1179:
4828:
4719:
1276:
6280:
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
4257:
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
5305:, 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.
5816:
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
424:
5387:
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
6247:
2651:
1428:
5083:
5540:
3351:
2932:
3469:
3189:
5828:
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
769:
5608:
3770:
6272:
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
5663:
5382:
1360:
4898:
4789:
2069:
1269:
6156:
6652:
5303:
5165:
4990:
4930:
4821:
4682:
4646:
4594:
4436:
4324:
4149:
4090:
3978:
3636:
3544:
3264:
3224:
2964:
2499:
2390:
2245:
2145:
2105:
1840:
1804:
1757:
1725:
1670:
1527:
1113:
977:
912:
495:
273:
217:
165:
7847:
6823:
5245:
5426:
3093:
3021:
1231:
5125:
1325:
1169:
1085:
6944:
3597:
6869:
6345:
5929:
5779:
3689:
3427:
2467:
1029:
839:
659:
126:
6182:
6116:
6896:
6748:
6696:
6617:
6586:
6555:
6513:
6478:
6451:
6424:
6070:
6043:
5999:
5960:
5896:
5853:
5192:
4709:
4544:
4517:
4463:
4292:
4255:
4212:
4185:
4117:
4050:
4014:
3943:
3907:
3871:
3831:
3508:
3120:
3048:
2800:
2526:
2354:
2213:
699:
5271:
4365:
515:
463:
185:
3800:
3381:
2595:
1570:
1469:
6372:
5806:
5733:
5453:
2769:
2566:
2180:
866:
581:
4958:
2020:
945:
2416:
2303:
241:
5313:
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
4684:
whose equilibrium measure is absolutely continuous with respect to
Lebesgue measure are the Lattès examples. That is, for all non-Lattès endomorphisms,
2108:
5669:
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
6993:
289:
7521:
6968:
2966:
to itself, the equilibrium measure can be much more complicated, as one sees already in complex dimension 1 from pictures of Julia sets.
6199:
2600:
1377:
4998:
7896:
7458:
6378:
has absolute value less than 1, at least one has absolute value greater than 1, and none has absolute value equal to 1. (Thus
5470:
7914:
7872:
7796:
7686:
7499:
7466:
7431:
7366:
Berteloot, François; Dupont, Christophe (2005), "Une caractérisation des endomorphismes de Lattès par leur mesure de Green",
4492:
with respect to that measure, by
Fornaess and Sibony. It follows, for example, that for almost every point with respect to
1031:, the Julia set is the unit circle. For other polynomial mappings, the Julia set is often highly irregular, for example a
7890:
7452:
6706:
of a smooth projective rational surface with positive topological entropy (hence simple action on cohomology) such that
4469:
invariant measure of maximal entropy, by Briend and Duval. This is another way to say that the most chaotic behavior of
3273:
3641:
Another characterization of the equilibrium measure (due to Briend and Duval) is as follows. For each positive integer
3194:
One striking characterization of the equilibrium measure is that it describes the asymptotics of almost every point in
1371:
8025:
7995:
7965:
7587:
7368:
7349:
7314:
2812:
5673:
of absolute value greater than 1. Many projective varieties do not have such automorphisms, but (for example) many
3436:
3129:
7655:(2010), "Dynamics in several complex variables: endomorphisms of projective spaces and polynomial-like mappings",
6906:. The same holds for the subset of saddle periodic points, because both sets of periodic points grow at a rate of
947:
with complex coefficients, or more generally by a rational function.) Namely, there is always a compact subset of
1807:
715:
6005:
has some chaotic behavior, and the most chaotic behavior is concentrated on the small Julia set. At least when
5552:
4368:
1439:
915:
3694:
7854:
7788:
7712:
6268:
are defined by taking the quotient space by a finite group of an abelian surface with automorphism, and then
4609:
1692:
Complex dynamics has been effectively developed in any dimension. This section focuses on the mappings from
6963:
6903:
5624:
5343:
4092:. A more algebraic proof of this Zariski density was given by Najmuddin Fakhruddin. Another consequence of
3480:
1330:
7519:; Dupont, Christophe (2020), "Automorphisms of surfaces: Kummer rigidity and measure of maximal entropy",
4834:
4725:
6825:. Consider the probability measure which is evenly distributed on the isolated periodic points of period
2028:
1236:
6121:
2267:.) This measure was defined by Hans Brolin (1965) for polynomials in one variable, by Alexandre Freire,
8111:
8017:
7987:
7939:
7491:
7415:
6625:
5276:
5138:
4963:
4903:
4794:
4655:
4619:
4567:
4409:
4297:
4122:
4063:
3951:
3609:
3517:
3237:
3197:
2937:
2472:
2363:
2218:
2118:
2078:
1813:
1777:
1730:
1698:
1643:
1474:
1090:
950:
885:
468:
246:
190:
138:
7823:
4936:
In dimension 1, more is known about the "irregularity" of the equilibrium measure. Namely, define the
8106:
8101:
6780:
4391:
521:
6998:
5205:
8047:
7710:(2010), "Super-potentials for currents on compact Kähler manifolds and dynamics of automorphisms",
6530:
and its inverse map are ergodic and, more strongly, mixing with respect to the equilibrium measure
5399:
3061:
2989:
2357:
1693:
1185:
612:
5091:
1282:
1178:
1126:
1042:
7610:
6988:
6909:
3873:, and so one gets the same limit measure by averaging only over the repelling periodic points in
3557:
1847:
133:
6832:
6308:
5901:
5742:
3652:
3390:
2430:
992:
802:
622:
89:
7451:(2010), "Quelques aspects des systèmes dynamiques polynomiaux: existence, exemples, rigidité",
7008:
6193:
6161:
6095:
430:
6874:
6717:
6665:
6595:
6564:
6533:
6491:
6456:
6429:
6393:
6048:
6012:
5968:
5938:
5874:
5831:
5170:
4687:
4522:
4495:
4441:
4261:
4224:
4190:
4154:
4095:
4019:
3983:
3912:
3876:
3840:
3809:
3486:
3098:
3026:
2778:
2504:
2323:
2191:
668:
8116:
5250:
4341:
1629:
500:
448:
170:
4827:
4372:
3775:
3356:
2574:
1540:
1448:
8068:
8035:
8005:
7975:
7924:
7882:
7806:
7773:
7743:
7696:
7641:
7597:
7552:
7509:
7476:
7441:
7399:
7359:
7324:
6350:
5784:
5711:
5692:, which includes the case of a smooth complex projective variety. Say that an automorphism
5431:
5199:
4489:
2662:
2535:
2158:
882:
showed in the late 1910s that much of this story extends to any complex algebraic map from
844:
550:
8045:(1990), "Parabolic orbifolds and the dimension of the maximal measure for rational maps",
4943:
1856:
921:
8:
7021:
6973:
6249:, has spectral radius greater than 1, and so it gives a positive-entropy automorphism of
5338:
4331:
2984:
2395:
2282:
2186:
1036:
57:
8086:
7943:
7900:
7813:
7780:
7759:
7721:
7664:
7619:
7530:
7377:
3802:. Consider the probability measure which is evenly distributed on the points of period
1764:
226:
86:
A simple example that shows some of the main issues in complex dynamics is the mapping
6978:
6390:
with simple action on cohomology, the saddle periodic points are dense in the support
5689:
8121:
8021:
7991:
7982:
Morosawa, Shunsuke; Nishimura, Yasuichiro; Taniguchu, Masahiko; Ueda, Tetsuo (2000),
7961:
7910:
7868:
7792:
7682:
7583:
7495:
7462:
7427:
7345:
7310:
7031:
5821:. In fact, every automorphism that preserves a metric has topological entropy zero.)
5809:
4718:
2272:
2268:
438:
53:
49:
33:
8056:
7953:
7858:
7731:
7674:
7629:
7575:
7563:
7559:
7540:
7419:
7387:
7302:
6958:
5674:
4613:
37:
29:
7735:
6080:
Some abelian varieties have an automorphism of positive entropy. For example, let
5194:
is equal to the
Hausdorff dimension of its support (the Julia set) if and only if
8064:
8031:
8001:
7971:
7920:
7878:
7851:
Laminations and foliations in dynamics, geometry and topology (Stony Brook, 1998)
7802:
7769:
7739:
7703:
7692:
7660:
7648:
7637:
7605:
7593:
7571:
7548:
7505:
7472:
7437:
7407:
7395:
7355:
7341:
7320:
6699:
6277:
5456:
4597:
3514:
goes to infinity. In more detail: only finitely many closed complex subspaces of
3227:
3055:
2306:
2276:
1435:
1172:
61:
8081:
7678:
3606:
to be the unique largest totally invariant closed complex subspace not equal to
7863:
7750:
Fakhruddin, Najmuddin (2003), "Questions on self maps of algebraic varieties",
7335:
7003:
6714:
is conjugate to an irrational rotation. Points in that open set never approach
6085:
3430:
1116:
616:
600:
69:
52:
is iterated. In geometric terms, that amounts to iterating a mapping from some
7889:
Guedj, Vincent (2010), "Propriétés ergodiques des applications rationnelles",
7633:
7579:
7306:
8095:
7817:
7707:
7652:
7298:
6983:
5318:
4057:
2314:
2310:
1767:
to itself. Note, however, that many varieties have no interesting self-maps.
1275:
220:
65:
7412:
Frontiers in complex dynamics: in celebration of John Milnor's 80th birthday
4556:
7516:
7483:
7448:
7331:
7026:
5813:
5464:
4601:
2529:
1760:
879:
875:
7957:
7423:
6698:
of the equilibrium measure. For example, Eric
Bedford, Kyounghee Kim, and
6072:
assigns zero mass to all sets of sufficiently small
Hausdorff dimension.)
3948:
The equilibrium measure gives zero mass to any closed complex subspace of
3837:
goes to infinity. Moreover, most periodic points are repelling and lie in
2532:(the standard measure, scaled to have total measure 1) on the unit circle
8042:
7931:
6658:
with simple action on cohomology, there can be a nonempty open subset of
4649:
4215:
2112:
1617:
1601:
1445:
of the Fatou set is pre-periodic, meaning that there are natural numbers
1434:, the complement of the Julia set, where the dynamics is "tame". Namely,
419:{\displaystyle z,\;f(z)=z^{2},\;f(f(z))=z^{4},f(f(f(z)))=z^{8},\;\ldots }
7130:
Milnor (2006), Theorem 5.2 and problem 14-2; Fornaess (1996), Chapter 3.
8060:
7544:
6269:
5678:
5670:
4652:, François Berteloot, and Christophe Dupont, the only endomorphisms of
1363:
529:
45:
7175:
Zdunik (1990), Theorem 2; Berteloot & Dupont (2005), introduction.
7948:
7905:
7764:
7391:
7382:
6488:(or even just the saddle periodic points contained in the support of
4712:
1612:
is conjugate to an irrational rotation of the open unit disk; or (4)
981:
608:
81:
7337:
Iteration of rational functions: complex analytic dynamical systems
4831:
A random sample from the equilibrium measure of the non-Lattès map
3226:
when followed backward in time, by Jean-Yves Briend, Julien Duval,
437:| is less than 1, then the orbit converges to 0, in fact more than
8016:, London Mathematical Society Lecture Note Series, vol. 274,
7726:
7669:
7624:
7535:
6662:
on which neither forward nor backward orbits approach the support
7608:(2012), "Dynamics of automorphisms on compact Kähler manifolds",
7486:(2014), "Dynamics of automorphisms of compact complex surfaces",
4485:
1032:
7261:
Dinh & Sibony (2010), "Super-potentials ...", Theorem 4.4.2.
