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Tuesday 24 May 2022Wandering domains in transcendental dynamics: topology and dynamics

David Marti-Pete - University of Liverpool

LSI Seminar Room B 13:30-14:30

For a transcendental entire or meromorphic function, the Fatou set is the largest open set on which its iterates are defined and form a normal family. A wandering domain is a connected component of the Fatou set which is not eventually periodic. The first example of a transcendental entire function with a wandering domain was constructed by Baker in the 1970s. 

Wandering domains, which do not exist for rational maps, play an important role in transcendental dynamics and in the last decade there has been a resurgence in their interest. For example, Bishop proved that the Julia sets of transcendental entire functions can have Hausdorff dimension 1 by constructing a function with wandering domains. 

Wandering domains are very diverse in terms of both their topology (simply connected or multiply connected) and their dynamics (escaping, oscillating or, perhaps, even have bounded orbit). Recently, Boc Thaler proved the surprising result that every bounded regular domain such that its closure has a connected complement is the wandering domain of some transcendental entire function. Inspired by this result, together with Rempe and Waterman, we were able to obtain wandering domains that form Lakes of Wada. 

In this talk, I will describe the main topological and dynamical properties of wandering domains (and their boundaries) and give an overview of the current open questions.

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