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Thursday 01 Dec 2022Transcendental entire functions with Cantor bouquet Julia sets

Leticia Pardo-Simon - University of Manchester

Harrison 103 15:30-16:30

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 particular topological structure known as 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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