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Dynamical behavior of two permutable entire functions

Kin-Keung Poon, Chung-Chun Yang (1998)

Annales Polonici Mathematici

We show that two permutable transcendental entire functions may have different dynamical properties, which is very different from the rational functions case.

Dynamical properties of some classes of entire functions

A. Eremenko, M. Yu Lyubich (1992)

Annales de l'institut Fourier

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.

Dynamics of quadratic polynomials : complex bounds for real maps

Mikhail Lyubich, Michael Yampolsky (1997)

Annales de l'institut Fourier

We prove complex bounds for infinitely renormalizable real quadratic maps with essentially bounded combinatorics. This is the last missing ingredient in the problem of complex bounds for all infinitely renormalizable real quadratics. One of the corollaries is that the Julia set of any real quadratic map z ↦ z 2 + c , c ∈ [ - 2 , 1 / 4 ] , is locally connected.

Exact asymptotics of nonlinear difference equations with levels 1 and 1 +

G.K Immink (2008)

Annales de la faculté des sciences de Toulouse Mathématiques

We study a class of nonlinear difference equations admitting a 1 -Gevrey formal power series solution which, in general, is not 1 - (or Borel-) summable. Using right inverses of an associated difference operator on Banach spaces of so-called quasi-functions, we prove that this formal solution can be lifted to an analytic solution in a suitable domain of the complex plane and show that this analytic solution is an accelero-sum of the formal power series.

Exceptional values of meromorphic functions and of their derivatives on annuli

Yuxian Chen, Zhaojun Wu (2012)

Annales Polonici Mathematici

This paper is devoted to exceptional values of meromorphic functions and of their derivatives on annuli. Some facts on exceptional values for meromorphic functions in the complex plane which were established by Singh, Gopalakrishna and Bhoosnurmath [Math. Ann. 191 (1971), 121-142, and Ann. Polon. Math. 35 (1977/78), 99-105] will be considered on annuli.

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