1168:
1163:
1159:
1155:
1152:
1149:
1146:
1143:
1128:Mandelbrot set
1114:
1110:
1107:
1087:
1084:
1079:
1075:
1071:
1068:
1065:
1062:
1059:
1029:
1025:
1021:
1018:
1015:
1012:
1009:
977:
972:
969:
947:
944:
941:
938:
912:
907:
904:
866:
862:
841:
838:
835:
832:
829:
824:
820:
771:
768:
765:
762:
759:
756:
753:
750:
747:
744:
741:
738:
735:
732:
701:
698:
695:
690:
686:
661:
658:
655:
652:
649:
644:
640:
612:measure theory
583:
580:
576:
572:
568:
517:
495:
490:
487:
465:
442:absolute value
438:
437:
426:
422:
417:
413:
409:
406:
403:
400:
397:
394:
391:
388:
385:
382:
379:
376:
371:
367:
363:
360:
357:
354:
351:
348:
345:
342:
338:
333:
329:
325:
322:
319:
316:
313:
309:
306:
273:
268:
265:
243:
217:
212:
209:
187:
165:
160:
157:
126:
122:
118:
115:
112:
109:
106:
91:Main article:
88:
85:
77:p-adic numbers
28:
24:
14:
13:
10:
9:
6:
4:
3:
2:
8145:
8134:
8131:
8129:
8126:
8124:
8121:
8119:
8116:
8114:
8111:
8110:
8108:
8099:
8096:
8094:
8091:
8090:
8086:
8081:
8077:
8073:
8069:
8065:
8061:
8060:
8055:
8051:
8048:
8044:
8040:
8038:0-521-77476-4
8034:
8030:
8026:
8021:
8018:
8014:
8010:
8008:0-521-66258-3
8004:
8000:
7996:
7991:
7988:
7984:
7980:
7978:0-691-12488-4
7974:
7970:
7966:
7961:
7956:
7952:
7948:
7944:
7940:
7937:
7933:
7929:
7923:
7918:
7913:
7909:
7905:
7904:
7898:
7895:
7891:
7887:
7881:
7876:
7871:
7867:
7863:
7860:(examples)",
7845:
7830:
7826:
7822:
7819:
7815:
7811:
7805:
7801:
7797:
7793:
7789:
7786:
7782:
7777:
7772:
7768:
7764:
7759:
7756:
7752:
7748:
7744:
7739:
7734:
7730:
7726:
7725:
7720:
7716:
7712:
7709:
7705:
7701:
7695:
7691:
7687:
7682:
7677:
7673:
7669:
7665:
7661:
7657:
7654:
7650:
7646:
7642:
7637:
7632:
7628:
7624:
7623:
7618:
7613:
7610:
7606:
7602:
7600:0-387-97942-5
7596:
7592:
7588:
7584:
7580:
7576:
7572:
7568:
7565:
7561:
7557:
7553:
7548:
7543:
7539:
7535:
7534:
7529:
7528:Cantat, Serge
7525:
7522:
7518:
7514:
7508:
7504:
7500:
7496:
7495:Cantat, Serge
7492:
7489:
7485:
7481:
7475:
7471:
7467:
7466:
7461:
7460:Cantat, Serge
7457:
7454:
7450:
7446:
7440:
7436:
7432:
7428:
7424:
7420:
7415:
7412:
7408:
7404:
7400:
7395:
7390:
7386:
7382:
7381:
7375:
7372:
7368:
7364:
7362:0-387-97589-6
7358:
7354:
7350:
7349:
7344:
7343:Beardon, Alan
7340:
7337:
7333:
7329:
7327:3-528-06520-6
7323:
7319:
7315:
7311:
7310:Vieweg Verlag
7307:
7302:
7301:
7297:
7287:
7284:
7278:
7275:
7269:
7266:
7260:
7257:
7251:
7248:
7242:
7240:
7238:
7234:
7228:
7225:
7219:
7216:
7210:
7207:
7201:
7198:
7192:
7189:
7183:
7180:
7174:
