Multiply connected wandering domains of meromorphic functions: internal dynamics and connectivity

Abstract

We discuss how the nine-way classification scheme devised by Benini et al. for the dynamics of simply connected wandering domains of entire functions, based on the long-term behaviour of the hyperbolic distance between iterates of pairs of points and also the distance between orbits and the domains' boundaries, carries over to the general case of multiply connected wandering domains of meromorphic functions. Most strikingly, we see that not all pairs of points in such a wandering domain behave in the same way relative to the hyperbolic distance, and that the connectivity of the wandering domain greatly influences its possible internal dynamics. After illustrating our results with the well-studied case of Baker wandering domains, we further illustrate the diversity of multiply connected wandering domains in general by constructing a meromorphic function with a wandering domain without eventual connectivity. Finally, we show that an analogue of the "convergence to the boundary" classification of Benini et al. does hold in general, and add new information about how this convergence takes place.

Multiply connected wandering domains of meromorphic functions: the pursuit of uniform internal dynamics

“…The first steps were taken by Bergweiler, Rippon, and Stallard in 2013 [8], who described the behaviour of multiply connected wandering domains of entire functions (recall that the connectivity of a domain Ω ⊂ C is its number of complementary components). Their methods made extensive use of the peculiarities of multiply connected wandering domains of entire functions, but more recent work examined the internal dynamics of wandering domains in terms of the hyperbolic metric -which is always an available tool when talking about Fatou components -for both simply [7] and multiply connected wandering domains [13]. This is the point of view that we adopt in this work as well.…”

“…A central result of [7] is that, if U and all its iterates are simply connected, then the answer is independent of our particular choice of z and w; in other words, all pairs of points behave in the same way. On the other hand, if U is multiply connected, then it was shown in [13] that the answer may depend on the chosen pair -and, in particular, that all possible long-term behaviours can co-exist in the same domain.…”

“…An immediate question, then, is: how complicated can this co-existence be? The observed cases in [13] are all "well-behaved", in the sense that there are dynamically defined, smooth laminations of U that determine how each pair of points behave. In particular, every non-empty open subset of U contains pairs of points exhibiting all the behaviours present in U .…”

“…Theorem 1.1 is meaningless for wandering domains whose iterates are all simply connected (Benini et al's results for simply connected wandering domains, despite being stated only for entire functions, generalise to meromorphic functions). It is also vacuous for multiply connected wandering domains with finite eventual connectivity (see [13]), and so it is of most interest for infinitely connected wandering domains of meromorphic functions.…”