Wandering domains for composition of entire functions

C. Bishop has constructed an example of an entire function f in Eremenko-Lyubich class with at least two grand orbits of oscillating wandering domains. In this paper we show that his example has exactly two such orbits, that is, f has no unexpected wandering domains. We apply this result to the classical problem of relating the Julia sets of composite functions with the Julia set of its members. More precisely, we show the existence of two entire maps f and g in Eremenko-Lyubich class such that the Fatou set of f compose with g has a wandering domain, while all Fatou components of f or g are preperiodic. This complements a result of A. Singh and results of W. Bergweiler and A. Hinkkanen related to this problem.Comment: 21 pages, 3 figure

Invariants en dinàmica complexa

Usant com a fil conductor el mètode de Newton per a polinomis complexos, aquesta lliçó pretén mostrar els diferents comportaments que poden tenir les òrbites d?un sistema dinàmic generat per la iteració d?una funció analítica del pla complex. Veurem com aquestes òrbites s?agrupen en conjunts invariants amb dinàmiques molt variades, separats per fronteres fractals amb propietats remarcables, tant topològiques com dinàmiques.Using Newton’s method for complex polynomials as a conducting theme, this lecture tries to show the possible asymptotic behaviours of orbits under iteration of holomorphic maps. We see how these orbits form invariant sets with different possible dynamics, separated by fractal boundaries with amazing topological and dynamical properties

Dynamics of projectable functions: Towards an atlas of wandering domains for a family of Newton maps

We present a one-parameter family $F_\lambda$ of transcendental entire functions with zeros, whose Newton's method yields wandering domains, coexisting with the basins of the roots of $F_\lambda$. Wandering domains for Newton maps of zero-free functions have been built before by, e.g., Buff and R\"uckert based on the lifting method. This procedure is suited to our Newton maps as members of the class of projectable functions (or maps of the cylinder), i.e. transcendental meromorphic functions $f(z)$ in the complex plane that are semiconjugate, via the exponential, to some map $g(w)$, which may have at most a countable number of essential singularities. In this paper we make a systematic study of the general relation (dynamical and otherwise) between $f$ and $g$, and inspect the extension of the logarithmic lifting method of periodic Fatou components to our context, especially for those $g$ of finite-type. We apply these results to characterize the entire functions with zeros whose Newton's method projects to some map $g$ which is defined at both $0$ and $\infty$. The family $F_\lambda$ is the simplest in this class, and its parameter space shows open sets of $\lambda$-values in which the Newton map exhibits wandering or Baker domains, in both cases regions of initial conditions where Newton's root-finding method fails.Comment: 34 pages, 9 figure

Boundary dynamics in unbounded Fatou components

We study the behaviour of a transcendental entire map $f\colon \mathbb{C}\to\mathbb{C}$ on an unbounded invariant Fatou component $U$, assuming that infinity is accessible from $U$. It is well-known that $U$ is simply connected. Hence, by means of a Riemann map $\varphi\colon\mathbb{D}\to U$ and the associated inner function, the boundary of $U$ is described topologically in terms of the disjoint union of clusters sets, each of them consisting of one or two connected components in $\mathbb{C}$. Moreover, under more precise assumptions on the distribution of singular values, it is proven that periodic and escaping boundary points are dense in $\partial U$, being all periodic boundary points accessible from $U$. Finally, under the same conditions, the set of singularities of $g$ is shown to have zero Lebesgue measure