7171:
7165:
7162:
7156:
7153:
7147:
7144:
7138:
7135:
7129:
7126:
7120:
7118:
7114:
7108:
7105:
7099:
7096:
7090:
7087:
7081:
7079:
7075:
7069:
7066:
7060:
7057:
7050:
7044:
7041:
7039:
7036:
7034:
7031:
7030:
7028:
7027:
7021:
7018:
7016:
7013:
7011:
7008:
7006:
7003:
7001:
6998:
6996:
6995:Schwarz lemma
6993:
6991:
6988:
6986:
6983:
6981:
6978:
6976:
6973:
6971:
6968:
6967:
6965:
6964:
6960:
6958:
6942:
6932:
6928:
6916:
6912:
6894:
6890:
6869:
6866:
6860:
6852:
6848:
6839:
6821:
6817:
6813:
6810:
6807:
6801:
6795:
6787:
6783:
6779:
6775:
6771:
6766:
6764:
6745:
6737:
6733:
6724:
6720:
6716:
6712:
6693:
6685:
6681:
6672:
6668:
6650:
6631:
6615:
6611:
6602:
6584:
6580:
6571:
6553:
6549:
6540:
6536:
6531:
6529:
6511:
6507:
6498:
6494:
6476:
6472:
6449:
6445:
6421:
6413:
6409:
6400:
6396:
6392:
6388:
6370:
6366:
6345:
6342:
6336:
6328:
6324:
6315:
6311:
6307:
6303:
6295:
6293:
6290:
6286:
6282:
6278:
6273:
6271:
6267:
6263:
6245:
6239:
6234:
6227:
6222:
6216:
6206:
6202:
6198:
6182:
6179:
6176:
6148:
6145:
6139:
6136:
6116:
6113:
6110:
6102:
6098:
6095:be a complex
6094:
6086:
6084:
6068:
6064:
6040:
6032:
6028:
6019:
6015:
5996:
5988:
5984:
5976:
5958:
5954:
5945:
5927:
5923:
5919:
5916:
5894:
5890:
5881:
5877:
5876:Green measure
5873:
5870:, called the
5869:
5851:
5847:
5838:
5833:
5831:
5826:
5822:
5804:
5800:
5776:
5768:
5765:
5762:
5758:
5749:
5731:
5727:
5718:
5714:
5710:
5706:
5702:
5699:be a compact
5698:
5693:
5691:
5687:
5683:
5679:
5655:
5652:
5644:
5640:
5631:
5627:
5608:
5603:
5599:
5595:
5592:
5587:
5579:
5573:
5567:
5560:
5559:
5558:
5556:
5532:
5529:
5521:
5518:
5514:
5510:
5504:
5496:
5493:
5490:
5486:
5477:
5473:
5469:
5451:
5447:
5426:
5423:
5420:
5417:
5414:
5406:
5402:
5396:
5374:
5371:
5363:
5359:
5351:
5347:
5343:
5339:
5335:
5331:
5327:
5326:automorphisms
5319:
5317:
5301:
5271:
5268:
5265:
5243:
5240:
5236:
5232:
5226:
5220:
5212:
5208:
5190:
5186:
5163:
5145:
5141:
5122:
5116:
5111:
5107:
5083:
5077:
5074:
5068:
5062:
5059:
5053:
5047:
5042:
5038:
5028:
5022:
5016:
5013:
5006:
5005:
5004:
4988:
4958:
4950:
4928:
4896:
4892:
4887:
4881:
4873:
4870:
4867:
4861:
4855:
4849:
4840:
4819:
4787:
4783:
4778:
4772:
4764:
4761:
4758:
4752:
4746:
4740:
4731:
4727:
4725:
4707:
4703:
4680:
4662:
4644:
4626:
4622:
4618:
4614:
4610:
4592:
4574:
4570:
4569:
4560:
4558:
4542:
4538:
4515:
4511:
4502:
4498:
4494:
4490:
4485:
4483:
4479:
4461:
4457:
4434:
4416:
4412:
4408:
4404:
4400:
4396:
4392:
4387:
4385:
4381:
4365:
4362:
4359:
