Transcendental Entire Functions with Cantor Bouquet Julia Sets

In the study of the dynamics of a transcendental entire function f, we aim to describe its locus of chaotic behaviour, known as its Julia set and denoted by J(f). For many such f, the Julia set is a collection of unbounded curves that escape to infinity under iteration and form a Cantor bouquet, i.e., a subset of the complex plane ambiently homeomorphic to a straight brush. We show that there exists f whose Julia set J(f) is a collection of escaping curves, but J(f) is not a Cantor bouquet. On the other hand, we prove for certain f that if J(f) contains an absorbing Cantor bouquet, that is, a Cantor bouquet to which all escaping points are eventually mapped, then J(f) must be a Cantor bouquet. This is joint work with L. Rempe.

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