6382:
is expanding in some directions and contracting at others, near
1727:
to itself, the richest source of examples. The main results for
595:
is chaotic, in various ways. For example, for almost all points
7295:
A history of complex dynamics: from Schröder to Fatou and Julia
4722:
A random sample from the equilibrium measure of the Lattès map
3806:. Then these measures also converge to the equilibrium measure
2970:
7981:
7234:
Dinh & Sibony (2010), "Super-potentials ...", section 4.4.
6761:
describes the distribution of the isolated periodic points of
6242:{\displaystyle {\begin{pmatrix}2&1\\1&1\end{pmatrix}}}
5167:
of degree greater than 1, Zdunik showed that the dimension of
2646:{\displaystyle f\colon \mathbf {CP} ^{n}\to \mathbf {CP} ^{n}}
1423:{\displaystyle f\colon \mathbf {CP} ^{1}\to \mathbf {CP} ^{1}}
1039:
is not an integer. This occurs even for mappings as simple as
4519:, its forward orbit is uniformly distributed with respect to
2807:
1687:
918:
greater than 1. (Such a mapping may be given by a polynomial
7112:
Dinh & Sibony (2010), "Dynamics ...", Proposition 1.2.3.
6757:
At least in complex dimension 2, the equilibrium measure of
4473:
is concentrated on the support of the equilibrium measure.
4151:
is that each point has zero mass. As a result, the support
445:| is greater than 1, then the orbit converges to the point
5078:{\displaystyle \dim(\mu )=\inf\{\dim _{H}(Y):\mu (Y)=1\},}
4119:
giving zero mass to closed complex subspaces not equal to
3980:
that is not the whole space. Since the periodic points in
2975:
A basic property of the equilibrium measure is that it is
2255:, that describes the most chaotic part of the dynamics of
7405:
7091:
Dinh & Sibony (2010), "Dynamics ...", Theorem 1.4.13.
7073:
Dinh & Sibony (2010), "Dynamics ...", Theorem 1.7.11.
7148:
Dinh & Sibony (2010), "Dynamics ...", Theorem 1.6.3.
7139:
Dinh & Sibony (2010), "Dynamics ...", Theorem 1.7.1.
7082:
Dinh & Sibony (2010), "Dynamics ...", Theorem 1.4.1.
5535:{\displaystyle H^{p,p}(X)\subset H^{2p}(X,\mathbf {C} )}
5308:
132:
to itself. It is helpful to view this as a map from the
6622:
A notable difference with the case of endomorphisms of
6257:
is the Haar measure (the standard
Lebesgue measure) on
6208:
3909:. There may also be repelling periodic points outside
7826:
6912:
6877:
6835:
6783:
6720:
6668:
6628:
6598:
6567:
6536:
6494:
6459:
6432:
6396:
6353:
6311:
6202:
6164:
6124:
6098:
6051:
6015:
5971:
5941:
5904:
5877:
5834:
5787:
5745:
5714:
5627:
5555:
5473:
5434:
5402:
5346:
5279:
5253:
5208:
5173:
5141:
5094:
5001:
4966:
4946:
4906:
4837:
4797:
4728:
4690:
4658:
4622:
4570:
4525:
4498:
4444:
4412:
4344:
4300:
4264:
4227:
4193:
4157:
4125:
4098:
4066:
4022:
3986:
3954:
3915:
3879:
3843:
3812:
3778:
3697:
3655:
3612:
3560:
3520:
3489:
3439:
3393:
3359:
3276:
3240:
3200:
3132:
3101:
3064:
3029:
2992:
2940:
2815:
2781:
2665:
2603:
2577:
2538:
2507:
2475:
2433:
2398:
2366:
2326:
2285:
2221:
2194:
2161:
2121:
2081:
2031:
1859:
1816:
1780:
1733:
1701:
1646:
1543:
1477:
1451:
1380:
1333:
1285:
1239:
1188:
1129:
1093:
1045:
995:
953:
924:
888:
847:
805:
718:
671:
625:
553:
503:
471:
451:
292:
249:
229:
193:
173:
141:
92:
7892:
Quelques aspects des systèmes dynamiques polynomiaux
7454:
Quelques aspects des systèmes dynamiques polynomiaux
6769:
or an iterate, which are ignored here.) Namely, let
1529:. Therefore, to analyze the dynamics on a component
6045:has positive Hausdorff dimension. (More precisely,
7841:
6938:
6890:
6863:
6817:
6742:
6690:
6646:
6611:
6580:
6549:
6507:
6480:vanishes on closed complex subspaces not equal to
6472:
6445:
6418:
6366:
6339:
6241:
6176:
6150:
6110:
6064:
6037:
5993:
5954:
5923:
5890:
5847:
5800:
5773:
5727:
5657:
5602:
5534:
5447:
5420:
5376:
5297:
5265:
5239:
5186:
5159:
5119:
5077:
4984:
4952:
4932:, but the equilibrium measure is highly irregular.
4924:
4892:
4815:
4783:
4703:
4676:
4640:
4588:
4538:
4511:
4457:
4430:
4359:
4318:
4286:
4249:
4206:
4179:
4143:
4111:
4084:
4044:
4008:
3972:
3937:
3901:
3865:
3825:
3794:
3764:
3683:
3630:
3591:
3538:
3502:
3463:
3421:
3375:
3346:{\displaystyle (1/d^{rn})(f^{r})^{*}(\delta _{z})}
3345:
3258:
3218:
3183:
3114:
3087:
3042:
3015:
2958:
2926:
2794:
2763:
2645:
2589:
2560:
2520:
2493:
2461:
2410:
2384:
2348:
2297:
2239:
2207:
2174:
2151:is greater than 1; then the degree of the mapping
2139:
2099:
2063:
2014:
1834:
1798:
1751:
1719:
1664:
1628:is conjugate to an irrational rotation of an open
1564:
1521:
1463:
1422:
1354:
1319:
1263:
1225:
1163:
1107:
1079:
1023:
971:
939:
906:
860:
833:
763:
693:
653:
575:
509:
497:, again more than exponentially fast. (Here 0 and
489:
457:
418:
267:
235:
211:
179:
159:
120:
75:
8093:
7659:, Lecture Notes in Mathematics, vol. 1998,
6902:goes to infinity, by Eric Bedford, Lyubich, and
5617:is also the logarithm of the spectral radius of
5572:
5273:. (In the latter cases, the Julia set is all of
5020:
4476:Finally, one can say more about the dynamics of
7558:
7365:
6773:be an automorphism of a compact Kähler surface
6592:are both uniformly distributed with respect to
6347:, at least one eigenvalue of the derivative of
6196:has absolute value greater than 2, for example
5127:denotes the Hausdorff dimension of a Borel set
2927:{\displaystyle \{:|z_{1}|=\cdots =|z_{n}|=1\}.}
7812:
5333:to itself. The case of main interest is where
4338:is always greater than zero, in fact equal to
615:on the circle. There are also infinitely many
540:fixed point means one where the derivative of
7752:Journal of the Ramanujan Mathematical Society
7603:
6765:. (There may also be complex curves fixed by
3464:{\displaystyle E\subsetneq \mathbf {CP} ^{n}}
3184:{\displaystyle f^{*}\mu _{f}=\deg(f)\mu _{f}}
1632:. (Note that the "backward orbit" of a point
779:can be considered chaotic, since points near
429:behave, qualitatively? The answer is: if the
40:mapping. This article focuses on the case of
7522:Journal of the European Mathematical Society
7515:
5781:has only one eigenvalue with absolute value
5069:
5023:
4992:(or more generally on a smooth manifold) by
2971:Characterizations of the equilibrium measure
2918:
2816:
7488:Frontiers in complex dynamics (Banff, 2011)
6969:Infinite compositions of analytic functions
4604:. In this case, the equilibrium measure of
4480:on the support of the equilibrium measure:
2934:For more general holomorphic mappings from
7749:
7702:
7647:
7243:Cantat & Dupont (2020), section 1.2.1.
7157:Berteloot & Dupont (2005), Théorème 1.
7108:
7106:
7100:Fornaess & Sibony (2001), Theorem 4.3.
6994:Carathéodory's theorem (conformal mapping)
6710:has a Siegel disk, on which the action of
6301:periodic point if, for a positive integer
3645:, the number of periodic points of period
1688:The equilibrium measure of an endomorphism
1592:approach a fixed point in the boundary of
764:{\displaystyle f(f(\cdots (f(z))\cdots ))}
412:
328:
299:
7947:
7904:
7862:
7763:
7725:
7668:
7623:
7534:
7381:
7292:
7252:Cantat & Dupont (2020), Main Theorem.
7220:De Thélin & Dinh (2012), Theorem 1.2.
7069:
7067:
6871:). Then this measure converges weakly to
6557:. It follows that for almost every point
6484:. It follows that the periodic points of
6284:
5603:{\displaystyle h(f)=\max _{p}\log d_{p}.}
4052:, it follows that the periodic points of
8014:The Mandelbrot set, theme and variations
7779:
7230:
7228:
7226:
4826:
4717:
3765:{\displaystyle (d^{r(n+1)}-1)/(d^{r}-1)}
1274:
1177:
223:.) The basic question is: given a point
7330:
7103:
6281:automorphism to be somewhat irregular.
6075:
5735:takes its maximum value, the action of
4214:has no isolated points, and so it is a
1576:contains an attracting fixed point for
1372:classification of the possible dynamics
8094:
8041:
7930:
7482:
7447:
7064:
4330:has some chaotic behavior is that the
8082:Gallery of dynamics (Curtis McMullen)
7888:
7785:Dynamics in several complex variables
7223:
6588:, the forward and backward orbits of
5658:{\displaystyle H^{*}(X,\mathbf {C} )}
5377:{\displaystyle H^{*}(X,\mathbf {Z} )}
5309:Automorphisms of projective varieties
1355:{\displaystyle c\doteq 0.383-0.0745i}
4893:{\displaystyle f(z)=(z-2)^{4}/z^{4}}
4784:{\displaystyle f(z)=(z-2)^{2}/z^{2}}
4596:obtained from an endomorphism of an
4326:.) Another way to make precise that
2392:; this is simply the Julia set when
2317:in any dimension (around 1994). The
872:has absolute value greater than 1.)
8011:
3353:which is evenly distributed on the
3270:, consider the probability measure
2064:{\displaystyle f_{0},\ldots ,f_{n}}
1264:{\displaystyle a\doteq -0.5+0.866i}
619:on the circle, meaning points with
591:. At these points, the dynamics of
13:
7410:; Sutherland, Scott, eds. (2014),
6777:with positive topological entropy
6526:with simple action on cohomology,
6151:{\displaystyle GL(2,\mathbf {Z} )}
5542:. Then the topological entropy of
3230:, and Sibony. Namely, for a point
1842:to itself, for a positive integer
504:
452:
174:
14:
8133:
8075:
7369:Commentarii Mathematici Helvetici
7121:Fakhruddin (2003), Corollary 5.3.