4356:
4348:
4344:
4340:
4322:
4289:
4281:
4277:
4252:
4244:
4240:
4230:
4228:
4210:
4206:
4182:
4174:
4170:
4147:
4115:
4111:
4088:
4070:
4069:Zariski dense
4066:
4047:
4039:
4035:
4027:are dense in
4011:
4003:
3999:
3976:
3957:
3940:
3932:
3928:
3904:
3896:
3892:
3868:
3860:
3856:
3847:
3829:
3825:
3816:
3798:
3795:
3791:
3767:
3764:
3759:
3755:
3747:
3740:
3737:
3729:
3726:
3723:
3717:
3713:
3689:
3686:
3680:
3672:
3668:
3659:
3655:
3650:
3634:
3616:
3613:
3597:
3594:
3588:
3580:
3577:
3573:
3564:
3560:
3542:
3524:
3506:
3502:
3493:
3489:
3485:
3467:
3454:
3451:
3443:
3427:
3424:
3418:
3410:
3406:
3397:
3379:
3376:
3372:
3346:
3342:
3333:
3323:
3319:
3307:
3304:
3300:
3295:
3291:
3280:
3262:
3244:
3240:
3222:
3203:
3187:
3183:
3176:
3170:
3167:
3164:
3159:
3155:
3149:
3145:
3136:
3118:
3114:
3091:
3087:
3081:
3077:
3068:
3064:
3046:
3042:
3019:
3015:
3009:
3005:
2997:
2993:
2989:
2981:
2962:
2932:
2926:
2923:
2913:
2909:
2900:
2897:
2894:
2884:
2880:
2871:
2863:
2859:
2855:
2852:
2849:
2844:
2840:
2836:
2833:
2820:
2817:-dimensional
2816:
2798:
2794:
2785:
2769:
2761:
2756:
2752:
2748:
2745:
2742:
2737:
2732:
2728:
2721:
2710:
2706:
2702:
2699:
2696:
2691:
2687:
2677:
2670:
2669:
2668:
2667:
2649:
2631:
2618:
2615:
2595:
2592:
2589:
2581:
2566:
2563:
2555:
2542:
2524:
2520:
2497:
2465:
2461:
2457:
2451:
2445:
2437:
2436:
2432:
2430:
2416:
2413:
2410:
2388:
2370:
2351:
2343:
2339:
2331:
2327:
2326:Nessim Sibony
2323:
2322:John Fornaess
2319:
2303:
2300:
2297:
2289:
2285:
2281:
2277:
2273:
2272:Green measure
2269:
2265:
2261:
2243:
2211:
2207:
2199:
2194:
2178:
2174:
2165:
2161:
2143:
2126:mapping from
2125:
2121:
2103:
2085:
2067:
2063:
2059:
2056:
2053:
2048:
2044:
2012:
2008:
2004:
2001:
1998:
1993:
1989:
1980:
1976:
1972:
1969:
1966:
1958:
1954:
1950:
1947:
1944:
1939:
1935:
1926:
1922:
1915:
1904:
1900:
1896:
1893:
1890:
1885:
1881:
1871:
1864:
1863:
1862:
1860:
1856:
1838:
1820:
1802:
1784:
1779:
1777:
1773:
1772:rational maps
1755:
1723:
1706:
1698:
1696:
1694:
1690:
1686:
1668:
1650:
1646:
1642:
1638:
1634:
1630:
1626:
1622:
1618:
1614:
1610:
1606:
1602:
1598:
1594:
1590:
1586:
1570:
1567:
1561:
1555:
1547:
1543:
1524:
1516:
1512:
1508:
1502:
1494:
1490:
1469:
1466:
1463:
1455:
1452:
1448:
1444:
1426:
1408:
1395:
1392:
1384:
1376:
1360:
1357:
1354:
1351:
1348:
1345:
1325:
1322:
1317:
1313:
1309:
1303:
1297:
1288:
1269:
1266:
1263:
1260:
1257:
1254:
1251:
1231:
1228:
1225:
1220:
1216:
1212:
1206:
1200:
1191:
1187:
1185:
1169:
1166:
