abstract: I study the iteration of holomorphic self-maps of C*, the complex plane with the origin removed, for which both zero and infinity are essential singularities. In this talk we will compare the dynamics of this class of functions with that of transcendental entire functions, which have only one essential singularity at infinity. The escaping set of these maps consists of the points whose orbit accumulates to zero and/or infinity under iteration following what we call essential itineraries, and the Julia set contains escaping points with every essential itinerary. The concept of essential itinerary leads to a partition of the escaping set into uncountably many disjoint sets, the boundary of each of which is the Julia set. Under certain hypotheses, each of these sets contains uncountably many curves to zero and infinity. We can use approximation theory to provide examples of functions with escaping Fatou components that have any prescribed essential itinerary. Finally, I will talk about the complex Arnold standard family which is an example of such functions.

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