6647:{\displaystyle \mathbf {CP} ^{n}}
6453:. On the other hand, the measure
5298:{\displaystyle \mathbf {CP} ^{1}}
5160:{\displaystyle \mathbf {CP} ^{1}}
4985:{\displaystyle \mathbf {CP} ^{1}}
4925:{\displaystyle \mathbf {CP} ^{1}}
4816:{\displaystyle \mathbf {CP} ^{1}}
4677:{\displaystyle \mathbf {CP} ^{n}}
4641:{\displaystyle \mathbf {CP} ^{n}}
4589:{\displaystyle \mathbf {CP} ^{n}}
4431:{\displaystyle \mathbf {CP} ^{n}}
4402:. For a holomorphic endomorphism
4319:{\displaystyle \mathbf {CP} ^{n}}
4144:{\displaystyle \mathbf {CP} ^{n}}
4085:{\displaystyle \mathbf {CP} ^{n}}
3973:{\displaystyle \mathbf {CP} ^{n}}
3691:), counted with multiplicity, is
3631:{\displaystyle \mathbf {CP} ^{n}}
3539:{\displaystyle \mathbf {CP} ^{n}}
3259:{\displaystyle \mathbf {CP} ^{n}}
3219:{\displaystyle \mathbf {CP} ^{n}}
2959:{\displaystyle \mathbf {CP} ^{n}}
2494:{\displaystyle \mathbf {CP} ^{1}}
2385:{\displaystyle \mathbf {CP} ^{n}}
2240:{\displaystyle \mathbf {CP} ^{n}}
2140:{\displaystyle \mathbf {CP} ^{n}}
2100:{\displaystyle \mathbf {CP} ^{n}}
2025:for some homogeneous polynomials
1835:{\displaystyle \mathbf {CP} ^{n}}
1799:{\displaystyle \mathbf {CP} ^{n}}
1759:have been extended to a class of
1752:{\displaystyle \mathbf {CP} ^{n}}
1720:{\displaystyle \mathbf {CP} ^{n}}
1665:{\displaystyle \mathbf {CP} ^{1}}
1533:, one can assume after replacing
1522:{\displaystyle f^{a}(U)=f^{b}(U)}
1108:{\displaystyle c\in \mathbf {C} }
972:{\displaystyle \mathbf {CP} ^{1}}
907:{\displaystyle \mathbf {CP} ^{1}}
544:has absolute value less than 1.)
490:{\displaystyle \mathbf {CP} ^{1}}
268:{\displaystyle \mathbf {CP} ^{1}}
212:{\displaystyle \mathbf {CP} ^{1}}
160:{\displaystyle \mathbf {CP} ^{1}}
56:to itself. The related theory of
7936:Dynamics in one complex variable
7842:{\displaystyle \mathbf {P} ^{2}}
7829:
6955:Dynamics in complex dimension 1
6634:
6631:
6141:
5648:
5525:
5367:
5285:
5282:
5198:is conjugate to a Lattès map, a
5147:
5144:
4972:
4969:
4912:
4909:
4803:
4800:
4711:assigns its full mass 1 to some
4664:
4661:
4628:
4625:
4576:
4573:
4418:
4415:
4378:For any continuous endomorphism
4306:
4303:
4131:
4128:
4072:
4069:
3960:
3957:
3618:
3615:
3526:
3523:
3451:
3448:
3246:
3243:
3206:
3203:
2946:
2943:
2633:
2630:
2615:
2612:
2481:
2478:
2372:
2369:
2227:
2224:
2182:, which is also greater than 1.
2127:
2124:
2087:
2084:
1822:
1819:
1786:
1783:
1739:
1736:
1707:
1704:
1652:
1649:
1588:in the sense that all points in
1410:
1407:
1392:
1389:
1279:The Julia set of the polynomial
1182:The Julia set of the polynomial
1101:
959:
956:
894:
891:
783:diverge exponentially fast from
547:On the other hand, suppose that
477:
474:
255:
252:
199:
196:
147:
144:
7273:
7264:
7255:
7246:
7237:
7214:
7205:
7202:Cantat (2010), sections 7 to 9.
7196:
7187:
7178:
7169:
7160:
7151:
7142:
7133:
6818:{\displaystyle h(f)=\log d_{1}}
4390:is equal to the maximum of the
2571:More generally, for an integer
2259:. (It has also been called the
1808:morphism of algebraic varieties
775:on the circle, the dynamics of
76:Dynamics in complex dimension 1
7897:Société Mathématique de France
7459:Société Mathématique de France
7124:
7115:
7094:
7085:
7076:
7055:
7046:
6927:
6913:
6852:
6846:
6793:
6787:
6737:
6731:
6685:
6679:
6413:
6407:
6328:
6322:
6145:
6131:
6032:
6026:
5988:
5982:
5768:
5762:
5652:
5638:
5565:
5559:
5529:
5515:
5496:
5490:
5371:
5357:
5240:{\displaystyle f(z)=z^{\pm d}}
5218:
5212:
5114:
5108:
5060:
5054:
5045:
5039:
5014:
5008:
4866:
4853:
4847:
4841:
4757:
4744:
4738:
4732:
4549:
4281:
4275:
4244:
4238:
4174:
4168:
4039:
4033:
4003:
3997:
3932:
3926:
3896:
3890:
3860:
3854:
3759:
3740:
3732:
3721:
3709:
3698:
3672:
3666:
3580:
3574:
3410:
3404:
3340:
3327:
3318:
3304:
3301:
3277:
3168:
3162:
2908:
2893:
2879:
2864:
2857:
2819:
2755:
2713:
2707:
2704:
2672:
2669:
2625:
2548:
2540:
2443:
2437:
2360:of the equilibrium measure in
2343:
2337:
2111:, this is the same thing as a
2009:
2006:
1974:
1952:
1920:
1907:
1901:
1898:
1866:
1863:
1553:
1547:
1516:
1510:
1494:
1488:
1402:
1295:
1289:
1198:
1192:
1139:
1133:
1119:is the set of complex numbers
1055:
1049:
1005:
999:
934:
928:
822:
816:
758:
755:
749:
746:
740:
734:
728:
722:
688:
682:
642:
636:
563:
555:
393:
390:
387:
381:
375:
369:
347:
344:
338:
332:
309:
303:
102:
96:
1:
7855:American Mathematical Society
7789:American Mathematical Society
7736:10.1090/S1056-3911-10-00549-7
7713:Journal of Algebraic Geometry
7657:Holomorphic dynamical systems
7286:
7211:Cantat (2014), section 2.4.3.
6253:. The equilibrium measure of
5704:if: there is only one number
5421:{\displaystyle 0\leq p\leq n}
5202:(up to sign), or a power map
4394:(or "metric entropy") of all
4386:, the topological entropy of
3088:{\displaystyle f^{*}\mu _{f}}
3016:{\displaystyle f_{*}\mu _{f}}
2775:Then the equilibrium measure
2075:that have no common zeros in
1846:. Such a mapping is given in
1620:, meaning that the action of
1604:, meaning that the action of
1226:{\displaystyle f(z)=z^{2}+az}
701:means the result of applying
167:to itself, by adding a point
8087:Surveys in Dynamical Systems
7270:Cantat (2010), Théorème 9.8.
7193:Cantat (2000), Théorème 2.2.
7166:Milnor (2006), problem 14-2.
6964:Complex quadratic polynomial
6654:is that for an automorphism
5681:do have such automorphisms.
5613:(The topological entropy of
5120:{\displaystyle \dim _{H}(Y)}
3479:, the measures just defined
1320:{\displaystyle f(z)=z^{2}+c}
1164:{\displaystyle f(z)=z^{2}+c}
1080:{\displaystyle f(z)=z^{2}+c}
989:is chaotic. For the mapping
536:is zero at those points. An
7:
7679:10.1007/978-3-642-13171-4_4
7279:Cantat (2014), Theorem 8.2.
7184:Milnor (2006), problem 5-3.
6949:
6939:{\displaystyle (d_{1})^{r}}
6426:of the equilibrium measure
5702:simple action on cohomology
5321:complex projective variety
3592:{\displaystyle f^{-1}(S)=S}
3483:to the equilibrium measure
2802:is the Haar measure on the
2421:
1680:, need not be contained in
1370:There is a rather complete
1123:such that the Julia set of
985:, on which the dynamics of
771:.) Even at periodic points
611:in the circle, and in fact
219:has the advantage of being
60:studies iteration over the
10:
8138:
8018:Cambridge University Press
7988:Cambridge University Press
7940:Princeton University Press
7492:Princeton University Press
7416:Princeton University Press
7293:Alexander, Daniel (1994),
7052:Milnor (2006), section 13.
7018:Related areas of dynamics
6864:{\displaystyle f^{r}(z)=z}
6702:constructed automorphisms
6340:{\displaystyle f^{r}(z)=z}
5924:{\displaystyle \log d_{p}}
5869:measure of maximal entropy
5774:{\displaystyle H^{p,p}(X)}
5463:acting by pullback on the
4900:. The Julia set is all of
4791:. The Julia set is all of
4438:, the equilibrium measure
4382:of a compact metric space
3684:{\displaystyle f^{r}(z)=z}
3422:{\displaystyle f^{r}(w)=z}
2501:, the equilibrium measure
2462:{\displaystyle f(z)=z^{2}}
2265:measure of maximal entropy
1024:{\displaystyle f(z)=z^{2}}
834:{\displaystyle f^{r}(z)=z}
791:. (The periodic points of
661:for some positive integer
654:{\displaystyle f^{r}(z)=z}
599:on the circle in terms of
121:{\displaystyle f(z)=z^{2}}
79:
7634:10.1016/j.aim.2012.01.014
7580:10.1007/978-1-4612-4364-9
7307:10.1007/978-3-663-09197-4
6184:integer matrices acts on
6177:{\displaystyle 2\times 2}
6111:{\displaystyle E\times E}
5337:acts nontrivially on the
4940:of a probability measure
4392:measure-theoretic entropy
4375:, and Feliks Przytycki.
3471:such that for all points
587:is on the unit circle in
187:to the complex numbers. (
128:from the complex numbers
8048:Inventiones Mathematicae
7061:Guedj (2010), Theorem B.
7039:
6891:{\displaystyle \mu _{f}}
6743:{\displaystyle J^{*}(f)}
6691:{\displaystyle J^{*}(f)}
6612:{\displaystyle \mu _{f}}
6581:{\displaystyle \mu _{f}}
6550:{\displaystyle \mu _{f}}
6508:{\displaystyle \mu _{f}}
6473:{\displaystyle \mu _{f}}
6446:{\displaystyle \mu _{f}}
6419:{\displaystyle J^{*}(f)}
6374:on the tangent space at
6065:{\displaystyle \mu _{f}}
6038:{\displaystyle J^{*}(f)}
5994:{\displaystyle J^{*}(f)}
5955:{\displaystyle \mu _{f}}
5891:{\displaystyle \mu _{f}}
5848:{\displaystyle \mu _{f}}
5621:on the whole cohomology
5187:{\displaystyle \mu _{f}}
4704:{\displaystyle \mu _{f}}
4539:{\displaystyle \mu _{f}}
4512:{\displaystyle \mu _{f}}
4458:{\displaystyle \mu _{f}}
4287:{\displaystyle J^{*}(f)}
4250:{\displaystyle J^{*}(f)}
4207:{\displaystyle \mu _{f}}
4180:{\displaystyle J^{*}(f)}
4112:{\displaystyle \mu _{f}}
4045:{\displaystyle J^{*}(f)}
4009:{\displaystyle J^{*}(f)}
3938:{\displaystyle J^{*}(f)}
3902:{\displaystyle J^{*}(f)}
3866:{\displaystyle J^{*}(f)}
3826:{\displaystyle \mu _{f}}
3599:), and one can take the
3503:{\displaystyle \mu _{f}}
3115:{\displaystyle \mu _{f}}
3043:{\displaystyle \mu _{f}}
2983:, in the sense that the
2795:{\displaystyle \mu _{f}}
2521:{\displaystyle \mu _{f}}
2349:{\displaystyle J^{*}(f)}
2208:{\displaystyle \mu _{f}}
2147:to itself.) Assume that
1694:complex projective space
694:{\displaystyle f^{r}(z)}
7611:Advances in Mathematics
6989:Riemann mapping theorem
6515:) are Zariski dense in
6386:.) For an automorphism
6092:be the abelian surface
5855:of maximal entropy for
5325:, meaning isomorphisms
5266:{\displaystyle d\geq 2}
4715:of Lebesgue measure 0.
