### Escaping dynamics of a class of transcendental functions

Ergodic Theory and Dynamical Systems Seminar

29th November 2018, 2:00 pm – 3:00 pm

Howard House, 4th Floor Seminar Room

When studying the dynamics under iteration of polynomials, the notion of external rays, that is, curves that escape uniformly to infinity, and their landing properties are powerful tools for understanding the structure and dynamics of their Julia sets. The existence of analogous curves for transcendental entire maps was conjectured by Eremenko in 1989. These ‘rays’ or ‘hairs’ have been proved to exist and land for some functions with bounded postsingular set by showing that their Julia set is structured as a “Cantor Bouquet. In this talk I will consider certain functions with unbounded postsingular set whose singular orbits escape at some minimum speed. In this setting, some hairs will split when they hit critical points. We show that the existence of a map on their parameter space whose Julia set is a Cantor Bouquet guarantees that such hairs, if maybe now with split ends, still land.

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