1161:
1157:
1153:
1147:
1141:
1133:
1129:
1108:
1105:
1085:
1082:
1077:
1073:
1069:
1063:
1057:
1049:
1045:
1027:
1023:
1019:
1013:
1007:
999:
995:
994:
975:
942:
936:
928:
925:to itself of
910:
892:
888:
884:
882:
864:
860:
839:
836:
830:
822:
818:
809:
805:
801:
797:
793:
789:
785:
763:
754:
748:
742:
736:
730:
722:
719:
715:
696:
688:
684:
675:
659:
656:
650:
642:
638:
629:
625:
621:
617:
613:
609:
605:
601:
597:
581:
578:
570:
556:
554:
550:
546:
542:
538:
534:
531:
493:
455:
451:
450:exponentially
447:
443:
424:
420:
415:
411:
407:
395:
389:
383:
377:
374:
369:
365:
361:
352:
346:
340:
336:
331:
327:
323:
317:
311:
307:
304:
297:
296:
295:
293:
292:forward orbit
289:
271:
241:
233:
215:
163:
146:
142:
124:
120:
116:
110:
104:
94:
86:
84:
82:
78:
74:
70:
66:
62:
58:
54:
50:
46:
42:
38:
34:
27:
19:
8128:Chaos theory
8063:
8057:
8054:Zdunik, Anna
8024:
7994:
7960:math/9201272
7946:
7943:Milnor, John
7917:math/0611302
7902:
7861:
7795:
7776:math/0212208
7766:
7762:
7728:
7722:
7667:
7626:
7620:
7578:
7537:
7531:
7498:
7464:
7422:
7394:math/0501034
7384:
7378:
7347:
7305:
7286:
7277:
7268:
7259:
7250:
7227:
7218:
7209:
7200:
7191:
7182:
7173:
7164:
7155:
7146:
7137:
7128:
7107:
7098:
7089:
7068:
7059:
7038:Chaos theory
6915:John Smillie
6910:
6837:
6785:
6781:
6777:
6773:
6769:
6767:
6762:
6722:
6718:
6714:
6670:
6666:
6632:
6600:
6569:
6538:
6534:
6532:
6527:
6496:
6492:
6398:
6394:
6390:
6386:
6313:
6309:
6308:is called a
6305:
6301:
6299:
6284:
6276:
6274:
6269:
6265:
6261:
6200:
6196:
6100:
6092:
6090:
6017:
6013:
5974:
5943:
5909:has entropy
5879:
5875:
5871:
5867:
5836:
5834:
5829:
5825:Serge Cantat
5747:
5716:
5712:
5708:
5704:
5696:
5694:
5677:
5629:
5625:
5623:
5554:
5471:
5404:
5400:
5397:
5345:
5341:
5337:
5333:
5325:
5323:
5206:
5143:
5139:
5098:
4948:
4946:
4616:
4613:finite group
4572:
4566:
4564:
4492:
4488:
4486:
4481:
4477:
4414:
4410:
4406:
4398:
4394:
4390:
4388:
4346:
4338:
4232:The support
4231:
4064:
3958:
3845:
3814:
3657:
3653:
3651:
3614:
3611:
3562:
3558:
3522:
3487:
3483:
3395:
3278:
3242:
3204:
3134:
3062:
3034:is equal to
2991:
2987:
2985:
2814:
2541:Haar measure
2329:
2318:John Hubbard
2284:Ricardo Mañé
2275:
2271:
2267:
2263:
2259:
2195:
2163:
2159:
2083:
2035:
1854:
1817:, meaning a
1782:
1780:
1702:
1692:
1688:
1684:
1683:that map to
1648:
1644:
1636:
1632:
1624:
1620:
1616:
1608:
1604:
1600:
1596:
1592:
1588:
1584:
1545:
1541:
1453:
1442:
1380:
1373:. This is a
1131:
997:
991:
891:Gaston Julia
887:Pierre Fatou