4398:-invariant measures on
4360:{\displaystyle n\log d}
3266:and a positive integer
3058:, the pullback measure
2185:Then there is a unique
1848:homogeneous coordinates
1640:, the set of points in
1374:of a rational function
795:on the unit circle are
603:, the forward orbit of
510:{\displaystyle \infty }
458:{\displaystyle \infty }
180:{\displaystyle \infty }
134:complex projective line
8012:Tan, Lei, ed. (2000),
7864:10.1090/conm/269/04329
7843:
6940:
6892:
6865:
6819:
6744:
6692:
6648:
6613:
6582:
6551:
6509:
6474:
6447:
6420:
6368:
6341:
6285:Saddle periodic points
6243:
6178:
6152:
6112:
6066:
6039:
5995:
5956:
5925:
5892:
5849:
5802:
5775:
5729:
5659:
5604:
5536:
5449:
5422:
5378:
5299:
5267:
5241:
5188:
5161:
5131:. For an endomorphism
5121:
5079:
4986:
4954:
4933:
4926:
4894:
4824:
4817:
4785:
4705:
4678:
4642:
4590:
4540:
4513:
4459:
4432:
4361:
4320:
4288:
4251:
4208:
4181:
4145:
4113:
4086:
4046:
4010:
3974:
3939:
3903:
3867:
3827:
3796:
3795:{\displaystyle d^{rn}}
3766:
3685:
3632:
3593:
3540:
3504:
3465:
3423:
3377:
3376:{\displaystyle d^{rn}}
3347:
3260:
3220:
3185:
3116:
3089:
3044:
3017:
2960:
2928:
2796:
2765:
2647:
2591:
2590:{\displaystyle d>1}
2562:
2522:
2495:
2463:
2412:
2386:
2350:
2305:(around 1983), and by
2299:
2241:
2209:
2176:
2141:
2101:
2065:
2016:
1836:
1800:
1774:be an endomorphism of
1753:
1721:
1676:under some iterate of
1666:
1566:
1565:{\displaystyle f(U)=U}
1523:
1465:
1464:{\displaystyle a<b}
1424:
1367:
1356:
1321:
1272:
1265:
1227:
1165:
1109:
1081:
1035:in the sense that its
1025:
973:
941:
908:
862:
835:
765:
695:
655:
577:
511:
491:
459:
420:
269:
237:
213:
181:
161:
122:
7958:10.1515/9781400835539
7844:
7820:(2001), "Dynamics of
7424:10.1515/9781400851317
6941:
6893:
6866:
6820:
6745:
6693:
6649:
6614:
6583:
6552:
6510:
6475:
6448:
6421:
6369:
6367:{\displaystyle f^{r}}
6342:
6244:
6179:
6153:
6113:
6067:
6040:
5996:
5957:
5926:
5893:
5850:
5803:
5801:{\displaystyle d_{p}}
5776:
5730:
5728:{\displaystyle d_{p}}
5660:
5605:
5537:
5450:
5448:{\displaystyle d_{p}}
5423:
5392:of complex dimension
5379:
5300:
5268:
5242:
5189:
5162:
5122:
5080:
4987:
4955:
4927:
4895:
4830:
4818:
4786:
4721:
4706:
4679:
4643:
4610:absolutely continuous
4591:
4541:
4514:
4460:
4433:
4362:
4321:
4289:
4252:
4209:
4182:
4146:
4114:
4087:
4047:
4011:
3975:
3940:
3904:
3868:
3828:
3797:
3767:
3686:
3633:
3594:
3541:
3505:
3466:
3424:
3378:
3348:
3261:
3221:
3186:
3117:
3095:is also defined, and
3090:
3045:
3018:
2961:
2929:
2797:
2766:
2764:{\displaystyle f()=.}
2648:
2592:
2563:
2561:{\displaystyle |z|=1}
2523:
2496:
2464:
2413:
2387:
2351:
2300:
2242:
2210:
2177:
2175:{\displaystyle d^{n}}
2142:
2102:
2066:
2017:
1837:
1801:
1754:
1722:
1667:
1567:
1524:
1466:
1425:
1357:
1322:
1278:
1266:
1228:
1181:
1166:
1110:
1082:
1026:
974:
942:
909:
863:
861:{\displaystyle f^{r}}
836:
766:
696:
656:
613:uniformly distributed
578:
576:{\displaystyle |z|=1}
512:
492:
460:
421:
270:
238:
214:
182:
162:
123:
18:Branch of mathematics
7984:Holomorphic dynamics
7824:
7663:, pp. 165–294,
7494:, pp. 463–514,
6910:
6875:
6833:
6781:
6750:under the action of
6718:
6666:
6626:
6596:
6565:
6534:
6522:For an automorphism
6492:
6457:
6430:
6394:
6351:
6309:
6266:Kummer automorphisms
6200:
6188:. Any group element
6162:
6122:
6096:
6076:Kummer automorphisms
6049:
6013:
5969:
5939:
5902:
5875:
5832:
5824:For an automorphism
5785:
5743:
5712:
5625:
5553:
5471:
5432:
5400:
5344:
5277:
5251:
5206:
5200:Chebyshev polynomial
5171:
5139:
5092:
4999:
4964:
4953:{\displaystyle \mu }
4944:
4904:
4835:
4795:
4726:
4688:
4656:
4620:
4568:
4523:
4496:
4488:and, more strongly,
4442:
4410:
4342:
4298:
4262:
4225:
4191:
4155:
4123:
4096:
4064:
4020:
3984:
3952:
3913:
3877:
3841:
3810:
3776:
3695:
3653:
3610:
3558:
3518:
3487:
3437:
3391:
3357:
3274:
3238:
3198:
3130:
3099:
3062:
3027:
2990:
2938:
2813:
2779:
2663:
2601:
2575:
2536:
2505:
2473:
2431:
2396:
2364:
2324:
2283:
2219:
2192:
2159:
2119:
2079:
2029:
2015:{\displaystyle f()=}
1857:
1814:
1778:
1731:
1699:
1644:
1541:
1475:
1449:
1378:
1331:
1283:
1237:
1186:
1127:
1091:
1043:
993:
951:
940:{\displaystyle f(z)}
922:
886:
845:
841:, the derivative of
803:
716:
669:
623:
551:
501:
469:
449:
290:
247:
227:
191:
171:
139:
90:
26:holomorphic dynamics
7899:, pp. 97–202,
7814:Fornaess, John Erik
7781:Fornaess, John Erik
7406:Bonifant, Araceli;
7022:Arithmetic dynamics
6999:Böttcher's equation
5871:). (In particular,
5861:equilibrium measure
5339:singular cohomology
4938:Hausdorff dimension
4560:is an endomorphism
4332:topological entropy
3772:, which is roughly
2985:pushforward measure
2754:
2730:
2411:{\displaystyle n=1}
2309:, Peter Papadopol,
2298:{\displaystyle n=1}
2249:equilibrium measure
2187:probability measure
2071:of the same degree
1537:by an iterate that
1440:connected component
1037:Hausdorff dimension
528:, meaning that the
58:arithmetic dynamics
8061:10.1007/BF01234434
7857:, pp. 47–85,
7839:
7604:de Thélin, Henry;
7461:, pp. 13–95,
6936:
6888:
6861:
6815:
6740:
6688:
6644:
6609:
6578:
6547:
6505:
6470:
6443:
6416:
6364:
6337:
6239:
6233:
6174:
6148:
6108:
6062:
6035:
5991:
5952:
5935:.) The support of
5921:
5888:
5845:
5798:
5771:
5725:
5655:
5600:
5580:
5532:
5445:
5418:
5374:
5295:
5263:
5237:
5184:
5157:
5117:
5075:
4982:
4950:
4934:
4922:
4890:
4825:
4813:
4781:
4701:
4674:
4638:
4586:
4536:
4509:
4455:
4428:
4373:Michał Misiurewicz
4357:
4316:
4284:
4247:
4204:
4177:
4141:
4109:
4082:
4042:
4006:
3970:
3935:
3899:
3863:
3823:
3792:
3762:
3681:
3628:
3589:
3536:
3500:
3461:
3429:. Then there is a
3419:
3373:
3343:
3256:
3216:
3181:
3126:in the sense that
3112:
3085:
3040:
3013:
2956:
2924:
2792:
2761:
2740:
2716:
2643:
2587:
2558:
2518:
2491:
2459:
2408:
2382:
2346:
2295:
2237:
2205:
2172:
2137:
2097:
2061:
2012:
1832:
1796:
1765:projective variety
1749:
1717:
1662:
1572:. Then either (1)
1562:
1519:
1461:
1420:
1368:
1352:
1317:
1273:
1261:
1223:
1161:
1105:
1077:
1021:
969:
937:
904:
858:
831:
761:
691:
651:
573:
507:
487:
455:
416:
265:
233:
209:
177:
157:
118:
42:algebraic dynamics
28:, is the study of
8112:Dynamical systems
7916:978-2-85629-338-6
7874:978-0-8218-1985-2
7798:978-0-8218-0317-2
7688:978-3-642-13170-7
7564:Gamelin, Theodore
7560:Carleson, Lennart
7501:978-0-691-15929-4
7468:978-2-85629-338-6
7433:978-0-691-15929-4
7032:Symbolic dynamics
6289:A periodic point
6118:. Then the group
5810:simple eigenvalue
5675:rational surfaces
5571:
4648:. Conversely, by
4600:by dividing by a
3548:totally invariant
3124:totally invariant
1438:showed that each
236:{\displaystyle z}
54:algebraic variety
50:rational function
30:dynamical systems
8129:
8107:Complex analysis
8102:Complex dynamics
8071:
8038:
8008:
7978:
7951:
7938:(3rd ed.),
7927:
7908:
7885:
7866:
7848:
7846:
7845:
7840:
7838:
7837:
7832:
7809:
7776:
7767:
7746:
7729:
7704:Dinh, Tien-Cuong
7699:
7672:
7649:Dinh, Tien-Cuong
7644:
7627:
7618:(5): 2640–2655,
7606:Dinh, Tien-Cuong
7600:
7568:Complex dynamics
7555:
7545:10.4171/JEMS/946
7538:
7529:(4): 1289–1351,
7512:
7479:
7444:
7408:Lyubich, Mikhail
7402:
7385:
7362:
7327:
7280:
7277:
7271:
7268:
7262:
7259:
7253:
7250:
7244:
7241:
7235:
7232:
7221:
7218:
7212:
7209:
7203:
7200:
7194:
7191:
7185:
7182:
7176:
7173:
7167:
7164:
7158:
7155:
7149:
7146:
7140:
7137:
7131:
7128:
7122:
7119:
7113:
7110:
7101:
7098:
7092:
7089:
7083:
7080:
7074:
7071:
7062:
7059:
7053:
7050:
6974:Montel's theorem
6959:Complex analysis
6945:
6943:
6942:
6937:
6935:
6934:
6925:
6924:
6897:
6895:
6894:
6889:
6887:
6886:
6870:
6868:
6867:
6862:
6845:
6844:
6824:
6822:
6821:
6816:
6814:
6813:
6754:or its inverse.
