Multiply connected wandering domains of entire functions

Abstract

The dynamical behaviour of a transcendental entire function in any periodic component of the Fatou set is well understood. Here we study the dynamical behaviour of a transcendental entire function $f$ in any multiply connected wandering domain $U$ of $f$. By introducing a certain positive harmonic function $h$ in $U$, related to harmonic measure, we are able to give the first detailed description of this dynamical behaviour. Using this new technique, we show that, for sufficiently large $n$, the image domains $U_n=f^n(U)$ contain large annuli, $C_n$, and that the union of these annuli acts as an absorbing set for the iterates of $f$ in $U$. Moreover, $f$ behaves like a monomial within each of these annuli and the orbits of points in $U$ settle in the long term at particular `levels' within the annuli, determined by the function $h$. We also discuss the proximity of $\partial U_n$ and $\partial C_n$ for large $n$, and the connectivity properties of the components of $U_n\setminus \overline{C_n}$. These properties are deduced from new results about the behaviour of an entire function which omits certain values in an annulus.

Cited by in corpus (11)

• Fatou components and singularities of meromorphic functions
• The size and topology of quasi-Fatou components of quasiregular maps
• Hollow quasi-Fatou components of quasiregular maps
• Multiply connected wandering domains of meromorphic functions: internal dynamics and connectivity
• The escaping set of transcendental self-maps of the punctured plane
• Bounded Fatou and Julia components of meromorphic functions
• Escaping Fatou components of transcendental self-maps of the punctured plane
• The escaping set in transcendental dynamics
• Lyapunov exponents and related concepts for entire functions
• Transcendental Julia Sets of Minimal Hausdorff Dimension
• Multiply connected wandering domains of meromorphic functions: the pursuit of uniform internal dynamics