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Title:
Constructing entire functions with wandering domains by a quasiconformal modification
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Abstract:
In 2015 Christopher Bishop constructed the first example of a bounded-type transcendental entire function with a wandering domain using a revolutionary new technique called quasiconfomal folding. It is easy to check that his method produces a function of infinite order. In a joint work with Mitsuhiro Shishikura, we constructed the first examples of functions in the Eremenko-Lyubich class of finite order with wandering domains. In Bishop's example, as well as in our construction, the wandering domains are of oscillating type, that is, with an unbounded non-escaping orbit. Our examples have order p/2 for any positive integer p and thus, since functions in the class B have order at least 1/2, we can achieve the smallest possible order. To build such functions, we performed a quasiconformal modification of the hyperbolic cosine map. In a work in progress with David Sixsmith, we also use this technique to construct transcendental entire functions with simply connected fast escaping wandering domains.
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