6749:
6747:
6746:
6741:
6730:
6729:
6697:
6695:
6694:
6689:
6678:
6677:
6653:
6651:
6650:
6645:
6643:
6642:
6637:
6618:
6616:
6615:
6610:
6608:
6607:
6587:
6585:
6584:
6579:
6577:
6576:
6561:with respect to
6556:
6554:
6553:
6548:
6546:
6545:
6514:
6512:
6511:
6506:
6504:
6503:
6479:
6477:
6476:
6471:
6469:
6468:
6452:
6450:
6449:
6444:
6442:
6441:
6425:
6423:
6422:
6417:
6406:
6405:
6373:
6371:
6370:
6365:
6363:
6362:
6346:
6344:
6343:
6338:
6321:
6320:
6248:
6246:
6245:
6240:
6238:
6237:
6183:
6181:
6180:
6175:
6157:
6155:
6154:
6149:
6144:
6117:
6115:
6114:
6109:
6071:
6069:
6068:
6063:
6061:
6060:
6044:
6042:
6041:
6036:
6025:
6024:
6000:
5998:
5997:
5992:
5981:
5980:
5961:
5959:
5958:
5953:
5951:
5950:
5931:with respect to
5930:
5928:
5927:
5922:
5920:
5919:
5897:
5895:
5894:
5889:
5887:
5886:
5854:
5852:
5851:
5846:
5844:
5843:
5808:, and this is a
5807:
5805:
5804:
5799:
5797:
5796:
5780:
5778:
5777:
5772:
5761:
5760:
5734:
5732:
5731:
5726:
5724:
5723:
5664:
5662:
5661:
5656:
5651:
5637:
5636:
5609:
5607:
5606:
5601:
5596:
5595:
5579:
5541:
5539:
5538:
5533:
5528:
5514:
5513:
5489:
5488:
5465:Hodge cohomology
5454:
5452:
5451:
5446:
5444:
5443:
5427:
5425:
5424:
5419:
5383:
5381:
5380:
5375:
5370:
5356:
5355:
5304:
5302:
5301:
5296:
5294:
5293:
5288:
5272:
5270:
5269:
5264:
5246:
5244:
5243:
5238:
5236:
5235:
5193:
5191:
5190:
5185:
5183:
5182:
5166:
5164:
5163:
5158:
5156:
5155:
5150:
5126:
5124:
5123:
5118:
5104:
5103:
5084:
5082:
5081:
5076:
5035:
5034:
4991:
4989:
4988:
4983:
4981:
4980:
4975:
4959:
4957:
4956:
4951:
4931:
4929:
4928:
4923:
4921:
4920:
4915:
4899:
4897:
4896:
4891:
4889:
4888:
4879:
4874:
4873:
4822:
4820:
4819:
4814:
4812:
4811:
4806:
4790:
4788:
4787:
4782:
4780:
4779:
4770:
4765:
4764:
4710:
4708:
4707:
4702:
4700:
4699:
4683:
4681:
4680:
4675:
4673:
4672:
4667:
4647:
4645:
4644:
4639:
4637:
4636:
4631:
4614:Lebesgue measure
4612:with respect to
4595:
4593:
4592:
4587:
4585:
4584:
4579:
4545:
4543:
4542:
4537:
4535:
4534:
4518:
4516:
4515:
4510:
4508:
4507:
4464:
4462:
4461:
4456:
4454:
4453:
4437:
4435:
4434:
4429:
4427:
4426:
4421:
4366:
4364:
4363:
4358:
4325:
4323:
4322:
4317:
4315:
4314:
4309:
4293:
4291:
4290:
4285:
4274:
4273:
4256:
4254:
4253:
4248:
4237:
4236:
4213:
4211:
4210:
4205:
4203:
4202:
4186:
4184:
4183:
4178:
4167:
4166:
4150:
4148:
4147:
4142:
4140:
4139:
4134:
4118:
4116:
4115:
4110:
4108:
4107:
4091:
4089:
4088:
4083:
4081:
4080:
4075:
4051:
4049:
4048:
4043:
4032:
4031:
4015:
4013:
4012:
4007:
3996:
3995:
3979:
3977:
3976:
3971:
3969:
3968:
3963:
3944:
3942:
3941:
3936:
3925:
3924:
3908:
3906:
3905:
3900:
3889:
3888:
3872:
3870:
3869:
3864:
3853:
3852:
3832:
3830:
3829:
3824:
3822:
3821:
3801:
3799:
3798:
3793:
3791:
3790:
3771:
3769:
3768:
3763:
3752:
3751:
3739:
3725:
3724:
3690:
3688:
3687:
3682:
3665:
3664:
3637:
3635:
3634:
3629:
3627:
3626:
3621:
3598:
3596:
3595:
3590:
3573:
3572:
3545:
3543:
3542:
3537:
3535:
3534:
3529:
3509:
3507:
3506:
3501:
3499:
3498:
3470:
3468:
3467:
3462:
3460:
3459:
3454:
3428:
3426:
3425:
3420:
3403:
3402:
3382:
3380:
3379:
3374:
3372:
3371:
3352:
3350:
3349:
3344:
3339:
3338:
3326:
3325:
3316:
3315:
3300:
3299:
3287:
3265:
3263:
3262:
3257:
3255:
3254:
3249:
3225:
3223:
3222:
3217:
3215:
3214:
3209:
3190:
3188:
3187:
3182:
3180:
3179:
3152:
3151:
3142:
3141:
3121:
3119:
3118:
3113:
3111:
3110:
3094:
3092:
3091:
3086:
3084:
3083:
3074:
3073:
3049:
3047:
3046:
3041:
3039:
3038:
3022:
3020:
3019:
3014:
3012:
3011:
3002:
3001:
2965:
2963:
2962:
2957:
2955:
2954:
2949:
2933:
2931:
2930:
2925:
2911:
2906:
2905:
2896:
2882:
2877:
2876:
2867:
2856:
2855:
2837:
2836:
2801:
2799:
2798:
2793:
2791:
2790:
2770:
2768:
2767:
2762:
2753:
2748:
2729:
2724:
2703:
2702:
2684:
2683:
2652:
2650:
2649:
2644:
2642:
2641:
2636:
2624:
2623:
2618:
2596:
2594:
2593:
2588:
2567:
2565:
2564:
2559:
2551:
2543:
2527:
2525:
2524:
2519:
2517:
2516:
2500:
2498:
2497:
2492:
2490:
2489:
2484:
2468:
2466:
2465:
2460:
2458:
2457:
2427:For the mapping
2417:
2415:
2414:
2409:
2391:
2389:
2388:
2383:
2381:
2380:
2375:
2355:
2353:
2352:
2347:
2336:
2335:
2304:
2302:
2301:
2296:
2246:
2244:
2243:
2238:
2236:
2235:
2230:
2214:
2212:
2211:
2206:
2204:
2203:
2181:
2179:
2178:
2173:
2171:
2170:
2146:
2144:
2143:
2138:
2136:
2135:
2130:
2106:
2104:
2103:
2098:
2096:
2095:
2090:
2070:
2068:
2067:
2062:
2060:
2059:
2041:
2040:
2021:
2019:
2018:
2013:
2005:
2004:
1986:
1985:
1973:
1972:
1951:
1950:
1932:
1931:
1919:
1918:
1897:
1896:
1878:
1877:
1841:
1839:
1838:
1833:
1831:
1830:
1825:
1805:
1803:
1802:
1797:
1795:
1794:
1789:
1758:
1756:
1755:
1750:
1748:
1747:
1742:
1726:
1724:
1723:
1718:
1716:
1715:
1710:
1671:
1669:
1668:
1663:
1661:
1660:
1655:
1571:
1569:
1568:
1563:
1528:
1526:
1525:
1520:
1509:
1508:
1487:
1486:
1470:
1468:
1467:
1462:
1429:
1427:
1426:
1421:
1419:
1418:
1413:
1401:
1400:
1395:
1361:
1359:
1358:
1353:
1326:
1324:
1323:
1318:
1310:
1309:
1270:
1268:
1267:
1262:
1232:
1230:
1229:
1224:
1213:
1212:
1170:
1168:
1167:
1162:
1154:
1153:
1114:
1112:
1111:
1106:
1104:
1086:
1084:
1083:
1078:
1070:
1069:
1030:
1028:
1027:
1022:
1020:
1019:
978:
976:
975:
970:
968:
967:
962:
946:
944:
943:
938:
913:
911:
910:
905:
903:
902:
897:
867:
865:
864:
859:
857:
856:
840:
838:
837:
832:
815:
814:
770:
768:
767:
762:
700:
698:
697:
692:
681:
680:
660:
658:
657:
652:
635:
634:
582:
580:
579:
574:
566:
558:
516:
514:
513:
508:
496:
494:
493:
488:
486:
485:
480:
464:
462:
461:
456:
425:
423:
422:
417:
408:
407:
362:
361:
324:
323:
274:
272:
271:
266:
264:
263:
258:
242:
240:
239:
234:
218:
216:
215:
210:
208:
207:
202:
186:
184:
183:
178:
166:
164:
163:
158:
156:
155:
150:
127:
125:
124:
119:
117:
116:
62:rational numbers
38:complex analytic
22:Complex dynamics
8137:
8136:
8132:
8131:
8130:
8128:
8127:
8126:
8092:
8091:
8078:
8028:
7998:
7968:
7917:
7875:
7833:
7828:
7827:
7825:
7822:
7821:
7799:
7689:
7661:Springer-Verlag
7590:
7572:Springer-Verlag
7502:
7469:
7434:
7352:
7342:Springer-Verlag
7317:
7289:
7284:
7283:
7278:
7274:
7269:
7265:
7260:
7256:
7251:
7247:
7242:
7238:
7233:
7224:
7219:
7215:
7210:
7206:
7201:
7197:
7192:
7188:
7183:
7179:
7174:
7170:
7165:
7161:
7156:
7152:
7147:
7143:
7138:
7134:
7129:
7125:
7120:
7116:
7111:
7104:
7099:
7095:
7090:
7086:
7081:
7077:
7072:
7065:
7060:
7056:
7051:
7047:
7042:
7004:Orbit portraits
6979:Poincaré metric
6952:
6930:
6926:
6920:
6916:
6911:
6908:
6907:
6882:
6878:
6876:
6873:
6872:
6840:
6836:
6834:
6831:
6830:
6809:
6805:
6782:
6779:
6778:
6725:
6721:
6719:
6716:
6715:
6700:Curtis McMullen
6673:
6669:
6667:
6664:
6663:
6638:
6630:
6629:
6627:
6624:
6623:
6603:
6599:
6597:
6594:
6593:
6572:
6568:
6566:
6563:
6562:
6541:
6537:
6535:
6532:
6531:
6499:
6495:
6493:
6490:
6489:
6464:
6460:
6458:
6455:
6454:
6437:
6433:
6431:
6428:
6427:
6401:
6397:
6395:
6392:
6391:
6358:
6354:
6352:
6349:
6348:
6316:
6312:
6310:
6307:
6306:
6287:
6232:
6231:
6226:
6220:
6219:
6214:
6204:
6203:
6201:
6198:
6197:
6163:
6160:
6159:
6140:
6123:
6120:
6119:
6097:
6094:
6093:
6078:
6056:
6052:
6050:
6047:
6046:
6020:
6016:
6014:
6011:
6010:
6009:is projective,
5976:
5972:
5970:
5967:
5966:
5964:small Julia set
5946:
5942:
5940:
5937:
5936:
5915:
5911:
5903:
5900:
5899:
5882:
5878:
5876:
5873:
5872:
5839:
5835:
5833:
5830:
5829:
5812:. For example,
5792:
5788:
5786:
5783:
5782:
5750:
5746:
5744:
5741:
5740:
5719:
5715:
5713:
5710:
5709:
5690:Kähler manifold
5647:
5632:
5628:
5626:
5623:
5622:
5591:
5587:
5575:
5554:
5551:
5550:
5524:
5506:
5502:
5478:
5474:
5472:
5469:
5468:
5457:spectral radius
5439:
5435:
5433:
5430:
5429:
5401:
5398:
5397:
5366:
5351:
5347:
5345:
5342:
5341:
5311:
5289:
5281:
5280:
5278:
5275:
5274:
5252:
5249:
5248:
5228:
5224:
5207:
5204:
5203:
5178:
5174:
5172:
5169:
5168:
5151:
5143:
5142:
5140:
5137:
5136:
5099:
5095:
5093:
5090:
5089:
5030:
5026:
5000:
4997:
4996:
4976:
4968:
4967:
4965:
4962:
4961:
4945:
4942:
4941:
4916:
4908:
4907:
4905:
4902:
4901:
4884:
4880:
4875:
4869:
4865:
4836:
4833:
4832:
4807:
4799:
4798:
4796:
4793:
4792:
4775:
4771:
4766:
4760:
4756:
4727:
4724:
4723:
4695:
4691:
4689:
4686:
4685:
4668:
4660:
4659:
4657:
4654:
4653:
4632:
4624:
4623:
4621:
4618:
4617:
4598:abelian variety
4580:
4572:
4571:
4569:
4566:
4565:
4552:
4530:
4526:
4524:
4521:
4520:
4503:
4499:
4497:
4494:
4493:
4449:
4445:
4443:
4440:
4439:
4422:
4414:
4413:
4411:
4408:
4407:
4343:
4340:
4339:
4310:
4302:
4301:
4299:
4296:
4295:
4269:
4265:
4263:
4260:
4259:
4232:
4228:
4226:
4223:
4222:
4198:
4194:
4192:
4189:
4188:
4162:
4158:
4156:
4153:
4152:
4135:
4127:
4126:
4124:
4121:
4120:
4103:
4099:
4097:
4094:
4093:
4076:
4068:
4067:
4065:
4062:
4061:
4027:
4023:
4021:
4018:
4017:
3991:
3987:
3985:
3982:
3981:
3964:
3956:
3955:
3953:
3950:
3949:
3920:
3916:
3914:
3911:
3910:
3884:
3880:
3878:
3875:
3874:
3848:
3844:
3842:
3839:
3838:
3817:
3813:
3811:
3808:
3807:
3783:
3779:
3777:
3774:
3773:
3747:
3743:
3735:
3705:
3701:
3696:
3693:
3692:
3660:
3656:
3654:
3651:
3650:
3622:
3614:
3613:
3611:
3608:
3607:
3601:exceptional set
3565:
3561:
3559:
3556:
3555:
3530:
3522:
3521:
3519:
3516:
3515:
3494:
3490:
3488:
3485:
3484:
3481:converge weakly
3455:
3447:
3446:
3438:
3435:
3434:
3398:
3394:
3392:
3389:
3388:
3364:
3360:
3358:
3355:
3354:
3334:
3330:
3321:
3317:
3311:
3307:
3292:
3288:
3283:
3275:
3272:
3271:
3250:
3242:
3241:
3239:
3236:
3235:
3228:Tien-Cuong Dinh
3210:
3202:
3201:
3199:
3196:
3195:
3175:
3171:
3147:
3143:
3137:
3133:
3131:
3128:
3127:
3106:
3102:
3100:
3097:
3096:
3079:
3075:
3069:
3065:
3063:
3060:
3059:
3056:finite morphism
3034:
3030:
3028:
3025:
3024:
3007:
3003:
2997:
2993:
2991:
2988:
2987:
2973:
2950:
2942:
2941:
2939:
2936:
2935:
2907:
2901:
2897:
2892:
2878:
2872:
2868:
2863:
2851:
2847:
2832:
2828:
2814:
2811:
2810:
2786:
2782:
2780:
2777:
2776:
2749:
2744:
2725:
2720:
2698:
2694:
2679:
2675:
2664:
2661:
2660:
2637:
2629:
2628:
2619:
2611:
2610:
2602:
2599:
2598:
2576:
2573:
2572:
2547:
2539:
2537:
2534:
2533:
2512:
2508:
2506:
2503:
2502:
2485:
2477:
2476:
2474:
2471:
2470:
2453:
2449:
2432:
2429:
2428:
2424:
2397:
2394:
2393:
2376:
2368:
2367:
2365:
2362:
2361:
2331:
2327:
2325:
2322:
2321:
2319:small Julia set
2284:
2281:
2280:
2277:Mikhail Lyubich
2231:
2223:
2222:
2220:
2217:
2216:
2199:
2195:
2193:
2190:
2189:
2166:
2162:
2160:
2157:
2156:
2131:
2123:
2122:
2120:
2117:
2116:
2091:
2083:
2082:
2080:
2077:
2076:
2055:
2051:
2036:
2032:
2030:
2027:
2026:
2000:
1996:
1981:
1977:
1968:
1964:
1946:
1942:
1927:
1923:
1914:
1910:
1892:
1888:
1873:
1869:
1858:
1855:
1854:
1826:
1818:
1817:
1815:
1812:
1811:
1790:
1782:
1781:
1779:
1776:
1775:
1743:
1735:
1734:
1732:
1729:
1728:
1711:
1703:
1702:
1700:
1697:
1696:
1690:
1656:
1648:
1647:
1645:
1642:
1641:
1542:
1539:
1538:
1504:
1500:
1482:
1478:
1476:
1473:
1472:
1450:
1447:
1446:
1436:Dennis Sullivan
1414:
1406:
1405:
1396:
1388:
1387:
1379:
1376:
1375:
1332:
1329:
1328:
1305:
1301:
1284:
1281:
1280:
1238:
1235:
1234:
1208:
1204:
1187:
1184:
1183:
1149:
1145:
1128:
1125:
1124:
1100:
1092:
1089:
1088:
1087:for a constant
1065:
1061:
1044:
1041:
1040:
1015:
1011:
994:
991:
990:
963:
955:
954:
952:
949:
948:
923:
920:
919:
898:
890:
889:
887:
884:
883:
852:
848:
846:
843:
842:
810:
806:
804:
801:
800:
787:upon iterating
717:
714:
713:
676:
672:
670:
667:
666:
630:
626:
624:
621:
620:
617:periodic points
583:, meaning that
562:
554:
552:
549:
548:
519:superattracting
502:
499:
498:
481:
473:
472:
470:
467:
466:
450:
447:
446:
403:
399:
357:
353:
319:
315:
291:
288:
287:
275:, how does its
259:
251:
250:
248:
245:
244:
228:
225:
224:
203:
195:
194:
192:
189:
188:
172:
169:
168:
151:
143:
142:
140:
137:
136:
112:
108:
91:
88:
87:
84:
78:
70:complex numbers
68:instead of the
19:
12:
11:
5:
8135:
8125:
8124:
8119:
8114:
8109:
8104:
8090:
8089:
8084:
8077:
8076:External links
8074:
8073:
8072:
8055:(3): 627–649,
8039:
8026:
8009:
7996:
7979:
7966:
7928:
7915:
7886:
7873:
7836:
7831:
7818:Sibony, Nessim
7810:
7797:
7777:
7758:(2): 109–122,
7747:
7720:(3): 473–529,
7708:Sibony, Nessim
7700:
7687:
7653:Sibony, Nessim
7645:
7601:
7588:
7556:
7513:
7500:
7480:
7467:
7445:
7432:
7403:
7392:10.4171/CMH/21
7376:(2): 433–454,
7363:
7350:
7328:
7315:
7288:
7285:
7282:
7281:
7272:
7263:
7254:
7245:
7236:
7222:
7213:
7204:
7195:
7186:
7177:
7168:
7159:
7150:
7141:
7132:
7123:
7114:
7102:
7093:
7084:
7075:
7063:
7054:
7044:
7043:
7041:
7038:
7037:
7036:
7035:
7034:
7029:
7024:
7015:
7014:
7013:
7012:
7006:
7001:
6996:
6991:
6986:
6981:
6976:
6971:
6966:
6961:
6951:
6948:
6933:
6929:
6923:
6919:
6915:
6885:
6881:
6860:
6857:
6854:
6851:
6848:
6843:
6839:
6829:(meaning that
6812:
6808:
6804:
6801:
6798:
6795:
6792:
6789:
6786:
6739:
6736:
6733:
6728:
6724:
6687:
6684:
6681:
6676:
6672:
6641:
6636:
6633:
6606:
6602:
6575:
6571:
6544:
6540:
6502:
6498:
6467:
6463:
6440:
6436:
6415:
6412:
6409:
6404:
6400:
6361:
6357:
6336:
6333:
6330:
6327:
6324:
6319:
6315:
6286:
6283:
6236:
6230:
6227:
6225:
6222:
6221:
6218:
6215:
6213:
6210:
6209:
6207:
6173:
6170:
6167:
6158:of invertible
6147:
6143:
6139:
6136:
6133:
6130:
6127:
6107:
6104:
6101:
6086:elliptic curve
6077:
6074:
6059:
6055:
6034:
6031:
6028:
6023:
6019:
6001:. Informally:
5990:
5987:
5984:
5979:
5975:
5962:is called the
5949:
5945:
5918:
5914:
5910:
5907:
5885:
5881:
5842:
5838:
5795:
5791:
5770:
5767:
5764:
5759:
5756:
5753:
5749:
5722:
5718:
5654:
5650:
5646:
5643:
5640:
5635:
5631:
5611:
5610:
5599:
5594:
5590:
5586:
5583:
5578:
5574:
5570:
5567:
5564:
5561:
5558:
5531:
5527:
5523:
5520:
5517:
5512:
5509:
5505:
5501:
5498:
5495:
5492:
5487:
5484:
5481:
5477:
5442:
5438:
5417:
5414:
5411:
5408:
5405:
5373:
5369:
5365:
5362:
5359:
5354:
5350:
5310:
5307:
5292:
5287:
5284:
5262:
5259:
5256:
5234:
5231:
5227:
5223:
5220:
5217:
5214:
5211:
5181:
5177:
5154:
5149:
5146:
5116:
5113:
5110:
5107:
5102:
5098:
5086:
5085:
5074:
5071:
5068:
5065:
5062:
5059:
5056:
5053:
5050:
5047:
5044:
5041:
5038:
5033:
5029:
5025:
5022:
5019:
5016:
5013:
5010:
5007:
5004:
4979:
4974:
4971:
4949:
4919:
4914:
4911:
4887:
4883:
4878:
4872:
4868:
4864:
4861:
4858:
4855:
4852:
4849:
4846:
4843:
4840:
4810:
4805:
4802:
4778:
4774:
4769:
4763:
4759:
4755:
4752:
4749:
4746:
4743:
4740:
4737:
4734:
4731:
4698:
4694:
4671:
4666:
4663:
4635:
4630:
4627:
4583:
4578:
4575:
4551:
4548:
4533:
4529:
4506:
4502:
4452:
4448:
4425:
4420:
4417:
4369:Mikhail Gromov
4356:
4353:
4350:
4347:
4313:
4308:
4305:
4283:
4280:
4277:
4272:
4268:
4246:
4243:
4240:
4235:
4231:
4201:
4197:
4176:
4173:
4170:
4165:
4161:
4138:
4133:
4130:
4106:
4102:
4079:
4074:
4071:
4041:
4038:
4035:
4030:
4026:
4005:
4002:
3999:
3994:
3990:
3967:
3962:
3959:
3934:
3931:
3928:
3923:
3919:
3898:
3895:
3892:
3887:
3883:
3862:
3859:
3856:
3851:
3847:
3820:
3816:
3789:
3786:
3782:
3761:
3758:
3755:
3750:
3746:
3742:
3738:
3734:
3731:
3728:
3723:
3720:
3717:
3714:
3711:
3708:
3704:
3700:
3680:
3677:
3674:
3671:
3668:
3663:
3659:
3649:(meaning that
3625:
3620:
3617:
3588:
3585:
3582:
3579:
3576:
3571:
3568:
3564:
3554:(meaning that
3533:
3528:
3525:
3497:
3493:
3458:
3453:
3450:
3445:
3442:
3431:Zariski closed
3418:
3415:
3412:
3409:
3406:
3401:
3397:
3370:
3367:
3363:
3342:
3337:
3333:
3329:
3324:
3320:
3314:
3310:
3306:
3303:
3298:
3295:
3291:
3286:
3282:
3279:
3253:
3248:
3245:
3213:
3208:
3205:
3178:
3174:
3170:
3167:
3164:
3161:
3158:
3155:
3150:
3146:
3140:
3136:
3109:
3105:
3082:
3078:
3072:
3068:
3037:
3033:
3010:
3006:
3000:
2996:
2972:
2969:
2968:
2967:
2953:
2948:
2945:
2923:
2920:
2917:
2914:
2910:
2904:
2900:
2895:
2891:
2888:
2885:
2881:
2875:
2871:
2866:
2862:
2859:
2854:
2850:
2846:
2843:
2840:
2835:
2831:
2827:
2824:
2821:
2818:
2789:
2785:
2773:
2772:
2771:
2760:
2757:
2752:
2747:
2743:
2739:
2736:
2733:
2728:
2723:
2719:
2715:
2712:
2709:
2706:
2701:
2697:
2693:
2690:
2687:
2682:
2678:
2674:
2671:
2668:
2655:
2654:
2653:be the mapping
2640:
2635:
2632:
2627:
2622:
2617:
2614:
2609:
2606:
2586:
2583:
2580:
2569:
2557:
2554:
2550:
2546:
2542:
2515:
2511:
2488:
2483:
2480:
2456:
2452:
2448:
2445:
2442:
2439:
2436:
2423:
2420:
2407:
2404:
2401:
2379:
2374:
2371:
2345:
2342:
2339:
2334:
2330:
2294:
2291:
2288:
2234:
2229:
2226:
2202:
2198:
2169:
2165:
2134:
2129:
2126:
2109:Chow's theorem
2094:
2089:
2086:
2058:
2054:
2050:
2047:
2044:
2039:
2035:
2023:
2022:
2011:
2008:
2003:
1999:
1995:
1992:
1989:
1984:
1980:
1976:
1971:
1967:
1963:
1960:
1957:
1954:
1949:
1945:
1941:
1938:
1935:
1930:
1926:
1922:
1917:
1913:
1909:
1906:
1903:
1900:
1895:
1891:
1887:
1884:
1881:
1876:
1872:
1868:
1865:
1862:
1829:
1824:
1821:
1793:
1788:
1785:
1746:
1741:
1738:
1714:
1709:
1706:
1689:
1686:
1659:
1654:
1651:
1561:
1558:
1555:
1552:
1549:
1546:
1518:
1515:
1512:
1507:
1503:
1499:
1496:
1493:
1490:
1485:
1481:
1460:
1457:
1454:
1417:
1412:
1409:
1404:
1399:
1394:
1391:
1386:
1383:
1351:
1348:
1345:
1342:
1339:
1336:
1316:
1313:
1308:
1304:
1300:
1297:
1294:
1291:
1288:
1260:
1257:
1254:
1251:
1248:
1245:
1242:
1222:
1219:
1216:
1211:
1207:
1203:
1200:
1197:
1194:
1191:
1160:
1157:
1152:
1148:
1144:
1141:
1138:
1135:
1132:
1117:Mandelbrot set
1103:
1099:
1096:
1076:
1073:
1068:
1064:
1060:
1057:
1054:
1051:
1048:
1018:
1014:
1010:
1007:
1004:
1001:
998:
966:
961:
958:
936:
933:
930:
927:
901:
896:
893:
855:
851:
830:
827:
824:
821:
818:
813:
809:
760:
757:
754:
751:
748:
745:
742:
739:
736:
733:
730:
727:
724:
721:
690:
687:
684:
679:
675:
650:
647:
644:
641:
638:
633:
629:
601:measure theory
572:
569:
565:
561:
557:
506:
484:
479:
476:
454:
431:absolute value
427:
426:
415:
411:
406:
402:
398:
395:
392:
389:
386:
383:
380:
377:
374:
371:
368:
365:
360:
356:
352:
349:
346:
343:
340:
337:
334:
331:
327:
322:
318:
314:
311:
308:
305:
302:
298:
295:
262:
257:
254:
232:
206:
201:
198:
176:
154:
149:
146:
115:
111:
107:
104:
101:
98:
95:
80:Main article:
77:
74:
66:p-adic numbers
17:
9:
6:
4:
3:
2:
8134:
8123:
8120:
8118:
8115:
8113:
8110:
8108:
8105:
8103:
8100:
8099:
8097:
8088:
8085:
8083:
8080:
8079:
8070:
8066:
8062:
8058:
8054:
8050:
8049:
8044:
8040:
8037:
8033:
8029:
8027:0-521-77476-4
8023:
8019:
8015:
8010:
8007:
8003:
7999:
7997:0-521-66258-3
7993:
7989:
7985:
7980:
7977:
7973:
7969:
7967:0-691-12488-4
7963:
7959:
7955:
7950:
7945:
7941:
7937:
7933:
7929:
7926:
7922:
7918:
7912:
7907:
7902:
7898:
7894:
7893:
7887:
7884:
7880:
7876:
7870:
7865:
7860:
7856:
7852:
7849:(examples)",
7834:
7819:
7815:
7811:
7808:
7804:
7800:
7794:
7790:
7786:
7782:
7778:
7775:
7771:
7766:
7761:
7757:
7753:
7748:
7745:
7741:
7737:
7733:
7728:
7723:
7719:
7715:
7714:
7709:
7705:
7701:
7698:
7694:
7690:
7684:
7680:
7676:
7671:
7666:
7662:
7658:
7654:
7650:
7646:
7643:
7639:
7635:
7631:
7626:
7621:
7617:
7613:
7612:
7607:
7602:
7599:
7595:
7591:
7589:0-387-97942-5
7585:
7581:
7577:
7573:
7569:
7565:
7561:
7557:
7554:
7550:
7546:
7542:
7537:
7532:
7528:
7524:
7523:
7518:
7517:Cantat, Serge
7514:
7511:
7507:
7503:
7497:
7493:
7489:
7485:
7484:Cantat, Serge
7481:
7478:
7474:
7470:
7464:
7460:
7456:
7455:
7450:
7449:Cantat, Serge
7446:
7443:
7439:
7435:
7429:
7425:
7421:
7417:
7413:
7409:
7404:
7401:
7397:
7393:
7389:
7384:
7379:
7375:
7371:
7370:
7364:
7361:
7357:
7353:
7351:0-387-97589-6
7347:
7343:
7339:
7338:
7333:
7332:Beardon, Alan
7329:
7326:
7322:
7318:
7316:3-528-06520-6
7312:
7308:
7304:
7300:
7299:Vieweg Verlag
7296:
7291:
7290:
7276:
7267:
7258:
7249:
7240:
7231:
7229:
7227:
7217:
7208:
7199:
7190:
7181:
7172:
7163:
7154:
7145:
7136:
7127:
7118:
7109:
7107:
7097:
7088:
7079:
7070:
7068:
7058:
7049:
7045:
7033:
7030:
7028:
7025:
7023:
7020:
7019:
7017:
7016:
7010:
7007:
7005:
7002:
7000:
6997:
6995:
6992:
6990:
6987:
6985:
6984:Schwarz lemma
6982:
6980:
6977:
6975:
6972:
6970:
6967:
6965:
6962:
6960:
6957:
6956:
6954:
6953:
6947:
6931:
6921:
6917:
6905:
6901:
6883:
6879:
6858:
6855:
6849:
6841:
6837:
6828:
6810:
6806:
6802:
6799:
6796:
6790:
6784:
6776:
6772:
6768:
6764:
6760:
6755:
6753:
6734:
6726:
6722:
6713:
6709:
6705:
6701:
6682:
6674:
6670:
6661:
6657:
6639:
6620:
6604:
6600:
6591:
6573:
6569:
6560:
6542:
6538:
6529:
6525:
6520:
6518:
6500:
6496:
6487:
6483:
6465:
6461:
6438:
6434:
6410:
6402:
6398:
6389:
6385:
6381:
6377:
6359:
6355:
6334:
6331:
6325:
6317:
6313:
6304:
6300:
6296:
6292:
6282:
6279:
6275:
6271:
6267:
6262:
6260:
6256:
6252:
6234:
6228:
6223:
6216:
6211:
6205:
6195:
6191:
6187:
6171:
6168:
6165:
6137:
6134:
6128:
6125:
6105:
6102:
6099:
6091:
6087:
6084:be a complex
6083:
6073:
6057:
6053:
6029:
6021:
6017:
6008:
6004:
5985:
5977:
5973:
5965:
5947:
5943:
5934:
5916:
5912:
5908:
5905:
5883:
5879:
5870:
5866:
5865:Green measure
5862:
5859:, called the
5858:
5840:
5836:
5827:
5822:
5820:
5815:
5811:
5793:
5789:
5765:
5757:
5754:
5751:
5747:
5738:
5720:
5716:
5707:
5703:
5699:
5695:
5691:
5688:be a compact
5687:
5682:
5680:
5676:
5672:
5668:
5644:
5641:
5633:
5629:
5620:
5616:
5597:
5592:
5588:
5584:
5581:
5576:
5568:
5562:
5556:
5549:
5548:
5547:
5545:
5521:
5518:
5510:
5507:
5503:
5499:
5493:
5485:
5482:
5479:
5475:
5466:
5462:
5458:
5440:
5436:
5415:
5412:
5409:
5406:
5403:
5395:
5391:
5385:
5363:
5360:
5352:
5348:
5340:
5336:
5332:
5328:
5324:
5320:
5316:
5315:automorphisms
5306:
5290:
5260:
5257:
5254:
5232:
5229:
5225:
5221:
5215:
5209:
5201:
5197:
5179:
5175:
5152:
5134:
5130:
5111:
5105:
5100:
5096:
5072:
5066:
5063:
5057:
5051:
5048:
5042:
5036:
5031:
5027:
5017:
5011:
5005:
5002:
4995:
4994:
4993:
4977:
4947:
4939:
4917:
4885:
4881:
4876:
4870:
4862:
4859:
4856:
4850:
4844:
4838:
4829:
4808:
4776:
4772:
4767:
4761:
4753:
4750:
4747:
4741:
4735:
4729:
4720:
4716:
4714:
4696:
4692:
4669:
4651:
4633:
4615:
4611:
4607:
4603:
4599:
4581:
4563:
4559:
4558:
4547:
4531:
4527:
4504:
4500:
4491:
4487:
4483:
4479:
4474:
4472:
4468:
4450:
4446:
4423:
4405:
4401:
4397:
4393:
4389:
4385:
4381:
4376:
4374:
4370:
4354:
4351:
4348:
4345:
4337:
4333:
4329:
4311:
4278:
4270:
4266:
4241:
4233:
4229:
4219:
4217:
4199:
4195:
4171:
4163:
4159:
4136:
4104:
4100:
4077:
4059:
4058:Zariski dense
4055:
4036:
4028:
4024:
4016:are dense in
4000:
3992:
3988:
3965:
3946:
3929:
3921:
3917:
3893:
3885:
3881:
3857:
3849:
3845:
3836:
3818:
3814:
3805:
3787:
