Jové Campabadal, AnnaFagella Rabionet, Núria2025-07-292025-07-292025-02-110002-9947https://hdl.handle.net/2445/222648We 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].42 p.application/pdfengcc-by-nc-nd (c) American Mathematical Society (AMS), 2025http://creativecommons.org/licenses/by-nc-nd/4.0/Funcions meromorfesSistemes dinàmics complexosMeromorphic functionsComplex dynamical systemsinfo:eu-repo/semantics/article7564112025-07-29info:eu-repo/semantics/openAccess