885:
880:
807:
803:
799:
795:
791:
787:
783:
720:
717:
713:
673:
615:
607:
603:
599:
595:
557:
552:
548:
544:
536:
533:fixed points
529:
453:
445:
439:
291:
287:
140:
96:
52:
43:obtained by
36:
32:
31:
26:
5690:K3 surfaces
4661:Anna Zdunik
4561:Lattès maps
4227:perfect set
2280:Artur Lopes
2124:holomorphic
1629:Herman ring
1613:Siegel disk
8107:Categories
7298:References
6316:such that
6281:blowing up
5719:such that
5682:eigenvalue
4568:Lattès map
4305:is all of
3061:. Because
1482:such that
1375:Cantor set
549:attracting
541:derivative
452:fast. If |
57:polynomial
55:, where a
7738:0804.0860
7681:0810.0811
7636:1009.5796
7547:1410.1202
6891:μ
6814:
6738:∗
6686:∗
6612:μ
6581:μ
6550:μ
6508:μ
6473:μ
6446:μ
6414:∗
6180:×
6114:×
6065:μ
6033:∗
5989:∗
5955:μ
5920:
5891:μ
5848:μ
5645:∗
5596:
5511:⊂
5424:≤
5418:≤
5364:∗
5269:≥
5241:±
5187:μ
5117:
5063:μ
5048:
5023:μ
5017:
4959:μ
4871:−
4762:−
4724:Borel set
4704:μ
4539:μ
4512:μ
4458:μ
4363:
4282:∗
4245:∗
4207:μ
4175:∗
4112:μ
4040:∗
4004:∗
3933:∗
3897:∗
3861:∗
3826:μ
3765:−
3738:−
3578:−
3503:μ
3455:⊊
3343:δ
3334:∗
3184:μ
3171:
3156:μ
3150:∗
3115:μ
3088:μ
3082:∗
3043:μ
3016:μ
3010:∗
2988:invariant
2898:⋯
2853:…
2795:μ
2746:…
2700:…
2637:→
2619::
2521:μ
2344:∗
2208:μ
2057:…
2002:…
1970:…
1948:…
1894:…
1774:from any
1597:parabolic
1443:Fatou set
1414:→
1396::
1355:−
1349:≐
1258:−
1255:≐
1184:connected
1109:∈
993:Julia set
808:repelling
764:⋯
743:⋯
516:∞
464:∞
425:…
186:∞
93:Julia set
45:iterating
8133:Fractals
7945:(2006),
7794:(1996),
7577:(1993),
7345:(1991),
6961:See also
6099:and let
5676:.) Thus
2433:Examples
676:. (Here
8080:1032883
8047:1765080
8017:1747010
7987:2193309
7936:2932434
7894:1810536
7818:1363948
7785:1995861
7755:2629598
7708:2648690
7653:2889139
7609:1230383
7564:4071328
7521:3289919
7488:2932433
7453:3289442
7411:2142250
7371:1128089
7336:1260930
7022:puzzles
6287:and is
5466:be the
4497:ergodic
4476:is the
3486:not in
3444:subset
3394:points
2539:is the
2369:support
2367:is the
1641:annulus
1441:in the
1044:fractal
723:times,
232:compact
75:or the
8078:
8045:
8035:
8015:
8005:
7985:
7975:
7934:
7924:
7892:
7882:
7816:
7806:
7783:
7753:
7706:
7696:
7651:
7607:
7597:
7562:
7519:
7509:
7486:
7476:
7451:
7441:
7409:
7369:
7359:
7334:
7324:
7020:Yoccoz
6310:saddle
6289:smooth
6203:whose
5478:group
5439:, let
5330:smooth
5099:where
4501:mixing
4478:unique
3561:under
2990:under
2608:, let
2324:, and
2286:, and
2258:, the
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