Dynamical properties of some classes of entire functions
Annales de l'Institut Fourier, Volume 42 (1992) no. 4, pp. 989-1020.

The paper is concerned with the dynamics of an entire transcendental function whose inverse has only finitely many singularities. It is rpoven that there are no escaping orbits on the Fatou set. Under some extra assumptions the set of escaping orbits has zero Lebesgue measure. If a function depends analytically on parameters then a periodic point as a function of parameters has only algebraic singularities. This yields the Structural Stability Theorem.

Cet article étudie la dynamique d’une fonction entière transcendante dont l’inverse n’a qu’un nombre fini de singularités. On prouve qu’il n’y a pas d’orbites qui tendent à l’infini sur l’ensemble de Fatou. Sous certaines conditions supplémentaires l’ensemble des orbites tendant vers l’infini est de mesure de Lebesgue nulle. Si une fonction dépend analytiquement de paramètres, alors un point périodique comme fonction des paramètres n’a seulement que des singularités algébriques. Ceci implique le théorèm’e de stabilité structurelle.

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Eremenko, A.; Lyubich, M. Yu. Dynamical properties of some classes of entire functions. Annales de l'Institut Fourier, Volume 42 (1992) no. 4, pp. 989-1020. doi : 10.5802/aif.1318. http://archive.numdam.org/articles/10.5802/aif.1318/

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