zbMATH — the first resource for mathematics

A direct proof of the density of repulsive cycles in Julia sets. (Une démonstration directe de la densité des cycles répulsifs dans l’ensemble de Julia.) (French) Zbl 1073.37522
Dolbeault, P. (ed.) et al., Complex analysis and geometry. Proceedings of the international conference in honor of Pierre Lelong on the occasion of his 85th birthday, Paris, France, September 22–26, 1997. Basel: Birkhäuser (ISBN 3-7643-6352-5/hbk). Prog. Math. 188, 221-222 (2000).
The density of repelling periodic cycles in the Julia set of a rational or entire function is important in complex dynamics. The reviewer’s proof of this in the transcendental entire case [Math. Z. 104, 252–256 (1968; Zbl 0172.09502)] used the three islands theorem of Ahlfors. Recent attempts to give a simpler proof use a renormalisation criterion of L. Zalcman for non-normal families and some more elementary results of value distribution. The authors give an elegant version depending on the above criterion and Picard’s theorem. A related simple argument has been published by D. Bargmann [Ergodic Theory Dyn. Syst. 19, No. 3, 553–558 (1999; Zbl 0942.37033)].
For the entire collection see [Zbl 0940.00031].

37F50 Small divisors, rotation domains and linearization in holomorphic dynamics
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets