Boundary dynamics in unbounded Fatou components.
We study the behaviour of a transcendental entire map $f: \mathbb{C} \rightarrow \mathbb{C}$ on an unbounded invariant Fatou component $U$, assuming that infinity is accessible from $U$. It is wellknown that $U$ is simply connected. Hence, by means of a Riemann map $\varphi: \mathbb{D} \rightarrow U...
| Autores: | , |
|---|---|
| Formato: | artículo |
| Estado: | Versión aceptada para publicación |
| Fecha de publicación: | 2025 |
| País: | España |
| Recursos: | Universidad de Barcelona |
| Repositorio: | Dipòsit Digital de la UB |
| OAI Identifier: | oai:diposit.ub.edu:2445/222648 |
| Acesso em linha: | https://hdl.handle.net/2445/222648 |
| Access Level: | acceso abierto |
| Palavra-chave: | Funcions meromorfes Sistemes dinàmics complexos Meromorphic functions Complex dynamical systems |
| Resumo: | We study the behaviour of a transcendental entire map $f: \mathbb{C} \rightarrow \mathbb{C}$ on an unbounded invariant Fatou component $U$, assuming that infinity is accessible from $U$. It is wellknown that $U$ is simply connected. Hence, by means of a Riemann map $\varphi: \mathbb{D} \rightarrow U$ and the associated inner function $g:=\varphi^{-1} \circ f \circ \varphi$, 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 $\mathbb{C}$, improving the results in [BD99; Bar08]. Moreover, under mild assumptions on the location of singular values in $U$ (allowing them even to accumulate at infinity, as long as they accumulate through accesses to $\infty)$, we show that periodic and escaping boundary points are dense in $\partial U$, and that all periodic boundary points accessible from $U$. Finally, under similar conditions, the set of singularities of $g$ is shown to have zero Lebesgue measure, strengthening substantially the results in [EFJS19; ERS20]. |
|---|