3784:
3780:
3756:
3753:
3748:
3744:
3736:
3729:
3726:
3718:
3715:
3712:
3706:
3702:
3678:
3675:
3669:
3661:
3657:
3648:
3644:
3639:
3623:
3605:
3602:
3586:
3583:
3577:
3569:
3566:
3562:
3553:
3549:
3531:
3513:
3495:
3491:
3482:
3478:
3474:
3456:
3443:
3440:
3432:
3416:
3413:
3407:
3399:
3395:
3386:
3368:
3365:
3361:
3335:
3331:
3322:
3312:
3308:
3296:
3293:
3289:
3284:
3280:
3269:
3251:
3233:
3229:
3211:
3192:
3176:
3172:
3165:
3159:
3156:
3153:
3148:
3144:
3138:
3134:
3125:
3107:
3103:
3080:
3076:
3070:
3066:
3057:
3053:
3035:
3031:
3008:
3004:
2998:
2994:
2986:
2982:
2978:
2951:
2921:
2915:
2912:
2902:
2898:
2889:
2886:
2883:
2873:
2869:
2860:
2852:
2848:
2844:
2841:
2838:
2833:
2829:
2825:
2822:
2809:
2806:-dimensional
2805:
2787:
2783:
2774:
2758:
2750:
2745:
2741:
2737:
2734:
2731:
2726:
2721:
2717:
2710:
2699:
2695:
2691:
2688:
2685:
2680:
2676:
2666:
2659:
2658:
2657:
2656:
2638:
2620:
2607:
2604:
2584:
2581:
2578:
2570:
2555:
2552:
2544:
2531:
2513:
2509:
2486:
2454:
2450:
2446:
2440:
2434:
2426:
2425:
2419:
2405:
2402:
2399:
2377:
2359:
2340:
2332:
2328:
2320:
2316:
2315:Nessim Sibony
2312:
2311:John Fornaess
2308:
2292:
2289:
2286:
2278:
2274:
2270:
2266:
2262:
2261:Green measure
2258:
2254:
2250:
2232:
2200:
2196:
2188:
2183:
2167:
2163:
2154:
2150:
2132:
2115:mapping from
2114:
2110:
2092:
2074:
2056:
2052:
2048:
2045:
2042:
2037:
2033:
2001:
1997:
1993:
1990:
1987:
1982:
1978:
1969:
1965:
1961:
1958:
1955:
1947:
1943:
1939:
1936:
1933:
1928:
1924:
1915:
1911:
1904:
1893:
1889:
1885:
1882:
1879:
1874:
1870:
1860:
1853:
1852:
1851:
1849:
1845:
1827:
1809:
1791:
1773:
1768:
1766:
1762:
1761:rational maps
1744:
1712:
1695:
1685:
1683:
1679:
1675:
1657:
1639:
1635:
1631:
1627:
1623:
1619:
1615:
1611:
1607:
1603:
1599:
1595:
1591:
1587:
1583:
1579:
1575:
1559:
1556:
1550:
1544:
1536:
1532:
1513:
1505:
1501:
1497:
1491:
1483:
1479:
1458:
1455:
1452:
1444:
1441:
1437:
1433:
1415:
1397:
1384:
1381:
1373:
1365:
1349:
1346:
1343:
1340:
1337:
1334:
1314:
1311:
1306:
1302:
1298:
1292:
1286:
1277:
1258:
1255:
1252:
1249:
1246:
1243:
1240:
1220:
1217:
1214:
1209:
1205:
1201:
1195:
1189:
1180:
1176:
1174:
1158:
1155:
1150:
1146:
1142:
1136:
1130:
1122:
1118:
1097:
1094:
1074:
1071:
1066:
1062:
1058:
1052:
1046:
1038:
1034:
1016:
1012:
1008:
1002:
996:
988:
984:
983:
964:
931:
925:
917:
914:to itself of
899:
881:
877:
873:
871:
853:
849:
828:
825:
819:
811:
807:
798:
794:
790:
786:
782:
778:
774:
752:
743:
737:
731:
725:
719:
711:
708:
704:
685:
677:
673:
664:
648:
645:
639:
631:
627:
618:
614:
610:
606:
602:
598:
594:
590:
586:
570:
567:
559:
545:
543:
539:
535:
531:
527:
523:
520:
482:
444:
440:
439:exponentially
436:
432:
413:
409:
404:
400:
396:
384:
378:
372:
366:
363:
358:
354:
350:
341:
335:
329:
325:
320:
316:
312:
306:
300:
296:
293:
286:
285:
284:
282:
281:forward orbit
278:
260:
230:
222:
204:
152:
135:
131:
113:
109:
105:
99:
93:
83:
73:
71:
67:
63:
59:
55:
51:
47:
43:
39:
35:
31:
27:
23:
16:
8117:Chaos theory
8052:
8046:
8043:Zdunik, Anna
8013:
7983:
7949:math/9201272
7935:
7932:Milnor, John
7906:math/0611302
7891:
7850:
7784:
7765:math/0212208
7755:
7751:
7717:
7711:
7656:
7615:
7609:
7567:
7526:
7520:
7487:
7453:
7411:
7383:math/0501034
7373:
7367:
7336:
7294:
7275:
7266:
7257:
7248:
7239:
7216:
7207:
7198:
7189:
7180:
7171:
7162:
7153:
7144:
7135:
7126:
7117:
7096:
7087:
7078:
7057:
7048:
7027:Chaos theory
6904:John Smillie
6899:
6826:
6774:
6770:
6766:
6762:
6758:
6756:
6751:
6711:
6707:
6703:
6659:
6655:
6621:
6589:
6558:
6527:
6523:
6521:
6516:
6485:
6481:
6387:
6383:
6379:
6375:
6302:
6298:
6297:is called a
6294:
6290:
6288:
6273:
6265:
6263:
6258:
6254:
6250:
6189:
6185:
6089:
6081:
6079:
6006:
6002:
5963:
5932:
5898:has entropy
5868:
5864:
5860:
5856:
5825:
5823:
5818:
5814:Serge Cantat
5736:
5705:
5701:
5697:
5693:
5685:
5683:
5666:
5618:
5614:
5612:
5543:
5460:
5393:
5389:
5386:
5334:
5330:
5326:
5322:
5314:
5312:
5195:
5132:
5128:
5087:
4937:
4935:
4605:
4602:finite group
4561:
4555:
4553:
4481:
4477:
4475:
4470:
4466:
4403:
4399:
4395:
4387:
4383:
4379:
4377:
4335:
4327:
4221:The support
4220:
4053:
3947:
3834:
3803:
3646:
3642:
3640:
3603:
3600:
3551:
3547:
3511:
3476:
3472:
3384:
3267:
3231:
3193:
3123:
3051:
3023:is equal to
2980:
2976:
2974:
2803:
2530:Haar measure
2318:
2307:John Hubbard
2273:Ricardo Mañé
2264:
2260:
2256:
2252:
2248:
2184:
2152:
2148:
2072:
2024:
1843:
1806:, meaning a
1771:
1769:
1691:
1681:
1677:
1673:
1672:that map to
1637:
1633:
1625:
1621:
1613:
1609:
1605:
1597:
1593:
1589:
1585:
1581:
1577:
1573:
1534:
1530:
1442:
1431:
1369:
1362:. This is a
1120:
986:
980:
880:Gaston Julia
876:Pierre Fatou
874:
869:
796:
792:
788:
784:
780:
776:
772:
709:
706:
702:
662:
604:
596:
592:
588:
584:
546:
541:
537:
533:
525:
522:fixed points
518:
442:
434:
428:
280:
276:
129:
85:
41:
32:obtained by
25:
21:
20:
15:
5679:K3 surfaces
4650:Anna Zdunik
4550:Lattès maps
4216:perfect set
2269:Artur Lopes
2113:holomorphic
1618:Herman ring
1602:Siegel disk
8096:Categories
7287:References
6305:such that
6270:blowing up
5708:such that
5671:eigenvalue
4557:Lattès map
4294:is all of
3050:. Because
1471:such that
1364:Cantor set
538:attracting
530:derivative
441:fast. If |
46:polynomial
44:, where a
7727:0804.0860
7670:0810.0811
7625:1009.5796
7536:1410.1202
6880:μ
6803:
6727:∗
6675:∗
6601:μ
6570:μ
6539:μ
6497:μ
6462:μ
6435:μ
6403:∗
6169:×
6103:×
6054:μ
6022:∗
5978:∗
5944:μ
5909:
5880:μ
5837:μ
5634:∗
5585:
5500:⊂
5413:≤
5407:≤
5353:∗
5258:≥
5230:±
5176:μ
5106:
5052:μ
5037:
5012:μ
5006:
4948:μ
4860:−
4751:−
4713:Borel set
4693:μ
4528:μ
4501:μ
4447:μ
4352:
4271:∗
4234:∗
4196:μ
4164:∗
4101:μ
4029:∗
3993:∗
3922:∗
3886:∗
3850:∗
3815:μ
3754:−
3727:−
3567:−
3492:μ
3444:⊊
3332:δ
3323:∗
3173:μ
3160:
3145:μ
3139:∗
3104:μ
3077:μ
3071:∗
3032:μ
3005:μ
2999:∗
2977:invariant
2887:⋯
2842:…
2784:μ
2735:…
2689:…
2626:→
2608::
2510:μ
2333:∗
2197:μ
2046:…
1991:…
1959:…
1937:…
1883:…
1763:from any
1586:parabolic
1432:Fatou set
1403:→
1385::
1344:−
1338:≐
1247:−
1244:≐
1173:connected
1098:∈
982:Julia set
797:repelling
753:⋯
732:⋯
505:∞
453:∞
414:…
175:∞
82:Julia set
34:iterating
8122:Fractals
7934:(2006),
7783:(1996),
7566:(1993),
7334:(1991),
6950:See also
6088:and let
5665:.) Thus
2422:Examples
665:. (Here
8069:1032883
8036:1765080
8006:1747010
7976:2193309
7925:2932434
7883:1810536
7807:1363948
7774:1995861
7744:2629598
7697:2648690
7642:2889139
7598:1230383
7553:4071328
7510:3289919
7477:2932433
7442:3289442
7400:2142250
7360:1128089
7325:1260930
7011:puzzles
6276:and is
5455:be the
4486:ergodic
4465:is the
3475:not in
3433:subset
3383:points
2528:is the
2358:support
2356:is the
1630:annulus
1430:in the
1033:fractal
712:times,
221:compact
64:or the
8067:
8034:
8024:
8004:
7994:
7974:
7964:
7923:
7913:
7881:
7871:
7805:
7795:
7772:
7742:
7695:
7685:
7640:
7596:
7586:
7551:
7508:
7498:
7475:
7465:
7440:
7430:
7398:
7358:
7348:
7323:
7313:
7009:Yoccoz
6299:saddle
6278:smooth
6192:whose
5467:group
5428:, let
5319:smooth
5088:where
4490:mixing
4467:unique
3550:under
2979:under
2597:, let
2313:, and
2275:, and
2247:, the
2107:. (By
1596:; (3)
1580:; (2)
1347:0.0745
1115:. The
979:, the
916:degree
7944:arXiv
7901:arXiv
7760:arXiv
7722:arXiv
7665:arXiv
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7531:arXiv
7378:arXiv
7040:Notes
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