Multiply connected wandering domains of entire functions

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 \(U_n \setminus \bar{C_n}\). These properties are deduced from new results about the behaviour of an entire function which omits certain values in an annulus.