Accesses to infinity from Fatou components

Abstract

We study the boundary behaviour of a meromorphic map $f:\mathbb{C}\to \hat{\mathbb{C}}$ on its invariant simply connected Fatou component $U$. To this aim, we develop the theory of accesses to boundary points of $U$ and their relation to the dynamics of $f$. In particular, we establish a correspondence between invariant accesses from $U$ to infinity or weakly repelling points of $f$ and boundary fixed points of the associated inner function on the unit disc. We apply our results to describe the accesses to infinity from invariant Fatou components of the Newton maps.

31 pages, 8 figures

References in corpus (2)

• Hyperbolic entire functions with bounded Fatou components
• Absorbing sets and Baker domains for holomorphic maps

Cited by in corpus (8)

• Fatou components and singularities of meromorphic functions
• Escaping points in the boundaries of Baker domains
• On connected preimages of simply-connected domains under entire functions
• A characterization of postcritically minimal Newton maps of complex exponential functions
• Fatou's associates
• Dynamics of projectable functions: Towards an atlas of wandering domains for a family of Newton maps
• Boundary behaviour of universal covering maps
• Holomorphic motions, natural families of entire maps, and multiplier-like objects for wandering domains